Finite Group Databasebeta

← #10896454

Subgroups — #10896454

Order 768.

604 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83812C211yesmaximal, characteristic
2#79077C211yesmaximal, characteristic, complemented
3C3×#1868C211yesmaximal, characteristic, complemented
4#72028C211yesmaximal, characteristic, complemented
5#70549C211yesmaximal, characteristic, complemented
6#70546C211yesmaximal, characteristic, complemented
7#70538C211yesmaximal, characteristic, complemented
8#70042C211yesmaximal, characteristic
9#70038C211yesmaximal, characteristic, complemented
10#70005C211yesmaximal, characteristic, complemented
11#70000C211yesmaximal, characteristic
12#69042C211yesmaximal, characteristic, complemented
13#69033C211yesmaximal, characteristic, complemented
14#69015C211yesmaximal, characteristic, complemented
15#69013C211yesmaximal, characteristic, complemented
16#29271—31nomaximal, Sylow
17C3×#568C2211yescharacteristic
18C2×#950C2211yescharacteristic
19#5745C2211yescharacteristic
20#5517C2211yescharacteristic
21C3×#424C2211yescharacteristic
22C3×#424C2211yescharacteristic
23C3×#410C2211yescharacteristic
24C3×#408C2211yescharacteristic
25#5236C2211yescharacteristic
26#5236C2211yescharacteristic
27C2×#837C2211yescharacteristic
28#5145—22no—
29#5144—22no—
30#5139C2211yescharacteristic
31C2×#825C2211yescharacteristic
32#5125C2211yescharacteristic
33#5124C2211yescharacteristic
34C2×#822C2211yescharacteristic
35C2×#822C2211yescharacteristic
36#5122C2211yescharacteristic
37#5118C2211yescharacteristic
38#5110C2211yescharacteristic
39#5102C2211yescharacteristic
40#5102C2211yescharacteristic
41C2×#815C2211yescharacteristic
42#5006C2211yescharacteristic
43#5005C2211yescharacteristic
44#4998C2211yescharacteristic
45#4997C2211yescharacteristic
46#4831—22no—
47#4828—22no—
48C3×#344C2211yescharacteristic
49C3×#344C2211yescharacteristic
50#4557C2211yescharacteristic
51#4556C2211yescharacteristic
52#4555C2211yescharacteristic
53#4533C2211yescharacteristic
54#4532C2211yescharacteristic
55#4531C2211yescharacteristic
56#2961—31no—
57#2957—31no—
58#2098—31no—
59#2098—31no—
60#2031—31no—
61#2031—31no—
62#1976—31no—
63#1975—31no—
64#1953—31no—
65#1946—31no—
66#1868—31no—
67#1868D311yescharacteristic, complemented
68#1861—31no—
69#1861—31no—
70#1833—31no—
71C2×C3×#41C2311yescharacteristic
72C2×#235C2311yescharacteristic
73C3×#131—22no—
74C3×#131—41no—
75C2×C3×M16—41no—
76C2×C3×M16C2311yescharacteristic
77C2×C3×M16C2311yescharacteristic
78C2×C3×M16C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×#117C2311yescharacteristic
81C3×#117C2311yescharacteristic
82#950—21no—
83#950—21no—
84#950—21no—
85#950—21no—
86C2×#216Q811yescharacteristic, complemented
87C2×#215Q811yescharacteristic, complemented
88C2×#214Q811yescharacteristic, complemented
89C2×#213Q811yescharacteristic, complemented
90#926C2311yescharacteristic
91#926C2311yescharacteristic
92C2×C4×Dic3C2311yescharacteristic
