Finite Group Databasebeta

← #10826076

Subgroups — #10826076

Order 768.

608 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83329C211yesmaximal, characteristic, complemented
2#78989C211yesmaximal, characteristic
3C3×#2031C211yesmaximal, characteristic, complemented
4#71842C211yesmaximal, characteristic, complemented
5#70404C211yesmaximal, characteristic, complemented
6#70379C211yesmaximal, characteristic
7#70370C211yesmaximal, characteristic, complemented
8#70043C211yesmaximal, characteristic
9#70040C211yesmaximal, characteristic, complemented
10#69016C211yesmaximal, characteristic, complemented
11#69014C211yesmaximal, characteristic, complemented
12#68751C211yesmaximal, characteristic, complemented
13#68740C211yesmaximal, characteristic, complemented
14#68700C211yesmaximal, characteristic, complemented
15#68697C211yesmaximal, characteristic, complemented
16#23834—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18#5822C2211yescharacteristic
19C2×#948C2211yescharacteristic
20C2×#907C2211yescharacteristic
21C3×#421C2211yescharacteristic
22C2×C3×#104C2211yescharacteristic
23C3×#410C2211yescharacteristic
24#5237C2211yescharacteristic
25#5233C2211yescharacteristic
26#5132C2211yescharacteristic
27#5124C2211yescharacteristic
28#5122C2211yescharacteristic
29#5024—22no—
30#5024—22no—
31#5006C2211yescharacteristic
32#5005C2211yescharacteristic
33#5002C2211yescharacteristic
34#4999C2211yescharacteristic
35#4999C2211yescharacteristic
36#4997C2211yescharacteristic
37#4997C2211yescharacteristic
38#4997C2211yescharacteristic
39C2×#805C2211yescharacteristic
40#4924C2211yescharacteristic
41C2×#803C2211yescharacteristic
42#4824—22no—
43#4823—22no—
44C3×#344C2211yescharacteristic
45C3×#339C2211yescharacteristic
46C3×H(3,ℤ/4)C2211yescharacteristic
47C3⋊#338C2211yescharacteristic
48#4558C2211yescharacteristic
49#4558C2211yescharacteristic
50#4556C2211yescharacteristic
51#4533C2211yescharacteristic
52#4533C2211yescharacteristic
53#4531C2211yescharacteristic
54#4531C2211yescharacteristic
55#4523C2211yescharacteristic
56#2961—31no—
57#2955—31no—
58#2103—31no—
59#2102—31no—
60#2080—31no—
61#2080—31no—
62#2053—31no—
63#2050—31no—
64#2031—31no—
65#2031D311yescharacteristic, complemented
66#1765—31no—
67#1765—31no—
68#1757—31no—
69#1757—31no—
70#1747—31no—
71C2×C3×#41C2311yescharacteristic
72C2×#235C2311yescharacteristic
73C2×C3×M16C2311yescharacteristic
74C2×C3×M16C2311yescharacteristic
75C3×C4⋊Q8—21no—
76C3×#125—21no—
77C3×C22⋊Q8—22no—
78C3×C4×Q8—22no—
79C3×#117C2311yescharacteristic
80C3×#117C2311yescharacteristic
81C3×#117C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83#951—21no—
84#950—22no—
85#949—21no—
86#948—21no—
87#948—21no—
88#948—21no—
89#948—21no—
90#942—21no—
91#932—21no—
92#926C2311yescharacteristic
93C2×#208—21no—
94C2×#208C2311yescharacteristic
95C2×#208C2311yescharacteristic
96C2×#208C2311yescharacteristic
97C2×#208C2311yescharacteristic
98C22×C3⋊C8—21no—
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101#907—21no—
102#907—21no—
103#907—21no—
104#907—21no—
105C2×ΓL(1,𝔽25)—21no—
106C2×D3×C8—21no—
107C3×#104—22no—
108C3×#104—21no—
109C3×#104—21no—
110C3×#99—22no—
111#836—21no—
112#835—21no—
113#833—21no—
114#832—21no—
115#824—21no—
116#823—21no—
117#822—22no—
118#815—21no—
119C8×Dic3—21no—
120#810—21no—
121#809—21no—
122#806—22no—
123#805—22no—
124#805—21no—
125#805—21no—
126C4⋊(C3⋊C8)—22no—
127C4⋊(C3⋊C8)—22no—
128C4⋊(C3⋊C8)—22no—
129#803—21no—
130#803—21no—
131#803—21no—
132#803—21no—
133#803D412yes—
134C4×C3⋊C8—21no—
135C4×C3⋊C8—21no—
136C4×C3⋊C8D412yes—
137#575—31no—
138C2×#131—31no—
139C2×#131—31no—
140#533C2×D311yescharacteristic
141#446—62no—
142#446—62no—
143#444—62no—
144#443—62no—
145#430—31no—
146#424—31no—
147#424—31no—
