Finite Group Databasebeta

← #33092546

Subgroups — #33092546

Order 18688.

1014 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988826C211yesmaximal, characteristic
2C2×#17988692C211yesmaximal, characteristic, complemented
3C22×#14556502C211yesmaximal, characteristic
4C4×#14556570C211yesmaximal, characteristic
5#22610801C211yesmaximal, characteristic, complemented
6#22610666C211yesmaximal, characteristic, complemented
7#22610624C211yesmaximal, characteristic, complemented
8#22610620C211yesmaximal, characteristic, complemented
9#22610355C211yesmaximal, characteristic
10#22610352C211yesmaximal, characteristic
11#22609954C211yesmaximal, characteristic
12#22609829C211yesmaximal, characteristic, complemented
13#22609764C211yesmaximal, characteristic, complemented
14#22609761C211yesmaximal, characteristic, complemented
15C4×C22⋊Q8×C73C211yesmaximal, characteristic
16C22×#11924018C2211yescharacteristic
17C23×#10596831C2211yescharacteristic
18C23×#10596831C2211yescharacteristic
19C2×#14556570C2211yescharacteristic
20C2×C4×Dic146C2211yescharacteristic
21#17988826C418yescomplemented
22#17988692—24no—
23C2×#14556524C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556515C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556514C2211yescharacteristic
31C2×#14556514C2211yescharacteristic
32C2×#14556502—24no—
33C2×#14556502C2211yescharacteristic
34C2×#14556502C2211yescharacteristic
35C2×#14556502C2211yescharacteristic
36C2×#14556502C2211yescharacteristic
37#17988493C2211yescharacteristic
38#17988485C2211yescharacteristic
39#17988471—21no—
40#17988460—21no—
41#17988447C2211yescharacteristic
42C4×#11924007—21no—
43C4×C4⋊Dic73C2211yescharacteristic
44C4×C22⋊Dic73—21no—
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51C2×C22⋊Q8×C73C2211yescharacteristic
52C2×C4×Q8×C73C2211yescharacteristic
53C22×C42×C73C2211yescharacteristic
54#391×C73C2211yescharacteristic
55#389×C73C2211yescharacteristic
56#387×C73—21no—
57#385×C73—21no—
58#380×C73C2211yescharacteristic
59C4×C4⋊C4×C73—21no—
60C4×C4⋊C4×C73C2211yescharacteristic
61C4×C22⋊C4×C73—21no—
62C22×Dic146C2311yescharacteristic
63C2×#11924018—22no—
64C2×#11924018—22no—
65C2×#11924018C2×C414yescomplemented
66C22×#10596831—22no—
67C22×#10596831—22no—
68C22×#10596831—22no—
69C22×#10596831—22no—
70C22×#10596831—22no—
71C22×#10596831C2×C414yes—
72C22×#10596831C2311yescharacteristic
73C22×#10596831C2311yescharacteristic
74C22×#10596831C2311yescharacteristic
75C22×#10596831C2311yescharacteristic
76#14556570—22no—
77#14556570C2×C414yes—
78C4×Dic146—22no—
79C4×Dic146—22no—
80C2×C4⋊Dic73C2311yescharacteristic
81C2×C4⋊Dic73C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×C22⋊Dic73—21no—
87C2×C22⋊Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90#14556524C2×C414yescomplemented
91#14556522—22no—
92#14556522—22no—
93#14556515—22no—
94#14556515—22no—
95#14556515C2×C414yescomplemented
96#14556514—22no—
