Finite Group Databasebeta

← #10696501

Subgroups — #10696501

Order 768.

526 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#4703C211yesmaximal, characteristic, complemented
2C2×#4714C211yesmaximal, characteristic
3#74255C211yesmaximal, characteristic
4C2×#4465C211yesmaximal, characteristic, complemented
5C2×#4464C211yesmaximal, characteristic
6#69555C211yesmaximal, characteristic, complemented
7C3×#1295C211yesmaximal, characteristic, complemented
8#10083—31nomaximal, Sylow
9C2×#875C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13C2×C3×#97C2211yescharacteristic
14C3×#382C2211yescharacteristic
15C2×#821—21no—
16C2×#816—21no—
17C2×#816—21no—
18C2×#816—21no—
19#4994—22no—
20C2×#803C2211yescharacteristic
21#4714—22no—
22#4714—22no—
23#4703—22no—
24#4703—22no—
25#4677—21no—
26#4674—21no—
27#4672—21no—
28#4656—21no—
29#4529—22no—
30#4465—22no—
31#4464—22no—
32C2×#491—31no—
33C2×#489—31no—
34#2820—31no—
35#1993—31no—
36C2×#330—31no—
37C2×#329—31no—
38#1295D311yescharacteristic, complemented
39C22×D12D411yescharacteristic
40C22×Dic6—21no—
41C22×Dic6D411yescharacteristic
42C22×C3×C8—21no—
43C22×C3×C8—21no—
44C2×C3×C4⋊C4D412yes—
45C2×C3×C42C2311yescharacteristic
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—22no—
49C2×(C2×D3)⋊C4—22no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—22no—
53C2×C4⋊Dic3D411yescharacteristic
54C2×C4⋊Dic3D411yescharacteristic
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C22×C3⋊C8—21no—
58#875—21no—
59#875—21no—
60#875—21no—
61#875—21no—
62#870—21no—
63#870—21no—
64#870—21no—
65#870—21no—
66#869—21no—
67#869—21no—
68#869—21no—
69#869D412yes—
70C3×#97—21no—
71C3×#97—21no—
72C3×#97D412yes—
73C3×#97D412yes—
74C3×#95—21no—
75C3×#95—21no—
76C3⋊#95—22no—
77#821—24no—
78#816—24no—
79#816—24no—
80#816—24no—
81#803—21no—
82#803—21no—
83C2×C4⋊Q8—31no—
84C2×#125—31no—
85C2×#124—31no—
86#491—62no—
87#491—62no—
88#489—62no—
89#489—62no—
90C2×#103—61no—
91C2×#103—61no—
92C2×#103—61no—
93C2×#102—61no—
94C2×#97—31no—
95C2×#97C2×D311yescharacteristic
96C2×#97C2×D311yescharacteristic
97#399—61no—
98#397—61no—
99#396—62no—
100#395—61no—
101#394—61no—
102#382C2×D311yescharacteristic
103#340—62no—
104#330—62no—
105#329—62no—
106C23×D3—21no—
107C22×C3×C4—22no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—41no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—42no—
118C2×Dic6—22no—
119C2×Dic6—21no—
120C2×Dic6—21no—
121C2×Dic6—42no—
122C2×Dic6—44no—
123C2×Dic6—41no—
124C2×C3×C8—22no—
125C2×C3×C8—22no—
126C2×C3×C8—42no—
127C2×C3×C8—42no—
128C3×C4⋊C4—42no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133(C2×D3)⋊C4—41no—
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—82no—
138(C2×D3)⋊C4—82no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—22no—
142C4⋊Dic3—22no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147C4⋊Dic3—82no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C22×Q8—31no—
156C22×Q8—61no—
157C22×D4—31no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—61no—
161C4⋊Q8—32no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165#125—61no—
166#125—61no—
167#125—61no—
168#125—61no—
169#124—61no—
170#124—61no—
171#124—61no—
172#124—61no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—62no—
180C2×C4⋊C4D1212yes—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—62no—
184C2×C22⋊C4—62no—
185C2×C42C22×D311yescharacteristic
186#103—64no—
187#103—64no—
188#103—64no—
189#102—64no—
190#97—21no—
191#97—21no—
