Finite Group Databasebeta

← #33122188

Subgroups — #33122188

Order 18688.

1012 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×#17988891C211yesmaximal, characteristic, complemented
2C2×#17988879C211yesmaximal, characteristic, complemented
3#22612613C211yesmaximal, characteristic
4#22611266C211yesmaximal, characteristic, complemented
5#22611256C211yesmaximal, characteristic, complemented
6#22611079C211yesmaximal, characteristic, complemented
7#22611069C211yesmaximal, characteristic, complemented
8#22610889C211yesmaximal, characteristic, complemented
9#22610884C211yesmaximal, characteristic, complemented
10#22610243C211yesmaximal, characteristic
11#22610238C211yesmaximal, characteristic
12#22610130C211yesmaximal, characteristic
13#22610119C211yesmaximal, characteristic
14#22609915C211yesmaximal, characteristic
15C2×#544×C73C211yesmaximal, characteristic
16C2×#14556570C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18#17988891—24no—
19#17988879—24no—
20C2×#14556524C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556524C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×C4⋊#10596831C2211yescharacteristic
25C2×C4⋊#10596831C2211yescharacteristic
26C2×#14556514C2211yescharacteristic
27C2×#14556514C2211yescharacteristic
28C2×#14556514C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×C22⋊#10596831C2211yescharacteristic
31C2×C22⋊#10596831C2211yescharacteristic
32#17988493C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988482—21no—
35#17988481—21no—
36#17988475—21no—
37#17988468—21no—
38#17988447C2211yescharacteristic
39C4×C4⋊Dic73C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988342C2211yescharacteristic
49#17988342C2211yescharacteristic
50#544×C73—24no—
51#544×C73—24no—
52C2×C4⋊Q8×C73—21no—
53C2×C4⋊Q8×C73C2211yescharacteristic
54C2×C4⋊Q8×C73C2211yescharacteristic
55C2×#125×C73—21no—
56C2×#125×C73C2211yescharacteristic
57C2×#125×C73C2211yescharacteristic
58C2×C22⋊Q8×C73—21no—
59C2×C22⋊Q8×C73—21no—
60C2×#117×C73C2211yescharacteristic
61C2×#117×C73C2211yescharacteristic
62C22×C4⋊C4×C73C2211yescharacteristic
63C22×Dic146C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68#14556570—22no—
69#14556570Q814yescomplemented
70C4×Dic146—22no—
71C4×Dic146—22no—
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×C22⋊Dic73—21no—
79C2×C22⋊Dic73—21no—
80C2×C4×Dic73—21no—
81C2×C4×Dic73—21no—
82#14556524—22no—
83#14556524—22no—
84#14556524—22no—
85#14556522—22no—
86C4⋊#10596831—22no—
87C4⋊#10596831—22no—
88#14556514—22no—
89#14556514—22no—
90#14556514—22no—
91#14556514—22no—
92C22⋊#10596831—22no—
93C22⋊#10596831—22no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99C22×Q8×C73—21no—
100C22×Q8×C73—21no—
101C23×C4×C73—21no—
102C23×C4×C73C2311yescharacteristic
103C4⋊Q8×C73—28no—
104C4⋊Q8×C73—22no—
105C4⋊Q8×C73—22no—
106C4⋊Q8×C73—22no—
107C4⋊Q8×C73D414yes—
108#125×C73—28no—
109#125×C73—22no—
110#125×C73—22no—
111#125×C73—22no—
112#125×C73D414yes—
113C22⋊Q8×C73—28no—
114C22⋊Q8×C73—28no—
115#117×C73—22no—
116#117×C73—22no—
117#117×C73—22no—
118#117×C73—22no—
119C2×C4⋊C4×C73—21no—
120C2×C4⋊C4×C73—21no—
121C2×C4⋊C4×C73—21no—
122C2×C4⋊C4×C73—21no—
123C2×C4⋊C4×C73—21no—
124C2×C4⋊C4×C73—21no—
125C2×C4⋊C4×C73—21no—
126C2×C4⋊C4×C73—22no—
127C2×C4⋊C4×C73—22no—
128C2×C4⋊C4×C73D411yescharacteristic
129C2×C4⋊C4×C73D411yescharacteristic
