Finite Group Databasebeta

← #34018642

Subgroups — #34018642

Order 18688.

624 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626748C211yesmaximal, characteristic
2Q8×#11924036C211yesmaximal, characteristic, complemented
3#22625543C211yesmaximal, characteristic, complemented
4#22625542C211yesmaximal, characteristic, complemented
5#22625539C211yesmaximal, characteristic, complemented
6#22625538C211yesmaximal, characteristic, complemented
7#22625526C211yesmaximal, characteristic, complemented
8#22625525C211yesmaximal, characteristic, complemented
9#22625523C211yesmaximal, characteristic, complemented
10#22625521C211yesmaximal, characteristic, complemented
11#22625505C211yesmaximal, characteristic, complemented
12#22625274C211yesmaximal, characteristic, complemented
13#22625249C211yesmaximal, characteristic
14#22625143C211yesmaximal, characteristic, complemented
15#22620016C211yesmaximal, characteristic
16#17989637C2211yescharacteristic
17#17989636—21no—
18#17989636—21no—
19#17989636C2211yescharacteristic
20#17989636C2211yescharacteristic
21#17989636C2211yescharacteristic
22Q8×#10596833C2211yescharacteristic
23Q8×#10596833C2211yescharacteristic
24#17989621C2211yescharacteristic
25#17989617—21no—
26#17989615—21no—
27#17989615C2211yescharacteristic
28#17989612—21no—
29#17989609C2211yescharacteristic
30#17989588—21no—
31#17989588C2211yescharacteristic
32#17989552—21no—
33C4×#11924038C2211yescharacteristic
34C4×#11924036—21no—
35C4×#11924036C2211yescharacteristic
36#17989451C2211yescharacteristic
37#17989450C2211yescharacteristic
38#17989449C2211yescharacteristic
39#17989448C2211yescharacteristic
40#17989437C2211yescharacteristic
41#17989437C2211yescharacteristic
42#17989437C2211yescharacteristic
43#17989437C2211yescharacteristic
44Q8⋊#10596833C2211yescharacteristic
45Q8⋊#10596833C2211yescharacteristic
46#17989435C2211yescharacteristic
47#17989435C2211yescharacteristic
48#17989429C2211yescharacteristic
49#17989429C2211yescharacteristic
50#17989428C2211yescharacteristic
51#17989428C2211yescharacteristic
52#17989122C2211yescharacteristic
53C4×Q8×D73C2211yescharacteristic
54#17988919C411yescharacteristic
55#17988902C411yescharacteristic
56#17988891C411yescharacteristic
57#17988810C411yescharacteristic
58#17988809C411yescharacteristic
59#17988744C411yescharacteristic
60#17988727C411yescharacteristic
61#17988681C2211yescharacteristic
62#17988679C2211yescharacteristic
63#17988674C2211yescharacteristic
64#17988672C2211yescharacteristic
65#17988639C2211yescharacteristic
66#479×C73C411yescharacteristic
67C2×#11924038—21no—
68C2×#11924038—21no—
69C2×#11924036—21no—
70C2×#11924036—21no—
71C2×#11924036D411yescharacteristic
72C2×#11924036D411yescharacteristic
73#14556658—21no—
74#14556658—21no—
75#14556658—21no—
76#14556658—21no—
77C4⋊#10596833—21no—
78C4⋊#10596833—21no—
79C4⋊#10596833—21no—
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84#14556648—21no—
85#14556648—21no—
86#14556648—21no—
87#14556646—21no—
88#14556646—21no—
89#14556646—21no—
90C4×#10596833—21no—
91C4×#10596833—21no—
92C4×#10596833—21no—
93C4×#10596833—21no—
94C4×#10596833—21no—
95C4×#10596833D411yescharacteristic
96C4×#10596833D411yescharacteristic
97C4×#10596833C2311yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C2×Q8×D73C2×C411yescharacteristic
102C2×Dic292C2×C411yescharacteristic
103C2×#11924018C2×C411yescharacteristic
104C2×#11924017C2×C411yescharacteristic
105C2×#11924015C2×C411yescharacteristic
106#14556586—21no—
