Finite Group Databasebeta

← #10785147

Subgroups — #10785147

Order 768.

598 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
1Q8×#215C211yesmaximal, characteristic, complemented
2#83704C211yesmaximal, characteristic, complemented
3#81022C211yesmaximal, characteristic
4#80492C211yesmaximal, characteristic
5C3×#1645C211yesmaximal, characteristic, complemented
6#71228C211yesmaximal, characteristic
7#71222C211yesmaximal, characteristic, complemented
8#71118C211yesmaximal, characteristic, complemented
9#71058C211yesmaximal, characteristic, complemented
10#68499C211yesmaximal, characteristic, complemented
11#68496C211yesmaximal, characteristic, complemented
12#68273C211yesmaximal, characteristic, complemented
13#68250C211yesmaximal, characteristic, complemented
14#68216C211yesmaximal, characteristic, complemented
15#68214C211yesmaximal, characteristic, complemented
16#19796—31nomaximal, Sylow
17C3×Q82C2211yescharacteristic
18#5726C2211yescharacteristic
19#5584C2211yescharacteristic
20Q8×D12C2211yescharacteristic
21C3×#487C2211yescharacteristic
22C3×#475C2211yescharacteristic
23C3×#462C2211yescharacteristic
24#5188—21no—
25#5187—21no—
26#5094—21no—
27#5093C2211yescharacteristic
28#5093C2211yescharacteristic
29#5089—21no—
30#5084—21no—
31#5084C2211yescharacteristic
32#5084C2211yescharacteristic
33#5080—21no—
34#5080C2211yescharacteristic
35#5079C2211yescharacteristic
36#5031C2211yescharacteristic
37#5030—21no—
38#5028—21no—
39C4×#215—21no—
40C4×#215C2211yescharacteristic
41C4×#215C2211yescharacteristic
42#5024C2211yescharacteristic
43Q8×C3⋊C8C2211yescharacteristic
44#5020—21no—
45#5020C2211yescharacteristic
46#4864—21no—
47#4862—21no—
48#4846C2211yescharacteristic
49#4841C2211yescharacteristic
50#4834C2211yescharacteristic
51#4834C2211yescharacteristic
52#4814—21no—
53#4806—21no—
54#4804—21no—
55D3×#103—21no—
56#4682C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×Q8⋊C8C2211yescharacteristic
59C3×Q8⋊C8C2211yescharacteristic
60#4547C2211yescharacteristic
61#4486C2211yescharacteristic
62#4482C2211yescharacteristic
63#4480C2211yescharacteristic
64#4479C2211yescharacteristic
65#4459C2211yescharacteristic
66#4459C2211yescharacteristic
67#4454C2211yescharacteristic
68Q8×SD16—31no—
69#3369—31no—
70#3302—31no—
71#3298—31no—
72#1718—31no—
73#1700—31no—
74#1683—31no—
75#1680—31no—
76#1671—31no—
77#1650—31no—
78#1649—31no—
79#1645D311yescharacteristic, complemented
80#1635—31no—
81#1582—31no—
82SD16⋊C8—31no—
83C2×#239—21no—
84C2×D3×Q8—21no—
85C2×C3×Q16—21no—
86C3×C4⋊Q8—21no—
87C3×C4⋊Q8—21no—
88C3×C4⋊Q8—41no—
89C3×C4⋊Q8C2311yescharacteristic
90C3×C4⋊Q8C2311yescharacteristic
91C3×C4×Q8—21no—
92C3×C4×Q8—21no—
93C3×C4×Q8C2311yescharacteristic
94C3×C4×Q8C2311yescharacteristic
95#946—21no—
96#946—21no—
97C2×#216—21no—
98C2×#215D411yescharacteristic
99C2×#215D411yescharacteristic
100C2×#204—21no—
101C2×D3×C8—21no—
102C2×D3×C8—21no—
103#898—41no—
104#897—21no—
105#897—41no—
106#895—21no—
107#894—41no—
108#892—21no—
109D3×C4⋊C4—21no—
110#875—21no—
111C4×D12—21no—
112C4×D12C2311yescharacteristic
113C4×D12C2311yescharacteristic
114#869—21no—
115#869C2311yescharacteristic
116#869C2311yescharacteristic
117C4×Dic6—21no—
118C3×#106—21no—
119C3×C4⋊C8C2311yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×#103—21no—
122C3×#103—21no—
123C3×#103—41no—
124C3×C4×C8C2311yescharacteristic
125#835—21no—
126#835—21no—
127#835—21no—
128#835D411yescharacteristic
129#835D411yescharacteristic
130#821—21no—
131(C2×D3)⋊C8—21no—
132(C2×D3)⋊C8—21no—
133#818—21no—
134#818—41no—
135#816—21no—
136#810—21no—
137#810—21no—
138#810—21no—
139#810—21no—
140#810—21no—
141#809—21no—
142#809—21no—
143#809D411yescharacteristic
144#809D411yescharacteristic
145#808—21no—
146#808—21no—
147#808—21no—
148#808—21no—
149#808—21no—
150#808D411yescharacteristic
151#808D411yescharacteristic
152C4⋊(C3⋊C8)—21no—
153C4⋊(C3⋊C8)—21no—
