Finite Group Databasebeta

← #10848596

Subgroups — #10848596

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86204C211yesmaximal, characteristic
2C3×#2868C211yesmaximal, characteristic, complemented
3#83209C211yesmaximal, characteristic
4#83198C211yesmaximal, characteristic, complemented
5#71795C214yesmaximal
6#71789C211yesmaximal, characteristic, complemented
7#71784C214yesmaximal, complemented
8#69396C211yesmaximal, characteristic, complemented
9C4×#805C211yesmaximal, characteristic
10#25589—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4⋊Q8C2214yes—
13C2×C3×C4⋊Q8C2211yescharacteristic
14#5816C2212yes—
15#5812C2212yes—
16C2×#942C2212yes—
17#5732—22no—
18#5731—22no—
19#5728—22no—
20#5721—22no—
21#5513—24no—
22C4×#235C2211yescharacteristic
23#5470C2211yescharacteristic
24C3×#384—24no—
25C3×C43C2211yescharacteristic
26C2×#836C2214yes—
27#5225C2212yes—
28#5225C2214yes—
29#5090—22no—
30#5083—22no—
31#5082—22no—
32#5079—22no—
33C2×#805C2212yes—
34#4924C2211yescharacteristic
35C4×#208C2211yescharacteristic
36#4523C2212yes—
37#4523C2214yes—
38#3509—31no—
39#3139—31no—
40#3039—31no—
41#2868D311yescharacteristic, complemented
42#2005—34no—
43#1994—31no—
44#1966—31no—
45#1886—34no—
46C4×#104—31no—
47C22×C3×Q8C2312yes—
48C22×C3×Q8C2312yes—
49C2×#235C2311yescharacteristic
50C3×C4⋊Q8—22no—
51C3×C4⋊Q8—21no—
52C3×C4⋊Q8—21no—
53C3×C4⋊Q8—21no—
54C3×C4⋊Q8—21no—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—28no—
57C3×C4⋊Q8—44no—
58C2×C3×C4⋊C4—28no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4D414yes—
61C2×C3×C42—22no—
62C2×C3×C42C2312yes—
63C2×C3×C42C2311yescharacteristic
64#947—22no—
65#946—24no—
66Q8×Dic3—24no—
67Q8×Dic3—24no—
68#944—22no—
69C2×#216—22no—
70#942—44no—
71C2×#215—22no—
72#929—24no—
73C4×#67—22no—
74#926—24no—
75#926C2312yes—
76#926C2311yescharacteristic
77#924—22no—
78C2×#208C2312yes—
79C2×#208C2312yes—
80#898—28no—
81#897—24no—
82#892—24no—
83#890—24no—
84#889—28no—
85#887—24no—
86C4×D12D412yes—
87#872—21no—
88C42×D3—21no—
89#870D412yes—
90#869D412yes—
91C4×Dic6—22no—
92C4×Dic6D412yes—
93#836—44no—
94#835—28no—
95#835—24no—
96#805—24no—
97C4⋊(C3⋊C8)—22no—
98#803—21no—
99C4×C3⋊C8—21no—
100C2×#137—32no—
101#557—62no—
102#556—62no—
103#552—64no—
104#541—62no—
105#536—32no—
106#533—31no—
107#533—62no—
108C2×C4⋊Q8C2×D314yes—
109C2×C4⋊Q8C2×D311yescharacteristic
110C2×C4⋊Q8C2×D311yescharacteristic
111#519—32no—
112C4×#41—31no—
113#491—62no—
114#487—62no—
115#462—62no—
116#461—62no—
117#423—31no—
118C2×#104—32no—
119#419—34no—
120#419—32no—
121C2×#101—34no—
122C4×M16—31no—
123#384—24no—
124C43C2×D311yescharacteristic
125#339—32no—
126#339—34no—
127C2×C3×Q8—22no—
128C2×C3×Q8—22no—
129C2×C3×Q8—24no—
130C2×C3×Q8—24no—
131C2×C3×Q8—22no—
132C2×C3×Q8—22no—
133C2×C3×Q8—24no—
134C2×C3×Q8—24no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4C2×D414yes—
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2411yescharacteristic
140C2×#67—21no—
141#235—22no—
142#235—22no—
143C2×D12C2×D411yescharacteristic
144C2×C4×D3—22no—
145C2×C4×D3—21no—
146C2×Dic6—22no—
147C2×Dic6C2×D411yescharacteristic
148C3×C4⋊C4—24no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—416no—
151C3×C4⋊C4—42no—
152C3×C4⋊C4—42no—
153C3×C42—28no—
154C3×C42—22no—
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D412yes—
158C3×C42#4114yes—
159#217—22no—
160#217—21no—
161#216—44no—
162#215—44no—
163(C2×D3)⋊C4—28no—
164(C2×D3)⋊C4—22no—
165C4⋊Dic3—28no—
166C4⋊Dic3—22no—
167C4⋊Dic3C2×D412yes—
168C4⋊Dic3C2×D414yes—
169Dic3⋊C4—28no—
170Dic3⋊C4—24no—
171Dic3⋊C4—22no—
172C4×Dic3—28no—
173C4×Dic3—22no—
174C4×Dic3—22no—
175C4×Dic3—21no—
176#208—22no—
177#208—22no—
178#208—24no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—22no—
181C2×#41—31no—
182C22×Q8C22×D312yes—
183C22×Q8C22×D312yes—
184#137—124no—
185C2×Q16—62no—
186C2×SD16—62no—
187C2×M16—32no—
188C2×M16—32no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—28no—
196C4⋊Q8—32no—
197C4⋊Q8—44no—
198C4⋊Q8—62no—
199#126—68no—
