Finite Group Databasebeta

← #33143143

Subgroups — #33143143

Order 18688.

946 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988891C211yesmaximal, characteristic, complemented
2C22×#14556514C211yesmaximal, characteristic
3C22⋊C4×Dic146C211yesmaximal, characteristic
4#22611482C211yesmaximal, characteristic, complemented
5#22611323C211yesmaximal, characteristic, complemented
6#22611309C211yesmaximal, characteristic, complemented
7#22611079C211yesmaximal, characteristic, complemented
8#22610198C211yesmaximal, characteristic, complemented
9#22610184C211yesmaximal, characteristic, complemented
10#22610119C211yesmaximal, characteristic
11#22610111C211yesmaximal, characteristic
12#22609954C211yesmaximal, characteristic
13#22609943C211yesmaximal, characteristic, complemented
14#22609865C211yesmaximal, characteristic
15C73×#2442C211yesmaximal, characteristic
16C23×Dic146C2211yescharacteristic
17C23×#10596831C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20#17988891—24no—
21C2×#14556524C2211yescharacteristic
22C2×#14556524C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×C4⋊#10596831C2211yescharacteristic
26C2×C4⋊#10596831C2211yescharacteristic
27C2×#14556514—24no—
28C2×#14556514C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556514C2211yescharacteristic
31C2×#14556514C2211yescharacteristic
32C2×#14556514C2211yescharacteristic
33C2×#14556514C2211yescharacteristic
34C2×#14556514C2211yescharacteristic
35C2×#14556514C2211yescharacteristic
36C2×C22⋊#10596831C2211yescharacteristic
37#17988493C2211yescharacteristic
38#17988493C2211yescharacteristic
39#17988491—21no—
40#17988491—21no—
41#17988447C2211yescharacteristic
42#17988445—21no—
43C22⋊C4×Dic73—21no—
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988342C2211yescharacteristic
49#17988342C2211yescharacteristic
50C2×#125×C73C2211yescharacteristic
51C22×C4⋊C4×C73C2211yescharacteristic
52#400×C73—21no—
53#398×C73—21no—
54#389×C73C2211yescharacteristic
55#389×C73C2211yescharacteristic
56#385×C73—21no—
57#382×C73C2211yescharacteristic
58#380×C73—21no—
59#380×C73C2211yescharacteristic
60C4×C22⋊C4×C73C2211yescharacteristic
61C22×Dic146—22no—
62C22×Dic146—22no—
63C22×Dic146—21no—
64C22×Dic146—22no—
65C22×Dic146—41no—
66C22×Dic146—41no—
67C22×Dic146Q814yes—
68C22×Dic146C2311yescharacteristic
69C22×Dic146C2311yescharacteristic
70C23×Dic73—21no—
71C22×#10596831—22no—
72C22×#10596831—22no—
73C22×#10596831—41no—
74C22×#10596831C2311yescharacteristic
75C22×#10596831C2311yescharacteristic
76C22×#10596831C2311yescharacteristic
77C22×#10596831C2311yescharacteristic
78C4×Dic146—22no—
79C4×Dic146—22no—
80C4×Dic146—41no—
81C4×Dic146—41no—
82C2×C4⋊Dic73C2311yescharacteristic
83C2×C4⋊Dic73C2311yescharacteristic
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×C22⋊Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89#14556524—22no—
90#14556524—41no—
91#14556522—22no—
92#14556522—41no—
93C4⋊#10596831—22no—
94C4⋊#10596831—41no—
95#14556514—22no—
96#14556514—22no—
97#14556514—22no—
98#14556514—22no—
99#14556514—22no—
100#14556514—41no—
101#14556514—41no—
102#14556514D414yes—
103C22⋊#10596831—22no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109C23×C4×C73—21no—
110C23×C4×C73C2311yescharacteristic
111#125×C73—22no—
112#125×C73—41no—
113C2×C4⋊C4×C73—22no—
114C2×C4⋊C4×C73—21no—
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73—21no—
117C2×C4⋊C4×C73—21no—
118C2×C4⋊C4×C73—21no—
119C2×C4⋊C4×C73—41no—
120C2×C4⋊C4×C73C2311yescharacteristic
121C2×C4⋊C4×C73C2311yescharacteristic
122C2×C4⋊C4×C73C2311yescharacteristic
123C2×C4⋊C4×C73C2311yescharacteristic
124C2×C22⋊C4×C73—21no—