107#14556579C2×C411yescharacteristic
108#14556576—21no—
109#14556571C2×C411yescharacteristic
110#14556566C2×C411yescharacteristic
111Q8×Dic73C2311yescharacteristic
112C4×Dic146—21no—
113C4×Dic146C2×C411yescharacteristic
114#14556556—21no—
115#14556556C2311yescharacteristic
116#14556556C2311yescharacteristic
117#14556553—21no—
118#14556553C2311yescharacteristic
119#14556549—21no—
120C4⋊C4×D73—21no—
121C4⋊C4×D73C2×C411yescharacteristic
122C42×D73—21no—
123C42×D73C2311yescharacteristic
124#14556524C2×C411yescharacteristic
125#14556522C2×C411yescharacteristic
126C4⋊#10596831C2×C411yescharacteristic
127#14556516C2×C411yescharacteristic
128#14556514C2×C411yescharacteristic
129#14556514C2×C411yescharacteristic
130#14556514C2×C411yescharacteristic
131#14556513C2×C411yescharacteristic
132#14556511C2×C411yescharacteristic
133#14556503C2×C411yescharacteristic
134#14556503C2×C411yescharacteristic
135C8×Dic73C2311yescharacteristic
136C8×Dic73C2311yescharacteristic
137#125×C73C2×C411yescharacteristic
138C4×Q8×C73C2×C411yescharacteristic
139#107×C73C2×C411yescharacteristic
140#106×C73C2×C411yescharacteristic
141C4⋊C8×C73C2×C411yescharacteristic
142#103×C73C2×C411yescharacteristic
143#103×C73C2×C411yescharacteristic
144#11924038—42no—
145#11924038—42no—
146#11924036—22no—
147#11924036—41no—
148#11924036—42no—
149#11924036—42no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—21no—
153C2×#10596833—21no—
154C2×#10596833—21no—
155C2×#10596833—21no—
156C2×#10596833C2×D411yescharacteristic
157C2×#10596833C2×D411yescharacteristic
158C2×#10596833C2×D411yescharacteristic
159C2×#10596833C2×D411yescharacteristic
160C2×Dic146—21no—
161C2×Dic146C22×C411yescharacteristic
162C2×Dic146C22×C411yescharacteristic
163Q8×D73—22no—
164Q8×D73—22no—
165C2×D292—21no—
166C2×D292C22×C411yescharacteristic
167C2×C4×D73—21no—
168C2×C4×D73—21no—
169C2×C4×D73—21no—
170C2×C4×D73C22×C411yescharacteristic
171C2×C4×D73C2×D411yescharacteristic
172C2×C4×D73#4111yescharacteristic
173Dic292—22no—
174#11924018—22no—
175#11924017—22no—
176#11924015—22no—
177C2×#10596831C22×C411yescharacteristic
178C2×#10596831C22×C411yescharacteristic
179C4⋊Dic73—21no—
180C4⋊Dic73—21no—
181C4⋊Dic73C22×C411yescharacteristic
182#11924007—21no—
183#11924007—21no—
184#11924007—21no—
185#11924007—21no—
186#11924005—21no—
187#11924005—21no—
188#11924005—21no—
189#11924005—21no—
190#11924005—21no—
191C4×Dic73—21no—
192C4×Dic73—21no—
193C4×Dic73—21no—
194C4×Dic73—21no—
195C4×Dic73C2×D411yescharacteristic
196C4×Dic73#4111yescharacteristic
197C4×Dic73C2411yescharacteristic
198C2×Q8×C73C22×C411yescharacteristic
199C2×C8×C73C22×C411yescharacteristic
200C2×C8×C73C22×C411yescharacteristic
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—21no—
204C4⋊C4×C73C22×C411yescharacteristic
205C4⋊C4×C73C22×C411yescharacteristic
206C4⋊C4×C73C22×C411yescharacteristic
207C42×C73—21no—
208C42×C73C22×C411yescharacteristic
209C22×D73—21no—
210C22×D73C4×D411yescharacteristic
211Q8×C73—21no—
212Q8×C73#13112yes—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73C4×D411yescharacteristic
218C2×C4×C73C4×D411yescharacteristic
219C2×C4×C73C23×C411yescharacteristic
220C2×Dic73—21no—
221C2×Dic73—21no—
222C2×Dic73—21no—
223C2×Dic73C4×D411yescharacteristic
224C2×Dic73C22×D411yescharacteristic
225C2×Dic73C2×#4111yescharacteristic
226D292—21no—
227D292—21no—
228D292—42no—
229C4×D73—24no—
230C4×D73—22no—
231C4×D73—21no—
232C4×D73—21no—
233C4×D73—21no—
234C4×D73—21no—
235C4×D73—42no—
