Finite Group Databasebeta

← #10697106

Subgroups — #10697106

Order 768.

574 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79312C211yesmaximal, characteristic
2#78538C211yesmaximal, characteristic
3C2×#4681C211yesmaximal, characteristic, complemented
4#70778C211yesmaximal, characteristic, complemented
5#69879C211yesmaximal, characteristic, complemented
6#69735C211yesmaximal, characteristic
7C3×#1278C211yesmaximal, characteristic, complemented
8#10170—31nomaximal, Sylow
9C2×#909—22no—
10#5477C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15C2×C3×C4×C8C2211yescharacteristic
16C2×#816—22no—
17#4924C2211yescharacteristic
18#4841—22no—
19#4738—22no—
20#4715—22no—
21#4699—21no—
22#4693—21no—
23#4693—21no—
24#4681—22no—
25#4681—21no—
26#4681—22no—
27#4554—22no—
28#4486—22no—
29#4459—22no—
30#3206—31no—
31#3152—31no—
32C2×#487—31no—
33#1997—31no—
34#1546—31no—
35#1492—31no—
36#1278D311yescharacteristic, complemented
37C2×#235—21no—
38C22×Dic6—21no—
39C22×Dic6D412yes—
40C2×C3×M16D412yes—
41C22×C3×C8—21no—
42C2×C3×C42C2311yescharacteristic
43#930—21no—
44#930—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×#208—21no—
50#909—44no—
51#909—44no—
52C2×Dic12—24no—
53C2×#204—24no—
54C2×#204—24no—
55#875—21no—
56#874D411yescharacteristic
57C4×D12—21no—
58C4×D12—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869D411yescharacteristic
67#869D412yes—
68C3×C4⋊C8D412yes—
69C3×C22⋊C8—21no—
70C3×#97—21no—
71C3×C4×C8—21no—
72C3×C4×C8—21no—
73C3×C4×C8—21no—
74C3×C4×C8D412yes—
75#816—28no—
76#816—24no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C2×#137—62no—
80#532—31no—
81C2×C4⋊Q8—31no—
82C2×C4⋊Q8—31no—
83#497—62no—
84#494—61no—
85#492—61no—
86#492—61no—
87#487—62no—
88#487—61no—
89#487—62no—
90#461—62no—
91#448—62no—
92#431C2×D311yescharacteristic
93#423—31no—
94C2×#103—62no—
95C4×M16C2×D311yescharacteristic
96C2×C4×C8C2×D311yescharacteristic
97#343—62no—
98#332—62no—
99D4⋊C8—62no—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C2×#67—41no—
104C22×Dic3—22no—
105C22×Dic3—41no—
106#235—42no—
107#235—42no—
108C2×D12—21no—
109C2×D12—21no—
110C2×D12—21no—
111C2×C4×D3—41no—
112C2×Dic6—22no—
113C2×Dic6—22no—
114C2×Dic6—22no—
115C2×Dic6—21no—
116C2×Dic6—44no—
117C2×Dic6—42no—
118C2×Dic6—42no—
119C2×Dic6—41no—
120C3×M16—22no—
121C3×M16SD1614yescomplemented
122C2×C3×C8—22no—
123C2×C3×C8—22no—
124C2×C3×C8—22no—
125C2×C3×C8—42no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130(C2×D3)⋊C4—41no—
131(C2×D3)⋊C4—41no—
132C4⋊Dic3—22no—
133C4⋊Dic3—21no—
134C4⋊Dic3—22no—
135C4⋊Dic3—21no—
136C4⋊Dic3—41no—
137C4⋊Dic3—41no—
138C4⋊Dic3—41no—
139C4⋊Dic3—41no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142#208—42no—
143C2×C3⋊C8—41no—
144Dic12—48no—
145#204—48no—
146#204—84no—
147C2×#41—61no—
148C22×Q8—32no—
149C22×Q8—61no—
150#137—124no—
151#137—124no—
152C2×Q16—64no—
153C2×SD16—64no—
154C2×SD16—64no—
155C2×M16—61no—
156C2×M16D1212yes—
157C22×C8—21no—
158C4⋊Q8—32no—
159C4⋊Q8—31no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167#127—31no—
168#124—61no—
169#121—61no—
170#121—61no—
171C4×D4—61no—
172C4×D4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C42C22×D311yescharacteristic
178C4⋊C8—61no—
179C4⋊C8—61no—
180C4⋊C8D1212yes—
181#103—68no—
182#103—64no—
183C22⋊C8—21no—
184#97—21no—
185C4×C8—21no—
186C4×C8—21no—
187C4×C8—21no—
188C4×C8D1212yes—
189C23×C3#12111yescharacteristic