97#14556514—22no—
98#14556514C2×C414yescomplemented
99#14556502—22no—
100#14556502—22no—
101#14556502—22no—
102#14556502D414yes—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108C22×Q8×C73C2311yescharacteristic
109C23×C4×C73D411yescharacteristic
110C23×C4×C73D411yescharacteristic
111C23×C4×C73C2311yescharacteristic
112C22⋊Q8×C73—22no—
113C22⋊Q8×C73C2×C414yes—
114C4×Q8×C73—22no—
115C4×Q8×C73—22no—
116C2×C4⋊C4×C73—21no—
117C2×C4⋊C4×C73—21no—
118C2×C4⋊C4×C73—21no—
119C2×C4⋊C4×C73C2311yescharacteristic
120C2×C4⋊C4×C73C2311yescharacteristic
121C2×C4⋊C4×C73C2311yescharacteristic
122C2×C22⋊C4×C73—21no—
123C2×C22⋊C4×C73—21no—
124C2×C42×C73—22no—
125C2×C42×C73—22no—
126C2×C42×C73—21no—
127C2×C42×C73—21no—
128C2×C42×C73D411yescharacteristic
129C2×C42×C73D411yescharacteristic
130C2×C42×C73C2311yescharacteristic
131C2×C42×C73C2311yescharacteristic
132#95×C73—21no—
133#95×C73—21no—
134#95×C73—21no—
135#95×C73—21no—
136#95×C73—21no—
137C2×Dic146—21no—
138C2×Dic146—21no—
139C2×Dic146—21no—
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146C22×C412yes—
143C22×Dic73—21no—
144C22×Dic73—21no—
145C22×Dic73—21no—
146C22×Dic73—21no—
147#11924018—28no—
148C2×#10596831—28no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—28no—
156C2×#10596831C22×C412yes—
157C2×#10596831C22×C412yes—
158C2×#10596831C2×D412yes—
159C2×#10596831C2×D412yes—
160C2×#10596831#4112yes—
161C2×#10596831#4112yes—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—21no—
166C4⋊Dic73—21no—
167C4⋊Dic73C22×C412yes—
168#11924007—24no—
169#11924007—42no—
170#11924007—42no—
171C22⋊Dic73—24no—
172C22⋊Dic73—42no—
173C4×Dic73—42no—
174C4×Dic73—42no—
175C24×C73C2×D411yescharacteristic
176C2×Q8×C73—21no—
177C2×Q8×C73—21no—
178C2×Q8×C73—21no—
179C2×Q8×C73—21no—
180C2×Q8×C73—21no—
181C2×Q8×C73C22×C412yes—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—22no—
187C22×C4×C73—22no—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73—22no—
191C22×C4×C73—22no—
192C22×C4×C73—21no—
193C22×C4×C73—21no—
194C22×C4×C73—41no—
195C22×C4×C73—41no—
196C22×C4×C73C22×C412yes—
197C22×C4×C73C2×D411yescharacteristic
198C22×C4×C73C2×D411yescharacteristic
199C22×C4×C73C2×D411yescharacteristic
200C22×C4×C73C2×D411yescharacteristic
201C22×C4×C73C2×D411yescharacteristic
202C22×C4×C73#4112yes—
203C22×C4×C73C2411yescharacteristic
204C4⋊C4×C73—24no—
205C4⋊C4×C73—21no—
206C4⋊C4×C73—21no—
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73—21no—
210C4⋊C4×C73—42no—
211C4⋊C4×C73—42no—
212C4⋊C4×C73C22×C412yes—
213C22⋊C4×C73—24no—
214C22⋊C4×C73—42no—
215C42×C73—22no—
216C42×C73—22no—
217C42×C73—21no—
218C42×C73—21no—
219C42×C73—42no—
220C42×C73—42no—
221C42×C73C2×D412yes—
222C42×C73#4112yes—
223C23×C73—21no—
224C23×C73—21no—
225C23×C73—21no—
226C23×C73—21no—
227C23×C73C4×D412yes—
228C23×C73#13712yes—
229C23×C73#13712yes—
230C23×C73C22×D411yescharacteristic
231Q8×C73—22no—
232Q8×C73—22no—
233Q8×C73—41no—
