Finite Group Databasebeta

← #10692910

Subgroups — #10692910

Order 768.

512 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 93 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#74054C211yesmaximal, characteristic
2#73518C211yesmaximal, characteristic
3#69981C211yesmaximal, characteristic
4#69973C211yesmaximal, characteristic
5#69533C211yesmaximal, characteristic
6C3×#1284C211yesmaximal, characteristic
7#67616C211yesmaximal, characteristic
8#9460—31nomaximal, Sylow
9C22×C4⋊Dic3C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×#381C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14#5211C2211yescharacteristic
15#5207C2211yescharacteristic
16#4943—21no—
17#4943—21no—
18#4942—21no—
19Dic3×C22⋊C4—21no—
20#4936C412yescomplemented
21#4929C412yescomplemented
22#4659—22no—
23#4654—22no—
24#4653—22no—
25#4554—22no—
26#4467—22no—
27#4452—22no—
28#2503—31no—
29#2313—31no—
30#1821—31no—
31#1511—31no—
32#1511—31no—
33#1284—31no—
34#1284D311yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C23×Dic3D411yescharacteristic
37C23×Dic3D411yescharacteristic
38C22×Dic6C2×C411yescharacteristic
39C2×C3×M16—21no—
40C2×C3×C22⋊C4—21no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C42C2×C412yes—
43C2×#217—21no—
44C2×#217—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3C2×C411yescharacteristic
50C2×C4⋊Dic3C2×C412yes—
51C2×C4⋊Dic3D412yes—
52#924—21no—
53#924—21no—
54#924—21no—
55#924—21no—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—22no—
58C2×C4×Dic3—22no—
59C2×#208—21no—
60C3×C22⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×#95—21no—
63C3×#95—21no—
64C3×#95D412yes—
65C3⋊#95—22no—
66C3⋊#95—22no—
67C3⋊#95—21no—
68C3⋊#95—22no—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C3⋊#95D412yes—
73#830—21no—
74#830—21no—
75C2×C22⋊Q8—31no—
76C22×C4⋊C4—31no—
77#407—31no—
78#407C2×D311yescharacteristic
79#400—62no—
80#391—32no—
81#387—62no—
82#387—62no—
83#382—32no—
84#381—31no—
85#381—61no—
86#381—61no—
87#381C2×D311yescharacteristic
88#380—61no—
89C4×C22⋊C4—61no—
90C4×C22⋊C4C2×D311yescharacteristic
91#343—62no—
92#328—62no—
93#328—62no—
94C24×C3C2×D411yescharacteristic
95C22×C3×C4—22no—
96C22×C3×C4—22no—
97C22×C3×C4—42no—
98C22×C3×C4C22×C411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—22no—
111C22×Dic3—42no—
112C22×Dic3—42no—
113C22×Dic3C22⋊C412yes—
114C22×Dic3C22⋊C412yes—
115C2×Dic6—21no—
116C2×Dic6—21no—
117C2×Dic6—42no—
118C3×M16—42no—
119C2×C3×C8—41no—
120C3×C22⋊C4—42no—
121C3×C22⋊C4—42no—
122C3×C42—42no—
123#217—42no—
124#217—41no—
125#217—41no—
126C4⋊Dic3—22no—
127C4⋊Dic3—22no—
128C4⋊Dic3—22no—
129C4⋊Dic3—22no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132Dic3⋊C4—41no—
133Dic3⋊C4—41no—
134Dic3⋊C4—82no—
135C4×Dic3—82no—
136#208—42no—
137C2×C3⋊C8—41no—
138C22×Q8—31no—
139C23×C4—31no—
140C23×C4—31no—
141C23×C4C22×D311yescharacteristic
142C2×M16—21no—
143C2×M16—61no—
144C22⋊Q8—61no—
145C22⋊Q8—61no—
146C22⋊Q8—61no—
147C22⋊Q8—61no—
148C2×C4⋊C4—32no—
149C2×C4⋊C4—32no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4—62no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—61no—
160C2×C22⋊C4—61no—
161C2×C42—62no—
162C2×C42C4×D312yes—
163C22⋊C8—21no—
164C22⋊C8—21no—
165C22⋊C8—61no—
166C22⋊C8—61no—
167#95—21no—
168#95—21no—
169#95—32no—
170#95—62no—
171#95—62no—
172#95—62no—
173#95—62no—
174#95—62no—
175#95—62no—
176#95—61no—