125C2×C22⋊C4×C73D411yescharacteristic
126C2×C22⋊C4×C73D411yescharacteristic
127C2×C42×C73C2311yescharacteristic
128C2×C42×C73C2311yescharacteristic
129#95×C73—21no—
130#95×C73—21no—
131#95×C73—21no—
132#95×C73—21no—
133#95×C73—21no—
134#95×C73D411yescharacteristic
135#95×C73D411yescharacteristic
136C2×Dic146—21no—
137C2×Dic146—21no—
138C2×Dic146—21no—
139C2×Dic146—21no—
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146—21no—
143C2×Dic146—28no—
144C2×Dic146—24no—
145C2×Dic146—21no—
146C2×Dic146—44no—
147C2×Dic146—44no—
148C2×Dic146—44no—
149C2×Dic146—42no—
150C2×Dic146—42no—
151C2×Dic146C2×D412yes—
152C2×Dic146C2×Q812yes—
153C2×Dic146C2×Q812yes—
154C2×Dic146#4112yes—
155C22×Dic73—21no—
156C22×Dic73—24no—
157C22×Dic73—21no—
158C22×Dic73—21no—
159C22×Dic73—21no—
160C22×Dic73—42no—
161C22×Dic73—42no—
162C2×#10596831—21no—
163C2×#10596831—21no—
164C2×#10596831—21no—
165C2×#10596831—21no—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C2×#10596831—21no—
169C2×#10596831—21no—
170C2×#10596831—21no—
171C2×#10596831—21no—
172C2×#10596831—21no—
173C2×#10596831—44no—
174C2×#10596831C2×D412yes—
175C4⋊Dic73—21no—
176C4⋊Dic73—21no—
177C4⋊Dic73—21no—
178C4⋊Dic73—21no—
179#11924007—42no—
180#11924007—42no—
181C22⋊Dic73—42no—
182C4×Dic73—42no—
183C4×Dic73—42no—
184C24×C73C2×D411yescharacteristic
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73—21no—
192C22×C4×C73—21no—
193C22×C4×C73—42no—
194C22×C4×C73—42no—
195C22×C4×C73—81no—
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2×D411yescharacteristic
198C22×C4×C73C2×D411yescharacteristic
199C22×C4×C73C2×D411yescharacteristic
200C22×C4×C73C2×D411yescharacteristic
201C22×C4×C73C2×Q812yes—
202C22×C4×C73#4112yes—
203C22×C4×C73C2411yescharacteristic
204C4⋊C4×C73—21no—
205C4⋊C4×C73—21no—
206C4⋊C4×C73—21no—
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73—21no—
210C4⋊C4×C73—21no—
211C4⋊C4×C73—42no—
212C4⋊C4×C73—42no—
213C4⋊C4×C73—42no—
214C4⋊C4×C73—81no—
215C4⋊C4×C73—81no—
216C4⋊C4×C73C2×D412yes—
217C22⋊C4×C73—22no—
218C22⋊C4×C73—22no—
219C22⋊C4×C73—42no—
220C42×C73—21no—
221C42×C73—21no—
222C42×C73—21no—
223C42×C73—21no—
224C23×C73—21no—
225C23×C73—21no—
226C23×C73—21no—
227C23×C73—21no—
228C23×C73C22⋊Q812yes—
229C23×C73#13512yes—
230C23×C73#13712yes—
231C23×C73C22×D411yescharacteristic
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—22no—
235C2×C4×C73—22no—
236C2×C4×C73—22no—
237C2×C4×C73—22no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—22no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—21no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
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—84no—
263C2×C4×C73C22⋊Q812yes—
264C2×C4×C73C22×D411yescharacteristic
265C2×C4×C73C22×Q811yescharacteristic
266C2×C4×C73C2×#4111yescharacteristic
267C2×C4×C73C2×#4111yescharacteristic
268C2×C4×C73H(5,𝔽2)11yescharacteristic
269C2×C4×C73H(5,𝔽2)11yescharacteristic
270C2×Dic73—22no—
271C2×Dic73—22no—
272C2×Dic73—22no—
273C2×Dic73—22no—
274C2×Dic73—48no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283Dic146—22no—
284Dic146—22no—
285Dic146—22no—
286Dic146—22no—
287Dic146—22no—
288Dic146—48no—
289Dic146—41no—
290Dic146—41no—
291Dic146C22⋊Q814yes—
292#10596831—22no—
293#10596831—41no—
294#10596831—41no—
295#10596831—41no—
296C22×C73—22no—
297C22×C73—22no—
298C22×C73—22no—
299C22×C73—22no—
300C22×C73—22no—
301C22×C73—22no—
302C22×C73#47914yescomplemented
303C22×C73C2×C22⋊Q811yescharacteristic
304C22×C73D4211yescharacteristic
305C22×C73#54611yescharacteristic
306C22×C73C2×#13511yescharacteristic
307C22×C73C2×#13711yescharacteristic
308C22×C73#57711yescharacteristic