130C2×C4⋊C4×C73C2311yescharacteristic
131C2×C4⋊C4×C73C2311yescharacteristic
132C2×C4⋊C4×C73C2311yescharacteristic
133C2×C4⋊C4×C73C2311yescharacteristic
134C2×C22⋊C4×C73—21no—
135C2×C22⋊C4×C73D411yescharacteristic
136C2×C22⋊C4×C73D411yescharacteristic
137C2×C42×C73—21no—
138C2×C42×C73C2311yescharacteristic
139C2×C42×C73C2311yescharacteristic
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146—21no—
143C2×Dic146—21no—
144C2×Dic146—21no—
145C2×Dic146C2×Q812yes—
146C22×Dic73—21no—
147C22×Dic73—21no—
148C22×Dic73—21no—
149C22×Dic73—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—21no—
166C4⋊Dic73—21no—
167C4⋊Dic73C2×Q812yes—
168#11924007—24no—
169#11924007—42no—
170#11924007—81no—
171C22⋊Dic73—24no—
172C22⋊Dic73—81no—
173C4×Dic73—42no—
174C4×Dic73—81no—
175C24×C73C2×D411yescharacteristic
176C2×Q8×C73—24no—
177C2×Q8×C73—24no—
178C2×Q8×C73—81no—
179C2×Q8×C73—81no—
180C2×Q8×C73—81no—
181C2×Q8×C73—81no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73—24no—
191C22×C4×C73—81no—
192C22×C4×C73—81no—
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73C2×D411yescharacteristic
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2×D411yescharacteristic
198C22×C4×C73C2×Q812yes—
199C22×C4×C73#4112yes—
200C22×C4×C73C2411yescharacteristic
201C4⋊C4×C73—24no—
202C4⋊C4×C73—24no—
203C4⋊C4×C73—24no—
204C4⋊C4×C73—24no—
205C4⋊C4×C73—24no—
206C4⋊C4×C73—24no—
207C4⋊C4×C73—24no—
208C4⋊C4×C73—22no—
209C4⋊C4×C73—22no—
210C4⋊C4×C73—21no—
211C4⋊C4×C73—21no—
212C4⋊C4×C73—21no—
213C4⋊C4×C73—21no—
214C4⋊C4×C73C2×D412yes—
215C4⋊C4×C73C2×D412yes—
216C4⋊C4×C73C2×D412yes—
217C4⋊C4×C73C2×D412yes—
218C22⋊C4×C73—24no—
219C22⋊C4×C73—22no—
220C22⋊C4×C73—22no—
221C42×C73—24no—
222C42×C73—21no—
223C42×C73—21no—
224C42×C73C2×D412yes—
225C42×C73C2×D412yes—
226C23×C73—21no—
227C23×C73—21no—
228C23×C73—21no—
229C23×C73—21no—
230C23×C73—21no—
231C23×C73—21no—
232C23×C73C22⋊Q812yes—
233C23×C73C22×D411yescharacteristic
234Q8×C73—84no—
235Q8×C73—84no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—22no—
243C2×C4×C73—22no—
244C2×C4×C73—22no—
245C2×C4×C73—22no—
246C2×C4×C73—21no—
247C2×C4×C73—22no—
248C2×C4×C73—21no—
249C2×C4×C73—22no—
250C2×C4×C73—22no—
251C2×C4×C73—22no—
252C2×C4×C73—21no—
253C2×C4×C73—21no—
254C2×C4×C73—21no—
255C2×C4×C73—21no—
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—84no—
265C2×C4×C73#12012yes—
266C2×C4×C73#12012yes—
267C2×C4×C73C22⋊Q812yes—
268C2×C4×C73C22×D411yescharacteristic
269C2×C4×C73C22×D411yescharacteristic
270C2×C4×C73C22×Q811yescharacteristic
271C2×C4×C73C2×#4111yescharacteristic
272C2×C4×C732−1+411yescharacteristic
273C2×C4×C732−1+411yescharacteristic
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—41no—
291#10596831—41no—
292#10596831—41no—
293#10596831—41no—
294C22×C73—22no—
295C22×C73—22no—
296C22×C73—22no—
297C22×C73—41no—
298C22×C73—41no—
299C22×C73—41no—
300C22×C73—41no—
301C22×C73C2×#12011yescharacteristic
302C22×C73C2×C22⋊Q811yescharacteristic
303C22×C73#53611yescharacteristic
304C22×C73#57611yescharacteristic
305C22×C73#57711yescharacteristic
306C22×C73#57811yescharacteristic
307C22×C73#57811yescharacteristic
308C4×C73—22no—
309C4×C73—22no—
310C4×C73—22no—
311C4×C73—41no—
312C4×C73—42no—
313C4×C73—41no—
314C4×C73—42no—
315C4×C73—42no—
316C4×C73—42no—
317C4×C73#54811yescharacteristic
318C4×C73#54811yescharacteristic
319C4×C73D4×Q811yescharacteristic
