Finite Group Databasebeta

← #10764102

Subgroups — #10764102

Order 768.

900 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86215C211yesmaximal, characteristic
2#79294C211yesmaximal, characteristic
3#79052C211yesmaximal, characteristic
4C3×#1549C211yesmaximal, characteristic, complemented
5#71773C211yesmaximal, characteristic, complemented
6#70777C211yesmaximal, characteristic, complemented
7#70511C211yesmaximal, characteristic, complemented
8#68431C212yesmaximal, complemented
9#68422C212yesmaximal, complemented
10#68420C212yesmaximal, complemented
11#68419C212yesmaximal, complemented
12#17777—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14Q8×#67—22no—
15#5730—21no—
16#5729C414yes—
17#5728C2212yes—
18#5727—21no—
19D3×C4⋊Q8C2212yes—
20#5722—21no—
21#5721—21no—
22#5511—22no—
23#5504C412yes—
24#5500C412yes—
25C4×#235C2211yescharacteristic
26#5470C2211yescharacteristic
27C3×#431C2211yescharacteristic
28C3×#423C2211yescharacteristic
29#5128—22no—
30#4925C2211yescharacteristic
31#4924C2211yescharacteristic
32#4838—21no—
33#4837C2212yes—
34#4837C2212yes—
35#4833C2212yes—
36#4833C2212yes—
37#4830—21no—
38#4729—22no—
39#4688—21no—
40C8×Dic6—21no—
41C3×#332C2212yes—
42C3×#332C2212yes—
43#4547C2212yes—
44#4547C2212yes—
45#4486C2211yescharacteristic
46#4486C2212yes—
47#4486C2211yescharacteristic
48#4482C2212yes—
49#4482C2211yescharacteristic
50#4482C2211yescharacteristic
51#3503—31no—
52#2991—31no—
53#2991—31no—
54#1561—31no—
55#1561—32no—
56#1549—32no—
57#1549D311yescharacteristic, complemented
58#1546—32no—
59#1546—31no—
60#1544—31no—
61#1544—32no—
62C22×C3×Q8—21no—
63C2×D3×Q8—22no—
64C2×#235C2×C411yescharacteristic
65C2×C3×M16—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8C2×C412yes—
69C3×C4⋊Q8C2312yes—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2×C412yes—
72C2×C3×C42C2311yescharacteristic
73#946—22no—
74#946—24no—
75Q8×Dic3—22no—
76#944—24no—
77#944—22no—
78C4×#67—24no—
79C4×#67D412yes—
80C4×#67D412yes—
81#926C2×C412yes—
82#926C2×C411yescharacteristic
83#924—21no—
84#924—21no—
85#924—24no—
86C2×#208—21no—
87C2×ΓL(1,𝔽25)—22no—
88C2×ΓL(1,𝔽25)—22no—
89#897—22no—
90#896—24no—
91#893—22no—
92#892C2×C414yes—
93D3×C4⋊C4—22no—
94D3×C4⋊C4C2×C414yes—
95#890—22no—
96#890C2×C414yes—
97#889—24no—
98#888—22no—
99#888—22no—
100#888C2×C414yes—
101#887—22no—
102#887—22no—
103C4×D12D411yescharacteristic
104C4×D12D411yescharacteristic
105#872C2×C412yes—
106C42×D3C2312yes—
107#870C2×C412yes—
108#869C2311yescharacteristic
109#869C2311yescharacteristic
110C4×Dic6—21no—
111C4×Dic6—21no—
112C4×Dic6D411yescharacteristic
113C4×Dic6D411yescharacteristic
114C3×C4⋊C8—21no—
115C3×C4⋊C8C2312yes—
116C3×C4⋊C8C2311yescharacteristic
117C3×C4⋊C8C2311yescharacteristic
118C3×C22⋊C8—21no—
119C3×C4×C8—21no—
120#830—21no—
121(C2×D3)⋊C8—22no—
122(C2×D3)⋊C8—22no—
123Dic3⋊C8—22no—
124Dic3⋊C8—22no—
125C8×Dic3—22no—
126C8×Dic3—22no—
127C4⋊(C3⋊C8)—21no—
128C4⋊(C3⋊C8)C2312yes—
129C4⋊(C3⋊C8)C2311yescharacteristic
130C4⋊(C3⋊C8)C2311yescharacteristic
131C4×C3⋊C8—21no—
132Q82—61no—
133#557—61no—
134#554—62no—
135#554—61no—
136D4×Q8—61no—
137D4×Q8—62no—
138#544—34no—
139#544—32no—
140#533—32no—
141#533—31no—
142C2×C4⋊Q8—32no—
143C2×C4⋊Q8C2×D311yescharacteristic
144#520—32no—
145C4×#41—31no—
146C8×Q8—61no—
147C8×Q8—61no—
148#436—61no—
149#436—62no—
150#436—61no—
151#436—62no—
152#431—31no—
153#431—32no—
154#431—32no—
155#431C2×D311yescharacteristic
156#423—31no—
157#423—32no—
158#423—32no—
159#423C2×D311yescharacteristic
160#332—32no—