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156#803—21no—
157C4×C3⋊C8—21no—
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160Q82C2×D311yescharacteristic
161#556—31no—
162D4×Q8—31no—
163#547—31no—
164#500—31no—
165#498—31no—
166#498—61no—
167#489—61no—
168#487—31no—
169#487—31no—
170#487—31no—
171#487C2×D311yescharacteristic
172#485—61no—
173#483—61no—
174#476—31no—
175#476—31no—
176#476—61no—
177#475—31no—
178#475—61no—
179#475—61no—
180#475C2×D311yescharacteristic
181#474—31no—
182#467—61no—
183#465—61no—
184#464—61no—
185#462—31no—
186#462C2×D311yescharacteristic
187#461—31no—
188#452—61no—
189#448—61no—
190#446—31no—
191C8×Q8—31no—
192#441—61no—
193C4×SD16—31no—
194C4×SD16—31no—
195C4×SD16—61no—
196C8×D4—31no—
197C8×D4—31no—
198#416—61no—
199C2×#103—61no—
200#335—31no—
201#334—31no—
202#333C2×D311yescharacteristic
203#332—31no—
204#332—31no—
205#332—31no—
206Q8⋊C8C2×D311yescharacteristic
207Q8⋊C8C2×D311yescharacteristic
208D4⋊C8—31no—
209D4⋊C8—31no—
210#322—31no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—41no—
214C2×C3×Q8C2×D411yescharacteristic
215C2×C3×Q8C2×D411yescharacteristic
216#239—42no—
217#239—81no—
218D3×Q8—42no—
219D3×Q8—81no—
220C2×D12—41no—
221C2×D12C2×D411yescharacteristic
222C2×D12C2×D411yescharacteristic
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—41no—
226C2×C4×D3—41no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—41no—
230C3×Q16—81no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—41no—
241C3×C4⋊C4—41no—
242C3×C4⋊C4C2×D411yescharacteristic
243C3×C4⋊C4C2×D411yescharacteristic
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—41no—
247C3×C42C2411yescharacteristic
248#216—42no—
249#215—22no—
250#215—22no—
251(C2×D3)⋊C4—21no—
252(C2×D3)⋊C4—21no—
253(C2×D3)⋊C4—41no—
254(C2×D3)⋊C4—41no—
255(C2×D3)⋊C4—41no—
256(C2×D3)⋊C4—41no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—41no—
262C4⋊Dic3C2×D411yescharacteristic
263C4⋊Dic3C2×D411yescharacteristic
264Dic3⋊C4—41no—
265Dic3⋊C4—41no—
266C4×Dic3—41no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271C2×C3⋊C8C2×D411yescharacteristic
272C2×C3⋊C8C2×D411yescharacteristic
273#204—81no—
274D3×C8—81no—
275D3×C8—81no—
276C2×#41—61no—
277C22×Q8—61no—
278C2×Q16—21no—
279C2×Q16—61no—
280C2×SD16—31no—
281C2×SD16—31no—
282C2×SD16—61no—
283C22×C8—61no—
284C22×C8—61no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—31no—
288C4⋊Q8—31no—
289C4⋊Q8—41no—
290C4⋊Q8—61no—
291C4⋊Q8C22×D311yescharacteristic
292C4⋊Q8C22×D311yescharacteristic
293#126—121no—
294#124—61no—
295#123—121no—
296C22⋊Q8—61no—
297C22⋊Q8—61no—
298C22⋊Q8—61no—
299C22⋊Q8—121no—
300#121—61no—
301C4×Q8—21no—
302C4×Q8—21no—
303C4×Q8—61no—
304C4×Q8C22×D311yescharacteristic
305C4×Q8C22×D311yescharacteristic
306C4×D4—31no—
307C4×D4—31no—
308C4×D4—61no—
309#117—61no—
310C2×C4⋊C4—61no—
311#107—61no—
312#107—121no—
313#106—21no—
314#106—31no—
315#106—31no—
316#106—61no—
317#106—61no—
318#106—61no—
319#106—61no—
320#106—61no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—61no—
324C4⋊C8—61no—
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327#103—21no—
328#103—21no—
329#103—31no—
330#103—31no—
331#103—41no—
332#103—61no—
333#103—61no—
334#103—61no—
335#103—61no—
336#103—61no—
337#103—61no—
338#103—61no—
339#103—61no—
340#103—61no—
341#102—31no—
342#102—31no—
343#102—61no—
344#102—61no—
345#102—61no—
346C22⋊C8—61no—
347C22⋊C8—61no—
348#97—61no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8—61no—
352C4×C8C22×D311yescharacteristic
353C22×D3—21no—
354C22×D3—21no—
355C3×Q8—21no—
356C3×Q8—21no—
357C3×Q8—21no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—81no—
361C3×Q8—161no—
362C3×Q8#13512yes—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4C22×D411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4H(5,𝔽2)11yescharacteristic
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376D12—21no—
377D12—21no—
