Finite Group Databasebeta

← #10697902

Subgroups — #10697902

Order 768.

640 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C3×#3229C211yesmaximal, characteristic, complemented
2#83355C211yesmaximal, characteristic
3#83251C211yesmaximal, characteristic
4#71839C211yesmaximal, characteristic, complemented
5#71773C211yesmaximal, characteristic, complemented
6#70774C211yesmaximal, characteristic
7#67736C211yesmaximal, characteristic, complemented
8#10294—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#947—22no—
12C2×#874C2211yescharacteristic
13C3×#498—21no—
14C3×#487—21no—
15C3×#483—22no—
16C3×#475—22no—
17C3×#431C2211yescharacteristic
18#5088—21no—
19#5085—21no—
20#5046—22no—
21#5027—22no—
22#4988—22no—
23#4988—22no—
24C42×Dic3C2211yescharacteristic
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27#4547—22no—
28#4532—22no—
29#4485—22no—
30#3229D311yescharacteristic, complemented
31#3208—31no—
32#2858—31no—
33#1994—31no—
34#1546—31no—
35#1545—31no—
36#1276—31no—
37C22×C3×Q8—21no—
38C22×D12—21no—
39C22×D12—21no—
40C22×D12D412yes—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×#117—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D412yes—
55C2×C3×C42C2311yescharacteristic
56#947—44no—
57#947—44no—
58C2×#215—24no—
59C2×(C2×D3)⋊C4—28no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3—21no—
62C2×C4×Dic3D412yes—
63C2×#208—21no—
64#874—21no—
65#874—21no—
66#874—21no—
67#874—21no—
68C3×#106—24no—
69C3×C4⋊C8D412yes—
70C3×#103—24no—
71C3×C22⋊C8—21no—
72C3×C4×C8—21no—
73#830—21no—
74#809—24no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D412yes—
78C4×C3⋊C8—21no—
79#533C2×D311yescharacteristic
80C2×C4⋊Q8C2×D311yescharacteristic
81C2×#127—31no—
82C2×#124—62no—
83#498—21no—
84#492—61no—
85#487—21no—
86#486—61no—
87#483—22no—
88#475—22no—
89#460—62no—
90#450—62no—
91#431—31no—
92#431C2×D311yescharacteristic
93#423—31no—
94#390—62no—
95#390—62no—
96C43—31no—
97#339—62no—
98#332—62no—
99#331—62no—
100C23×D3—22no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C2×C3×Q8—42no—
106C2×C3×Q8—42no—
107C22×C3×C4—22no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×Dic3—22no—
112C2×D12—22no—
113C2×D12—22no—
114C2×D12—44no—
115C2×D12—44no—
116C2×D12—41no—
117C2×D12—41no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—42no—
121C2×D12—42no—
122C2×C3×C8—22no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C42—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139#215—84no—
140(C2×D3)⋊C4—416no—
141C4×Dic3—22no—
142C4×Dic3—42no—
143C4×Dic3—42no—
144C4×Dic3SD1614yescomplemented
145#208—42no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—41no—
148C22×Q8—21no—
149C22×D4—32no—
150C22×D4—61no—
151C22×D4—61no—
152C2×SD16—64no—
153C2×M16—61no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8#6711yescharacteristic
159C4⋊Q8#6711yescharacteristic
160#127—61no—
161#127—61no—
162#127—61no—
163#127—61no—
164#125—21no—
165#124—124no—
166#124—124no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169C4×Q8—21no—
170C4×Q8—21no—
171#117—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4#6712yes—
174C2×C22⋊C4—68no—
175C2×C42—32no—
176C2×C42—61no—
177C2×C42—61no—
178C2×C42C22×D311yescharacteristic
179#106—24no—
180C4⋊C8—32no—
181C4⋊C8—61no—
182C4⋊C8—61no—
183C4⋊C8#6712yes—
184#103—24no—
185#102—64no—
186C22⋊C8—21no—
187C22⋊C8—61no—
188C4×C8—21no—
189C4×C8—61no—
190C23×C3#12011yescharacteristic
191C22×D3—44no—
