Finite Group Databasebeta

← #10849422

Subgroups — #10849422

Order 768.

984 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86227C211yesmaximal, characteristic
2#83284C211yesmaximal, characteristic, complemented
3#79061C211yesmaximal, characteristic, complemented
4#73307C211yesmaximal, characteristic, complemented
5C3×#1962C211yesmaximal, characteristic, complemented
6#70525C211yesmaximal, characteristic, complemented
7#70364C212yesmaximal, complemented
8#70363C212yesmaximal, complemented
9#70362C212yesmaximal, complemented
10#70361C212yesmaximal, complemented
11C4×#805C211yesmaximal, characteristic, complemented
12#25668—31nomaximal, Sylow
13C3×#532C2211yescharacteristic
14#5833C2212yes—
15#5827C2212yes—
16#5731—21no—
17#5719—21no—
18#5680—22no—
19#5601—24no—
20#5565—22no—
21#5541—22no—
22C4×#235C2211yescharacteristic
23C2×#870C2211yescharacteristic
24C2×#870C2211yescharacteristic
25C3×#423C2211yescharacteristic
26C2×C3×#104C2212yes—
27C2×C3×#104C2212yes—
28C3×C43C2211yescharacteristic
29#5232C2212yes—
30#5232C2212yes—
31#5133C2212yes—
32#5121C2212yes—
33#5082—21no—
34#5071—21no—
35#5056—22no—
36#4936C2212yes—
37#4935C2212yes—
38#4928—24no—
39C2×#805C2211yescharacteristic
40C2×#805C2211yescharacteristic
41C4×#208C2211yescharacteristic
42#4847—22no—
43#4829—22no—
44#4554C2212yes—
45#4554C2212yes—
46#4523C2211yescharacteristic
47#4523C2211yescharacteristic
48#3512—31no—
49#3142—31no—
50#3080—31no—
51#2859—31no—
52#2040—32no—
53#2039—32no—
54#1998—31no—
55#1962D311yescharacteristic, complemented
56#1860—32no—
57#1859—32no—
58C4×#104—31no—
59C2×C3×#41C2312yes—
60C2×#235C2311yescharacteristic
61C2×C3×M16C2312yes—
62C3×C4⋊Q8D411yescharacteristic
63C3×#127D411yescharacteristic
64C3×#124D412yes—
65C3×#121—22no—
66C3×C4×D4—22no—
67C2×C3×C42—22no—
68C2×C3×C42C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×C3×C42C2311yescharacteristic
71#946—22no—
72Q8×Dic3—22no—
73#939—24no—
74#939—24no—
75#938—22no—
76#935—24no—
77D4×Dic3—22no—
78D4×Dic3—24no—
79#930—22no—
80C4×#67—22no—
81#926—22no—
82#926C2311yescharacteristic
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3—24no—
85C2×C4⋊Dic3C2312yes—
86C2×C4⋊Dic3C2312yes—
87#924—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4D414yes—
90C2×#208C2311yescharacteristic
91C2×#208C2311yescharacteristic
92#898—24no—
93#889—24no—
94#883—24no—
95#67⋊C4—24no—
96#879—24no—
97#877—24no—
98C4×D12D412yes—
99#872—21no—
100C42×D3—21no—
101#870—21no—
102#870—21no—
103#870—22no—
104#870—21no—
105#870—21no—
106#870—22no—
107C4×Dic6D412yes—
108C3×C4⋊C8—22no—
109C3×#104—24no—
110C3×#104—24no—
111#835—22no—
112#835—22no—
113#832—22no—
114#832—22no—
115C3⋊#95—24no—
116#821—22no—
117#818—22no—
118#817—22no—
119#816—22no—
120#805—22no—
121#805—22no—
122#803—21no—
123C4×C3⋊C8—21no—
124#553—64no—
125#553—62no—
126#552—62no—
127#541—61no—
128#540—61no—
129#537—32no—
130#532C2×D311yescharacteristic
131#529—62no—
132C2×#125—31no—
133C2×#125—31no—
134#520—32no—
135C4×#41—31no—
136#491—61no—
137#490—61no—
138#488—62no—
139#479—62no—
140#478—62no—
141#428—32no—
142#423C2×D311yescharacteristic
143C2×#104—31no—
144C2×#104—31no—
145C2×#104C2×D312yes—
146C2×#104C2×D312yes—
147#417—32no—
148#417—32no—
149#417—32no—
150C4×M16—31no—
151#389—64no—
152#384—32no—
153#382—32no—
154C43C2×D311yescharacteristic
155#343—32no—
156#343—32no—
157#339—31no—
158#339—31no—
159C3×#41—22no—
