Finite Group Databasebeta

← #10793775

Subgroups — #10793775

Order 768.

998 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82511C211yesmaximal, characteristic, complemented
2C4×#950C211yesmaximal, characteristic
3C2×#4823C212yesmaximal
4#78516C211yesmaximal, characteristic, complemented
5C3×#1759C211yesmaximal, characteristic, complemented
6#70369C212yesmaximal, complemented
7C4×#822C211yesmaximal, characteristic, complemented
8#69434C211yesmaximal, characteristic, complemented
9#69430C211yesmaximal, characteristic, complemented
10#68742C212yesmaximal, complemented
11#68739C212yesmaximal, complemented
12#20513—31nomaximal, Sylow
13C3×C4×#41C2211yescharacteristic
14C2×#950C2211yescharacteristic
15C2×#948C2212yes—
16C2×#907C2212yes—
17C4×#235C2211yescharacteristic
18C2×C3×#104C2212yes—
19C3×C4×M16C2211yescharacteristic
20C3×#376C2211yescharacteristic
21#5233C2211yescharacteristic
22C2×#822C2212yes—
23#5122C2212yes—
24C4×#216C412yescomplemented
25C4×#215C412yescomplemented
26C4×#214C412yescomplemented
27C4×#213C412yescomplemented
28#5002C2211yescharacteristic
29#4998C2211yescharacteristic
30#4997—22no—
31#4997C2212yes—
32C2×#805C2212yes—
33C4×#208—21no—
34C4×#208C2211yescharacteristic
35C2×#803—21no—
36C2×#803C2212yes—
37C2×#803C2211yescharacteristic
38C2×C4×C3⋊C8C2212yes—
39C2×C4×C3⋊C8C2211yescharacteristic
40#4823C4116yescomplemented
41#4707—21no—
42D3×#97—21no—
43#4689—22no—
44C3×#339C2212yes—
45#4558C2212yes—
46#4523C2212yes—
47C2×#444—32no—
48C4×#135—31no—
49#2875—31no—
50#2875—31no—
51C4×#108—31no—
52#1773—32no—
53#1764—32no—
54#1764—32no—
55#1759—31no—
56#1759D311yescharacteristic, complemented
57#1726—31no—
58C2×C3×#41C2311yescharacteristic
59C2×#235C2311yescharacteristic
60C2×C3×M16C2312yes—
61C3×C4×Q8C2×C412yes—
62C3×C4×D4—22no—
63C3×C4×D4C2×C412yes—
64C3×#117—21no—
65C3×#117C2311yescharacteristic
66C2×C3×C42—21no—
67C2×C3×C42C2312yes—
68C2×C3×C42C2311yescharacteristic
69#950C2×C418yes—
70#948—24no—
71#948C2×C418yescomplemented
72C2×#216C2×C411yescharacteristic
73C2×#215C2×C411yescharacteristic
74C2×#214C2×C411yescharacteristic
75C2×#213C2×C411yescharacteristic
76C4×#67—22no—
77#926C2311yescharacteristic
78C2×#208—22no—
79C2×#208C2312yes—
80C22×C3⋊C8—22no—
81C22×C3⋊C8C2311yescharacteristic
82C22×C3⋊C8C2311yescharacteristic
83C22×C3⋊C8C2312yes—
84#907—24no—
85#907C2×C418yescomplemented
86C2×ΓL(1,𝔽25)—22no—
87C2×D3×C8—22no—
88C4×D12C2×C412yes—
89#872—21no—
90C42×D3—21no—
91C4×Dic6C2×C412yes—
92C3×#104C2×C418yescomplemented
93C3×#97—21no—
94C3×C4×C8—21no—
95C3×#95—22no—
96#835C2×C412yes—
97#832C2×C412yes—
98#822C2×C418yescomplemented
99#815—22no—
100C8×Dic3—22no—
101#810C2×C412yes—
102#809C2×C412yes—
103#808C2×C412yes—
104#807C2×C412yes—
105#805C2×C418yescomplemented
106#803—21no—
107#803—22no—
108#803C2×C418yescomplemented
109#803D411yescharacteristic
110#803D411yescharacteristic
111#803D412yes—
112C4×C3⋊C8—21no—
113C4×C3⋊C8—22no—
114C4×C3⋊C8—24no—
115C4×C3⋊C8C2×C412yes—
116C4×C3⋊C8C2×C412yes—
117C2×#135—31no—
118C2×#131—32no—
119C2×#131—32no—
120C4×#41—31no—
121C4×#41C2×D311yescharacteristic
122#444—316no—
123C4×Q16—32no—
124C4×SD16—32no—
125C4×SD16—32no—
126C4×D8—32no—
127C2×#108—32no—
128#427—31no—
129C2×#104—32no—
130C2×#104C2×D312yes—
131#416—31no—
132#416—31no—
133#405—32no—
134#405—32no—
135#405—62no—
136#405—62no—
137C4×M16—31no—
138C4×M16—61no—