309C22×C73#57811yescharacteristic
310C4×C73—22no—
311C4×C73—41no—
312C4×C73—41no—
313C4×C73—42no—
314C4×C73—41no—
315C4×C73—41no—
316C4×C73—81no—
317C4×C73—81no—
318C4×C73—81no—
319C4×C73C2×C22⋊Q811yescharacteristic
320C4×C73#54311yescharacteristic
321C4×C73#54611yescharacteristic
322C4×C73#55111yescharacteristic
323Dic73—24no—
324Dic73—42no—
325Dic73—81no—
326Dic73—81no—
327#25957—731nomaximal, Sylow
328C2×C73—24no—
329C2×C73#243711yescharacteristic
330C2×C73C2×#47911yescharacteristic
331C2×C73#308211yescharacteristic
332C2×C73#327011yescharacteristic
333C2×C73#328211yescharacteristic
334C2×C73#330011yescharacteristic
335C2×C73#332011yescharacteristic
336C2×#479—731no—
337C22×#103—731no—
338#2442D7311yescharacteristic
339Q8×C22⋊C4—731no—
340#2097—731no—
341#2068—731no—
342#2066—731no—
343#2026—731no—
344#1865—731no—
345#1863—731no—
346#1852—731no—
347#1850—731no—
348#1807—731no—
349#1805—731no—
350#1787—731no—
351C73#2595711yescharacteristic, complemented, Sylow
352C23×Q8—731no—
353C23×C8—731no—
354C2×#125C2×D7311yescharacteristic
355C2×C4×Q8—731no—
356C2×C4×Q8—731no—
357C22×C4⋊C4C2×D7311yescharacteristic
358#479—1464no—
359C2×#107—731no—
360C2×#107—731no—
361C2×#106—731no—
362C2×#106—731no—
363C2×C4⋊C8—731no—
364C2×C4⋊C8—731no—
365C2×#103—731no—
366C2×#103—731no—
367C2×#103—731no—
368C2×#103—731no—
369C2×#103—731no—
370C2×#103—731no—
371C2×#103—731no—
372C2×#103—731no—
373C2×#103—1464no—
374C2×C22⋊C8—731no—
375#400—21no—
376#398—21no—
377#391—731no—
378#391—731no—
379#391—1461no—
380#391—1461no—
381#389C2×D7311yescharacteristic
382#389C2×D7311yescharacteristic
383#385—21no—
384#382C2×D7311yescharacteristic
385#380—21no—
386#380—731no—
387#380—1461no—
388#380C2×D7311yescharacteristic
389C4×C22⋊C4—1461no—
390C4×C22⋊C4C2×D7311yescharacteristic
391#340—731no—
392#340—731no—
393#340—731no—
394#340—731no—
395#336—731no—
396#336—731no—
397C22×Q8—2921no—
398C22×Q8—2921no—
399C22×Q8—734no—
400C22×Q8—731no—
401C22×Q8—731no—
402C22×Q8—1462no—
403C22×Q8—1462no—
404C22×Q8—1461no—
405C22×Q8—1462no—
406C23×C4—21no—
407C23×C4—1461no—
408C23×C4C22×D7311yescharacteristic
409C22×C8—2921no—
410C22×C8—731no—
411C22×C8—731no—
412C22×C8—731no—
413C22×C8—731no—
414C22×C8—1462no—
415C22×C8—1462no—
416#125—22no—
417#125—41no—
418C4×Q8—2921no—
419C4×Q8—2921no—
420C4×Q8—1462no—
421C4×Q8—1462no—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—41no—
429C2×C4⋊C4—731no—
430C2×C4⋊C4—731no—
431C2×C4⋊C4—1461no—
432C2×C4⋊C4—1461no—
433C2×C4⋊C4C22×D7311yescharacteristic
434C2×C4⋊C4C22×D7311yescharacteristic
435C2×C4⋊C4C22×D7311yescharacteristic
436C2×C4⋊C4C22×D7311yescharacteristic
437C2×C22⋊C4—21no—
438C2×C22⋊C4—1461no—
439C2×C22⋊C4C22⋊D7311yescharacteristic
440C2×C22⋊C4C22⋊D7311yescharacteristic
441C2×C42—1461no—
442C2×C42—1461no—
443C2×C42C22×D7311yescharacteristic
444C2×C42C22×D7311yescharacteristic
445#107—2921no—
446#107—1462no—
447#106—2921no—
448#106—1462no—
449C4⋊C8—2921no—
450C4⋊C8—1462no—
451#103—2921no—
452#103—2921no—
453#103—734no—
454#103—1462no—
455#103—1462no—
456#103—1462no—
457#103—1462no—
458#103—1462no—
459C22⋊C8—1462no—
460#95—21no—
461#95—21no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—1461no—
466#95—1461no—
467#95—1461no—
468#95—1461no—
469#95—1461no—
470#95C22⋊D7311yescharacteristic
471#95C22⋊D7311yescharacteristic
472C24C2×C22⋊D7311yescharacteristic
473C2×Q8—2924no—
474C2×Q8—2924no—
475C2×Q8—2924no—
476C2×Q8—2922no—
477C2×Q8—2922no—
478C2×Q8—732no—
479C2×Q8—732no—
480C2×Q8—732no—
481C2×Q8—732no—
482C2×Q8—1468no—
483C2×Q8—1461no—
484C2×Q8—1461no—
485C2×Q8—1461no—
486C2×Q8—1461no—
487C2×Q8—1464no—
488C2×Q8—1461no—
489C2×Q8—1461no—
