Finite Group Databasebeta

← #10696874

Subgroups — #10696874

Order 768.

484 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 83 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#83370C211yesmaximal, characteristic, complemented
2#83357C211yesmaximal, characteristic
3C2×#4714C211yesmaximal, characteristic
4C3×#1531C211yesmaximal, characteristic, complemented
5#70828C211yesmaximal, characteristic
6#70823C211yesmaximal, characteristic, complemented
7#67662C211yesmaximal, characteristic, complemented
8#10137—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#875C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C2×C3×#97C2211yescharacteristic
14C2×#821—21no—
15C2×#816—21no—
16#5093—21no—
17#5092—21no—
18#5089—21no—
19#5086—21no—
20#5050—21no—
21#5050—22no—
22#5046—21no—
23#5031—22no—
24#5028—22no—
25#4926C2211yescharacteristic
26#4925C2211yescharacteristic
27#4714—22no—
28#4714—22no—
29C3×#330—22no—
30#4482—22no—
31#4468—22no—
32#3215—31no—
33#3209—31no—
34C2×#489—31no—
35#1556—31no—
36#1531D311yescharacteristic, complemented
37#1469—31no—
38#1302—31no—
39C22×D12D411yescharacteristic
40C22×Dic6—21no—
41C22×Dic6D411yescharacteristic
42C22×C3×C8—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×#117—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#216—22no—
53C2×#216—24no—
54C2×#215—22no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59#875—21no—
60#875—21no—
61#875—21no—
62#875—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869D412yes—
67C3×C4⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#97—21no—
70C3×#97—21no—
71C3×#97D412yes—
72#830—21no—
73#830D412yes—
74#821—24no—
75#816—24no—
76#810—22no—
77#810—24no—
78#809—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81#803—21no—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84C2×C4⋊Q8—31no—
85C2×#124—31no—
86#497—61no—
87#494—61no—
88#489—62no—
89#489—61no—
90#489—62no—
91#487—61no—
92#464—62no—
93#462—62no—
94#451—62no—
95#451—61no—
96#450—61no—
97#432—31no—
98#432C2×D311yescharacteristic
99#431—31no—
100C2×#103—61no—
101C2×#102—61no—
102C2×#97C2×D311yescharacteristic
103#332—62no—
104#330—22no—
105#323—62no—
106C23×D3—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C22×Dic3—21no—
113C22×Dic3—41no—
114C2×D12—21no—
115C2×D12—21no—
116C2×D12—42no—
117C2×Dic6—22no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C2×Dic6—42no—
121C2×Dic6—41no—
122C2×Dic6—82no—
123C2×C3×C8—22no—
124C2×C3×C8—42no—
125C2×C3×C8—41no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C42—41no—
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138#216—84no—
139#216—82no—
140#215—82no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145C4⋊Dic3—22no—
146C4⋊Dic3—22no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C22×Q8—31no—
152C22×Q8—61no—
153C22×D4—31no—
154C2×Q16—64no—
155C2×Q16—62no—
156C2×SD16—62no—
157C22×C8—21no—
158C4⋊Q8—32no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162C4⋊Q8D1211yescharacteristic
163C4⋊Q8D1211yescharacteristic
164#125—21no—
165#124—61no—
166#124—61no—
167#124—61no—
168#124—61no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173#117—21no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C22⋊C4—61no—
177C2×C22⋊C4—61no—
178C2×C42C22×D311yescharacteristic
179C4⋊C8—21no—
180C4⋊C8—32no—
181C4⋊C8—61no—
182#103—64no—
183#103—62no—
184#103—64no—
185#102—62no—
186#102—64no—
187C22⋊C8—21no—
188C22⋊C8—32no—
189C22⋊C8—61no—
190#97—21no—
191#97—21no—
192#97—61no—
193#97#6712yes—
194C4×C8—61no—
195C23×C3#12111yescharacteristic
196C22×D3—41no—
197C22×D3—42no—
198C22×D3—81no—
199C22×D3—81no—
200C3×Q8—42no—
201C3×Q8—42no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12711yescharacteristic
213C2×Dic3—42no—
214C2×Dic3—41no—
215C2×Dic3—82no—
216C2×Dic3—81no—
217D12—42no—
218D12—42no—
219Dic6—44no—
220Dic6—42no—
221Dic6—42no—
222Dic6—82no—
223C3×C8—42no—
224C3×C8—81no—
225C3⋊C8—42no—
226C3⋊C8—42no—
227C24—61no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—61no—
231C2×Q8—62no—
232C2×Q8—61no—
233C2×Q8—122no—
234C2×Q8—121no—
235C2×Q8—242no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—122no—
239C22×C4—61no—
240C22×C4—121no—
241C22×C4#23511yescharacteristic
242C22×C4D3×D411yescharacteristic
243C22×C4D3×D411yescharacteristic
244Q16—244no—
245Q16—242no—
246SD16—242no—
247C2×C8—22no—
248C2×C8—42no—
249C2×C8—41no—
250C2×C8—62no—
251C2×C8—62no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—62no—
259C4⋊C4—62no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C22⋊C4—41no—
263C22⋊C4—121no—
264C22⋊C4—121no—
265C22⋊C4—121no—
266C22⋊C4—121no—
267C42—41no—
268C42C2×D1211yescharacteristic
269C42D3×D411yescharacteristic
270C42D3×D411yescharacteristic
271C42C2×#6711yescharacteristic
272C22×C3—21no—
273C22×C3—21no—
274C22×C3#39211yescharacteristic
275C22×C3#47312yes—
276C2×D3—81no—
277C2×D3—81no—
278C2×D3—81no—
279C2×D3—162no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—81no—
286C3×C4—81no—
287C3×C4#45512yes—
288Dic3—81no—
289Dic3—82no—
290C23—121no—
291C23—122no—
292C23—241no—
293C23—241no—
294C23#88411yescharacteristic
295Q8—42no—
296Q8—42no—
297Q8—124no—
298Q8—122no—
299Q8—122no—
300Q8—242no—
301D4—122no—
302D4—122no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—121no—
309C2×C4—122no—
310C2×C4—242no—
311C2×C4—241no—
312C2×C4#88211yescharacteristic
313C2×C4#88211yescharacteristic
314C2×C4#88411yescharacteristic
315C2×C4#93011yescharacteristic
316C2×C4#93711yescharacteristic
317C2×C4#94011yescharacteristic
318C8—42no—
319C8—81no—
320C8—122no—
321C8—122no—
322C2×C3—22no—
323C2×C3#200411yescharacteristic
324C2×C3#200811yescharacteristic
325C2×C3#202011yescharacteristic
326D3—161no—
327C22—21no—
328C22—21no—
329C22—241no—
330C22—241no—
331C22—241no—
332C22—482no—
333C22#475512yes—
334C22#496011yescharacteristic
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—81no—
341C4—81no—
342C4—242no—
343C4—241no—
344C4#482512yes—
345C3#1013711yescharacteristic, complemented, Sylow
346C2—22no—
347C2—481no—
348C2#7012511yescharacteristic
349C2#7038611yescharacteristic
350C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1531C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1531D3
hypocenterC3#10137
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1531

Complements

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

NH ≅ G/Ntype
1. #83370C2semidirect
4. C3×#1531C2semidirect
6. #70823C2semidirect
7. #67662C2semidirect
36. #1531D3semidirect
345. C3#10137semidirect

Maximal subgroups

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

subgroupindex
#833702
#833572
C2×#47142
C3×#15312
#708282
#708232
#676622
#101373