Finite Group Databasebeta

← #10764104

Subgroups — #10764104

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#86281C211yesmaximal, characteristic
2#79052C211yesmaximal, characteristic
3#79052C211yesmaximal, characteristic
4C3×#1561C211yesmaximal, characteristic, complemented
5#71829C211yesmaximal, characteristic, complemented
6#70508C211yesmaximal, characteristic, complemented
7#70506C211yesmaximal, characteristic, complemented
8#68431C212yesmaximal, complemented
9#68422C212yesmaximal, complemented
10#68420C212yesmaximal, complemented
11#68419C212yesmaximal, complemented
12#17777—31nomaximal, Sylow
13C3×#533C2211yescharacteristic
14#5728C2212yes—
15D3×C4⋊Q8C2212yes—
16#5689C414yes—
17#5635—22no—
18#5632—22no—
19#5579—21no—
20Q8×D12—21no—
21#5567—21no—
22Q8×Dic6—21no—
23#5523C412yes—
24#5519C412yes—
25C4×#235C2211yescharacteristic
26C2×#869C2211yescharacteristic
27C3×#423C2211yescharacteristic
28C3×#423C2211yescharacteristic
29#5128—22no—
30#5128—22no—
31#4925C2211yescharacteristic
32#4925C2211yescharacteristic
33#4838—21no—
34#4838—21no—
35#4837C2212yes—
36#4837C2212yes—
37#4833C2212yes—
38#4833C2212yes—
39#4830—21no—
40#4830—21no—
41C3×#332C2212yes—
42C3×#332C2212yes—
43#4547C2211yescharacteristic
44#4547C2211yescharacteristic
45#4547C2211yescharacteristic
46#4547C2211yescharacteristic
47#4486C2212yes—
48#4486C2212yes—
49#4482C2212yes—
50#4482C2212yes—
51#3503—31no—
52#2991—31no—
53#2991—31no—
54#1561—32no—
55#1561D311yescharacteristic, complemented
56#1549—31no—
57#1549—32no—
58#1546—31no—
59#1546—32no—
60#1544—32no—
61#1544—31no—
62C2×D3×Q8—22no—
63C2×#235C2×C411yescharacteristic
64C22×Dic6—21no—
65C2×C3×M16—21no—
66C2×C3×M16—21no—
67C3×C4⋊Q8C2311yescharacteristic
68C3×C4⋊Q8C2311yescharacteristic
69C3×#125C2×C412yes—
70C3×C22⋊Q8—21no—
71C3×C22⋊Q8—21no—
72C3×C4×Q8—21no—
73C3×C4×Q8—21no—
74C3×#117C2×C412yes—
75C2×C3×C42C2311yescharacteristic
76#946—22no—
77Q8×Dic3—22no—
78#944—22no—
79C4×#67D412yes—
80C4×#67D412yes—
81#926C2×C411yescharacteristic
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3C2×C412yes—
84C2×ΓL(1,𝔽25)—22no—
85C2×ΓL(1,𝔽25)—22no—
86#897—22no—
87#897—24no—
88#896—24no—
89#892C2×C414yes—
90D3×C4⋊C4—22no—
91D3×C4⋊C4C2×C414yes—
92#890C2×C414yes—
93#889—24no—
94#888—24no—
95#888—22no—
96#888C2×C414yes—
97#887—22no—
98#67⋊C4—24no—
99#878—24no—
100C4×D12—22no—
101C4×D12D411yescharacteristic
102C4×D12D411yescharacteristic
103#872C2×C412yes—
104C42×D3C2312yes—
105#870—22no—
106#869—21no—
107#869—21no—
108#869—22no—
109#869C2×C412yes—
110#869C2312yes—
111C4×Dic6—22no—
112C4×Dic6D411yescharacteristic
113C4×Dic6D411yescharacteristic
114C3×C4⋊C8—21no—
115C3×C4⋊C8—21no—
116C3×C4⋊C8C2312yes—
117C3×C4⋊C8C2312yes—
118#830—21no—
119#830—21no—
120(C2×D3)⋊C8—22no—
121(C2×D3)⋊C8—22no—
122Dic3⋊C8—22no—
123Dic3⋊C8—22no—
124C8×Dic3—22no—
125C8×Dic3—22no—
126C4⋊(C3⋊C8)C2311yescharacteristic
127C4⋊(C3⋊C8)C2311yescharacteristic
128C4⋊(C3⋊C8)C2311yescharacteristic
129C4⋊(C3⋊C8)C2311yescharacteristic
130C4×C3⋊C8—21no—
131C4×C3⋊C8—21no—
132Q82—61no—
133#557—61no—
134#554—61no—
135#554—62no—
136D4×Q8—61no—
137D4×Q8—62no—
138#544—34no—
139#544—32no—
140#533—32no—
141#533C2×D311yescharacteristic
142C2×C4⋊Q8—32no—
143C2×C4⋊Q8—31no—
144#520—32no—
145C4×#41—31no—