160C3×#41—22no—
161C3×#41—22no—
162C3×#41—22no—
163C2×C3×Q8C2×D412yes—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4C2×D412yes—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2411yescharacteristic
172C2×#67—21no—
173C2×#67—24no—
174C22×Dic3—24no—
175C22×Dic3—22no—
176C22×Dic3C2×D414yes—
177#235—21no—
178#235—21no—
179#235—21no—
180#235—21no—
181C2×D12C2×D411yescharacteristic
182C2×C4×D3—21no—
183C2×C4×D3—22no—
184C2×Dic6C2×D411yescharacteristic
185C3×M16—22no—
186C3×M16—22no—
187C2×C3×C8—22no—
188C3×C4⋊C4—22no—
189C3×C22⋊C4—22no—
190C3×C22⋊C4—22no—
191C3×C42—21no—
192C3×C42—21no—
193C3×C42—44no—
194C3×C42C2×D411yescharacteristic
195C3×C42C2×D411yescharacteristic
196C3×C42C2×D412yes—
197C3×C42#4112yes—
198C3×C42#4112yes—
199#217—24no—
200#217—24no—
201#217—21no—
202(C2×D3)⋊C4—24no—
203(C2×D3)⋊C4—22no—
204C4⋊Dic3—24no—
205C4⋊Dic3—22no—
206C4⋊Dic3—22no—
207C4⋊Dic3—22no—
208C4⋊Dic3—22no—
209C4⋊Dic3—48no—
210C4⋊Dic3C2×D412yes—
211Dic3⋊C4—24no—
212Dic3⋊C4—22no—
213Dic3⋊C4—24no—
214Dic3⋊C4—24no—
215Dic3⋊C4—42no—
216Dic3⋊C4—42no—
217C4×Dic3—24no—
218C4×Dic3—22no—
219C4×Dic3—21no—
220#208—21no—
221#208—21no—
222#208—21no—
223#208—21no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8—21no—
226C2×#41—31no—
227C2×#41C22×D312yes—
228C2×M16—31no—
229C2×M16—31no—
230C2×M16C22×D312yes—
231C4⋊Q8D1211yescharacteristic
232#127D1211yescharacteristic
233#126—64no—
234#126—64no—
235#125—64no—
236#125—61no—
237#125—61no—
238#125—61no—
239#125—61no—
240#125—62no—
241#125—62no—
242#124D1212yes—
243#123—64no—
244#123—64no—
245C22⋊Q8—62no—
246C22⋊Q8—62no—
247#121—22no—
248#121—62no—
249#121—64no—
250#121—64no—
251#121—62no—
252C4×Q8—32no—
253C4×Q8—62no—
254C4×D4—22no—
255C4×D4—32no—
256C4×D4—64no—
257C4×D4—64no—
258C4×D4—62no—
259C4×D4—62no—
260#117—31no—
261#117—64no—
262#117—62no—
263#117—61no—
264C2×C4⋊C4—34no—
265C2×C4⋊C4—32no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—64no—
269C2×C4⋊C4—62no—
270C2×C42—22no—
271C2×C42—61no—
272C2×C42C22×D311yescharacteristic
273C2×C42C22×D311yescharacteristic
274C2×C42C22×D311yescharacteristic
275#107—62no—
276#106—62no—
277C4⋊C8—22no—
278#104—24no—
279#104—24no—
280#104—62no—
281#104—62no—
282#103—62no—
283#103—62no—
284#103—62no—
285#102—62no—
286#102—62no—
287#102—62no—
288#97—61no—
289C4×C8—61no—
290#95—64no—
291C23×C3—22no—
292C23×C3C22×D411yescharacteristic
293C22×D3—21no—
294C3×Q8—22no—
295C3×Q8—22no—
296C3×D4—22no—
297C3×D4—22no—
298C3×D4—42no—
299C3×D4—42no—
300C3×D4—82no—
301C2×C3×C4—24no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4#12012yes—
310C2×C3×C4#12012yes—
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C42−1+411yescharacteristic
316C2×C3×C42−1+411yescharacteristic
317#67—41no—
318#67—41no—
319#67—84no—
320C2×Dic3—22no—
321C2×Dic3—21no—
322C2×Dic3—24no—
323C2×Dic3—44no—
324C2×Dic3—44no—
325C2×Dic3—42no—
326C2×Dic3—84no—
327D12—21no—
328D12—21no—
329C4×D3—41no—
330C4×D3—41no—
331C4×D3—82no—
332Dic6—21no—
333Dic6—21no—
334C3×C8—42no—
335C3⋊C8—41no—
336C3⋊C8—41no—
337#41—22no—
338#41—22no—
339#41—22no—
340#41—22no—
341#41—61no—
342#41—61no—
343#41—61no—
344#41—61no—
345C2×Q8—31no—
346C2×Q8C2×D1212yes—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—31no—
350C2×D4—61no—
351C2×D4—64no—
352C2×D4C2×D1212yes—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—34no—
357C22×C4—64no—
