Finite Group Databasebeta

← #10840296

Subgroups — #10840296

Order 768.

638 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82543C211yesmaximal, characteristic, complemented
2C2×#4748C211yesmaximal, characteristic
3C3×#1829C211yesmaximal, characteristic, complemented
4#71651C211yesmaximal, characteristic
5#70078C211yesmaximal, characteristic, complemented
6#70076C211yesmaximal, characteristic
7#70073C211yesmaximal, characteristic, complemented
8#69032C211yesmaximal, characteristic, complemented
9#69030C211yesmaximal, characteristic, complemented
10#69029C211yesmaximal, characteristic, complemented
11#69022C211yesmaximal, characteristic, complemented
12#69019C211yesmaximal, characteristic, complemented
13#69015C211yesmaximal, characteristic, complemented
14#69014C211yesmaximal, characteristic, complemented
15#69013C211yesmaximal, characteristic, complemented
16#24987—31nomaximal, Sylow
17C3×#518C2211yescharacteristic
18C22×#214C2211yescharacteristic
19C2×#932C2211yescharacteristic
20C2×#909C2211yescharacteristic
21C2×#907C2211yescharacteristic
22#5516C2211yescharacteristic
23C2×C3×#100C2211yescharacteristic
24C2×C3×#100C2211yescharacteristic
25#5215C2211yescharacteristic
26#5215C2211yescharacteristic
27#5139C2211yescharacteristic
28C2×#824C2211yescharacteristic
29C2×#824C2211yescharacteristic
30#5132C2211yescharacteristic
31#5125C2211yescharacteristic
32#5124C2211yescharacteristic
33C2×#822C2211yescharacteristic
34C2×#822C2211yescharacteristic
35#5122C2211yescharacteristic
36#5120C2211yescharacteristic
37#5018—22no—
38C4×#214—22no—
39#5006C2211yescharacteristic
40#5005C2211yescharacteristic
41#5004C2211yescharacteristic
42#4998C2211yescharacteristic
43#4997C2211yescharacteristic
44#4748—22no—
45#4748—22no—
46C3×#344C2211yescharacteristic
47C3×#344C2211yescharacteristic
48C3×#344C2211yescharacteristic
49C3×#344C2211yescharacteristic
50#4559C2211yescharacteristic
51#4557C2211yescharacteristic
52#4556C2211yescharacteristic
53#4556C2211yescharacteristic
54#4533C2211yescharacteristic
55#4531C2211yescharacteristic
56C2×#471—31no—
57#2949—31no—
58#2082—31no—
59#2081—31no—
60#2049—31no—
61#2048—31no—
62#1977—31no—
63#1953—31no—
64#1905—31no—
65#1904—31no—
66#1833—31no—
67#1830—31no—
68#1829D311yescharacteristic, complemented
69#1814—31no—
70#1757—31no—
71C22×C3×D4C2311yescharacteristic
72C2×#235C2311yescharacteristic
73C22×Dic6C2311yescharacteristic
74C2×C3×M16C2311yescharacteristic
75C2×C3×M16C2311yescharacteristic
76C2×C3×M16C2311yescharacteristic
77C2×C3×M16C2311yescharacteristic
78C3×C4×D4—22no—
79C3×C4×D4—41no—
80C3×#117C2311yescharacteristic
81C3×#117C2311yescharacteristic
82C2×C3×C22⋊C4—22no—
83C2×#214—21no—
84C2×#214—21no—
85C2×#214—21no—
86C2×#214—21no—
87C2×#214—21no—
88C2×#214D412yes—
89C2×#214D412yes—
90#932—22no—
91#932—41no—
92C2×#213—21no—
93#926C2311yescharacteristic