234Q8×C73—41no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—22no—
242C2×C4×C73—22no—
243C2×C4×C73—22no—
244C2×C4×C73—22no—
245C2×C4×C73—22no—
246C2×C4×C73—22no—
247C2×C4×C73—21no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—21no—
251C2×C4×C73—21no—
252C2×C4×C73—21no—
253C2×C4×C73—44no—
254C2×C4×C73—44no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73—41no—
260C2×C4×C73—41no—
261C2×C4×C73—41no—
262C2×C4×C73—41no—
263C2×C4×C73—41no—
264C2×C4×C73—41no—
265C2×C4×C73C4×D412yes—
266C2×C4×C73#12112yes—
267C2×C4×C73#12112yes—
268C2×C4×C73C23×C411yescharacteristic
269C2×C4×C73C22×D411yescharacteristic
270C2×C4×C73C2×#4111yescharacteristic
271C2×C4×C73C2×#4111yescharacteristic
272C2×C4×C73C2×#4111yescharacteristic
273C2×C4×C73C2×#4111yescharacteristic
274C2×Dic73—22no—
275C2×Dic73—22no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283C2×Dic73—41no—
284C2×Dic73—41no—
285C2×Dic73—41no—
286Dic146—22no—
287Dic146—22no—
288Dic146—41no—
289Dic146—41no—
290#10596831—22no—
291#10596831—22no—
292#10596831—22no—
293#10596831C4×D414yes—
294C22×C73—22no—
295C22×C73—22no—
296C22×C73—22no—
297C22×C73—22no—
298C22×C73—22no—
299C22×C73—22no—
300C22×C73#44114yescomplemented
301C22×C73C2×C4×D411yescharacteristic
302C22×C73C2×#12111yescharacteristic
303C22×C73#53211yescharacteristic
304C22×C73C2×#13711yescharacteristic
305C22×C73C2×#13711yescharacteristic
306C22×C73#57511yescharacteristic
307C22×C73#57511yescharacteristic
308C4×C73—22no—
309C4×C73—22no—
310C4×C73—22no—
311C4×C73—42no—
312C4×C73—41no—
313C4×C73—41no—
314C4×C73—42no—
315C4×C73—81no—
316C4×C73—81no—
317C4×C73C2×C4×D411yescharacteristic
318C4×C73C4×#4111yescharacteristic
319C4×C73#52511yescharacteristic
320C4×C73#52911yescharacteristic
321Dic73—42no—
322Dic73—42no—
323Dic73—81no—
324Dic73—81no—
325#21527—731nomaximal, Sylow
326C2×C73—24no—
327C2×C73C4×#12111yescharacteristic
328C2×C73C2×#44111yescharacteristic
329C2×C73#294311yescharacteristic
330C2×C73#305611yescharacteristic
331C2×C73#305611yescharacteristic
332C2×C73#318111yescharacteristic
333C2×C73#318111yescharacteristic
334C2×#467—731no—
335C2×#441—731no—
336C22×#97—731no—
337C4×C22⋊Q8—731no—
338C4×C22⋊Q8D7311yescharacteristic
339#1971—731no—
340#1943—731no—
341#1935—731no—
342#1935—731no—
343#1892—731no—
344#1892—731no—
345#1807—731no—
346#1776—731no—
347#1763—731no—
348#1763—731no—
349C73#2152711yescharacteristic, complemented, Sylow
350C22×Q16—731no—
351C23×C8—731no—
352C23×C8—731no—
353C2×C22⋊Q8—731no—
354C2×C22⋊Q8C2×D7311yescharacteristic
355C2×C4×Q8—731no—
356C2×C4×Q8C2×D7311yescharacteristic
357C22×C42C2×D7311yescharacteristic
358#467—738no—
359#441—1464no—
360C2×#107—731no—
361C2×#106—731no—
362C2×#106—731no—
363C2×#103—731no—
364C2×#103—731no—
365C2×#103—731no—
366C2×#103—731no—
367C2×#103—731no—
368C2×#103—731no—
369C2×#97—731no—
370C2×#97—731no—
371C2×#97—731no—
372C2×#97—731no—
373C2×#97—1464no—
374#391—731no—
375#391C2×D7311yescharacteristic