320C4×C73D4×Q811yescharacteristic
321Dic73—42no—
322Dic73—42no—
323Dic73—81no—
324Dic73—81no—
325#24311—731nomaximal, Sylow
326C2×C73—42no—
327C2×C73#240911yescharacteristic
328C2×C73#301111yescharacteristic
329C2×C73#301211yescharacteristic
330C2×C73#308211yescharacteristic
331C2×C73#308311yescharacteristic
332C2×C73#324211yescharacteristic
333C2×C73#324311yescharacteristic
334C2×#544D7311yescharacteristic
335C2×#479—731no—
336C2×#477—731no—
337#2323—731no—
338#2059—731no—
339#2057—731no—
340#2026—731no—
341#2024—731no—
342#1989—731no—
343#1988—731no—
344#1873—731no—
345#1872—731no—
346#1854—731no—
347#1852—731no—
348#1798—731no—
349C73#2431111yescharacteristic, complemented, Sylow
350#544—24no—
351#544—24no—
352C2×C4⋊Q8—21no—
353C2×C4⋊Q8C2×D7311yescharacteristic
354C2×C4⋊Q8C2×D7311yescharacteristic
355C2×#125—21no—
356C2×#125C2×D7311yescharacteristic
357C2×#125C2×D7311yescharacteristic
358C2×C22⋊Q8—21no—
359C2×C22⋊Q8—21no—
360C2×C22⋊Q8—731no—
361C2×C4×Q8—731no—
362C2×#117C2×D7311yescharacteristic
363C2×#117C2×D7311yescharacteristic
364C22×C4⋊C4C2×D7311yescharacteristic
365#479—1464no—
366#477—1464no—
367C2×#107—731no—
368C2×#107—731no—
369C2×#107—731no—
370C2×#106—731no—
371C2×C4⋊C8—731no—
372C2×C4⋊C8—731no—
373C2×#103—731no—
374C2×#103—731no—
375C2×#103—731no—
376C2×#103—731no—
377C2×C22⋊C8—731no—
378C2×C22⋊C8—731no—
379#391—731no—
380#389—731no—
381#389—1461no—
382#388—1461no—
383#388—1461no—
384#387—1461no—
385#380—731no—
386C4×C4⋊C4—731no—
387#340—731no—
388#340—731no—
389#340—731no—
390#340—731no—
391#340—731no—
392#340—731no—
393#340—731no—
394#340—731no—
395#336—731no—
396#336—731no—
397C22×Q8—21no—
398C22×Q8—21no—
399C22×Q8—731no—
400C23×C4—21no—
401C23×C4C22×D7311yescharacteristic
402C22×C8—731no—
403C22×C8—731no—
404C22×C8—731no—
405C22×C8—731no—
406C4⋊Q8—28no—
407C4⋊Q8—22no—
408C4⋊Q8—22no—
409C4⋊Q8—22no—
410C4⋊Q8C22⋊D7314yes—
411#125—28no—
412#125—22no—
413#125—22no—
414#125—22no—
415#125C22⋊D7314yes—
416C22⋊Q8—28no—
417C22⋊Q8—28no—
418C22⋊Q8—734no—
419C22⋊Q8—1462no—
420C4×Q8—1462no—
421C4×Q8—1462no—
422#117—22no—
423#117—22no—
424#117—22no—
425#117—22no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—22no—
434C2×C4⋊C4—22no—
435C2×C4⋊C4—731no—
436C2×C4⋊C4—731no—
437C2×C4⋊C4—731no—
438C2×C4⋊C4—1461no—
439C2×C4⋊C4—1461no—
440C2×C4⋊C4—1461no—
441C2×C4⋊C4C22⋊D7311yescharacteristic
442C2×C4⋊C4C22⋊D7311yescharacteristic
443C2×C4⋊C4C22×D7311yescharacteristic
444C2×C4⋊C4C22×D7311yescharacteristic
445C2×C4⋊C4C22×D7311yescharacteristic
446C2×C4⋊C4C22×D7311yescharacteristic
447C2×C22⋊C4—21no—
448C2×C22⋊C4—1461no—
449C2×C22⋊C4—1461no—
450C2×C22⋊C4C22⋊D7311yescharacteristic
451C2×C22⋊C4C22⋊D7311yescharacteristic
452C2×C42—21no—
453C2×C42—1461no—
454C2×C42—1461no—
455C2×C42C22×D7311yescharacteristic
456C2×C42C22×D7311yescharacteristic
457#107—1462no—
458#107—1462no—
459#107—1462no—
460#106—1462no—
461C4⋊C8—1462no—
462C4⋊C8—1462no—
463#103—1462no—
464#103—1462no—
465#103—1462no—
466#103—1462no—
467C22⋊C8—1462no—
468C22⋊C8—1462no—
469#95—1461no—
470#95—1461no—
471#95—1461no—
472#95—1461no—
473#95—1461no—
474C24C2×C22⋊D7311yescharacteristic
475C2×Q8—24no—
476C2×Q8—24no—
477C2×Q8—81no—
478C2×Q8—81no—
479C2×Q8—81no—
480C2×Q8—81no—
481C2×Q8—732no—
482C2×Q8—1461no—
483C2×Q8—1461no—
484C2×Q8—1461no—
485C2×Q8—1461no—