148#423—31no—
149#421—31no—
150#421C2×D311yescharacteristic
151C2×#104—31no—
152C2×#104C2×D311yescharacteristic
153#416—31no—
154#416—31no—
155#413—31no—
156#413—31no—
157#410—31no—
158#410C2×D311yescharacteristic
159#405—31no—
160#405—31no—
161#405—31no—
162#405—31no—
163C2×#97—31no—
164#344—31no—
165#344—31no—
166#344—31no—
167#344C2×D311yescharacteristic
168#341—31no—
169#341—31no—
170#339—31no—
171#339C2×D311yescharacteristic
172#338—31no—
173H(3,ℤ/4)—31no—
174H(3,ℤ/4)—31no—
175H(3,ℤ/4)C2×D311yescharacteristic
176C3×#41—21no—
177C3×#41—21no—
178C3×#41—21no—
179C3×#41—21no—
180C2×C3×Q8—22no—
181C2×C3×Q8#4111yescharacteristic
182C2×C3×D4—21no—
183C2×C3×D4#4111yescharacteristic
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4C2411yescharacteristic
187C2×#67—21no—
188#235—21no—
189#235—21no—
190#235—21no—
191#235—21no—
192C2×D12#4111yescharacteristic
193C2×C4×D3—21no—
194C2×Dic6#4111yescharacteristic
195C3×M16—21no—
196C3×M16—21no—
197C3×M16—21no—
198C3×M16—21no—
199C2×C3×C8—21no—
200C2×C3×C8—21no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C22⋊C4—21no—
207C3×C22⋊C4—21no—
208C3×C22⋊C4#4112yes—
209C3×C42—21no—
210C3×C42—21no—
211C3×C42—21no—
212C3×C42C2×D412yes—
213C3×C42#4111yescharacteristic
214C3×C42#4111yescharacteristic
215#217—21no—
216#216—22no—
217#215—22no—
218#214—22no—
219#213—22no—
220C4⋊Dic3—21no—
221C4×Dic3—21no—
222#208—21no—
223#208—21no—
224#208—21no—
225#208—21no—
226#208—21no—
227#208—21no—
228#208—21no—
229#208—21no—
230#208—41no—
231#208—41no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—21no—
236C2×C3⋊C8—21no—
237C2×C3⋊C8—21no—
238C2×C3⋊C8—41no—
239C2×C3⋊C8—41no—
240C2×C3⋊C8C2×D412yes—
241C2×C3⋊C8C2×D411yescharacteristic
242C2×C3⋊C8C2×D411yescharacteristic
243C2×C3⋊C8#4111yescharacteristic
244C2×C3⋊C8#4111yescharacteristic
245C2×C3⋊C8#4111yescharacteristic
246C2×C3⋊C8#4111yescharacteristic
247ΓL(1,𝔽25)—41no—
248ΓL(1,𝔽25)—41no—
249D3×C8—41no—
250D3×C8—41no—
251C2×#41—31no—
252C2×#41C22×D311yescharacteristic
253#137—61no—
254#137—61no—
255AGL(1,ℤ/8)—61no—
256AGL(1,ℤ/8)—61no—
257#135—62no—
258#131—61no—
259#131—61no—
260#131—61no—
261#131—61no—
262#131—61no—
263#131—61no—
264#131—61no—
265#131—61no—
266C2×M16—31no—
267C2×M16—31no—
268C2×M16—31no—
269C2×M16—31no—
270C2×M16—61no—
271C2×M16—61no—
272C2×M16C22×D311yescharacteristic
273C2×M16C22×D311yescharacteristic
274C22×C8—31no—
275C22×C8—31no—
276C22×C8—61no—
277C22×C8—61no—
278C4⋊Q8—21no—
279#125—21no—
280C22⋊Q8—22no—
281C4×Q8—22no—
282#117—31no—
283#117C22×D311yescharacteristic
284#117C22×D311yescharacteristic
285#117C22×D311yescharacteristic
286C2×C42C22×D311yescharacteristic
287#108—62no—
288C4⋊C8—62no—
289C4⋊C8—62no—
290C4⋊C8—62no—
291#104—22no—
292#104—21no—
293#104—21no—
294#104—62no—
295#104—61no—
296#104—61no—
297#103—61no—
298#103—61no—
299#102—61no—
300#102—61no—
301#101—61no—
302#101—61no—
303#100—61no—
304#100—61no—
305#99—22no—
306#99—62no—
307#97—32no—
308#97—61no—
309#97—61no—
310#97—61no—
311#97—61no—
312#97—61no—
313C4×C8—32no—
314C4×C8—61no—
315C4×C8—61no—
316C4×C8—61no—
317C23×C3—21no—
318C23×C3C2×#4111yescharacteristic
319C22×D3—21no—
320C3×Q8—21no—
321C3×Q8—21no—
322C3×Q8—82no—
323C3×D4—21no—
324C3×D4—21no—
325C3×D4—41no—
326C3×D4—41no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—81no—
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341C2×C3×C4C2×#4111yescharacteristic
342#67—41no—
343#67—41no—
344C2×Dic3—21no—
345C2×Dic3—41no—
346D12—21no—
347D12—21no—
348C4×D3—41no—
349C4×D3—41no—
350Dic6—21no—
351Dic6—21no—
352C3×C8—41no—
353C3×C8—41no—
354C3⋊C8—22no—
355C3⋊C8—41no—
356C3⋊C8—41no—
357C3⋊C8—41no—
358C3⋊C8—41no—
359C3⋊C8#12411yescharacteristic
360C3⋊C8#12411yescharacteristic
361C3⋊C8#12411yescharacteristic
362C3⋊C8#12411yescharacteristic