146C8×Q8—61no—
147C8×Q8—61no—
148#436—61no—
149#436—61no—
150#436—62no—
151#436—62no—
152#431—31no—
153#431—31no—
154#431—32no—
155#431—32no—
156#423—32no—
157#423—32no—
158#423C2×D311yescharacteristic
159#423C2×D311yescharacteristic
160#332—31no—
161#332—31no—
162#332—31no—
163#332—31no—
164#332—32no—
165#332—32no—
166#332—32no—
167#332—32no—
168#332C2×D312yes—
169#332C2×D312yes—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C22×C3×C4C22×C411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C2×#67C22⋊C412yes—
176C22×Dic3—22no—
177D3×Q8—82no—
178D3×Q8—82no—
179#235—24no—
180C2×D12C22⋊C411yescharacteristic
181C2×C4×D3—24no—
182C2×C4×D3C22×C412yes—
183C2×C4×D3C2×D412yes—
184C2×C4×D3C2×D412yes—
185C2×Dic6—22no—
186C2×Dic6—22no—
187C2×Dic6—24no—
188C2×Dic6—81no—
189C2×Dic6—81no—
190C2×Dic6C22⋊C411yescharacteristic
191C3×M16—81no—
192C3×M16—81no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4C22×C412yes—
200C3×C4⋊C4C22×C412yes—
201C3×C22⋊C4—22no—
202C3×C42—22no—
203C3×C42C22×C411yescharacteristic
204C3×C42C2×D411yescharacteristic
205C3×C42C2×D411yescharacteristic
206C3×C42C2411yescharacteristic
207#217C22⋊C412yes—
208(C2×D3)⋊C4—24no—
209(C2×D3)⋊C4C22⋊C412yes—
210(C2×D3)⋊C4C22⋊C412yes—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3—24no—
215C4⋊Dic3C22⋊C411yescharacteristic
216C4⋊Dic3C22⋊C411yescharacteristic
217C4⋊Dic3C22×C414yes—
218Dic3⋊C4—24no—
219Dic3⋊C4—24no—
220Dic3⋊C4—24no—
221Dic3⋊C4C22⋊C412yes—
222Dic3⋊C4C22⋊C412yes—
223C4×Dic3—24no—
224C4×Dic3C22×C412yes—
225C4×Dic3C2×D412yes—
226C4×Dic3C2×D412yes—
227C2×C3⋊C8—21no—
228C2×C3⋊C8—21no—
229C2×C3⋊C8—21no—
230C2×C3⋊C8—21no—
231ΓL(1,𝔽25)—82no—
232ΓL(1,𝔽25)—82no—
233C2×#41—31no—
234C22×Q8—62no—
235C22×Q8—61no—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16—62no—
239C2×M16—62no—
240C4⋊Q8—34no—
241C4⋊Q8—32no—
242C4⋊Q8—32no—
243C4⋊Q8—62no—
244C4⋊Q8—62no—
245C4⋊Q8—62no—
246C4⋊Q8—64no—
247C4⋊Q8—61no—
248C4⋊Q8—61no—
249C4⋊Q8C22×D311yescharacteristic
250C4⋊Q8C22×D311yescharacteristic
251#125—34no—
252#125—62no—
253#125—64no—
254#125C4×D312yes—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C22⋊Q8—64no—
258C22⋊Q8—64no—
259C22⋊Q8—64no—
260C22⋊Q8—62no—
261C22⋊Q8—62no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—31no—
265C4×Q8—31no—
266C4×Q8—62no—
267C4×Q8—62no—
268C4×Q8—62no—
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—34no—
278#117C4×D312yes—
279C2×C4⋊C4—32no—
280C2×C4⋊C4—34no—
281C2×C4⋊C4—62no—
282C2×C4⋊C4—61no—
283C2×C42—32no—
284C2×C42C22×D311yescharacteristic
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—31no—
288C4⋊C8—31no—
289C4⋊C8—31no—
290C4⋊C8—31no—
291C4⋊C8—62no—
292C4⋊C8—62no—
293C4⋊C8C22×D312yes—
294C4⋊C8C22×D312yes—
295C22⋊C8—62no—
296C22⋊C8—62no—
297C22⋊C8—61no—
298C22⋊C8—61no—
299C4×C8—62no—
300C4×C8—62no—
301C4×C8—61no—
302C4×C8—61no—
303C23×C3C2×C22⋊C411yescharacteristic
304C22×D3C2×C22⋊C412yes—
305C3×Q8—41no—
306C3×Q8—41no—
307C3×Q8—41no—
308C3×Q8—41no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4C2×C22⋊C411yescharacteristic
317C2×C3×C4C2×C22⋊C411yescharacteristic
318C2×C3×C4C2×C22⋊C411yescharacteristic
319C2×C3×C4C23×C411yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C22×D411yescharacteristic
322#67—24no—
323C2×Dic3—24no—
324C2×Dic3—82no—