93C2×#208C2311yescharacteristic
94C2×#208C2311yescharacteristic
95C2×#208C2311yescharacteristic
96C22×C3⋊C8C2311yescharacteristic
97#907—41no—
98D3×M16—41no—
99#901—22no—
100#890—21no—
101#888—21no—
102#878—22no—
103C4×Dic6—22no—
104C3×C4⋊C8—22no—
105C3×C4⋊C8—41no—
106C3×#99—22no—
107C3×C22⋊C8—41no—
108#837—22no—
109#837—21no—
110#837—21no—
111#837—41no—
112#825—22no—
113#825—21no—
114#825—21no—
115#822—21no—
116#822—21no—
117#822—21no—
118#822—21no—
119#822—41no—
120(C2×D3)⋊C8—41no—
121#819—41no—
122#815—21no—
123#815—21no—
124#815D412yes—
125Dic3⋊C8—22no—
126Dic3⋊C8—22no—
127Dic3⋊C8—41no—
128C8×Dic3D412yes—
129#808—21no—
130#807—21no—
131#806—22no—
132#805—41no—
133#803—21no—
134C4×C3⋊C8—21no—
135C2×#135—31no—
136#568—31no—
137#568C2×D311yescharacteristic
138#533—31no—
139#446—62no—
140#446—62no—
141#444—62no—
142#443—62no—
143#430—31no—
144#430—31no—
145C2×#108—31no—
146C2×#108—31no—
147#427—31no—
148#424—31no—
149#424—31no—
150#424C2×D311yescharacteristic
151#424C2×D311yescharacteristic
152#423—31no—
153#421—31no—
154#421—31no—
155#421—31no—
156#421—31no—
157C2×#104—31no—
158C2×#104—31no—
159#410—31no—
160#410C2×D311yescharacteristic
161#408—31no—
162#408C2×D311yescharacteristic
163#405—31no—
164#405—31no—
165C2×#97—31no—
166#344—31no—
167#344—31no—
168#344C2×D311yescharacteristic
169#344C2×D311yescharacteristic
170#339—31no—
171#339—31no—
172H(3,ℤ/4)—31no—
173H(3,ℤ/4)—31no—
174C3×#41—21no—
175C3×#41—21no—
176C3×#41—21no—
177C3×#41—21no—
178C2×C3×Q8C2×Q811yescharacteristic
179C2×C3×D4—21no—
180C2×C3×D4C2×Q811yescharacteristic
181C22×C3×C4—21no—
182C22×C3×C4C2411yescharacteristic
183C2×#67—21no—
184C22×Dic3—21no—
185#235—21no—
186#235—21no—
187#235—21no—
188#235—21no—
189C2×D12C2×Q811yescharacteristic
190C2×C4×D3—21no—
191C2×Dic6—22no—
192C2×Dic6C2×Q811yescharacteristic
193C3×M16—21no—
194C3×M16—21no—
195C3×M16—21no—
196C3×M16—21no—
197C3×M16—21no—
198C3×M16—21no—
199C3×M16—42no—
200C2×C3×C8—21no—
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—42no—
206C2×C3×C8C2×D412yes—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C22⋊C4—21no—
210C3×C22⋊C4#4112yes—
211C3×C42—21no—
212C3×C42—21no—
213#217—21no—
214#217—21no—
215#216—21no—
216#216—21no—
217#215—21no—
218#215—21no—
219#214—21no—
220#214—21no—
221#213—21no—
222#213—21no—
223C4⋊Dic3—22no—
224C4⋊Dic3—21no—
225C4⋊Dic3—21no—
226Dic3⋊C4—22no—
227Dic3⋊C4—22no—
228C4×Dic3—21no—
229C4×Dic3—21no—
230C4×Dic3C2×D412yes—
231C4×Dic3#4111yescharacteristic
232C4×Dic3#4111yescharacteristic
233#208—21no—
234#208—21no—
235#208—21no—
236#208—21no—
237#208—21no—
238#208—21no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8—21no—
241C2×C3⋊C8—21no—
242C2×C3⋊C8C2×D411yescharacteristic
243C2×C3⋊C8C2×D411yescharacteristic
244C2×C3⋊C8C2×Q811yescharacteristic
245C2×C3⋊C8C2×Q811yescharacteristic
246C2×C3⋊C8#4111yescharacteristic
247C2×C3⋊C8#4111yescharacteristic
248ΓL(1,𝔽25)—42no—
249D3×C8—42no—