139C4×M16—61no—
140C4×M16C2×D311yescharacteristic
141C2×#97—32no—
142C2×#97—31no—
143C2×#97—61no—
144C2×#97—61no—
145C2×C4×C8—31no—
146C2×C4×C8—32no—
147#376C2×D311yescharacteristic
148#341—32no—
149#339—32no—
150#339C2×D312yes—
151C3×#41—22no—
152C3×#41C22×C414yes—
153C2×C3×Q8C22×C411yescharacteristic
154C2×C3×D4—21no—
155C2×C3×D4C22×C411yescharacteristic
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4#4112yes—
160C22×C3×C4C2411yescharacteristic
161C2×#67—21no—
162#235—22no—
163#235C22×C414yes—
164C2×D12C22×C411yescharacteristic
165C2×C4×D3—21no—
166C2×C4×D3—22no—
167C2×Dic6C22×C411yescharacteristic
168C3×M16—22no—
169C3×M16C22×C414yes—
170C2×C3×C8—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4C22×C412yes—
173C3×C22⋊C4—21no—
174C3×C22⋊C4—22no—
175C3×C42—22no—
176C3×C42—21no—
177C3×C42—22no—
178C3×C42C22×C414yes—
179C3×C42C22×C412yes—
180C3×C42C2×D411yescharacteristic
181C3×C42C2×D411yescharacteristic
182#217—21no—
183#216C4214yes—
184#215C4214yes—
185#214C4214yes—
186#213C4214yes—
187(C2×D3)⋊C4—22no—
188C4⋊Dic3C22×C412yes—
189Dic3⋊C4—22no—
190C4×Dic3—22no—
191C4×Dic3—21no—
192#208—22no—
193#208—28no—
194#208C22×C414yes—
195C2×C3⋊C8—22no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—28no—
198C2×C3⋊C8C22×C412yes—
199C2×C3⋊C8C22×C411yescharacteristic
200C2×C3⋊C8C22×C411yescharacteristic
201C2×C3⋊C8C22×C414yes—
202C2×C3⋊C8C22×C414yes—
203C2×C3⋊C8C2×D412yes—
204C2×C3⋊C8C2×D412yes—
205C2×C3⋊C8#4114yes—
206ΓL(1,𝔽25)—28no—
207D3×C8—28no—
208C2×#41—31no—
209C2×#41C22×D311yescharacteristic
210#135—38no—
211C2×Q16—31no—
212C2×SD16—31no—
213C2×SD16—31no—
214C2×D8—31no—
215#131—38no—
216#131—38no—
217#131—64no—
218#131—64no—
219C2×M16—32no—
220C2×M16—62no—
221C2×M16—62no—
222C2×M16C22×D312yes—
223C22×C8—32no—
224C22×C8—31no—
225C22×C8—31no—
226C22×C8—62no—
227C22×C8—62no—
228C4×Q8—32no—
229C4×Q8C4×D312yes—
230C4×D4—22no—
231C4×D4—32no—
232C4×D4—62no—
233C4×D4C4×D312yes—
234#117—21no—
235#117—31no—
236#117—61no—
237#117C22×D311yescharacteristic
238C2×C42—21no—
239C2×C42—61no—
240C2×C42C22×D312yes—
241C2×C42C22×D311yescharacteristic
242#108—38no—
243#107—32no—
244#106—32no—
245#104—38no—
246#104C4×D318yescomplemented
247#103—32no—
248#103—32no—
249#102—32no—
250#102—32no—
251#97—21no—
252#97—38no—
253#97—31no—
254#97—31no—
255#97—32no—
256#97—62no—
257#97—62no—
258#97—61no—
259C4×C8—21no—
260C4×C8—32no—
261C4×C8—32no—
262C4×C8—64no—
263C4×C8—62no—
264C4×C8—62no—
265C4×C8—61no—
266#95—22no—
267C23×C3—21no—
268C23×C3C2×#4111yescharacteristic
269C22×D3—21no—
270C3×Q8—21no—
271C3×Q8C2×C4212yes—
272C3×D4—24no—
273C3×D4—21no—
274C3×D4C2×C4212yes—
275C2×C3×C4—21no—
276C2×C3×C4—24no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—42no—
280C2×C3×C4—44no—
281C2×C3×C4—82no—
282C2×C3×C4C4×D412yes—
283C2×C3×C4C4×D412yes—
284C2×C3×C4C23×C412yes—
285C2×C3×C4C23×C411yescharacteristic
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C2×#4112yes—
288#67—24no—
289C2×Dic3—21no—
290C2×Dic3—22no—
291D12—21no—
292D12C2×C4212yes—
293C4×D3—24no—
294C4×D3—82no—
295Dic6—21no—
296Dic6C2×C4212yes—
297C3×C8—24no—
298C3⋊C8—24no—
299C3⋊C8—22no—
300C3⋊C8C2×C4212yes—
301C3⋊C8C2×C4212yes—
302C3⋊C8C4×D414yes—
303C3⋊C8C4×D414yes—
304#41—22no—
305#41—34no—
306#41—62no—
307#41C2×C4×D314yes—
308C2×Q8—31no—
309C2×Q8C2×C4×D311yescharacteristic
310C2×D4—21no—
311C2×D4—31no—
312C2×D4—61no—