376#389—731no—
377#389C2×D7311yescharacteristic
378#387—21no—
379#387—1461no—
380#385—21no—
381#385—1461no—
382#380—731no—
383#380C2×D7311yescharacteristic
384C4×C4⋊C4—21no—
385C4×C4⋊C4—731no—
386C4×C4⋊C4—1461no—
387C4×C4⋊C4C2×D7311yescharacteristic
388C4×C22⋊C4—21no—
389C4×C22⋊C4—1461no—
390#340—731no—
391#340—731no—
392#340—731no—
393#340—731no—
394#340—731no—
395#340—731no—
396C22×Q8—731no—
397C22×Q8C22×D7311yescharacteristic
398C23×C4C22⋊D7311yescharacteristic
399C23×C4C22⋊D7311yescharacteristic
400C23×C4C22×D7311yescharacteristic
401C2×Q16—734no—
402C2×Q16—1462no—
403C2×Q16—1462no—
404C22×C8—731no—
405C22×C8—734no—
406C22×C8—731no—
407C22×C8—731no—
408C22×C8—731no—
409C22×C8—1462no—
410C22×C8—1462no—
411C22×C8—1462no—
412C22×C8—1462no—
413C22×C8—1462no—
414C22⋊Q8—22no—
415C22⋊Q8—734no—
416C22⋊Q8—1462no—
417C22⋊Q8C4×D7314yes—
418C4×Q8—22no—
419C4×Q8—22no—
420C4×Q8—1462no—
421C4×Q8—1462no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—731no—
426C2×C4⋊C4—731no—
427C2×C4⋊C4—731no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4—1461no—
430C2×C4⋊C4—1461no—
431C2×C4⋊C4C22×D7311yescharacteristic
432C2×C4⋊C4C22×D7311yescharacteristic
433C2×C4⋊C4C22×D7311yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—1461no—
437C2×C22⋊C4—1461no—
438C2×C42—22no—
439C2×C42—22no—
440C2×C42—21no—
441C2×C42—21no—
442C2×C42—1461no—
443C2×C42—1461no—
444C2×C42C22⋊D7311yescharacteristic
445C2×C42C22⋊D7311yescharacteristic
446C2×C42C22×D7311yescharacteristic
447C2×C42C22×D7311yescharacteristic
448#107—734no—
449#106—1462no—
450#106—1462no—
451#103—734no—
452#103—734no—
453#103—1462no—
454#103—1462no—
455#103—1462no—
456#103—1462no—
457#97—734no—
458#97—1462no—
459#97—1462no—
460#97—1462no—
461#95—21no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—1461no—
467#95—1461no—
468#95—1461no—
469#95—1461no—
470#95—1461no—
471C24C2×C22⋊D7311yescharacteristic
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—732no—
478C2×Q8—1461no—
479C2×Q8—1461no—
480C2×Q8—1461no—
481C2×Q8—1461no—
482C2×Q8—1461no—
483C2×Q8C2×C4×D7312yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—41no—
497C22×C4—41no—
498C22×C4—1461no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4—1461no—
502C22×C4C2×C4×D7312yes—
503C22×C4C2×C22⋊D7311yescharacteristic
504C22×C4C2×C22⋊D7311yescharacteristic
505C22×C4C2×C22⋊D7311yescharacteristic
506C22×C4C2×C22⋊D7311yescharacteristic
507C22×C4C2×C22⋊D7311yescharacteristic
508C22×C4#1192402812yes—
509C22×C4C23×D7311yescharacteristic
510Q16—1468no—
511C2×C8—732no—
512C2×C8—732no—
513C2×C8—732no—
514C2×C8—732no—
515C2×C8—732no—
516C2×C8—732no—
517C2×C8—1468no—
518C2×C8—1468no—
519C2×C8—1461no—
520C2×C8—1461no—
521C2×C8—1461no—
522C2×C8—1461no—
523C2×C8—1461no—
524C2×C8—1461no—
525C4⋊C4—24no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—2922no—
534C4⋊C4—2922no—
535C4⋊C4—732no—
536C4⋊C4—1461no—
537C4⋊C4—1461no—
538C4⋊C4—1461no—