250C2×#41—31no—
251C2×#41C22×D311yescharacteristic
252#135—61no—
253#135—61no—
254#135—61no—
255#135—61no—
256C2×Q16—31no—
257C2×SD16—31no—
258C2×SD16—31no—
259C2×D8—31no—
260#131—22no—
261#131—41no—
262#131—62no—
263#131—121no—
264C2×M16—31no—
265C2×M16—31no—
266C2×M16—31no—
267C2×M16—41no—
268C2×M16—121no—
269C2×M16C22×D311yescharacteristic
270C2×M16C22×D311yescharacteristic
271C2×M16C22×D311yescharacteristic
272C22×C8—31no—
273C22×C8C22×D311yescharacteristic
274C4⋊Q8—61no—
275#125—61no—
276C22⋊Q8—62no—
277C4×Q8—62no—
278#117—31no—
279#117—31no—
280#117C22×D311yescharacteristic
281#117C22×D311yescharacteristic
282C2×C42—31no—
283#108—61no—
284#108—61no—
285#108—61no—
286#108—61no—
287#108—121no—
288#108—121no—
289#107—61no—
290#106—61no—
291C4⋊C8—22no—
292C4⋊C8—41no—
293C4⋊C8—62no—
294C4⋊C8—62no—
295C4⋊C8—121no—
296#104—61no—
297#104—61no—
298#104—62no—
299#104—61no—
300#104—61no—
301#104—62no—
302#104—121no—
303#104—121no—
304#99—22no—
305#99—62no—
306C22⋊C8—41no—
307C22⋊C8—121no—
308#97—32no—
309#97—61no—
310#97—61no—
311#97—61no—
312C4×C8—32no—
313C4×C8—61no—
314C23×C3—21no—
315C23×C3C2×#4111yescharacteristic
316C22×D3—21no—
317C3×Q8—21no—
318C3×Q8—21no—
319C3×D4—21no—
320C3×D4—21no—
321C3×D4—41no—
322C3×D4—41no—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C22×Q811yescharacteristic
330C2×C3×C4C2×#4111yescharacteristic
331C2×C3×C4C2×#4111yescharacteristic
332C2×C3×C4H(5,𝔽2)11yescharacteristic
333C2×C3×C4H(5,𝔽2)11yescharacteristic
334#67—41no—
335#67—41no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—22no—
339C2×Dic3—21no—
340C2×Dic3—41no—
341C2×Dic3—81no—
342D12—21no—
343D12—21no—
344C4×D3—41no—
345C4×D3—41no—
346Dic6—21no—
347Dic6—21no—
348Dic6—82no—
349C3×C8—22no—
350C3×C8—41no—
351C3×C8—41no—
352C3×C8—41no—
353C3⋊C8—41no—
354C3⋊C8—41no—
355C3⋊C8—41no—
356C3⋊C8C22⋊Q811yescharacteristic
357C3⋊C8C22⋊Q811yescharacteristic
358C3⋊C8C22⋊Q811yescharacteristic
359C3⋊C8C22⋊Q811yescharacteristic
360#41—21no—
361#41—21no—
362#41—21no—
363#41—21no—
364#41—61no—
365#41—61no—
366#41—61no—
367#41—61no—
368C2×Q8—31no—
369C2×Q8—62no—
370C2×Q8D3×Q811yescharacteristic
371C2×D4—21no—
372C2×D4—31no—
373C2×D4—61no—
374C2×D4D3×Q811yescharacteristic
375C22×C4—21no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4C23×D311yescharacteristic
379Q16—61no—
380Q16—61no—
381SD16—61no—
382SD16—61no—
383SD16—61no—
384SD16—61no—
385D8—61no—
386D8—61no—
387M16—21no—
388M16—21no—
389M16—21no—
390M16—21no—
391M16—21no—
392M16—21no—
393M16—42no—
394M16—61no—
395M16—61no—
396M16—61no—
397M16—61no—
398M16—61no—
399M16—61no—
400M16—122no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—31no—
407C2×C8—31no—
408C2×C8—31no—
409C2×C8—31no—
410C2×C8—31no—
411C2×C8—31no—
412C2×C8—42no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—122no—
417C2×C8D3×D412yes—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—62no—
421C4⋊C4—62no—
