Finite Group Databasebeta

← #10876298

Subgroups — #10876298

Order 768.

720 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#84120C211yesmaximal, characteristic
2#78649C211yesmaximal, characteristic, complemented
3C3×#1794C211yesmaximal, characteristic
4#72145C211yesmaximal, characteristic
5#70405C212yesmaximal, complemented
6#70401C212yesmaximal, complemented
7#70294C212yesmaximal, complemented
8#70288C212yesmaximal, complemented
9#70044C211yesmaximal, characteristic, complemented
10#70039C211yesmaximal, characteristic, complemented
11#70001C211yesmaximal, characteristic, complemented
12#27599—31nomaximal, Sylow
13C3×#585C2211yescharacteristic
14Dic3×#41C2211yescharacteristic
15#5819C2211yescharacteristic
16C3×#421C2212yes—
17C3×#416C2212yes—
18C3×#410C2211yescharacteristic
19C3×#408C2211yescharacteristic
20#5237C2212yes—
21#5237C2211yescharacteristic
22#5236—22no—
23#5236C2212yes—
24C2×#837—22no—
25C2×#837C2212yes—
26#5234C2211yescharacteristic
27#5234C2212yes—
28#5124C2212yescomplemented
29#5119C2211yescharacteristic, complemented
30Dic3×M16C2211yescharacteristic, complemented
31#5102C2211yescharacteristic, complemented
32C2×#815C2211yescharacteristic, complemented
33#4999C2212yescomplemented
34#4720—22no—
35#4719—22no—
36#4552C2212yescomplemented
37#4533C2212yescomplemented
38#4532C2212yescomplemented
39#4531C2212yescomplemented
40#2923—31no—
41#2901—31no—
42#2098—32no—
43#2080—32no—
44#1938—32no—
45#1932—32no—
46#1868—31no—
47#1858—31no—
48#1827—31no—
49#1794D311yescharacteristic
50#1791—31no—
51C3×2−1+4—22no—
52C3×2−1+4—21no—
53C3×H(5,𝔽2)—22no—
54C3×H(5,𝔽2)—22no—
55C3×H(5,𝔽2)—21no—
56C2×C3×#41—22no—
57C2×C3×#41—42no—
58C2×C3×#41C2312yes—
59C2×C3×#41C2311yescharacteristic
60C2×C3×M16C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C3×#117C2312yes—
63#948—42no—
64Q8×Dic3—21no—
65D4×Dic3—21no—
66D4×Dic3—22no—
67#926D412yes—
68#926C2311yescharacteristic
69C2×C4×Dic3D412yes—
70C2×C4×Dic3C2311yescharacteristic
71C2×#208—22no—
72C2×#208C2312yes—
73C2×#208C2312yes—
74C2×#208C2311yescharacteristic
75C22×C3⋊C8C2311yescharacteristic
76C3×#104—42no—
77C3×#103—22no—
78C3×#102—22no—
79C3×#99—22no—
80C3×C22⋊C8—22no—
81#837—22no—
82#837—24no—
83#837—22no—
84#836—21no—
85#836—22no—
86#833—21no—
87#833—22no—
88#830—22no—
89#830—42no—
90#822—42no—
91#815—21no—
92#815—21no—
93#815—21no—
94Dic3⋊C8—22no—
95Dic3⋊C8—22no—
96C8×Dic3—21no—
97C4⋊(C3⋊C8)—42no—
98#585C2×D311yescharacteristic
99#568—31no—
100C4×#41—31no—
101#433—62no—
102#432—62no—
103#430—32no—
104#424—31no—
105#424—31no—
106#424—32no—
107#421—32no—
108#421—62no—
109#421C2×D312yes—
110C2×#104—32no—
111C2×#104—62no—
112#416C2×D312yes—
113#413—32no—
114#413—31no—
115#410C2×D311yescharacteristic
116#408—31no—
117#408—32no—
118#408C2×D311yescharacteristic
119C4×M16—31no—
120C2×#97—31no—
121#344—32no—
122#341—32no—
123#339—32no—
124H(3,ℤ/4)—32no—
125C3×#41—22no—