161#332—32no—
162#332—31no—
163#332—31no—
164#332—31no—
165#332—32no—
166#332—31no—
167#332—32no—
168#332C2×D312yes—
169#332C2×D312yes—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—81no—
173C2×C3×Q8—81no—
174C22×C3×C4—22no—
175C22×C3×C4C22×C411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C2×#67C22⋊C412yes—
179D3×Q8—82no—
180D3×Q8—82no—
181#235—24no—
182C2×D12C22⋊C411yescharacteristic
183C2×C4×D3—24no—
184C2×C4×D3C22×C412yes—
185C2×C4×D3C2×D412yes—
186C2×C4×D3C2×D412yes—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—24no—
190C2×Dic6C22⋊C411yescharacteristic
191C3×M16—81no—
192C2×C3×C8—21no—
193C2×C3×C8—22no—
194C2×C3×C8—21no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4C22×C414yes—
199C3×C42C22×C411yescharacteristic
200C3×C42C2×D411yescharacteristic
201C3×C42C2×D411yescharacteristic
202C3×C42C2411yescharacteristic
203#217—22no—
204#217C22⋊C412yes—
205(C2×D3)⋊C4—24no—
206(C2×D3)⋊C4C22⋊C412yes—
207(C2×D3)⋊C4C22⋊C412yes—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—24no—
211C4⋊Dic3C22⋊C411yescharacteristic
212C4⋊Dic3C22⋊C411yescharacteristic
213C4⋊Dic3C22×C412yes—
214C4⋊Dic3C22×C412yes—
215Dic3⋊C4—22no—
216Dic3⋊C4—22no—
217Dic3⋊C4—24no—
218Dic3⋊C4—24no—
219Dic3⋊C4—24no—
220Dic3⋊C4C22⋊C412yes—
221Dic3⋊C4C22⋊C412yes—
222C4×Dic3—22no—
223C4×Dic3—24no—
224C4×Dic3C22×C412yes—
225C4×Dic3C2×D412yes—
226C4×Dic3C2×D412yes—
227#208—81no—
228C2×C3⋊C8—21no—
229C2×C3⋊C8—21no—
230C2×C3⋊C8—22no—
231ΓL(1,𝔽25)—82no—
232ΓL(1,𝔽25)—82no—
233C2×#41—31no—
234C22×Q8—21no—
235C22×Q8—62no—
236C2×M16—21no—
237C2×M16—62no—
238C2×M16—62no—
239C2×M16—61no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—34no—
243C4⋊Q8—31no—
244C4⋊Q8—31no—
245C4⋊Q8—62no—
246C4⋊Q8—62no—
247C4⋊Q8—62no—
248C4⋊Q8—64no—
249C4⋊Q8C4×D312yes—
250C4⋊Q8C22×D312yes—
251#125—34no—
252#125—32no—
253#125—64no—
254#125—62no—
255C22⋊Q8—64no—
256C22⋊Q8—64no—
257C22⋊Q8—64no—
258C22⋊Q8—62no—
259C22⋊Q8—62no—
260C22⋊Q8—61no—
261C22⋊Q8—61no—
262C4×Q8—31no—
263C4×Q8—31no—
264C4×Q8—62no—
265C4×Q8—62no—
266C4×Q8—62no—
267C4×Q8—61no—
268C4×Q8—61no—
269C4×D4—32no—
270C4×D4—32no—
271C4×D4—31no—
272C4×D4—31no—
273C4×D4—62no—
274C4×D4—64no—
275#117—31no—
276#117—32no—
277#117—32no—
278#117—34no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—34no—
281C2×C4⋊C4—62no—
282C2×C4⋊C4C4×D312yes—
283C2×C42—32no—
284C2×C42C22×D311yescharacteristic
285C4⋊C8—21no—
286C4⋊C8—32no—
287C4⋊C8—31no—
288C4⋊C8—31no—
289C4⋊C8—62no—
290C4⋊C8—62no—
291C4⋊C8—61no—
292C4⋊C8C22×D311yescharacteristic
293C4⋊C8C22×D312yes—
294C4⋊C8C22×D311yescharacteristic
295C22⋊C8—21no—
296C22⋊C8—62no—
297C22⋊C8—62no—
298C22⋊C8—61no—
299C4×C8—21no—
300C4×C8—62no—
301C4×C8—62no—
302C4×C8—61no—
303C23×C3C2×C22⋊C411yescharacteristic
304C22×D3C2×C22⋊C412yes—
305C3×Q8—42no—
306C3×Q8—42no—
307C3×Q8—82no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—22no—
313C2×C3×C4—24no—
314C2×C3×C4—82no—
315C2×C3×C4C2×C22⋊C411yescharacteristic
316C2×C3×C4C2×C22⋊C411yescharacteristic
317C2×C3×C4C2×C22⋊C411yescharacteristic
318C2×C3×C4C23×C411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321#67—24no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3C2×C22⋊C412yes—
325C2×Dic3C2×C22⋊C412yes—
326C2×Dic3C2×C22⋊C412yes—
327D12—22no—
328C4×D3—22no—
329C4×D3—22no—
330C4×D3—22no—
331C4×D3—22no—
332C4×D3—24no—
333C4×D3—84no—
334Dic6—41no—
335Dic6—41no—
336Dic6—41no—
337Dic6—41no—
338Dic6—84no—
339Dic6#10114yes—
340C3×C8—41no—
341C3×C8—42no—
342C3×C8—41no—
343C3⋊C8—42no—
344C3⋊C8—41no—
