Finite Group Databasebeta

← #10690130

Subgroups — #10690130

Order 768.

458 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 105 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
2#74161C211yesmaximal, characteristic
3#69828C211yesmaximal, characteristic
4C2×#4465C211yesmaximal, characteristic
5#69555C211yesmaximal, characteristic
6C3×#1295C211yesmaximal, characteristic
7#67659C211yesmaximal, characteristic
8#9004—31nomaximal, Sylow
9C2×#870C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C2×C3×#97C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13C3×#382C2211yescharacteristic
14C2×#816—21no—
15C2×#816—21no—
16#4978—21no—
17#4976—21no—
18#4936C2211yescharacteristic
19C2×#803C2211yescharacteristic
20#4703—22no—
21#4703C414yescomplemented
22#4655—21no—
23#4654—21no—
24#4653—21no—
25#4647—21no—
26#4551—21no—
27#4551—21no—
28#4529—21no—
29#4529—21no—
30#4465—22no—
31C2×#491—31no—
32#2382—31no—
33#1993—31no—
34#1993—31no—
35C2×#330—31no—
36#1295—31no—
37#1295D311yescharacteristic
38C22×Dic6—21no—
39C22×C3×C8—21no—
40C22×C3×C8—21no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C42C2311yescharacteristic
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3D411yescharacteristic
47C2×C4⋊Dic3D411yescharacteristic
48C2×C4⋊Dic3D411yescharacteristic
49C2×C4⋊Dic3D411yescharacteristic
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3—21no—
55C2×C4×Dic3—21no—
56C22×C3⋊C8—21no—
57#870—21no—
58#870—21no—
59#870C2×C412yes—
60#870D412yes—
61#869—21no—
62#869—21no—
63#869—21no—
64#869C2×C412yes—
65C3×#97—21no—
66C3×#97—21no—
67C3×#97—21no—
68C3×#97C2×C412yes—
69C3×#95—21no—
70C3×#95—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77#816—24no—
78#816—24no—
79#803—21no—
80#803—21no—
81C2×C4⋊Q8—31no—
82C2×#125—31no—
83#491—34no—
84#491—62no—
85C2×#103—61no—
86C2×#103—61no—
87C2×#97—31no—
88C2×#97C2×D311yescharacteristic
89C2×#97C2×D311yescharacteristic
90#389—61no—
91#389—61no—
92#387—61no—
93#387—61no—
94#387—61no—
95#387—61no—
96#382—31no—
97#382C2×D311yescharacteristic
98#340—61no—
99#340—61no—
100#340—61no—
101#340—61no—
102#330—62no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—41no—
113C2×Dic6—22no—
114C2×Dic6—42no—
115C2×Dic6—42no—
116C2×Dic6—41no—
117C2×C3×C8—22no—
118C2×C3×C8—22no—
119C2×C3×C8—41no—
120C2×C3×C8—41no—
121C2×C3×C8—41no—
122C2×C3×C8—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C42C22×C411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42#4111yescharacteristic
128C3×C42#4111yescharacteristic
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—22no—
132C4⋊Dic3—22no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137C4⋊Dic3C22⋊C412yes—
138C4⋊Dic3C22⋊C412yes—
139Dic3⋊C4—22no—
140Dic3⋊C4—22no—
141Dic3⋊C4—22no—
142Dic3⋊C4—22no—
143Dic3⋊C4—81no—
144Dic3⋊C4—81no—
145C4×Dic3—81no—
146C4×Dic3—81no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—41no—
149C2×C3⋊C8—41no—
150C22×Q8—61no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—61no—
154C4⋊Q8—32no—
155C4⋊Q8—61no—
156C4⋊Q8—61no—
157C4⋊Q8—61no—
158#125—32no—
159#125—32no—
160#125—61no—
161#125—61no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—31no—
165C2×C4⋊C4—31no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4D1211yescharacteristic
173C2×C4⋊C4D1211yescharacteristic
174C2×C42—61no—
175C2×C42—61no—
176C2×C42C22×D311yescharacteristic
177#103—64no—
178#103—64no—
179#97—21no—
180#97—21no—
181#97—21no—
182#97—61no—
183#97—61no—