486C2×Q8—1461no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—24no—
496C22×C4—81no—
497C22×C4—81no—
498C22×C4—1461no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4—1461no—
502C22×C4C2×C22⋊D7311yescharacteristic
503C22×C4C2×C22⋊D7311yescharacteristic
504C22×C4C2×C22⋊D7311yescharacteristic
505C22×C4C2×C22⋊D7311yescharacteristic
506C22×C4C2×C22⋊D7311yescharacteristic
507C22×C4Q8×D7312yes—
508C22×C4#1192402912yes—
509C22×C4C23×D7311yescharacteristic
510C2×C8—1461no—
511C2×C8—1461no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8—1461no—
521C2×C8—1461no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—2922no—
536C4⋊C4—5841no—
537C4⋊C4—732no—
538C4⋊C4—1461no—
539C4⋊C4—1461no—
540C4⋊C4—1461no—
541C4⋊C4—1461no—
542C4⋊C4—1464no—
543C4⋊C4—1461no—
544C4⋊C4C2×C22⋊D7312yes—
545C4⋊C4C2×C22⋊D7312yes—
546C4⋊C4C2×C22⋊D7312yes—
547C4⋊C4C2×C22⋊D7312yes—
548C22⋊C4—24no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—5841no—
552C22⋊C4—1464no—
553C42—24no—
554C42—21no—
555C42—21no—
556C42—2922no—
557C42—5841no—
558C42C2×C22⋊D7312yes—
559C42C2×C22⋊D7312yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23#1455656712yes—
567C23C22×C22⋊D7311yescharacteristic
568Q8—84no—
569Q8—84no—
570Q8—2921no—
571Q8—2921no—
572Q8—1462no—
573Q8—1462no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
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—84no—
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×C4#1455656012yes—
616C2×C4#1455656012yes—
617C2×C4#1455656712yes—
618C2×C4C22×C22⋊D7311yescharacteristic
619C2×C4C22×C22⋊D7311yescharacteristic
620C2×C4C2×Q8×D7311yescharacteristic
621C2×C4C2×#1192402911yescharacteristic
622C2×C4#1455663911yescharacteristic
623C2×C4#1455663911yescharacteristic
624C8—2921no—
625C8—2921no—
626C8—2921no—
627C8—2921no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22C2×#1455656011yescharacteristic
636C22C2×#1455656711yescharacteristic
637C22#1798916111yescharacteristic
638C22#1798939011yescharacteristic
639C22#1798939811yescharacteristic
640C22#1798940211yescharacteristic
641C22#1798940211yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—41no—
646C4—42no—
647C4—41no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—2922no—
652C4—2922no—
653C4—5841no—
654C4—5841no—
655C4#1798924811yescharacteristic
656C4#1798924811yescharacteristic
657C4Q8×C22⋊D7311yescharacteristic
658C4Q8×C22⋊D7311yescharacteristic
659C2—42no—
660C2#2261329011yescharacteristic
661C2#2261922611yescharacteristic
662C2#2261923511yescharacteristic
663C2#2261976711yescharacteristic
664C2#2261977811yescharacteristic
665C2#2262119911yescharacteristic
666C2#2262121411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#544×C73C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenterC2×#544D73
hypocenterC73#24311
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
1. C2×#17988891C2semidirect
2. C2×#17988879C2semidirect
4. #22611266C2semidirect
5. #22611256C2semidirect
6. #22611079C2semidirect
7. #22611069C2semidirect
8. #22610889C2semidirect
9. #22610884C2semidirect
69. #14556570Q8semidirect
349. C73#24311semidirect

Maximal subgroups

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

subgroupindex
C2×#179888912
C2×#179888792
#226126132
#226112662
#226112562
#226110792
#226110692
#226108892
#226108842
#226102432
#226102382
#226101302
#226101192
#226099152
C2×#544×C732
#2431173