539C4⋊C4—1461no—
540C4⋊C4—1464no—
541C4⋊C4—1461no—
542C4⋊C4C2×C4×D7312yes—
543C22⋊C4—24no—
544C22⋊C4—42no—
545C22⋊C4—2922no—
546C22⋊C4—1464no—
547C42—22no—
548C42—22no—
549C42—21no—
550C42—21no—
551C42—42no—
552C42—42no—
553C42—2922no—
554C42—2922no—
555C42D4×D7312yes—
556C42#1192402712yes—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23C4×C22⋊D7312yes—
562C23#1455662112yes—
563C23#1455662212yes—
564C23C22×C22⋊D7311yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569Q8—2921no—
570Q8—2921no—
571Q8—1462no—
572Q8—1462no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—2921no—
604C2×C4—2921no—
605C2×C4—2921no—
606C2×C4—2921no—
607C2×C4—2921no—
608C2×C4—2921no—
609C2×C4—2921no—
610C2×C4—2921no—
611C2×C4—2921no—
612C2×C4—2921no—
613C2×C4—1462no—
614C2×C4—1462no—
615C2×C4C4×C22⋊D7312yes—
616C2×C4#1455656312yes—
617C2×C4#1455656312yes—
618C2×C4C22×C4×D7311yescharacteristic
619C2×C4C2×D4×D7311yescharacteristic
620C2×C4#41×D7311yescharacteristic
621C2×C4#41×D7311yescharacteristic
622C2×C4C2×#1192402711yescharacteristic
623C2×C4C2×#1192402811yescharacteristic
624C8—734no—
625C8—1462no—
626C8—1462no—
627C8—1462no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22#1798869214yescomplemented
635C22C2×C4×C22⋊D7311yescharacteristic
636C22C2×#1455656311yescharacteristic
637C22#1798913311yescharacteristic
638C22C2×#1455662111yescharacteristic
639C22C2×#1455662211yescharacteristic
640C22#1798938511yescharacteristic
641C22#1798938511yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—42no—
646C4—41no—
647C4—41no—
648C4—42no—
649C4—81no—
650C4—81no—
651C4—2922no—
652C4—2922no—
653C4—5841no—
654C4—5841no—
655C4C4×D4×D7311yescharacteristic
656C4C4×#1192402711yescharacteristic
657C4#1798909611yescharacteristic
658C4#1798911711yescharacteristic
659C2—24no—
660C2C4×#1455656311yescharacteristic
661C2C2×#1798869211yescharacteristic
662C2#2261872911yescharacteristic
663C2#2261955711yescharacteristic
664C2#2261956311yescharacteristic
665C2#2262063411yescharacteristic
666C2#2262064111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC4×C22⋊Q8×C73C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
cosocleC22×C4×C73C24
hypercenterC4×C22⋊Q8D73
hypocenterC73#21527
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#17988692C2semidirect
5. #22610801C2semidirect
6. #22610666C2semidirect
7. #22610624C2semidirect
8. #22610620C2semidirect
12. #22609829C2semidirect
13. #22609764C2semidirect
14. #22609761C2semidirect
21. #17988826C4semidirect
65. C2×#11924018C2×C4semidirect
90. #14556524C2×C4semidirect
95. #14556515C2×C4semidirect
98. #14556514C2×C4semidirect
300. C22×C73#441semidirect
349. C73#21527semidirect
634. C22#17988692semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179888262
C2×#179886922
C22×#145565022
C4×#145565702
#226108012
#226106662
#226106242
#226106202
#226103552
#226103522
#226099542
#226098292
#226097642
#226097612
C4×C22⋊Q8×C732
#2152773