184#97C4×D312yes—
185#95—21no—
186#95—21no—
187#95—61no—
188#95—61no—
189#95—61no—
190#95—61no—
191#95—61no—
192#95—61no—
193C23×C3#12011yescharacteristic
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4C2×C22⋊C411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12311yescharacteristic
212C2×Dic3—22no—
213C2×Dic3—22no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—42no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—81no—
226C2×Dic3—81no—
227Dic6—44no—
228Dic6—82no—
229C3×C8—42no—
230C3×C8—42no—
231C3⋊C8—42no—
232C2×Q8—62no—
233C2×Q8—122no—
234C2×Q8—122no—
235C2×Q8—121no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—121no—
243C22×C4C2×D1211yescharacteristic
244C22×C4D3×D411yescharacteristic
245C22×C4D3×D411yescharacteristic
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—62no—
253C2×C8—121no—
254C2×C8—121no—
255C4⋊C4—32no—
256C4⋊C4—32no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—62no—
260C4⋊C4—62no—
261C4⋊C4—62no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—62no—
265C4⋊C4—62no—
266C4⋊C4—62no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C4⋊C4—241no—
272C4⋊C4—241no—
273C42—241no—
274C42—241no—
275C42C2×C4×D311yescharacteristic
276C42#23511yescharacteristic
277C42D3×D411yescharacteristic
278C42#23911yescharacteristic
279C22×C3#38611yescharacteristic
280C22×C3#45411yescharacteristic
281C22×C3#45411yescharacteristic
282C22×C3#45511yescharacteristic
283C22×C3#45511yescharacteristic
284C22×C3#45611yescharacteristic
285C22×C3#45611yescharacteristic
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—81no—
293C3×C4—81no—
294Dic3—42no—
295Dic3—42no—
296Dic3—82no—
297Dic3—81no—
298Dic3—81no—
299C23#88211yescharacteristic
300Q8—124no—
301Q8—242no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—62no—
315C2×C4—62no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—122no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—241no—
328C2×C4—241no—
329C2×C4C4×D1211yescharacteristic
330C2×C4D3×C22⋊C411yescharacteristic
331C2×C4#88411yescharacteristic
332C2×C4#89311yescharacteristic
333C2×C4#89411yescharacteristic
334C2×C4#89511yescharacteristic
335C8—42no—
336C8—42no—
337C8—122no—
338C2×C3#166211yescharacteristic
339C2×C3#166311yescharacteristic
340C2×C3#166411yescharacteristic
341C2×C3#166411yescharacteristic
342C2×C3#188211yescharacteristic
343C2×C3#188211yescharacteristic
344C2×C3#188811yescharacteristic
345C22#466511yescharacteristic
346C22#474611yescharacteristic
347C22#475211yescharacteristic
348C22#482511yescharacteristic
349C22#482611yescharacteristic
350C22#482711yescharacteristic
351C22#482711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—81no—
359C4—81no—
360C4—122no—
361C4—122no—
362C4—241no—
363C4—241no—
364C4—242no—
365C3#900411yescharacteristic, complemented, Sylow
366C2#6835811yescharacteristic
367C2#6835911yescharacteristic
368C2#6837611yescharacteristic
369C2#6837811yescharacteristic
370C2#6876611yescharacteristic
371C2#6877511yescharacteristic
372C2#6900411yescharacteristic
subgroupG/H
centerC23#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1295C2
socle, prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1295D3
hypocenterC3#9004
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
21. #4703C4semidirect
365. C3#9004semidirect

Maximal subgroups

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

subgroupindex
C2×#47032
#741612
#698282
C2×#44652
#695552
C3×#12952
#676592
#90043