94#926C2311yescharacteristic
95C2×#208C2311yescharacteristic
96C2×#208C2311yescharacteristic
97C2×#208C2311yescharacteristic
98C22×C3⋊C8C2311yescharacteristic
99#909—22no—
100#909—41no—
101#907—21no—
102#907—21no—
103#907—21no—
104#907—21no—
105C2×Dic12—21no—
106C2×#204—21no—
107C2×ΓL(1,𝔽25)—21no—
108C2×D3×C8—21no—
109#887—41no—
110#877—22no—
111C4×Dic6—22no—
112C3×#100—22no—
113C3×#100—41no—
114#832—22no—
115#832—41no—
116#825—41no—
117#824—22no—
118#824—21no—
119#824—41no—
120#823—21no—
121#822—21no—
122#822—21no—
123#822—21no—
124#822—21no—
125#822—41no—
126#819—41no—
127#816—41no—
128#815—21no—
129C8×Dic3—21no—
130#810—22no—
131#808—21no—
132#807—21no—
133#806—22no—
134#805—41no—
135#803—21no—
136C4×C3⋊C8—21no—
137C2×#137—31no—
138C2×AGL(1,ℤ/8)—31no—
139C22×SD16—31no—
140C2×#131—31no—
141#519—31no—
142#518C2×D311yescharacteristic
143#471—62no—
144#471—62no—
145#440—62no—
146C4×SD16—62no—
147#430—31no—
148#430—31no—
149C2×#108—31no—
150C2×#108—31no—
151#427—31no—
152#421—31no—
153#421—31no—
154#419—31no—
155#419—31no—
156#418—31no—
157#418—31no—
158#413—31no—
159C2×#101—31no—
160C2×#101—31no—
161C2×#100C2×D311yescharacteristic
162C2×#100C2×D311yescharacteristic
163#410—31no—
164#405—31no—
165#405—31no—
166#344—31no—
167#344—31no—
168#344—31no—
169#344—31no—
170#344C2×D311yescharacteristic
171#344C2×D311yescharacteristic
172#344C2×D311yescharacteristic
173#344C2×D311yescharacteristic
174H(3,ℤ/4)—31no—
175H(3,ℤ/4)—31no—
176C24×C3—21no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4C2×D412yes—
183C22×C3×C4—42no—
184C22×C3×C4C2411yescharacteristic
185C2×#67—21no—
186C22×Dic3—21no—
187#235—21no—
188#235—21no—
189#235—21no—
190#235—21no—
191C2×D12#4111yescharacteristic
192C2×C4×D3—21no—
193C2×Dic6—21no—
194C2×Dic6—21no—
195C2×Dic6—21no—
196C2×Dic6—21no—
197C2×Dic6—21no—
198C2×Dic6C2×D412yes—
199C2×Dic6#4111yescharacteristic
200C3×M16—21no—
201C3×M16—21no—
202C3×M16—21no—
203C3×M16—21no—
204C3×M16—21no—
205C3×M16—21no—
206C3×M16—21no—
207C3×M16—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C22⋊C4—21no—
215C3×C22⋊C4—42no—
216C3×C22⋊C4#4112yes—
217C3×C42—21no—
218C3×C42—21no—
219#217—21no—
220#217—21no—
221#214—22no—
222#214—22no—
223#214—22no—
224#214—22no—
225#214—42no—
226#213—42no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229Dic3⋊C4—42no—
230C4×Dic3—21no—
231C4×Dic3—21no—
232C4×Dic3—42no—
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⋊C8#4111yescharacteristic
245C2×C3⋊C8#4111yescharacteristic
246C2×C3⋊C8#4111yescharacteristic
247C2×C3⋊C8#4111yescharacteristic
248Dic12—42no—
249#204—42no—
250ΓL(1,𝔽25)—41no—
251ΓL(1,𝔽25)—41no—
252D3×C8—41no—
253D3×C8—41no—
254C2×#41—31no—
255C22×Q8—31no—
256C22×D4C22×D311yescharacteristic
257#137—62no—
258#137—121no—
259AGL(1,ℤ/8)—62no—
260AGL(1,ℤ/8)—121no—
261C2×Q16—61no—
262C2×SD16—32no—
263C2×SD16—32no—
264C2×SD16—61no—