192#97—61no—
193#97—61no—
194#97D1212yes—
195#97D1212yes—
196#95—21no—
197#95—21no—
198#95—62no—
199C23×C3#12011yescharacteristic
200C22×D3—41no—
201C22×D3—42no—
202C22×D3—41no—
203C22×D3—42no—
204C22×D3—41no—
205C2×C3×C4—22no—
206C2×C3×C4—21no—
207C2×C3×C4—22no—
208C2×C3×C4—21no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12711yescharacteristic
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—42no—
221C2×Dic3—41no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—82no—
227D12—42no—
228D12—81no—
229Dic6—44no—
230Dic6—42no—
231Dic6—82no—
232Dic6—81no—
233C3×C8—42no—
234C3×C8—42no—
235C3⋊C8—42no—
236C24—61no—
237C2×Q8—61no—
238C2×Q8—62no—
239C2×Q8—61no—
240C2×Q8—122no—
241C2×Q8—124no—
242C2×Q8—121no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—122no—
246C22×C4—22no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—121no—
251C22×C4C2×D1211yescharacteristic
252C22×C4D3×D411yescharacteristic
253C22×C4D3×D411yescharacteristic
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—62no—
259C2×C8—121no—
260C2×C8—121no—
261C4⋊C4—42no—
262C4⋊C4—62no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—62no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C4⋊C4—242no—
275C22⋊C4—121no—
276C22⋊C4—121no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C22⋊C4—242no—
280C22⋊C4—242no—
281C42C2×D1211yescharacteristic
282C42C2×D1211yescharacteristic
283C42D3×D411yescharacteristic
284C42#23911yescharacteristic
285C22×C3#39211yescharacteristic
286C22×C3#45412yes—
287C22×C3#45511yescharacteristic
288C22×C3#45512yes—
289C22×C3#45611yescharacteristic
290C2×D3—82no—
291C2×D3—82no—
292C2×D3—81no—
293C2×D3—81no—
294C2×D3—81no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—82no—
302Dic3—82no—
303Dic3—81no—
304Dic3—81no—
305Dic3—81no—
306C23—121no—
307C23—121no—
308C23—121no—
309C23—122no—
310C23—122no—
311C23#88211yescharacteristic
312Q8—122no—
313Q8—124no—
314Q8—241no—
315Q8—242no—
316D4—122no—
317D4—241no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—122no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4—242no—
334C2×C4#87411yescharacteristic
335C2×C4#88211yescharacteristic
336C2×C4#88211yescharacteristic
337C2×C4#89511yescharacteristic
338C2×C4#89511yescharacteristic
339C2×C4#89511yescharacteristic
340C8—42no—
341C8—42no—
342C8—122no—
343C2×C3#166212yes—
344C2×C3#166412yes—
345C2×C3#200311yescharacteristic
346C2×C3#200411yescharacteristic
347C2×C3#200711yescharacteristic
348D3—161no—
349C22—242no—
350C22—242no—
351C22—241no—
352C22—241no—
353C22—241no—
354C22#467111yescharacteristic
355C22#474611yescharacteristic
356C22#475211yescharacteristic
357C22#482512yes—
358C22#482712yes—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—82no—
366C4—241no—
367C4—241no—
368C4—241no—
369C4—242no—
370C3#1008311yescharacteristic, complemented, Sylow
371C2—481no—
372C2#6835812yes—
373C2#6837812yes—
374C2#6876211yescharacteristic
375C2#6876711yescharacteristic
376C2#6900911yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1295C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenter#1295D3
hypocenterC3#10083
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
1. C2×#4703C2semidirect
4. C2×#4465C2semidirect
6. #69555C2semidirect
7. C3×#1295C2semidirect
38. #1295D3semidirect
370. C3#10083semidirect

Maximal subgroups

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

subgroupindex
C2×#47032
C2×#47142
#742552
C2×#44652
C2×#44642
#695552
C3×#12952
#100833