200#125—32no—
201#125—64no—
202#125—68no—
203#124—62no—
204#123—64no—
205C22⋊Q8—64no—
206C22⋊Q8—64no—
207C22⋊Q8—62no—
208C4×Q8—32no—
209C4×Q8—64no—
210C4×Q8—64no—
211C4×Q8—64no—
212C4×Q8—62no—
213C4×D4—32no—
214C4×D4—62no—
215#117—31no—
216#117—32no—
217#117—64no—
218#117—64no—
219#117—61no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4#6714yes—
223C2×C42—22no—
224C2×C42—61no—
225C2×C42C22×D312yes—
226C2×C42C22×D311yescharacteristic
227C4⋊C8—62no—
228#104—64no—
229#103—64no—
230#103—68no—
231#101—124no—
232#97—61no—
233C4×C8—61no—
234C23×C3C22×D411yescharacteristic
235C22×D3—21no—
236C3×Q8—42no—
237C3×Q8—42no—
238C3×Q8—44no—
239C3×Q8—42no—
240C3×Q8—42no—
241C3×Q8—44no—
242C2×C3×C4—28no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—28no—
247C2×C3×C4—42no—
248C2×C3×C4—44no—
249C2×C3×C4#12012yes—
250C2×C3×C4#12012yes—
251C2×C3×C4C22×D411yescharacteristic
252C2×C3×C4C22×D411yescharacteristic
253C2×C3×C4C2×#4112yes—
254C2×C3×C4H(5,𝔽2)12yes—
255#67—42no—
256C2×Dic3—22no—
257C2×Dic3—24no—
258C2×Dic3—21no—
259D12—21no—
260D12—21no—
261C4×D3—42no—
262C4×D3—44no—
263Dic6—21no—
264Dic6—21no—
265Dic6—44no—
266C3⋊C8—42no—
267C3⋊C8—42no—
268#41—62no—
269#41—62no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—31no—
279C2×Q8—62no—
280C2×D4—31no—
281C2×D4—61no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—62no—
285C22×C4—61no—
286C22×C4C2×#6714yes—
287C22×C4C2×#6712yes—
288C22×C4C23×D311yescharacteristic
289Q16—124no—
290SD16—124no—
291M16—62no—
292M16—62no—
293M16—64no—
294C2×C8—62no—
295C2×C8—62no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—32no—
299C4⋊C4—34no—
300C4⋊C4—416no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—68no—
304C4⋊C4—68no—
305C4⋊C4—62no—
306C4⋊C4—62no—
307C4⋊C4—64no—
308C22⋊C4—68no—
309C22⋊C4—61no—
310C22⋊C4—62no—
311C22⋊C4—62no—
312C42—28no—
313C42—22no—
314C42—68no—
315C42—62no—
316C42—61no—
317C42—62no—
318C42D3×D411yescharacteristic
319C42D3×D411yescharacteristic
320C42D3×D412yes—
321C42#23714yes—
322C22×C3—22no—
323C22×C3C2×#12011yescharacteristic
324C22×C3#54612yes—
325C2×D3—41no—
326C2×D3—42no—
327C3×C4—28no—
328C3×C4—22no—
329C3×C4—44no—
330C3×C4—82no—
331C3×C4#45614yes—
332C3×C4#53211yescharacteristic
333C3×C4#53511yescharacteristic
334C3×C4#55012yes—
335Dic3—41no—
336Dic3—44no—
337Dic3—42no—
338C23—61no—
339C23C22×#6711yescharacteristic
340Q8—42no—
341Q8—42no—
342Q8—44no—
343Q8—42no—
344Q8—42no—
345Q8—44no—
346Q8—61no—
347Q8—61no—
348Q8—124no—
349D4—61no—
350D4—61no—
351D4—122no—
352C2×C4—28no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—28no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—62no—
360C2×C4—64no—
361C2×C4—61no—
362C2×C4—122no—
363C2×C4—124no—
364C2×C4#93712yes—
365C2×C4#93712yes—
366C2×C4C2×D3×D411yescharacteristic
367C2×C4C2×D3×D411yescharacteristic
368C2×C4C2×#23712yes—
369C2×C4#100412yes—
370C8—122no—
371C8—122no—
372C2×C3—22no—
373C2×C3#243111yescharacteristic
374C2×C3C2×#45612yes—
375D3—81no—
376C22—22no—
377C22—122no—
378C22—121no—
379C22C2×#93711yescharacteristic
380C22#580612yes—
381C4—28no—
382C4—22no—
383C4—44no—
384C4—82no—
385C4—122no—
386C4—124no—
387C4—121no—
388C4#509514yes—
389C4#571511yescharacteristic
390C4#571611yescharacteristic
391C4#571812yes—
392C3#2558911yescharacteristic, complemented, Sylow
393C2—22no—
394C2—241no—
395C2#8318711yescharacteristic
396C2C2×#509512yes—
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2868C2
socleC22×C3C2×#120
prime radicalC2×#235C23
hypercenter#2868D3
hypocenterC3#25589
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2868

Complements

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

NH ≅ G/Ntype
2. C3×#2868C2semidirect
4. #83198C2semidirect
6. #71789C2semidirect
7. #71784C2semidirect
8. #69396C2semidirect
41. #2868D3semidirect
392. C3#25589semidirect

Maximal subgroups

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

subgroupindex
#862042
C3×#28682
#832092
#831982
#717952
#717892
#717842
#693962
C4×#8052
#255893