Finite Group Databasebeta

← #10697839

Subgroups — #10697839

Order 768.

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

normal subgroups only

This group has 626 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#79269C211yesmaximal, characteristic
3#73311C211yesmaximal, characteristic, complemented
4#70780C211yesmaximal, characteristic, complemented
5#70779C211yesmaximal, characteristic, complemented
6#69395C211yesmaximal, characteristic
7C3×#1276C211yesmaximal, characteristic, complemented
8#10283—31nomaximal, Sylow
9#5477C2211yescharacteristic
10C2×#869—22no—
11C2×#869C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C3×C43C2211yescharacteristic
16#4935—22no—
17#4935—22no—
18C2×#805—22no—
19#4924C2211yescharacteristic
20#4842—22no—
21#4841—22no—
22#4738—22no—
23#4735—22no—
24#4694—21no—
25#4693—21no—
26#4681—21no—
27#4680—21no—
28#4482—22no—
29#4481—22no—
30#3227—31no—
31#3206—31no—
32#2868—31no—
33#1962—31no—
34#1549—31no—
35#1548—31no—
36#1276D311yescharacteristic, complemented
37C2×#235—21no—
38C22×Dic6—21no—
39C22×Dic6D412yes—
40C2×C3×C42—21no—
41C2×C3×C42—21no—
42C2×C3×C42D412yes—
43C2×C3×C42C2311yescharacteristic
44#930—21no—
45#930—21no—
46C2×C4⋊Dic3—28no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3D412yes—
51C2×#208—21no—
52C2×#204—24no—
53#875—21no—
54#874D411yescharacteristic
55C4×D12—21no—
56C4×D12—21no—
57#869—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869—48no—
66#869D411yescharacteristic
67C3×C4⋊C8D412yes—
68C3×C4⋊C8D412yes—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72C3×C4×C8—21no—
73#821—24no—
74#817—24no—
75#816—24no—
76#805—44no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79#532—31no—
80C2×C4⋊Q8—31no—
81C2×C4⋊Q8—31no—
82C2×C4⋊Q8—62no—
83#498—61no—
84#492—61no—
85#487—61no—
86#486—61no—
87#481—62no—
88#476—62no—
89#461—62no—
90#448—62no—
91#431C2×D311yescharacteristic
92#431C2×D311yescharacteristic
93#423—31no—
94C2×#104—62no—
95#384—62no—
96#384—62no—
97C43C2×D311yescharacteristic
98#333—62no—
99#332—62no—
100C22×C3×C4—22no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C2×#67—41no—
105C22×Dic3—22no—
106C22×Dic3—22no—
107#235—42no—
108#235—42no—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12—21no—
112C2×C4×D3—41no—
113C2×Dic6—22no—
114C2×Dic6—21no—
115C2×Dic6—22no—
116C2×Dic6—44no—
117C2×Dic6—44no—
118C2×Dic6—41no—
119C2×Dic6—41no—
120C2×C3×C8—22no—
121C2×C3×C8—22no—
122C3×C42—24no—
123C3×C42—24no—
124C3×C42—22no—
125C3×C42SD1614yescomplemented
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130(C2×D3)⋊C4—41no—
131(C2×D3)⋊C4—41no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—22no—
135C4⋊Dic3—22no—
136C4⋊Dic3—416no—
137C4⋊Dic3—41no—
138C4⋊Dic3—41no—
139C4⋊Dic3—41no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143#208—42no—
144C2×C3⋊C8—41no—
145#204—84no—
146C2×#41—61no—
147C22×Q8—32no—
148C22×Q8—61no—
149C2×SD16—64no—
150C2×M16—61no—
151C4⋊Q8—31no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
156C4⋊Q8—61no—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160C4⋊Q8—128no—
161#127—31no—
162#124—61no—
163#121—61no—
164#121—61no—
165C4×D4—61no—
166C4×D4—61no—
167C2×C4⋊C4—32no—
168C2×C4⋊C4—68no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—61no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42D1212yes—
175C2×C42C22×D311yescharacteristic
176#106—64no—
177C4⋊C8—61no—
178C4⋊C8—61no—
179C4⋊C8D1212yes—
180C4⋊C8D1212yes—
181#104—124no—
182#103—64no—
183#102—64no—
184C22⋊C8—21no—
185C22⋊C8—21no—
186C4×C8—21no—