313C2×D4C2×C4×D311yescharacteristic
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—61no—
318C22×C4—62no—
319C22×C4#23712yes—
320C22×C4C23×D311yescharacteristic
321Q16—34no—
322SD16—34no—
323SD16—34no—
324D8—34no—
325M16—22no—
326M16—34no—
327M16—68no—
328M16—68no—
329M16—62no—
330M16C2×C4×D314yes—
331C2×C8—22no—
332C2×C8—31no—
333C2×C8—31no—
334C2×C8—34no—
335C2×C8—34no—
336C2×C8—32no—
337C2×C8—32no—
338C2×C8—34no—
339C2×C8—32no—
340C2×C8—68no—
341C2×C8—68no—
342C2×C8—62no—
343C2×C8—62no—
344C4⋊C4—22no—
345C4⋊C4—32no—
346C4⋊C4—62no—
347C4⋊C4C2×C4×D312yes—
348C22⋊C4—21no—
349C22⋊C4—22no—
350C22⋊C4—61no—
351C22⋊C4—62no—
352C42—22no—
353C42—21no—
354C42—22no—
355C42—62no—
356C42—61no—
357C42C2×C4×D314yes—
358C42C2×C4×D312yes—
359C42D3×D411yescharacteristic
360C42D3×D411yescharacteristic
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—42no—
364C22×C3C2×C4×D411yescharacteristic
365C22×C3C4×#4112yes—
366C2×D3—22no—
367C2×D3—42no—
368C3×C4—22no—
369C3×C4—22no—
370C3×C4—22no—
371C3×C4—44no—
372C3×C4—42no—
373C3×C4C22×C4211yescharacteristic
374C3×C4C2×C4×D412yes—
375C3×C4C4×#4111yescharacteristic
376Dic3—22no—
377Dic3—42no—
378C23—21no—
379C23—61no—
380C23C2×#23711yescharacteristic
381Q8—21no—
382Q8—32no—
383Q8—61no—
384Q8C42×D312yes—
385D4—24no—
386D4—21no—
387D4—32no—
388D4—64no—
389D4—61no—
390D4C42×D312yes—
391C2×C4—21no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—61no—
398C2×C4—64no—
399C2×C4—62no—
400C2×C4—82no—
401C2×C4—242no—
402C2×C4#67⋊C412yes—
403C2×C4#67⋊C412yes—
404C2×C4C22×C4×D311yescharacteristic
405C2×C4C22×C4×D312yes—
406C2×C4C2×D3×D411yescharacteristic
407C2×C4D3×#4112yes—
408C8—24no—
409C8—32no—
410C8—32no—
411C8—34no—
412C8—34no—
413C8—62no—
414C8—64no—
415C2×C3—22no—
416C2×C3—42no—
417C2×C3C42×D411yescharacteristic
418C2×C3#295712yes—
419D3—42no—
420C22—22no—
421C22—22no—
422C22—42no—
423C22—62no—
424C22—122no—
425C22C2×#67⋊C411yescharacteristic
426C22C4×#23712yes—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—44no—
431C4—42no—
432C4—62no—
433C4—122no—
434C4C2×C42×D311yescharacteristic
435C4#552211yescharacteristic
436C4C4×D3×D412yes—
437C3#2051311yescharacteristic, complemented, Sylow
438C2—22no—
439C2—42no—
440C2—122no—
441C2C4×#67⋊C411yescharacteristic
442C2#7899012yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC3×C4C22×C42
FrattiniC22×C4C23×D3
FittingC3×#1759C2
socleC22×C3C2×C4×D4
prime radicalC2×#950C22
cosocleC22×C3×C4C24
hypercenter#1759D3
hypocenterC3#20513
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1759

Complements

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

NH ≅ G/Ntype
1. #82511C2semidirect
4. #78516C2semidirect
5. C3×#1759C2semidirect
6. #70369C2semidirect
7. C4×#822C2semidirect
8. #69434C2semidirect
9. #69430C2semidirect
10. #68742C2semidirect
11. #68739C2semidirect
24. C4×#216C4semidirect
25. C4×#215C4semidirect
26. C4×#214C4semidirect
27. C4×#213C4semidirect
40. #4823C4semidirect
56. #1759D3semidirect
71. #948C2×C4semidirect
85. #907C2×C4semidirect
92. C3×#104C2×C4semidirect
98. #822C2×C4semidirect
105. #805C2×C4semidirect
108. #803C2×C4semidirect
246. #104C4×D3semidirect
437. C3#20513semidirect

Maximal subgroups

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

subgroupindex
#825112
C4×#9502
C2×#48232
#785162
C3×#17592
#703692
C4×#8222
#694342
#694302
#687422
#687392
#205133