Finite Group Databasebeta

← #10696800

Subgroups — #10696800

Order 768.

420 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 81 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×#3237C211yesmaximal, characteristic
2#83342C211yesmaximal, characteristic
3#83326C211yesmaximal, characteristic
4#71827C211yesmaximal, characteristic
5#71825C211yesmaximal, characteristic
6#70831C211yesmaximal, characteristic
7#67754C211yesmaximal, characteristic
8#10127—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#533C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#501—21no—
13C3×#491—21no—
14C3×#485—22no—
15C3×#477—21no—
16C3×#475—21no—
17C3×#432C2211yescharacteristic
18#5082—21no—
19#5081—21no—
20#5081—21no—
21#5081—21no—
22#5065—22no—
23#5044—21no—
24#5043—21no—
25#5022—22no—
26#4995—21no—
27#4995—21no—
28#4926C2211yescharacteristic
29#4925C2211yescharacteristic
30C4×#208C2211yescharacteristic
31#4535—22no—
32#4535—22no—
33#4469—22no—
34#3237D311yescharacteristic
35#3236—31no—
36#3166—31no—
37#1497—31no—
38#1485—31no—
39#1470—31no—
40#1291—31no—
41C3×C4⋊Q8D411yescharacteristic
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×#117—21no—
56C3×#117—21no—
57C2×C3×C42C2311yescharacteristic
58C2×C4⋊Dic3D411yescharacteristic
59C2×C4⋊Dic3D411yescharacteristic
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—21no—
62C2×#208—21no—
63#870—21no—
64#870—21no—
65#870—21no—
66#870—21no—
67C3×#107—22no—
68C3×#106—22no—
69C3×C4⋊C8—21no—
70C3×#103—22no—
71C3×#103—22no—
72C3×C22⋊C8D412yes—
73C3×#97—21no—
74#835—22no—
75#835—22no—
76#830—21no—
77#830—21no—
78#808—22no—
79#808—22no—
80#808—22no—
81#807—22no—
82#807—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)D412yes—
86#803—21no—
87#803—21no—
88C4×C3⋊C8—21no—
89C4×C3⋊C8D412yes—
90#533C2×D311yescharacteristic
91#533C2×D311yescharacteristic
92C2×#125—31no—
93#501—21no—
94#501—61no—
95#501—61no—
96#501—61no—
97#499—62no—
98#491—21no—
99#491—61no—
100#485—22no—
101#484—61no—
102#483—61no—
103#479—62no—
104#477—21no—
105#475—21no—
106#432—31no—
107#432C2×D311yescharacteristic
108#431—31no—
109#428—61no—
110#428—61no—
111C4×M16—31no—
112Q8⋊C8—62no—
113Q8⋊C8—62no—
114#324—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C22×Dic3—21no—
123C22×Dic3—21no—
124C2×C3×C8—22no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C42—41no—
140C3×C42—41no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153#208—24no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—41no—
157C2×M16—61no—
158C4⋊Q8#6711yescharacteristic
159C4⋊Q8#6711yescharacteristic
160C4⋊Q8#6711yescharacteristic
161C4⋊Q8#6711yescharacteristic
162#125—21no—
163#125—21no—
164#125—61no—
165#125—61no—
166#125—61no—
167#125—61no—
168C22⋊Q8—21no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176#117—21no—
177#117—21no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C42C22×D311yescharacteristic
183#107—22no—
184#107—62no—
185#107—62no—
186#107—62no—
187#106—22no—
188#106—62no—
189#106—62no—
190#106—62no—
191C4⋊C8—21no—
192C4⋊C8—32no—
193C4⋊C8—61no—
194#103—22no—
195#103—22no—
196#103—62no—
197#103—62no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200C22⋊C8#6712yes—
201#97—21no—
202#97—61no—
203#97—61no—
204C4×C8—32no—
205C4×C8—61no—
206C23×C3#12111yescharacteristic
207C3×Q8—42no—
208C3×Q8—42no—
209C3×Q8—42no—
210C3×Q8—42no—
211C2×C3×C4—22no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12711yescharacteristic
224C2×Dic3—42no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—81no—
228C3×C8—42no—
229C3⋊C8—24no—
230C3⋊C8—42no—
231C3⋊C8—81no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4D3×D411yescharacteristic
239C22×C4#23711yescharacteristic
240C22×C4C2×#6711yescharacteristic
241M16—64no—
242C2×C8—22no—
243C2×C8—62no—
244C2×C8—62no—
245C2×C8—121no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C42—41no—
269C42—41no—
270C42D3×D411yescharacteristic
271C42D3×D411yescharacteristic
272C42C2×#6711yescharacteristic
273C42C2×#6711yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3#39211yescharacteristic
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—81no—
284C3×C4—81no—
285C3×C4—81no—
286C3×C4—81no—
287C3×C4#45612yes—
288Dic3—81no—
289Dic3—81no—
290C23#93811yescharacteristic
291Q8—42no—
292Q8—42no—
293Q8—42no—
294Q8—42no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—122no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—241no—
306C2×C4#93711yescharacteristic
307C2×C4#93711yescharacteristic
308C2×C4#93911yescharacteristic
309C2×C4#93911yescharacteristic
310C2×C4#94011yescharacteristic
311C2×C4#95311yescharacteristic
312C8—42no—
313C8—64no—
314C8—122no—
315C8—241no—
316C2×C3—41no—
317C2×C3#200511yescharacteristic
318C2×C3#200911yescharacteristic
319C2×C3#201711yescharacteristic
320C22—21no—
321C22—21no—
322C22—21no—
323C22#522311yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—241no—
334C4—241no—
335C4#520512yes—
336C3#1012711yescharacteristic, complemented, Sylow
337C2—41no—
338C2#7186911yescharacteristic
339C2#7210811yescharacteristic
340C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3237C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#3237D3
hypocenterC3#10127
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3237

Complements

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

NH ≅ G/Ntype
336. C3#10127semidirect

Maximal subgroups

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

subgroupindex
C3×#32372
#833422
#833262
#718272
#718252
#708312
#677542
#101273