192C22×D3—44no—
193C22×D3—42no—
194C22×D3—44no—
195C3×Q8—42no—
196C3×Q8—42no—
197C3×Q8—82no—
198C3×Q8—81no—
199C3×Q8—81no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—44no—
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12711yescharacteristic
212C2×C3×C4C2×SD1612yes—
213C2×C3×C4AGL(1,ℤ/8)12yes—
214C2×Dic3—24no—
215C2×Dic3—42no—
216C2×Dic3—42no—
217D12—44no—
218D12—82no—
219D12—82no—
220D12—82no—
221D12—82no—
222D12—81no—
223D12—81no—
224C3×C8—42no—
225C3⋊C8—42no—
226C3⋊C8—81no—
227C24—62no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×D4—62no—
235C2×D4—62no—
236C2×D4—124no—
237C2×D4—124no—
238C2×D4—122no—
239C2×D4—122no—
240C2×D4—121no—
241C2×D4—121no—
242C2×D4—121no—
243C2×D4—121no—
244C22×C4—22no—
245C22×C4—62no—
246C22×C4D3×D411yescharacteristic
247C22×C4D3×D411yescharacteristic
248C22×C4C2×#6711yescharacteristic
249SD16—244no—
250M16—122no—
251C2×C8—22no—
252C2×C8—62no—
253C2×C8—121no—
254C4⋊C4—21no—
255C4⋊C4—22no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—22no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—1216no—
266C42—34no—
267C42—41no—
268C42—62no—
269C42—122no—
270C42—122no—
271C42D3×D411yescharacteristic
272C42#23711yescharacteristic
273C42C2×#6711yescharacteristic
274C42C2×#6711yescharacteristic
275C22×C3—21no—
276C22×C3—21no—
277C22×C3#39211yescharacteristic
278C22×C3#45012yes—
279C2×D3—88no—
280C2×D3—82no—
281C2×D3—82no—
282C2×D3—82no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—81no—
286C3×C4—81no—
287C3×C4—82no—
288C3×C4#45412yes—
289C3×C4#45512yes—
290C3×C4#46012yes—
291C3×C4#49211yescharacteristic
292C3×C4#49611yescharacteristic
293Dic3—44no—
294C23—124no—
295C23—124no—
296C23—124no—
297C23—122no—
298C23#93711yescharacteristic
299Q8—42no—
300Q8—42no—
301Q8—82no—
302Q8—81no—
303Q8—81no—
304D4—124no—
305D4—242no—
306D4—242no—
307D4—242no—
308D4—241no—
309D4—241no—
310D4—242no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—44no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—64no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4D3×SD1612yes—
321C2×C4#91412yes—
322C2×C4#93711yescharacteristic
323C2×C4#93811yescharacteristic
324C2×C4#93911yescharacteristic
325C2×C4#93911yescharacteristic
326C2×C4#94011yescharacteristic
327C2×C4#95311yescharacteristic
328C8—42no—
329C8—122no—
330C8—241no—
331C2×C3—22no—
332C2×C3#200311yescharacteristic
333C2×C3#200411yescharacteristic
334C2×C3#201411yescharacteristic
335D3—162no—
336C22—21no—
337C22—21no—
338C22—248no—
339C22—242no—
340C22—242no—
341C22—242no—
342C22#516912yes—
343C22#522311yescharacteristic
344C4—21no—
345C4—21no—
346C4—82no—
347C4—81no—
348C4—81no—
349C4—124no—
350C4#516312yes—
351C4#517511yescharacteristic
352C4#517611yescharacteristic
353C4#520212yes—
354C4#524512yes—
355C3#1029411yescharacteristic, complemented, Sylow
356C2—22no—
357C2—482no—
358C2#7186711yescharacteristic
359C2#7205611yescharacteristic
360C2#7210611yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3229C2
socleC22×C3#392
prime radicalC2×#874C22
hypercenter#3229D3
hypocenterC3#10294
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
1. C3×#3229C2semidirect
4. #71839C2semidirect
5. #71773C2semidirect
7. #67736C2semidirect
30. #3229D3semidirect
144. C4×Dic3SD16semidirect
355. C3#10294semidirect

Maximal subgroups

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

subgroupindex
C3×#32292
#833552
#832512
#718392
#717732
#707742
#677362
#102943