Finite Group Databasebeta

← #10696899

Subgroups — #10696899

Order 768.

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

normal subgroups only

This group has 472 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#83223C211yesmaximal, characteristic, complemented
2C2×#5089C211yesmaximal, characteristic
3#79351C211yesmaximal, characteristic
4C3×#1558C211yesmaximal, characteristic, complemented
5#70828C211yesmaximal, characteristic
6#70813C211yesmaximal, characteristic, complemented
7#67751C211yesmaximal, characteristic, complemented
8#10141—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#875C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14#5094—22no—
15#5093—21no—
16#5089—22no—
17#5089—22no—
18#5080—21no—
19#4971—22no—
20C2×#810—21no—
21C2×#809—21no—
22#4925C2211yescharacteristic
23C2×#803C2211yescharacteristic
24#4845—22no—
25#4739—21no—
26#4731—21no—
27#4715—21no—
28#4714—21no—
29C3×#333—22no—
30#4468—22no—
31#4465—22no—
32#3229—31no—
33#3215—31no—
34C2×#489—31no—
35#1558D311yescharacteristic, complemented
36#1531—31no—
37#1469—31no—
38#1302—31no—
39C22×C3×Q8—21no—
40C22×D12D411yescharacteristic
41C22×Dic6D411yescharacteristic
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51#926—21no—
52#924—21no—
53#924—21no—
54C22×C3⋊C8—21no—
55C2×Dic12—22no—
56C2×#204—22no—
57#875—21no—
58#875—21no—
59#875—21no—
60#875—21no—
61#870—21no—
62#869D411yescharacteristic
63#869D411yescharacteristic
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8D412yes—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#97—21no—
71C3×#97—21no—
72#830D412yes—
73#821—22no—
74#816—22no—
75#810—24no—
76#810—24no—
77#809—24no—
78#808—24no—
79C4⋊(C3⋊C8)—21no—
80#803—21no—
81#803D412yes—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#124—31no—
86#498—61no—
87#497—61no—
88#489—61no—
89#489—62no—
90#489—62no—
91#487—61no—
92#483—62no—
93#475—62no—
94#464—62no—
95#451—61no—
96#450—61no—
97#432C2×D311yescharacteristic
98#432C2×D311yescharacteristic
99#431—31no—
100C2×#103—61no—
101C2×#102—61no—
102C2×#97—31no—
103#333—22no—
104#330—62no—
105#323—62no—
106C23×D3—21no—
107C2×C3×Q8—22no—
108C2×C3×Q8—41no—
109C2×C3×Q8—82no—
110C22×C3×C4—41no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C22×Dic3—21no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—42no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—42no—
123C2×C3×C8—22no—
124C2×C3×C8—41no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133#217—41no—
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144C4×Dic3—41no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—42no—
148Dic12—82no—
149#204—82no—
150C22×Q8—21no—
151C22×Q8—31no—
152C22×D4—31no—
153C2×Q16—62no—
154C2×SD16—62no—
155C22×C8—61no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—31no—
160C4⋊Q8—31no—
161C4⋊Q8D1212yes—
162#125—61no—
163#124—61no—
164#124—61no—
165#124—61no—
166#124—61no—
167C22⋊Q8—61no—
168C22⋊Q8—61no—
169C4×Q8—61no—
170C4×Q8—61no—
171#117—61no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C22⋊C4—61no—
175C2×C22⋊C4—61no—
176C2×C42C22×D311yescharacteristic
177#106—64no—
178C4⋊C8—21no—
179C4⋊C8—61no—
180C4⋊C8#6712yes—
181#103—64no—
182#103—64no—
183#103—62no—
184#102—64no—
185#102—62no—
186C22⋊C8—21no—
187C22⋊C8—21no—
188C22⋊C8—32no—
189#97—21no—
190#97—21no—
191#97—32no—
192#97—61no—
193C4×C8—61no—
194C23×C3#12111yescharacteristic
195C22×D3—41no—
196C22×D3—42no—
197C22×D3—81no—
198C22×D3—81no—
199C3×Q8—82no—
200C3×Q8—82no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—82no—
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12711yescharacteristic
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—81no—
216D12—42no—
217D12—42no—
218Dic6—42no—
219Dic6—42no—
220Dic6—42no—
221Dic6—42no—
222C3×C8—42no—
223C3×C8—81no—
224C3⋊C8—42no—
225C3⋊C8—42no—
226C24—61no—
227C2×Q8—22no—
228C2×Q8—41no—
229C2×Q8—61no—
230C2×Q8—61no—
231C2×Q8—61no—
232C2×Q8—61no—
233C2×Q8—82no—
234C2×Q8—122no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—122no—
238C22×C4—41no—
239C22×C4—61no—
240C22×C4#23511yescharacteristic
241C22×C4D3×D411yescharacteristic
242C22×C4D3×D411yescharacteristic
243Q16—242no—
244SD16—242no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—62no—
248C2×C8—62no—
249C2×C8—122no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C22⋊C4—121no—
261C22⋊C4—121no—
262C22⋊C4—121no—
263C22⋊C4—121no—
264C22⋊C4—121no—
265C42—121no—
266C42C2×D1211yescharacteristic
267C42D3×D411yescharacteristic
268C42D3×D411yescharacteristic
269C42C2×#6711yescharacteristic
270C22×C3—21no—
271C22×C3—21no—
272C22×C3#39211yescharacteristic
273C22×C3#47112yes—
274C2×D3—81no—
275C2×D3—81no—
276C2×D3—81no—
277C2×D3—162no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—82no—
284C3×C4#45512yes—
285Dic3—81no—
286Dic3—81no—
287Dic3—81no—
288C23—121no—
289C23—122no—
290C23—241no—
291C23—241no—
292C23#88411yescharacteristic
293Q8—82no—
294Q8—82no—
295Q8—122no—
296Q8—122no—
297Q8—122no—
298Q8—122no—
299D4—122no—
300D4—122no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—82no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—241no—
310C2×C4#88211yescharacteristic
311C2×C4#88211yescharacteristic
312C2×C4#88411yescharacteristic
313C2×C4#93011yescharacteristic
314C2×C4#93711yescharacteristic
315C2×C4#94011yescharacteristic
316C8—42no—
317C8—81no—
318C8—122no—
319C8—122no—
320C2×C3—22no—
321C2×C3#200411yescharacteristic
322C2×C3#200911yescharacteristic
323C2×C3#201911yescharacteristic
324D3—161no—
325C22—21no—
326C22—21no—
327C22—241no—
328C22—241no—
329C22—241no—
330C22—482no—
331C22#474812yes—
332C22#496011yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—82no—
339C4—241no—
340C4—241no—
341C4—241no—
342C4#520212yes—
343C3#1014111yescharacteristic, complemented, Sylow
344C2—22no—
345C2—481no—
346C2#7009111yescharacteristic
347C2#7034411yescharacteristic
348C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1558C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1558D3
hypocenterC3#10141
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
1. #83223C2semidirect
4. C3×#1558C2semidirect
6. #70813C2semidirect
7. #67751C2semidirect
35. #1558D3semidirect
343. C3#10141semidirect

Maximal subgroups

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

subgroupindex
#832232
C2×#50892
#793512
C3×#15582
#708282
#708132
#677512
#101413