Finite Group Databasebeta

← #10696805

Subgroups — #10696805

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
1#79321C211yesmaximal, characteristic
2#79320C211yesmaximal, characteristic
3#78507C211yesmaximal, characteristic
4#70831C211yesmaximal, characteristic
5#70803C211yesmaximal, characteristic
6#69856C211yesmaximal, characteristic
7C3×#1291C211yesmaximal, characteristic
8#10127—31nomaximal, Sylow
9#5470C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5121—21no—
16#5121—21no—
17#4925C2211yescharacteristic
18#4851—21no—
19#4850—21no—
20#4829—22no—
21#4723—22no—
22#4722—21no—
23#4721—21no—
24#4703—21no—
25#4703—21no—
26#4702—21no—
27#4702—21no—
28#4702—21no—
29#4702—21no—
30#4683—22no—
31#4484—22no—
32#4469—22no—
33#4456—22no—
34#3237—31no—
35#3236—31no—
36#3166—31no—
37#1497—31no—
38#1485—31no—
39#1470—31no—
40#1291D311yescharacteristic
41C2×C3×M16—21no—
42C2×C3×C42C2311yescharacteristic
43#926—21no—
44#926—21no—
45C2×C4⋊Dic3D411yescharacteristic
46C2×C4⋊Dic3D411yescharacteristic
47#924—21no—
48#924—21no—
49#924—21no—
50#924—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53#870—21no—
54#870—21no—
55#870—21no—
56#870—21no—
57#870—21no—
58#870—21no—
59#869D411yescharacteristic
60#869D411yescharacteristic
61#869D411yescharacteristic
62#869D411yescharacteristic
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69C3×C22⋊C8—21no—
70C3×C22⋊C8D412yes—
71C3×#97—21no—
72C3×#97—21no—
73C3×#97—21no—
74C3×C4×C8D412yes—
75#830—21no—
76#818—22no—
77#818—22no—
78#818—22no—
79#818—22no—
80#817—22no—
81#817—22no—
82#817—22no—
83#817—22no—
84#816—22no—
85#816—22no—
86#816—22no—
87#816—22no—
88C4⋊(C3⋊C8)—21no—
89C4×C3⋊C8—21no—
90#533—31no—
91#533—31no—
92C2×#125—31no—
93#501—61no—
94#501—61no—
95#501—61no—
96#501—61no—
97#499—62no—
98#491—61no—
99#491—61no—
100#485—62no—
101#484—61no—
102#483—61no—
103#479—62no—
104#477—61no—
105#475—61no—
106#432C2×D311yescharacteristic
107#432C2×D311yescharacteristic
108#431—31no—
109#428—61no—
110#428—61no—
111C4×M16C2×D311yescharacteristic
112Q8⋊C8—62no—
113Q8⋊C8—62no—
114#324—62no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C22×Dic3—21no—
119C22×Dic3—21no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C3×M16—24no—
125C2×C3×C8—22no—
126C2×C3×C8—22no—
127C2×C3×C8—22no—
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132#217—41no—
133#217—41no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154C4×Dic3—41no—
155C4×Dic3—41no—
156C2×C3⋊C8—41no—
157C2×M16—21no—
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160C4⋊Q8—31no—
161C4⋊Q8—31no—
162#125—61no—
163#125—61no—
164#125—61no—
165#125—61no—
166#125—61no—
167#125—61no—
168C22⋊Q8—61no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C22⋊Q8—61no—
172C4×Q8—61no—
173C4×Q8—61no—
174C4×Q8—61no—
175C4×Q8—61no—
176#117—61no—
177#117—61no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C42C22×D311yescharacteristic
183#107—62no—
184#107—62no—
185#107—62no—
186#107—62no—
187#106—62no—
188#106—62no—
189#106—62no—
190#106—62no—
191C4⋊C8—21no—
192C4⋊C8—61no—
193C4⋊C8D1212yes—
194#103—62no—
195#103—62no—
196#103—62no—
197#103—62no—
198C22⋊C8—21no—
199C22⋊C8—61no—
200C22⋊C8D1212yes—
201#97—21no—
202#97—21no—
203#97—21no—
204C4×C8—61no—
205C4×C8D1212yes—
206C23×C3#12111yescharacteristic
207C2×C3×C4—22no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12711yescharacteristic
216C2×Dic3—42no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—81no—
224Dic6—42no—
225Dic6—42no—
226Dic6—42no—
227Dic6—42no—
228C3×C8—24no—
229C3×C8—42no—
230C3×C8—42no—
231C3⋊C8—81no—
232C2×Q8—61no—
233C2×Q8—61no—
234C2×Q8—61no—
235C2×Q8—61no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4C2×D1211yescharacteristic
239C22×C4D3×D411yescharacteristic
240C22×C4#23911yescharacteristic
241M16—24no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—121no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C22⋊C4—121no—
267C22⋊C4—121no—
268C42—121no—
269C42—121no—
270C42C2×D1211yescharacteristic
271C42C2×D1211yescharacteristic
272C42D3×D411yescharacteristic
273C42D3×D411yescharacteristic
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#45612yes—
284Dic3—81no—
285Dic3—81no—
286Dic3—81no—
287Dic3—81no—
288Dic3—81no—
289Dic3—81no—
290C23#89511yescharacteristic
291Q8—122no—
292Q8—122no—
293Q8—122no—
294Q8—122no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—122no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—241no—
306C2×C4#87411yescharacteristic
307C2×C4#88211yescharacteristic
308C2×C4#88211yescharacteristic
309C2×C4#88211yescharacteristic
310C2×C4#89511yescharacteristic
311C2×C4#89511yescharacteristic
312C8—24no—
313C8—42no—
314C8—42no—
315C8—241no—
316C2×C3—41no—
317C2×C3#200511yescharacteristic
318C2×C3#200911yescharacteristic
319C2×C3#201711yescharacteristic
320C22—21no—
321C22—21no—
322C22—21no—
323C22#467111yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—241no—
330C4—241no—
331C4—241no—
332C4—241no—
333C4—241no—
334C4—241no—
335C4#482612yes—
336C3#1012711yescharacteristic, complemented, Sylow
337C2—41no—
338C2#6872711yescharacteristic
339C2#6876811yescharacteristic
340C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1291C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1291D3
hypocenterC3#10127
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1291

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
#793212
#793202
#785072
#708312
#708032
#698562
C3×#12912
#101273