126C3×#41—22no—
127C3×#41—24no—
128C3×#41—22no—
129C3×#41—22no—
130C3×#41—22no—
131C3×#41—22no—
132C3×#41—44no—
133C3×#41—42no—
134C3×#41—42no—
135C3×#41—42no—
136C3×#41—42no—
137C2×C3×Q8—22no—
138C2×C3×Q8—42no—
139C2×C3×Q8#4111yescharacteristic
140C2×C3×Q8#4112yes—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—42no—
144C2×C3×D4—44no—
145C2×C3×D4—42no—
146C2×C3×D4#4111yescharacteristic
147C2×C3×D4#4112yes—
148C2×C3×D4#4112yes—
149C22×C3×C4—22no—
150C22×C3×C4—42no—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2411yescharacteristic
153C22×Dic3—22no—
154C22×Dic3—21no—
155C3×M16—22no—
156C2×C3×C8—21no—
157C2×C3×C8—21no—
158C2×C3×C8—22no—
159C3×C4⋊C4—22no—
160C3×C22⋊C4—22no—
161C3×C42—22no—
162#217—22no—
163#217—21no—
164C4⋊Dic3—22no—
165C4⋊Dic3—21no—
166C4×Dic3—22no—
167C4×Dic3—21no—
168C4×Dic3C2×D411yescharacteristic
169C4×Dic3C2×D412yes—
170C4×Dic3C2×D411yescharacteristic
171#208—22no—
172#208—22no—
173#208—22no—
174#208—22no—
175#208—22no—
176#208—42no—
177C2×C3⋊C8—21no—
178C2×C3⋊C8—21no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—42no—
1832−1+4—22no—
1842−1+4—21no—
185H(5,𝔽2)—22no—
186H(5,𝔽2)—22no—
187H(5,𝔽2)—21no—
188C2×#41—22no—
189C2×#41—42no—
190C2×#41C22×D312yes—
191C2×#41C22×D311yescharacteristic
192#131—122no—
193C2×M16—32no—
194C2×M16—32no—
195C2×M16—31no—
196C2×M16—62no—
197C2×M16C22×D311yescharacteristic
198C22×C8—31no—
199C22×C8C22×D311yescharacteristic
200C4×Q8—61no—
201C4×D4—61no—
202C4×D4—62no—
203#117—32no—
204#117—31no—
205#117C22×D312yes—
206C2×C42—32no—
207C2×C42—31no—
208#108—122no—
209C4⋊C8—62no—
210C4⋊C8—62no—
211C4⋊C8—122no—
212#104—42no—
213#104—62no—
214#104—62no—
215#104—64no—
216#103—22no—
217#102—22no—
218#101—61no—
219#101—62no—
220#100—61no—
221#100—62no—
222#99—22no—
223C22⋊C8—22no—
224C22⋊C8—62no—
225C22⋊C8—122no—
226#97—61no—
227#97—61no—
228#97—61no—
229C4×C8—61no—
230C23×C3—22no—
231C23×C3—42no—
232C23×C3#12312yes—
233C23×C3C2×#4111yescharacteristic
234C3×Q8—22no—
235C3×Q8—22no—
236C3×Q8—22no—
237C3×Q8—42no—
238C3×D4—24no—
239C3×D4—22no—
240C3×D4—22no—
241C3×D4—22no—
242C3×D4—42no—
243C3×D4—44no—
244C3×D4—42no—
245C3×D4—42no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—44no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4#12312yes—
255C2×C3×C4C22×D411yescharacteristic
256C2×C3×C4C2×#4111yescharacteristic
257C2×C3×C4C2×#4112yes—
258C2×C3×C4H(5,𝔽2)12yes—
259C2×Dic3—21no—
260C2×Dic3—22no—
261C2×Dic3—21no—
262C2×Dic3—81no—
263C2×Dic3—82no—
264C3×C8—41no—
265C3×C8—41no—
266C3⋊C8—41no—
267C3⋊C8—41no—
268C3⋊C8—42no—
269C3⋊C8—42no—
270#41—22no—
271#41—22no—
272#41—24no—
273#41—22no—
274#41—22no—
275#41—22no—
276#41—22no—
277#41—44no—
278#41—42no—
279#41—42no—
280#41—42no—
281#41—42no—
282C2×Q8—22no—
283C2×Q8—42no—
284C2×Q8#23511yescharacteristic
285C2×Q8#23712yes—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—42no—
289C2×D4—44no—
290C2×D4—42no—