177#95D1212yes—
178C23×C3—22no—
179C23×C3—22no—
180C23×C3—41no—
181C23×C3C2×C22⋊C411yescharacteristic
182C23×C3#12011yescharacteristic
183C23×C3#12311yescharacteristic
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4C4×D411yescharacteristic
197C2×C3×C4C4×D411yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×Dic3—22no—
201C2×Dic3—22no—
202C2×Dic3—42no—
203C2×Dic3—42no—
204C2×Dic3—42no—
205C2×Dic3—42no—
206C2×Dic3—42no—
207C2×Dic3—42no—
208C2×Dic3—42no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3—81no—
214C2×Dic3—81no—
215Dic6—81no—
216Dic6—81no—
217C3×C8—81no—
218C3⋊C8—81no—
219C24D3×D411yescharacteristic
220C2×Q8—61no—
221C2×Q8—61no—
222C2×Q8—122no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—32no—
226C22×C4—32no—
227C22×C4—42no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—62no—
235C22×C4—122no—
236C22×C4—122no—
237C22×C4C2×C4×D311yescharacteristic
238C22×C4C2×D1211yescharacteristic
239C22×C4#23511yescharacteristic
240C22×C4D3×D411yescharacteristic
241C22×C4D3×D411yescharacteristic
242C22×C4#23911yescharacteristic
243M16—42no—
244M16—122no—
245C2×C8—41no—
246C2×C8—121no—
247C4⋊C4—62no—
248C4⋊C4—62no—
249C4⋊C4—62no—
250C4⋊C4—62no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C4⋊C4—242no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—122no—
259C22⋊C4—121no—
260C22⋊C4—121no—
261C42—42no—
262C42—242no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—44no—
271C22×C3#38611yescharacteristic
272C22×C3#41012yes—
273C22×C3#45512yes—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—82no—
279C3×C4—82no—
280Dic3—42no—
281Dic3—42no—
282Dic3—82no—
283Dic3—81no—
284Dic3—81no—
285C23—22no—
286C23—22no—
287C23—41no—
288C23D3×C22⋊C411yescharacteristic
289C23#88211yescharacteristic
290C23#89411yescharacteristic
291Q8—241no—
292Q8—241no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—62no—
306C2×C4—62no—
307C2×C4—122no—
308C2×C4—122no—
309C2×C4—122no—
310C2×C4—122no—
311C2×C4—122no—
312C2×C4—122no—
313C2×C4—122no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—122no—
317C2×C4—122no—
318C2×C4—241no—
319C2×C4—241no—
320C2×C4C4×D1211yescharacteristic
321C2×C4#88411yescharacteristic
322C2×C4#89311yescharacteristic
323C2×C4#89511yescharacteristic
324C8—81no—
325C8—241no—
326C2×C3—22no—
327C2×C3—22no—
328C2×C3—41no—
329C2×C3#188511yescharacteristic
330C2×C3#189911yescharacteristic
331C2×C3#191211yescharacteristic
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—44no—
340C22#466511yescharacteristic
341C22#474012yes—
342C22#482512yes—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—82no—
348C4—82no—
349C4—122no—
350C4—122no—
351C4—242no—
352C4—241no—
353C4—241no—
354C3#946011yescharacteristic, complemented, Sylow
355C2—22no—
356C2—22no—
357C2—41no—
358C2#6870811yescharacteristic
359C2#6875911yescharacteristic
360C2#6895411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1284C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1284D3
hypocenterC3#9460
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1284

Complements

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

NH ≅ G/Ntype
20. #4936C4semidirect
21. #4929C4semidirect
354. C3#9460semidirect

Maximal subgroups

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

subgroupindex
#740542
#735182
#699812
#699732
#695332
C3×#12842
#676162
#94603