358C22×C4—61no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4D3×D412yes—
362C22×C4C23×D311yescharacteristic
363M16—22no—
364M16—22no—
365M16—61no—
366M16—61no—
367M16—61no—
368M16—61no—
369C2×C8—22no—
370C2×C8—61no—
371C2×C8—61no—
372C4⋊C4—22no—
373C4⋊C4—32no—
374C4⋊C4—64no—
375C4⋊C4—64no—
376C4⋊C4—64no—
377C4⋊C4—64no—
378C4⋊C4—62no—
379C4⋊C4—62no—
380C4⋊C4—62no—
381C4⋊C4—62no—
382C4⋊C4—62no—
383C4⋊C4—128no—
384C4⋊C4—122no—
385C4⋊C4—122no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—64no—
389C22⋊C4—64no—
390C22⋊C4—64no—
391C22⋊C4—61no—
392C22⋊C4—62no—
393C42—21no—
394C42—21no—
395C42—44no—
396C42—64no—
397C42—62no—
398C42—61no—
399C42C2×D1211yescharacteristic
400C42C2×D1211yescharacteristic
401C42D3×D412yes—
402C42#23712yes—
403C42#23712yes—
404C22×C3—22no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3C2×#12011yescharacteristic
409C22×C3#54811yescharacteristic
410C22×C3#54811yescharacteristic
411C2×D3—41no—
412C2×D3—42no—
413C3×C4—24no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—42no—
417C3×C4—42no—
418C3×C4#53211yescharacteristic
419C3×C4#53611yescharacteristic
420C3×C4#54711yescharacteristic
421C3×C4#54711yescharacteristic
422Dic3—41no—
423Dic3—42no—
424Dic3—44no—
425Dic3—82no—
426C23—22no—
427C23—61no—
428C23C2×D3×D411yescharacteristic
429Q8—22no—
430Q8—22no—
431Q8—61no—
432Q8—61no—
433D4—22no—
434D4—22no—
435D4—42no—
436D4—42no—
437D4—61no—
438D4—61no—
439D4—82no—
440D4—121no—
441D4—121no—
442D4—244no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—62no—
452C2×C4—61no—
453C2×C4—64no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—124no—
457C2×C4—124no—
458C2×C4—122no—
459C2×C4—242no—
460C2×C4—244no—
461C2×C4#88212yes—
462C2×C4#88212yes—
463C2×C4C22×D1211yescharacteristic
464C2×C4C2×D3×D411yescharacteristic
465C2×C4C2×#23711yescharacteristic
466C2×C4C2×#23711yescharacteristic
467C2×C4#101011yescharacteristic
468C2×C4#101011yescharacteristic
469C8—42no—
470C8—121no—
471C8—121no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—82no—
475C2×C3#243211yescharacteristic
476C2×C3#301812yes—
477D3—81no—
478C22—22no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—122no—
483C22—121no—
484C22C2×#88211yescharacteristic
485C22#559811yescharacteristic
486C22#559811yescharacteristic
487C4—24no—
488C4—22no—
489C4—22no—
490C4—42no—
491C4—42no—
492C4—124no—
493C4—122no—
494C4—121no—
495C4—242no—
496C4#555511yescharacteristic
497C4#555511yescharacteristic
498C4#567511yescharacteristic
499C4#571611yescharacteristic
500C3#2566811yescharacteristic, complemented, Sylow
501C2—21no—
502C2—21no—
503C2—82no—
504C2—241no—
505C2#7644511yescharacteristic
506C2#7898612yes—
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1962C2
socleC22×C3C2×#120
prime radical#86227C2
hypercenter#1962D3
hypocenterC3#25668
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1962

Complements

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

NH ≅ G/Ntype
2. #83284C2semidirect
3. #79061C2semidirect
4. #73307C2semidirect
5. C3×#1962C2semidirect
6. #70525C2semidirect
7. #70364C2semidirect
8. #70363C2semidirect
9. #70362C2semidirect
10. #70361C2semidirect
11. C4×#805C2semidirect
55. #1962D3semidirect
500. C3#25668semidirect

Maximal subgroups

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

subgroupindex
#862272
#832842
#790612
#733072
C3×#19622
#705252
#703642
#703632
#703622
#703612
C4×#8052
#256683