291C2×D4#23511yescharacteristic
292C2×D4#23712yes—
293C2×D4#23712yes—
294C22×C4—22no—
295C22×C4—42no—
296C22×C4—62no—
297C22×C4—61no—
298C22×C4D3×D412yes—
299C22×C4C23×D311yescharacteristic
300M16—22no—
301M16—62no—
302M16—62no—
303M16—62no—
304M16—62no—
305M16—62no—
306M16—122no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—61no—
311C2×C8—61no—
312C2×C8—62no—
313C2×C8—62no—
314C2×C8—62no—
315C2×C8—122no—
316C4⋊C4—22no—
317C4⋊C4—62no—
318C4⋊C4—61no—
319C22⋊C4—22no—
320C22⋊C4—62no—
321C22⋊C4—61no—
322C42—22no—
323C42—31no—
324C42—31no—
325C42—32no—
326C42—62no—
327C42—61no—
328C22×C3—22no—
329C22×C3—41no—
330C22×C3—42no—
331C22×C3—42no—
332C22×C3—82no—
333C22×C3—162no—
334C22×C3C2×#12311yescharacteristic
335C22×C3#55112yes—
336C3×C4—22no—
337C3×C4—42no—
338C3×C4—82no—
339C3×C4#52511yescharacteristic
340C3×C4#53511yescharacteristic
341C3×C4#55112yes—
342Dic3—42no—
343Dic3—41no—
344Dic3—41no—
345C23—22no—
346C23—42no—
347C23#88312yes—
348C23C2×#23711yescharacteristic
349Q8—22no—
350Q8—22no—
351Q8—22no—
352Q8—42no—
353D4—24no—
354D4—22no—
355D4—22no—
356D4—22no—
357D4—42no—
358D4—44no—
359D4—42no—
360D4—42no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—62no—
372C2×C4—241no—
373C2×C4—242no—
374C2×C4#88312yes—
375C2×C4C2×#23511yescharacteristic
376C2×C4C2×D3×D411yescharacteristic
377C2×C4C2×#23712yes—
378C2×C4#100412yes—
379C8—41no—
380C8—41no—
381C8—122no—
382C8—121no—
383C8—121no—
384C8—122no—
385C2×C3—21no—
386C2×C3—22no—
387C2×C3—42no—
388C2×C3—82no—
389C2×C3#248211yescharacteristic
390C22—22no—
391C22—41no—
392C22—42no—
393C22—42no—
394C22—82no—
395C22—162no—
396C22C2×#88311yescharacteristic
397C22#558712yes—
398C4—22no—
399C4—42no—
400C4—82no—
401C4—121no—
402C4—121no—
403C4—122no—
404C4#549512yes—
405C4#558611yescharacteristic
406C4#559211yescharacteristic
407C3#2759911yescharacteristic, complemented, Sylow
408C2—22no—
409C2—21no—
410C2—42no—
411C2—82no—
412C2#7383211yescharacteristic
subgroupG/H
centerC4#5592
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1794C2
socleC2×C3#2482
prime radicalC3×#585C22
hypercenter#1794D3
hypocenterC3#27599
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1794

Complements

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

NH ≅ G/Ntype
2. #78649C2semidirect
5. #70405C2semidirect
6. #70401C2semidirect
7. #70294C2semidirect
8. #70288C2semidirect
9. #70044C2semidirect
10. #70039C2semidirect
11. #70001C2semidirect
28. #5124C22semidirect
29. #5119C22semidirect
30. Dic3×M16C22semidirect
31. #5102C22semidirect
32. C2×#815C22semidirect
33. #4999C22semidirect
36. #4552C22semidirect
37. #4533C22semidirect
38. #4532C22semidirect
39. #4531C22semidirect
407. C3#27599semidirect

Maximal subgroups

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

subgroupindex
#841202
#786492
C3×#17942
#721452
#704052
#704012
#702942
#702882
#700442
#700392
#700012
#275993