363#41—21no—
364#41—21no—
365#41—21no—
366#41—21no—
367#41—61no—
368#41—61no—
369#41—61no—
370#41—61no—
371C2×Q8—22no—
372C2×Q8—31no—
373C2×Q8#23911yescharacteristic
374C2×D4—21no—
375C2×D4—31no—
376C2×D4—61no—
377C2×D4#23911yescharacteristic
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—61no—
381C22×C4C23×D311yescharacteristic
382Q16—62no—
383SD16—62no—
384SD16—62no—
385D8—62no—
386M16—21no—
387M16—21no—
388M16—21no—
389M16—21no—
390M16—61no—
391M16—61no—
392M16—61no—
393M16—61no—
394M16—61no—
395M16—61no—
396M16—61no—
397M16—61no—
398M16—121no—
399M16—121no—
400M16—121no—
401M16—121no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—31no—
405C2×C8—31no—
406C2×C8—31no—
407C2×C8—31no—
408C2×C8—31no—
409C2×C8—31no—
410C2×C8—32no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—121no—
418C2×C8—121no—
419C2×C8—121no—
420C2×C8—121no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—61no—
427C22⋊C4—21no—
428C22⋊C4—21no—
429C22⋊C4—61no—
430C22⋊C4#23712yes—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—61no—
435C42D3×D412yes—
436C42#23711yescharacteristic
437C42#23711yescharacteristic
438C22×C3—41no—
439C22×C3—41no—
440C22×C3—81no—
441C22×C3#52511yescharacteristic
442C22×C3#52911yescharacteristic
443C22×C3#52911yescharacteristic
444C2×D3—41no—
445C2×D3—81no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—42no—
449C3×C4—42no—
450C3×C4—41no—
451C3×C4—81no—
452C3×C4#52511yescharacteristic
453C3×C4C2×#12411yescharacteristic
454C3×C4#52911yescharacteristic
455C3×C4#52911yescharacteristic
456Dic3—41no—
457Dic3—81no—
458C23—21no—
459C23—61no—
460C23C2×#23711yescharacteristic
461Q8—21no—
462Q8—21no—
463Q8—61no—
464Q8—61no—
465Q8—82no—
466D4—21no—
467D4—21no—
468D4—41no—
469D4—41no—
470D4—61no—
471D4—61no—
472D4—121no—
473D4—121no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—61no—
483C2×C4—81no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4C2×D3×D411yescharacteristic
488C2×C4C2×#23711yescharacteristic
489C2×C4C2×#23911yescharacteristic
490C2×C4D3×#4111yescharacteristic
491C2×C4D3×#4111yescharacteristic
492C2×C4D3×#4111yescharacteristic
493C8—31no—
494C8—31no—
495C8—31no—
496C8—31no—
497C8—41no—
498C8—41no—
499C8—62no—
500C8—121no—
501C8—121no—
502C8—121no—
503C8—121no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—81no—
508C2×C3#239811yescharacteristic
509D3—81no—
510C22—41no—
511C22—41no—
512C22—81no—
513C22—121no—
514C22—241no—
515C22#558611yescharacteristic
516C22#562411yescharacteristic
517C22#568011yescharacteristic
518C4—41no—
519C4—41no—
520C4—42no—
521C4—42no—
522C4—41no—
523C4—81no—
524C4—121no—
525C4—241no—
526C4#562411yescharacteristic
527C4#562711yescharacteristic
528C4D3×#12411yescharacteristic
529C4#568011yescharacteristic
530C3#2383411yescharacteristic, complemented, Sylow
531C2—21no—
532C2—21no—
533C2—21no—
534C2—81no—
535C2—241no—
536C2#7677911yescharacteristic
subgroupG/H
centerC4D3×#124
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2031C2
socleC2×C3#2398
prime radical#5822C22
hypercenter#2031D3
hypocenterC3#23834
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2031

Complements

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

NH ≅ G/Ntype
1. #83329C2semidirect
3. C3×#2031C2semidirect
4. #71842C2semidirect
5. #70404C2semidirect
7. #70370C2semidirect
9. #70040C2semidirect
10. #69016C2semidirect
11. #69014C2semidirect
12. #68751C2semidirect
13. #68740C2semidirect
14. #68700C2semidirect
15. #68697C2semidirect
65. #2031D3semidirect
530. C3#23834semidirect

Maximal subgroups

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

subgroupindex
#833292
#789892
C3×#20312
#718422
#704042
#703792
#703702
#700432
#700402
#690162
#690142
#687512
#687402
#687002
#686972
#238343