236C4×D73—41no—
237C4×D73—41no—
238Dic146—24no—
239Dic146—21no—
240Dic146—21no—
241Dic146—21no—
242Dic146#13112yes—
243#10596833—21no—
244#10596833—21no—
245#10596833—21no—
246#10596833—41no—
247#10596833—41no—
248#10596833—41no—
249#10596833—41no—
250#10596833—41no—
251#10596833—41no—
252#10596833#13712yes—
253C8×C73—21no—
254C8×C73—21no—
255#10596831—21no—
256#10596831—21no—
257C22×C73C2×C4×D411yescharacteristic
258C2×D73—21no—
259C2×D73—21no—
260C2×D73—41no—
261C2×D73—41no—
262C2×D73—41no—
263C2×D73#44412yes—
264C4×C73—22no—
265C4×C73—21no—
266C4×C73—21no—
267C4×C73—41no—
268C4×C73—41no—
269C4×C73—41no—
270C4×C73C2×#13111yescharacteristic
271C4×C73C2×#13111yescharacteristic
272Dic73—21no—
273Dic73—22no—
274Dic73—21no—
275Dic73—41no—
276Dic73—41no—
277Dic73C2×#13711yescharacteristic
278Dic73#57511yescharacteristic
279#17915—731nomaximal, Sylow
280C2×C73#292911yescharacteristic
281C2×C73C4×#13711yescharacteristic
282C2×C73C2×#44411yescharacteristic
283D73—22no—
284D73—81no—
285#3108—731no—
286#2966—731no—
287C2×C8×Q8—731no—
288#1584—731no—
289#1584—731no—
290#1583—731no—
291#1583—731no—
292#1579—731no—
293#1579—731no—
294SD16⋊C8—731no—
295SD16⋊C8—731no—
296#1573—731no—
297#1491—731no—
298#1483—731no—
299#1451—731no—
300C73#1791511yescharacteristic, complemented, Sylow
301#529—731no—
302C2×C4×Q8—731no—
303#483—731no—
304#480—731no—
305#479—731no—
306#479#6440811yescharacteristic
307#464—731no—
308#464—731no—
309#451—731no—
310#448—731no—
311#446—731no—
312C8×Q8—731no—
313C8×Q8—731no—
314C8×Q8—731no—
315C8×Q8—731no—
316C8×Q8—731no—
317C8×Q8—1461no—
318C8×Q8—1461no—
319C4×Q16—731no—
320C4×Q16—731no—
321C4×SD16—731no—
322C4×SD16—731no—
323#436—731no—
324#435—1461no—
325C8×D4—731no—
326C8×D4—1461no—
327#433—1461no—
328#432—731no—
329#431—731no—
330C2×C4⋊C8—731no—
331C2×C4⋊C8—1461no—
332#405—1461no—
333C4×M16—731no—
334C2×C4×C8—731no—
335C2×C4×C8—1461no—
336#335—731no—
337#335—731no—
338#334—731no—
339#334—731no—
340Q8⋊C8—731no—
341Q8⋊C8—731no—
342Q8⋊C8—731no—
343Q8⋊C8—731no—
344Q8⋊C8—731no—
345Q8⋊C8—731no—
346D4⋊C8—731no—
347D4⋊C8—731no—
348#322—731no—
349#322—731no—
350#322—731no—
351#322—731no—
352C22×Q8—731no—
353C2×Q16—731no—
354C2×Q16—731no—
355C2×SD16—731no—
356C2×SD16—731no—
357C2×M16—1461no—
358C2×M16—1461no—
359C22×C8—731no—
360C22×C8—731no—
361C22×C8—1461no—
362C22×C8—1461no—
363#126—1461no—
364#125C2×#6440811yescharacteristic
365#124—731no—
366#123—1461no—
367C22⋊Q8—731no—
368#121—731no—
369C4×Q8—731no—
370C4×Q8—731no—
371C4×Q8—731no—
372C4×Q8—731no—
373C4×Q8—1461no—
374C4×Q8—1461no—
375C4×Q8C2×#6440811yescharacteristic
376C4×D4—731no—
377C4×D4—1461no—
378#117—1461no—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—1461no—
381C2×C42—731no—
382C2×C42—1461no—
383#107—731no—
384#107C2×#6440811yescharacteristic
385#106—731no—
386#106C2×#6440811yescharacteristic
387C4⋊C8—731no—
388C4⋊C8—731no—
389C4⋊C8—731no—
390C4⋊C8—731no—
391C4⋊C8—731no—
392C4⋊C8—1461no—
393C4⋊C8—1461no—
394C4⋊C8—1461no—
395C4⋊C8—1461no—
396C4⋊C8—1461no—
397C4⋊C8—1461no—
398C4⋊C8—1461no—
399C4⋊C8C2×#6440811yescharacteristic
400#103—731no—
401#103—731no—
402#103—731no—
403#103—731no—
404#103C2×#6440811yescharacteristic
405#103C2×#6440811yescharacteristic
406#102—731no—
407#102—731no—
408C22⋊C8—731no—
409C22⋊C8—731no—
410C22⋊C8—1461no—
411C22⋊C8—1461no—
412C22⋊C8—1461no—
413#97—1461no—
414#97—1461no—
415#97—1461no—
416C4×C8—731no—
417C4×C8—731no—
418C4×C8—731no—
419C4×C8—731no—
420C4×C8—731no—