325C2×Dic3C2×C22⋊C412yes—
326C2×Dic3C2×C22⋊C412yes—
327C2×Dic3C2×C22⋊C412yes—
328D12—22no—
329C4×D3—22no—
330C4×D3—22no—
331C4×D3—22no—
332C4×D3—22no—
333C4×D3—24no—
334C4×D3—84no—
335Dic6—42no—
336Dic6—42no—
337Dic6—84no—
338Dic6—82no—
339Dic6#10114yes—
340C3×C8—42no—
341C3×C8—42no—
342C3⋊C8—41no—
343C3⋊C8—41no—
344C3⋊C8—41no—
345C3⋊C8—41no—
346#41—64no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—31no—
350C2×Q8—64no—
351C2×Q8—62no—
352C2×Q8—62no—
353C2×Q8—241no—
354C2×Q8—241no—
355C2×Q8—242no—
356C2×Q8—242no—
357C2×D4—32no—
358C2×D4—31no—
359C22×C4—32no—
360C22×C4—32no—
361C22×C4—32no—
362C22×C4—64no—
363C22×C4—62no—
364C22×C4C2×C4×D311yescharacteristic
365C22×C4D3×D411yescharacteristic
366C22×C4D3×D411yescharacteristic
367M16—81no—
368M16—81no—
369M16—242no—
370M16—242no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—61no—
374C2×C8—61no—
375C2×C8—61no—
376C2×C8—61no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—32no—
382C4⋊C4—31no—
383C4⋊C4—32no—
384C4⋊C4—31no—
385C4⋊C4—34no—
386C4⋊C4—64no—
387C4⋊C4—64no—
388C4⋊C4—64no—
389C4⋊C4—64no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4—62no—
393C4⋊C4C2×C4×D312yes—
394C4⋊C4C2×C4×D312yes—
395C22⋊C4—22no—
396C22⋊C4—32no—
397C22⋊C4—32no—
398C22⋊C4—32no—
399C22⋊C4—64no—
400C42—22no—
401C42—32no—
402C42—32no—
403C42—32no—
404C42—64no—
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—42no—
420C3×C4—42no—
421C3×C4C2×#10112yes—
422C3×C4#57611yescharacteristic
423C3×C4#57711yescharacteristic
424C3×C4#57711yescharacteristic
425C3×C4#57811yescharacteristic
426Dic3—22no—
427Dic3—22no—
428Dic3—44no—
429Dic3C2×#10114yes—
430C23—32no—
431C23D3×C22⋊C411yescharacteristic
432Q8—34no—
433Q8—41no—
434Q8—41no—
435Q8—41no—
436Q8—41no—
437Q8—122no—
438Q8—122no—
439Q8—242no—
440Q8—244no—
441D4—64no—
442D4—62no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—32no—
451C2×C4—32no—
452C2×C4—32no—
453C2×C4—64no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—64no—
459C2×C4—244no—
460C2×C4—242no—
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—42no—
469C8—121no—
470C8—121no—
471C8—121no—
472C8—121no—
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—42no—
490C4—42no—
491C4—62no—
492C4—62no—
493C4—124no—
494C4D3×#10112yes—
495C4#577411yescharacteristic
496C4#577511yescharacteristic
497C4#577811yescharacteristic
498C4#577911yescharacteristic
499C3#1777711yescharacteristic, complemented, Sylow
500C2—41no—
501C2—122no—
502C2C2×D3×#10111yescharacteristic
503C2#7888211yescharacteristic
504C2#7888311yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1561C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#235C2×C4
cosocleC3×C42C24
hypercenter#1561D3
hypocenterC3#17777
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

Complements

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

NH ≅ G/Ntype
4. C3×#1561C2semidirect
5. #71829C2semidirect
6. #70508C2semidirect
7. #70506C2semidirect
8. #68431C2semidirect
9. #68422C2semidirect
10. #68420C2semidirect
11. #68419C2semidirect
55. #1561D3semidirect
499. C3#17777semidirect

Maximal subgroups

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

subgroupindex
#862812
#790522
C3×#15612
#718292
#705082
#705062
#684312
#684222
#684202
#684192
#177773