378D12—21no—
379D12—82no—
380D12#13712yes—
381C4×D3—41no—
382C4×D3—41no—
383C4×D3—41no—
384C4×D3—41no—
385C4×D3—82no—
386C4×D3—82no—
387Dic6—41no—
388Dic6—41no—
389Dic6—41no—
390Dic6—41no—
391Dic6—82no—
392C3×C8—41no—
393C3×C8—41no—
394C3×C8—41no—
395C3⋊C8—21no—
396C3⋊C8—21no—
397C3⋊C8—21no—
398C3⋊C8—21no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401#41—122no—
402#41—241no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—41no—
406C2×Q8—61no—
407C2×Q8—61no—
408C2×Q8—121no—
409C2×Q8—122no—
410C2×Q8—241no—
411C2×Q8D3×D411yescharacteristic
412C2×Q8D3×D411yescharacteristic
413C2×D4—31no—
414C2×D4—31no—
415C2×D4—121no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—121no—
419C22×C4—121no—
420Q16—81no—
421Q16—122no—
422SD16—62no—
423SD16—62no—
424SD16—241no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—31no—
429C2×C8—31no—
430C2×C8—31no—
431C2×C8—31no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—241no—
435C2×C8—241no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—31no—
443C4⋊C4—31no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—121no—
451C4⋊C4—121no—
452C4⋊C4—121no—
453C4⋊C4D3×D411yescharacteristic
454C4⋊C4D3×D411yescharacteristic
455C22⋊C4—61no—
456C22⋊C4—61no—
457C22⋊C4—121no—
458C22⋊C4—121no—
459C22⋊C4—121no—
460C22⋊C4—121no—
461C42—21no—
462C42—21no—
463C42—41no—
464C42—121no—
465C42C23×D311yescharacteristic
466C22×C3D4211yescharacteristic
467C2×D3—22no—
468C2×D3—41no—
469C2×D3—41no—
470C2×D3—41no—
471C2×D3—41no—
472C2×D3—41no—
473C3×C4—22no—
474C3×C4—21no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—81no—
479C3×C4—81no—
480C3×C4C2×#13511yescharacteristic
481C3×C4C2×#13711yescharacteristic
482C3×C4#57711yescharacteristic
483C3×C4#57711yescharacteristic
484Dic3—41no—
485Dic3—41no—
486Dic3—81no—
487Dic3—81no—
488C23—61no—
489C23—61no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—41no—
494Q8—41no—
495Q8—81no—
496Q8—121no—
497Q8—121no—
498Q8—121no—
499Q8—121no—
500Q8—161no—
501Q8—242no—
502Q8#91912yes—
503D4—32no—
504D4—61no—
505D4—61no—
506D4—61no—
507D4—242no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—242no—
523C2×C4—242no—
524C2×C4C2×D3×D411yescharacteristic
525C2×C4C2×D3×D411yescharacteristic
526C2×C4#100411yescharacteristic
527C8—41no—
528C8—41no—
529C8—41no—
530C8—61no—
531C8—61no—
532C8—61no—
533C8—61no—
534C8—121no—
535C8—121no—
536C2×C3#326611yescharacteristic
537C2×C3#328211yescharacteristic
538C2×C3#328911yescharacteristic
539D3—42no—
540D3—81no—
541C22—62no—
542C22—121no—
543C22—121no—
544C22—121no—
545C22—121no—
546C22—121no—
547C22#559011yescharacteristic
548C4—22no—
549C4—21no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—81no—
554C4—81no—
555C4—121no—
556C4—121no—
557C4—241no—
558C4—241no—
559C4#576211yescharacteristic
560C4C2×#91911yescharacteristic
561C4D3×#13711yescharacteristic
562C4#577811yescharacteristic
563C3#1979611yescharacteristic, complemented, Sylow
564C2—122no—
565C2—241no—
566C2#8006111yescharacteristic
567C2#8020711yescharacteristic
568C2#8029511yescharacteristic
subgroupG/H
centerC22#5590
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1645C2
socleC22×C3D42
prime radical#5726C22
hypercenter#1645D3
hypocenterC3#19796
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1645

Complements

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

NH ≅ G/Ntype
1. Q8×#215C2semidirect
2. #83704C2semidirect
5. C3×#1645C2semidirect
7. #71222C2semidirect
8. #71118C2semidirect
9. #71058C2semidirect
10. #68499C2semidirect
11. #68496C2semidirect
12. #68273C2semidirect
13. #68250C2semidirect
14. #68216C2semidirect
15. #68214C2semidirect
79. #1645D3semidirect
563. C3#19796semidirect

Maximal subgroups

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

subgroupindex
Q8×#2152
#837042
#810222
#804922
C3×#16452
#712282
#712222
#711182
#710582
#684992
#684962
#682732
#682502
#682162
#682142
#197963