190C22×D3—41no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12711yescharacteristic
199C2×C3×C4C2×SD1612yes—
200C2×C3×C4AGL(1,ℤ/8)12yes—
201#67—82no—
202C2×Dic3—42no—
203C2×Dic3—41no—
204C2×Dic3—42no—
205C2×Dic3—82no—
206C2×Dic3—82no—
207D12—42no—
208D12—42no—
209D12—42no—
210C4×D3—81no—
211C4×D3—81no—
212Dic6—44no—
213Dic6—44no—
214Dic6—42no—
215Dic6—44no—
216Dic6—81no—
217Dic6—81no—
218C3×C8—24no—
219C3×C8—42no—
220C3×C8—42no—
221C3⋊C8—81no—
222#41—122no—
223#41—122no—
224C2×Q8—62no—
225C2×Q8—62no—
226C2×Q8—61no—
227C2×Q8—62no—
228C2×Q8—124no—
229C2×Q8—121no—
230C2×Q8—122no—
231C2×Q8—122no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—121no—
236C22×C4—62no—
237C22×C4—121no—
238C22×C4—121no—
239C22×C4C2×D1211yescharacteristic
240C22×C4D3×D411yescharacteristic
241C22×C4#23911yescharacteristic
242Q16—128no—
243SD16—128no—
244SD16—244no—
245M16—22no—
246M16—122no—
247M16#20414yescomplemented
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—42no—
252C2×C8—121no—
253C4⋊C4—62no—
254C4⋊C4—61no—
255C4⋊C4—62no—
256C4⋊C4—61no—
257C4⋊C4—121no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C22⋊C4—121no—
264C22⋊C4—121no—
265C42C2×D1211yescharacteristic
266C42C2×D1211yescharacteristic
267C42D3×D411yescharacteristic
268C42D3×D411yescharacteristic
269C22×C3—21no—
270C22×C3—21no—
271C22×C3#39211yescharacteristic
272C22×C3#46012yes—
273C2×D3—81no—
274C2×D3—81no—
275C2×D3—81no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4#45012yes—
279C3×C4#45512yes—
280C3×C4#47312yes—
281C3×C4#49211yescharacteristic
282C3×C4#49611yescharacteristic
283Dic3—82no—
284Dic3—82no—
285Dic3—81no—
286C23—121no—
287C23#89511yescharacteristic
288Q8—124no—
289Q8—124no—
290Q8—122no—
291Q8—124no—
292Q8—241no—
293Q8—241no—
294D4—122no—
295D4—122no—
296D4—122no—
297D4—242no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—121no—
301C2×C4—122no—
302C2×C4—122no—
303C2×C4—242no—
304C2×C4—241no—
305C2×C4—241no—
306C2×C4—242no—
307C2×C4#87411yescharacteristic
308C2×C4#88211yescharacteristic
309C2×C4#88211yescharacteristic
310C2×C4#88211yescharacteristic
311C2×C4#89511yescharacteristic
312C2×C4#89511yescharacteristic
313C2×C4C2×#20412yes—
314C2×C4#90812yes—
315C8—24no—
316C8—42no—
317C8—42no—
318C8—241no—
319C2×C3—22no—
320C2×C3#200411yescharacteristic
321C2×C3#201411yescharacteristic
322C2×C3#202011yescharacteristic
323D3—161no—
324C22—21no—
325C22—21no—
326C22—241no—
327C22—241no—
328C22—241no—
329C22#467111yescharacteristic
330C22#484812yes—
331C4—21no—
332C4—21no—
333C4—241no—
334C4—242no—
335C4—242no—
336C4#469311yescharacteristic
337C4#471211yescharacteristic
338C4#473112yes—
339C4#482512yes—
340C4#489612yes—
341C3#1017011yescharacteristic, complemented, Sylow
342C2—22no—
343C2—481no—
344C2#6867611yescharacteristic
345C2#6876211yescharacteristic
346C2#6894411yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1278C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1278D3
hypocenterC3#10170
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1278

Complements

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

NH ≅ G/Ntype
3. C2×#4681C2semidirect
4. #70778C2semidirect
5. #69879C2semidirect
7. C3×#1278C2semidirect
36. #1278D3semidirect
121. C3×M16SD16semidirect
247. M16#204semidirect
341. C3#10170semidirect

Maximal subgroups

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

subgroupindex
#793122
#785382
C2×#46812
#707782
#698792
#697352
C3×#12782
#101703