421C4×C8—731no—
422C4×C8—731no—
423C4×C8—731no—
424C4×C8—1461no—
425C4×C8—1461no—
426C4×C8—1461no—
427C4×C8—1461no—
428C4×C8—1461no—
429C2×Q8—731no—
430C2×Q8—731no—
431C2×Q8—1461no—
432C2×Q8—1462no—
433C2×Q8—1462no—
434C2×Q8C22×#6440811yescharacteristic
435C2×D4—731no—
436C2×D4—1461no—
437C22×C4—731no—
438C22×C4—731no—
439C22×C4—731no—
440C22×C4—1461no—
441C22×C4—1461no—
442C22×C4—1461no—
443Q16—1462no—
444Q16—1462no—
445SD16—1462no—
446SD16—1462no—
447M16—2922no—
448M16—2922no—
449C2×C8—2922no—
450C2×C8—2922no—
451C2×C8—2921no—
452C2×C8—731no—
453C2×C8—731no—
454C2×C8—731no—
455C2×C8—731no—
456C2×C8—731no—
457C2×C8—731no—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1461no—
461C2×C8—1461no—
462C2×C8—1461no—
463C2×C8—1461no—
464C2×C8—1462no—
465C2×C8C22×#6440811yescharacteristic
466C2×C8C22×#6440811yescharacteristic
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—731no—
471C4⋊C4—1461no—
472C4⋊C4—1461no—
473C4⋊C4—1461no—
474C4⋊C4—1461no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4C22×#6440811yescharacteristic
478C4⋊C4C22×#6440811yescharacteristic
479C4⋊C4C22×#6440811yescharacteristic
480C22⋊C4—1461no—
481C22⋊C4—1461no—
482C22⋊C4—1461no—
483C22⋊C4—1461no—
484C22⋊C4—1461no—
485C42—21no—
486C42—731no—
487C42—731no—
488C42—731no—
489C42—1461no—
490C42—1461no—
491C42—1461no—
492C42—1461no—
493C42C22×#6440811yescharacteristic
494C23—731no—
495C23—1461no—
496Q8—21no—
497Q8—732no—
498Q8—1464no—
499Q8—1461no—
500Q8—1461no—
501Q8—1461no—
502Q8#1455667912yes—
503D4—2922no—
504D4—1461no—
505D4—1461no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—2922no—
511C2×C4—2921no—
512C2×C4—2921no—
513C2×C4—731no—
514C2×C4—731no—
515C2×C4—731no—
516C2×C4—1461no—
517C2×C4—1461no—
518C2×C4—1462no—
519C2×C4—1464no—
520C2×C4—1461no—
521C2×C4—1461no—
522C2×C4—1461no—
523C2×C4—1461no—
524C2×C4—1461no—
525C2×C4D4×#6440811yescharacteristic
526C2×C4D4×#6440811yescharacteristic
527C2×C4C23×#6440811yescharacteristic
528C8—21no—
529C8—21no—
530C8—2921no—
531C8—2921no—
532C8—2921no—
533C8—2921no—
534C8—2921no—
535C8—2921no—
536C8—732no—
537C8—1461no—
538C8—1461no—
539C8—1461no—
540C8—1461no—
541C8—1461no—
542C22—2921no—
543C22—2921no—
544C22—2921no—
545C22—732no—
546C22—1461no—
547C22—1461no—
548C22C2×D4×#6440811yescharacteristic
549C4—22no—
550C4—21no—
551C4—21no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—2921no—
556C4—2921no—
557C4—731no—
558C4—731no—
559C4—1461no—
560C4—1462no—
561C4—1461no—
562C4D73⋊#13111yescharacteristic
563C4C2×#1455667911yescharacteristic
564C2—5841no—
565C2—1462no—
566C2#2262664711yescharacteristic
567C2#137×#6440811yescharacteristic
568C2D73⋊#44411yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#479×C73C4
socleC22×C73C2×C4×D4
prime radical#14556566C2×C4
cosocleC4×Dic73C24
hypercenter#479#64408
hypocenterC73#17915
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
2. Q8×#11924036C2semidirect
3. #22625543C2semidirect
4. #22625542C2semidirect
5. #22625539C2semidirect
6. #22625538C2semidirect
7. #22625526C2semidirect
8. #22625525C2semidirect
9. #22625523C2semidirect
10. #22625521C2semidirect
11. #22625505C2semidirect
12. #22625274C2semidirect
14. #22625143C2semidirect
300. C73#17915semidirect

Maximal subgroups

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

subgroupindex
#226267482
Q8×#119240362
#226255432
#226255422
#226255392
#226255382
#226255262
#226255252
#226255232
#226255212
#226255052
#226252742
#226252492
#226251432
#226200162
#1791573