422C4⋊C4—62no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C22⋊C4—21no—
426C22⋊C4—61no—
427C22⋊C4—61no—
428C22⋊C4#23512yes—
429C42—21no—
430C42—21no—
431C42—32no—
432C42—31no—
433C42—31no—
434C42—61no—
435C42—61no—
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—81no—
439C22×C3#52511yescharacteristic
440C22×C3#55411yescharacteristic
441C22×C3#55911yescharacteristic
442C2×D3—41no—
443C2×D3—81no—
444C3×C4—42no—
445C3×C4—41no—
446C3×C4—81no—
447C3×C4C2×C22⋊Q811yescharacteristic
448C3×C4#53511yescharacteristic
449C3×C4#55411yescharacteristic
450C3×C4#55911yescharacteristic
451Dic3—41no—
452Dic3—42no—
453Dic3—41no—
454Dic3—41no—
455Dic3—81no—
456C23—21no—
457C23—61no—
458C23C2×#23511yescharacteristic
459Q8—21no—
460Q8—21no—
461Q8—61no—
462Q8—61no—
463Q8—242no—
464D4—21no—
465D4—21no—
466D4—41no—
467D4—41no—
468D4—61no—
469D4—61no—
470D4—121no—
471D4—121no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—62no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—121no—
484C2×C4—241no—
485C2×C4C2×D3×D411yescharacteristic
486C2×C4#100411yescharacteristic
487C2×C4#100411yescharacteristic
488C2×C4C2×D3×Q811yescharacteristic
489C2×C4D3×#4111yescharacteristic
490C2×C4D3×#4111yescharacteristic
491C8—22no—
492C8—31no—
493C8—31no—
494C8—31no—
495C8—31no—
496C8—41no—
497C8—41no—
498C8—41no—
499C8—121no—
500C8—121no—
501C8—121no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—81no—
506C2×C3#258011yescharacteristic
507D3—81no—
508C22—41no—
509C22—41no—
510C22—81no—
511C22—121no—
512C22—241no—
513C22#551111yescharacteristic
514C22#554711yescharacteristic
515C22#555911yescharacteristic
516C4—42no—
517C4—41no—
518C4—81no—
519C4—121no—
520C4—121no—
521C4—122no—
522C4—121no—
523C4—241no—
524C4#559211yescharacteristic
525C4D3×C22⋊Q811yescharacteristic
526C4#563611yescharacteristic
527C4#566411yescharacteristic
528C3#2927111yescharacteristic, complemented, Sylow
529C2—21no—
530C2—21no—
531C2—21no—
532C2—81no—
533C2—241no—
534C2#7448711yescharacteristic
subgroupG/H
centerC4#5592
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1868C2
socleC2×C3#2580
prime radicalC2×#950C22
hypercenter#1868D3
hypocenterC3#29271
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1868

Complements

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

NH ≅ G/Ntype
2. #79077C2semidirect
3. C3×#1868C2semidirect
4. #72028C2semidirect
5. #70549C2semidirect
6. #70546C2semidirect
7. #70538C2semidirect
9. #70038C2semidirect
10. #70005C2semidirect
12. #69042C2semidirect
13. #69033C2semidirect
14. #69015C2semidirect
15. #69013C2semidirect
67. #1868D3semidirect
86. C2×#216Q8semidirect
87. C2×#215Q8semidirect
88. C2×#214Q8semidirect
89. C2×#213Q8semidirect
528. C3#29271semidirect

Maximal subgroups

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

subgroupindex
#838122
#790772
C3×#18682
#720282
#705492
#705462
#705382
#700422
#700382
#700052
#700002
#690422
#690332
#690152
#690132
#292713