490C2×Q8—1461no—
491C2×Q8—1461no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—81no—
503C22×C4—2922no—
504C22×C4—2922no—
505C22×C4—1461no—
506C22×C4—1461no—
507C22×C4—1464no—
508C22×C4—1461no—
509C22×C4—1461no—
510C22×C4C2×C22⋊D7311yescharacteristic
511C22×C4C2×C22⋊D7311yescharacteristic
512C22×C4C2×C22⋊D7311yescharacteristic
513C22×C4C2×C22⋊D7311yescharacteristic
514C22×C4C2×C22⋊D7311yescharacteristic
515C22×C4Q8×D7312yes—
516C22×C4#1192402912yes—
517C22×C4C23×D7311yescharacteristic
518C2×C8—2924no—
519C2×C8—732no—
520C2×C8—1461no—
521C2×C8—1461no—
522C2×C8—1461no—
523C2×C8—1461no—
524C2×C8—1461no—
525C2×C8—1461no—
526C2×C8—1461no—
527C2×C8—1461no—
528C2×C8—1461no—
529C2×C8—1461no—
530C2×C8—1461no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—81no—
542C4⋊C4—81no—
543C4⋊C4—2922no—
544C4⋊C4—2922no—
545C4⋊C4—1461no—
546C4⋊C4—1461no—
547C4⋊C4—1461no—
548C4⋊C4—1461no—
549C4⋊C4D4×D7312yes—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—42no—
553C22⋊C4—2922no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—2922no—
559C42—2922no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23#1455656712yes—
565C23#1455661112yes—
566C23#1455662112yes—
567C23C22×C22⋊D7311yescharacteristic
568Q8—2928no—
569Q8—2921no—
570Q8—2921no—
571Q8—734no—
572Q8—1462no—
573Q8—1462no—
574Q8—1462no—
575Q8—1462no—
576Q8—1462no—
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—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—84no—
608C2×C4—2921no—
609C2×C4—2921no—
610C2×C4—2928no—
611C2×C4—2921no—
612C2×C4—2921no—
613C2×C4—2921no—
614C2×C4—2921no—
615C2×C4—2921no—
616C2×C4—2921no—
617C2×C4—1462no—
618C2×C4—1462no—
619C2×C4—1462no—
620C2×C4—1462no—
621C2×C4#1455656712yes—
622C2×C4C2×D4×D7311yescharacteristic
623C2×C4C2×Q8×D7311yescharacteristic
624C2×C4#41×D7311yescharacteristic
625C2×C4C2×#1192402911yescharacteristic
626C2×C4#1455663611yescharacteristic
627C2×C4#1455663711yescharacteristic
628C8—2921no—
629C8—2921no—
630C8—2921no—
631C8—1462no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22#1798889114yescomplemented
639C22C2×#1455656711yescharacteristic
640C22D4×C22⋊D7311yescharacteristic
641C22#1798922511yescharacteristic
642C22C2×#1455661111yescharacteristic
643C22C2×#1455662111yescharacteristic
644C22#1798939811yescharacteristic
645C22#1798940211yescharacteristic
646C4—22no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—42no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—2922no—
656C4—5841no—
657C4—5841no—
658C4—1464no—
659C4C22⋊Q8×D7311yescharacteristic
660C4#1798921211yescharacteristic
661C4#1798922811yescharacteristic
662C4#1798926411yescharacteristic
663C2—24no—
664C2#2261359411yescharacteristic
665C2C2×#1798889111yescharacteristic
666C2#2261976711yescharacteristic
667C2#2262149711yescharacteristic
668C2#2262163411yescharacteristic
669C2#2262181211yescharacteristic
670C2#2262209611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2442C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#2442D73
hypocenterC73#25957
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
1. C2×#17988891C2semidirect
4. #22611482C2semidirect
5. #22611323C2semidirect
6. #22611309C2semidirect
7. #22611079C2semidirect
8. #22610198C2semidirect
9. #22610184C2semidirect
13. #22609943C2semidirect
302. C22×C73#479semidirect
351. C73#25957semidirect
638. C22#17988891semidirect

Maximal subgroups

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

subgroupindex
C2×#179888912
C22×#145565142
C22⋊C4×Dic1462
#226114822
#226113232
#226113092
#226110792
#226101982
#226101842
#226101192
#226101112
#226099542
#226099432
#226098652
C73×#24422
#2595773