345C3⋊C8—41no—
346#41—64no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—31no—
350C2×Q8—64no—
351C2×Q8—61no—
352C2×Q8—61no—
353C2×Q8—81no—
354C2×Q8—81no—
355C2×Q8—242no—
356C2×Q8—242no—
357C2×D4—32no—
358C2×D4—31no—
359C22×C4—22no—
360C22×C4—32no—
361C22×C4—32no—
362C22×C4—32no—
363C22×C4—64no—
364C22×C4C2×C4×D311yescharacteristic
365C22×C4D3×D411yescharacteristic
366C22×C4D3×D411yescharacteristic
367M16—81no—
368M16—242no—
369M16—242no—
370M16—241no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—22no—
374C2×C8—61no—
375C2×C8—61no—
376C2×C8—62no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—32no—
381C4⋊C4—31no—
382C4⋊C4—32no—
383C4⋊C4—31no—
384C4⋊C4—32no—
385C4⋊C4—32no—
386C4⋊C4—64no—
387C4⋊C4—64no—
388C4⋊C4—64no—
389C4⋊C4—64no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C4⋊C4C2×C4×D314yes—
395C22⋊C4—32no—
396C22⋊C4—32no—
397C22⋊C4—32no—
398C22⋊C4—64no—
399C22⋊C4—62no—
400C42—32no—
401C42—32no—
402C42—32no—
403C42—64no—
404C42—62no—
405C42C2×C4×D311yescharacteristic
406C42D3×D411yescharacteristic
407C42D3×D411yescharacteristic
408C42C23×D311yescharacteristic
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3C22×C22⋊C411yescharacteristic
413C2×D3—22no—
414C2×D3—22no—
415C2×D3—22no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—44no—
420C3×C4C2×#10112yes—
421C3×C4#57611yescharacteristic
422C3×C4#57711yescharacteristic
423C3×C4#57711yescharacteristic
424C3×C4#57811yescharacteristic
425Dic3—22no—
426Dic3—22no—
427Dic3—42no—
428Dic3—42no—
429Dic3C2×#10114yes—
430C23—32no—
431C23D3×C22⋊C411yescharacteristic
432Q8—34no—
433Q8—42no—
434Q8—42no—
435Q8—82no—
436Q8—121no—
437Q8—121no—
438Q8—121no—
439Q8—121no—
440Q8—244no—
441D4—64no—
442D4—62no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—32no—
450C2×C4—32no—
451C2×C4—32no—
452C2×C4—64no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—82no—
460C2×C4—244no—
461C2×C4D3×C22⋊C411yescharacteristic
462C2×C4D3×C22⋊C411yescharacteristic
463C2×C4D3×C22⋊C411yescharacteristic
464C2×C4C22×C4×D311yescharacteristic
465C2×C4C2×D3×D411yescharacteristic
466C2×C4C2×D3×D411yescharacteristic
467C8—42no—
468C8—41no—
469C8—41no—
470C8—121no—
471C8—121no—
472C8—122no—
473C2×C3—41no—
474C2×C3C22×#10111yescharacteristic
475C2×C3#289711yescharacteristic
476C2×C3#289811yescharacteristic
477D3—42no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—62no—
482C22—62no—
483C22—62no—
484C22C2×D3×C22⋊C411yescharacteristic
485C4—21no—
486C4—21no—
487C4—21no—
488C4—34no—
489C4—44no—
490C4—62no—
491C4—62no—
492C4—122no—
493C4—122no—
494C4D3×#10112yes—
495C4#576211yescharacteristic
496C4#576611yescharacteristic
497C4#577411yescharacteristic
498C4#577511yescharacteristic
499C3#1777711yescharacteristic, complemented, Sylow
500C2—41no—
501C2—122no—
502C2C2×D3×#10111yescharacteristic
503C2#7879011yescharacteristic
504C2#7892511yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1549C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#235C2×C4
cosocleC3×C42C24
hypercenter#1549D3
hypocenterC3#17777
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
4. C3×#1549C2semidirect
5. #71773C2semidirect
6. #70777C2semidirect
7. #70511C2semidirect
8. #68431C2semidirect
9. #68422C2semidirect
10. #68420C2semidirect
11. #68419C2semidirect
57. #1549D3semidirect
499. C3#17777semidirect

Maximal subgroups

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

subgroupindex
#862152
#792942
#790522
C3×#15492
#717732
#707772
#705112
#684312
#684222
#684202
#684192
#177773