265C2×SD16—61no—
266C2×SD16—61no—
267C2×SD16—61no—
268C2×SD16—61no—
269C2×SD16—61no—
270C2×D8—61no—
271#131—61no—
272#131—61no—
273#131—61no—
274#131—61no—
275C2×M16—31no—
276C2×M16—31no—
277C2×M16—31no—
278C2×M16—61no—
279C2×M16C22×D311yescharacteristic
280C2×M16C22×D311yescharacteristic
281C2×M16C22×D311yescharacteristic
282C2×M16C22×D311yescharacteristic
283C22×C8—31no—
284C22×C8—61no—
285C4×Q8—62no—
286C4×Q8—121no—
287C4×D4—22no—
288C4×D4—41no—
289#117—31no—
290#117—31no—
291#117—62no—
292#117C22×D311yescharacteristic
293#117C22×D311yescharacteristic
294C2×C22⋊C4—22no—
295#108—61no—
296#108—61no—
297#108—61no—
298#108—61no—
299#108—121no—
300#108—121no—
301#107—61no—
302#106—61no—
303#104—121no—
304#104—121no—
305#103—62no—
306#103—121no—
307#102—62no—
308#102—121no—
309#101—62no—
310#101—61no—
311#101—121no—
312#100—22no—
313#100—41no—
314#100—61no—
315#99—62no—
316#97—61no—
317#97—61no—
318C4×C8—61no—
319C4×C8—61no—
320C23×C3—22no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—161no—
324C23×C3C2×#4111yescharacteristic
325C22×D3—21no—
326C3×D4—22no—
327C3×D4—22no—
328C3×D4—41no—
329C3×D4—41no—
330C2×C3×C4—22no—
331C2×C3×C4—41no—
332C2×C3×C4—162no—
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C2×#4111yescharacteristic
335C2×C3×C4C2×#4111yescharacteristic
336C2×C3×C4C2×#4111yescharacteristic
337C2×C3×C4H(5,𝔽2)11yescharacteristic
338C2×C3×C4H(5,𝔽2)11yescharacteristic
339#67—41no—
340#67—41no—
341C2×Dic3—21no—
342C2×Dic3—22no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347D12—21no—
348D12—21no—
349C4×D3—41no—
350C4×D3—41no—
351Dic6—22no—
352Dic6—22no—
353Dic6—21no—
354Dic6—21no—
355Dic6—41no—
356Dic6—41no—
357C3×C8—41no—
358C3×C8—41no—
359C3×C8—41no—
360C3×C8—41no—
361C3⋊C8—41no—
362C3⋊C8—41no—
363C3⋊C8—41no—
364C3⋊C8#12311yescharacteristic
365C3⋊C8#12311yescharacteristic
366C3⋊C8#12311yescharacteristic
367C3⋊C8#12311yescharacteristic
368C24—21no—
369#41—61no—
370#41—61no—
371#41—61no—
372#41—61no—
373C2×Q8—32no—
374C2×Q8—31no—
375C2×Q8—61no—
376C2×Q8—61no—
377C2×Q8—61no—
378C2×Q8—61no—
379C2×Q8—61no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—31no—
386C2×D4—61no—
387C2×D4D3×D412yes—
388C22×C4—42no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4C23×D311yescharacteristic
392Q16—122no—
393SD16—62no—
394SD16—62no—
395SD16—62no—
396SD16—62no—
397SD16—122no—
398SD16—122no—
399D8—122no—
400M16—21no—
401M16—21no—
402M16—21no—
403M16—21no—
404M16—21no—
405M16—21no—
406M16—21no—
407M16—21no—
408M16—61no—
409M16—61no—
410M16—61no—
411M16—61no—
412M16—61no—
413M16—61no—
414M16—121no—
415M16—121no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—31no—
421C2×C8—31no—
422C2×C8—31no—
423C2×C8—31no—
424C2×C8—31no—
425C2×C8—31no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—121no—
430C2×C8—121no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—122no—