187C4×C8—21no—
188C23×C3#12011yescharacteristic
189C22×D3—41no—
190C2×C3×C4—24no—
191C2×C3×C4—24no—
192C2×C3×C4—24no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12011yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12711yescharacteristic
201C2×C3×C4C2×SD1612yes—
202C2×C3×C4#13712yes—
203#67—82no—
204C2×Dic3—44no—
205C2×Dic3—44no—
206C2×Dic3—42no—
207C2×Dic3—41no—
208C2×Dic3—42no—
209D12—42no—
210D12—42no—
211D12—42no—
212C4×D3—81no—
213C4×D3—81no—
214Dic6—44no—
215Dic6—42no—
216Dic6—82no—
217Dic6—82no—
218Dic6—82no—
219C3×C8—42no—
220C3×C8—42no—
221C3⋊C8—81no—
222#41—122no—
223#41—122no—
224C2×Q8—62no—
225C2×Q8—61no—
226C2×Q8—62no—
227C2×Q8—124no—
228C2×Q8—124no—
229C2×Q8—121no—
230C2×Q8—121no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—121no—
235C22×C4—22no—
236C22×C4—62no—
237C22×C4—62no—
238C22×C4—121no—
239C22×C4C2×D1211yescharacteristic
240C22×C4D3×D411yescharacteristic
241C22×C4D3×D411yescharacteristic
242SD16—244no—
243M16—122no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—121no—
247C4⋊C4—62no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—62no—
251C4⋊C4—1216no—
252C4⋊C4—121no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C4⋊C4—121no—
258C22⋊C4—121no—
259C22⋊C4—121no—
260C42—24no—
261C42—24no—
262C42—22no—
263C42#20414yescomplemented
264C42C2×D1211yescharacteristic
265C42C2×D1211yescharacteristic
266C42D3×D411yescharacteristic
267C42#23911yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3#39211yescharacteristic
271C22×C3#44812yes—
272C2×D3—81no—
273C2×D3—81no—
274C2×D3—81no—
275C3×C4—28no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4#45312yes—
279C3×C4#45612yes—
280C3×C4#46112yes—
281C3×C4#49211yescharacteristic
282C3×C4#49711yescharacteristic
283Dic3—82no—
284Dic3—82no—
285Dic3—81no—
286C23—121no—
287C23#88211yescharacteristic
288Q8—124no—
289Q8—122no—
290Q8—242no—
291Q8—242no—
292Q8—242no—
293D4—122no—
294D4—122no—
295D4—122no—
296D4—242no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—124no—
303C2×C4—124no—
304C2×C4—121no—
305C2×C4—122no—
306C2×C4—122no—
307C2×C4—241no—
308C2×C4—241no—
309C2×C4#87411yescharacteristic
310C2×C4#88211yescharacteristic
311C2×C4#88211yescharacteristic
312C2×C4#89511yescharacteristic
313C2×C4#89511yescharacteristic
314C2×C4#89511yescharacteristic
315C2×C4C2×#20412yes—
316C2×C4#90912yes—
317C8—42no—
318C8—42no—
319C8—241no—
320C2×C3—22no—
321C2×C3#200211yescharacteristic
322C2×C3#200511yescharacteristic
323C2×C3#201311yescharacteristic
324D3—161no—
325C22—21no—
326C22—21no—
327C22—241no—
328C22—241no—
329C22—241no—
330C22#467111yescharacteristic
331C22#473812yes—
332C4—28no—
333C4—21no—
334C4—21no—
335C4—241no—
336C4—242no—
337C4—242no—
338C4#469311yescharacteristic
339C4#471511yescharacteristic
340C4#482212yes—
341C4#482612yes—
342C4#484112yes—
343C3#1028311yescharacteristic, complemented, Sylow
344C2—22no—
345C2—481no—
346C2#6868711yescharacteristic
347C2#6876111yescharacteristic
348C2#6876811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1276C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1276D3
hypocenterC3#10283
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
3. #73311C2semidirect
4. #70780C2semidirect
5. #70779C2semidirect
7. C3×#1276C2semidirect
36. #1276D3semidirect
125. C3×C42SD16semidirect
263. C42#204semidirect
343. C3#10283semidirect

Maximal subgroups

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

subgroupindex
#793122
#792692
#733112
#707802
#707792
#693952
C3×#12762
#102833