436C22⋊C4—21no—
437C22⋊C4—42no—
438C22⋊C4—61no—
439C22⋊C4—61no—
440C22⋊C4#23512yes—
441C42—21no—
442C42—21no—
443C42—61no—
444C42—61no—
445C42—122no—
446C22×C3—41no—
447C22×C3—42no—
448C22×C3—82no—
449C22×C3—161no—
450C22×C3—161no—
451C22×C3#52911yescharacteristic
452C22×C3#54611yescharacteristic
453C22×C3#55511yescharacteristic
454C2×D3—41no—
455C2×D3—81no—
456C3×C4—42no—
457C3×C4—81no—
458C3×C4C2×#12311yescharacteristic
459C3×C4#54011yescharacteristic
460C3×C4#54611yescharacteristic
461C3×C4#55311yescharacteristic
462Dic3—42no—
463Dic3—41no—
464Dic3—81no—
465Dic3—81no—
466C23—22no—
467C23—41no—
468C23—41no—
469C23—61no—
470C23—161no—
471C23C2×#23511yescharacteristic
472Q8—62no—
473Q8—61no—
474Q8—62no—
475Q8—61no—
476Q8—121no—
477Q8—121no—
478D4—22no—
479D4—22no—
480D4—41no—
481D4—41no—
482D4—61no—
483D4—61no—
484D4—121no—
485D4—121no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—62no—
489C2×C4—61no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4—121no—
496C2×C4—162no—
497C2×C4C2×D3×D411yescharacteristic
498C2×C4#100411yescharacteristic
499C2×C4D3×#4111yescharacteristic
500C2×C4D3×#4111yescharacteristic
501C2×C4D3×#4111yescharacteristic
502C2×C4#100911yescharacteristic
503C8—31no—
504C8—31no—
505C8—31no—
506C8—31no—
507C8—41no—
508C8—41no—
509C8—41no—
510C8—41no—
511C8—121no—
512C8—121no—
513C8—121no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—82no—
518C2×C3#242011yescharacteristic
519D3—81no—
520C22—41no—
521C22—42no—
522C22—82no—
523C22—121no—
524C22—161no—
525C22—161no—
526C22—241no—
527C22#549011yescharacteristic
528C22#552811yescharacteristic
529C22#553211yescharacteristic
530C4—42no—
531C4—81no—
532C4—121no—
533C4—122no—
534C4—241no—
535C4—241no—
536C4D3×#12311yescharacteristic
537C4#565411yescharacteristic
538C4#566111yescharacteristic
539C4#566311yescharacteristic
540C3#2498711yescharacteristic, complemented, Sylow
541C2—21no—
542C2—21no—
543C2—21no—
544C2—82no—
545C2—241no—
546C2#7406611yescharacteristic
subgroupG/H
centerC2#74066
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1829C2
socleC2×C3#2420
prime radicalC2×#932C22
hypercenter#1829D3
hypocenterC3#24987
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1829

Complements

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

NH ≅ G/Ntype
1. #82543C2semidirect
3. C3×#1829C2semidirect
5. #70078C2semidirect
7. #70073C2semidirect
8. #69032C2semidirect
9. #69030C2semidirect
10. #69029C2semidirect
11. #69022C2semidirect
12. #69019C2semidirect
13. #69015C2semidirect
14. #69014C2semidirect
15. #69013C2semidirect
68. #1829D3semidirect
540. C3#24987semidirect

Maximal subgroups

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

subgroupindex
#825432
C2×#47482
C3×#18292
#716512
#700782
#700762
#700732
#690322
#690302
#690292
#690222
#690192
#690152
#690142
#690132
#249873