Finite Group Databasebeta

← #10692868

Subgroups — #10692868

Order 768.

468 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 95 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#83239C211yesmaximal, characteristic
2#79314C211yesmaximal, characteristic
3C3×#1558C211yesmaximal, characteristic
4#71777C211yesmaximal, characteristic
5#70830C211yesmaximal, characteristic
6#67750C211yesmaximal, characteristic
7#67749C211yesmaximal, characteristic
8#9452—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#944C412yes—
11C2×C4×Dic6C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14#5229C412yes—
15#4983—21no—
16#4983—21no—
17#4972—21no—
18#4972—21no—
19C4×C4⋊Dic3C2211yescharacteristic
20#4926C2211yescharacteristic
21#4926C2211yescharacteristic
22#4828—22no—
23#4719—21no—
24#4718—21no—
25#4701—22no—
26C3×#333—22no—
27#4546—22no—
28#4469—21no—
29#4469—21no—
30#2983—31no—
31#2546—31no—
32#1558—31no—
33#1558D311yescharacteristic
34#1467—31no—
35#1285—31no—
36#1285—31no—
37C22×C3×Q8C2×C412yes—
38C22×Dic6C2×C411yescharacteristic
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C42C2311yescharacteristic
46#944—44no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3C2×C411yescharacteristic
49C2×Dic3⋊C4—22no—
50C2×Dic3⋊C4—22no—
51C2×Dic3⋊C4D411yescharacteristic
52C2×Dic3⋊C4D411yescharacteristic
53C2×C4×Dic3C2×C412yes—
54C2×C4×Dic3D411yescharacteristic
55C2×C4×Dic3D411yescharacteristic
56C4×Dic6—21no—
57C4×Dic6—21no—
58C4×Dic6—21no—
59C4×Dic6—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8D412yes—
62C3×C22⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×#97—21no—
65C3×#97—21no—
66C3⋊#95—21no—
67C3⋊#95—22no—
68C3⋊#95—22no—
69#830—21no—
70#830—21no—
71#815—22no—
72Dic3⋊C8—22no—
73Dic3⋊C8—22no—
74C8×Dic3—22no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D412yes—
77#803—21no—
78#803—21no—
79C2×C4⋊Q8—32no—
80C2×C4⋊Q8C2×D311yescharacteristic
81C2×C4×Q8—31no—
82#446—62no—
83#446—62no—
84#432—31no—
85#432—31no—
86#432—61no—
87#432C2×D311yescharacteristic
88#432C2×D311yescharacteristic
89#431—61no—
90#398—61no—
91#398—61no—
92#391—32no—
93#389—61no—
94#389—61no—
95C4×C4⋊C4—31no—
96#333—22no—
97#333—62no—
98#324—61no—
99#324—61no—
100C2×C3×Q8—22no—
101C2×C3×Q8—22no—
102C2×C3×Q8—42no—
103C2×C3×Q8—42no—
104C22×C3×C4—22no—
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×Dic3—22no—
109C22×Dic3C22⋊C411yescharacteristic
110C22×Dic3C22⋊C411yescharacteristic
111C22×Dic3C22⋊C411yescharacteristic
112C22×Dic3C22⋊C411yescharacteristic
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×Dic6—22no—
116C2×Dic6—41no—
117C2×C3×C8—22no—
118C2×C3×C8—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42#4111yescharacteristic
126C3×C42#4111yescharacteristic
127C4⋊Dic3—24no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130Dic3⋊C4—21no—
131Dic3⋊C4—21no—
132Dic3⋊C4—21no—
133Dic3⋊C4—21no—
134Dic3⋊C4—44no—
135Dic3⋊C4—44no—
136C4×Dic3—21no—
137C4×Dic3—21no—
138C4×Dic3—21no—
139C4×Dic3—21no—
140C4×Dic3—42no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—41no—
143C22×Q8—31no—
144C22×Q8C4×D312yes—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—124no—
150C4×Q8—61no—
151C4×Q8—61no—
152C4×Q8—61no—
153C4×Q8—61no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—62no—
161C2×C4⋊C4—62no—
162C2×C42—31no—
163C2×C42—31no—
164C2×C42—32no—
165C2×C42C22×D311yescharacteristic
166C4⋊C8—21no—
167C4⋊C8—32no—
168C4⋊C8—62no—
169C4⋊C8—62no—
170C4⋊C8—61no—
171C4⋊C8#6712yes—
172C22⋊C8—21no—
173C22⋊C8—21no—
174C22⋊C8—61no—
175C22⋊C8—61no—
176#97—21no—
177#97—21no—
178#97—62no—
179#97—61no—
180#97—61no—
181C4×C8—62no—
182#95—62no—
183#95—62no—
184#95—61no—
185C23×C3C2×C22⋊C411yescharacteristic
186C3×Q8—82no—
187C3×Q8—82no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—22no—
191C2×C3×C4—41no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4C4×D411yescharacteristic
196C2×C3×C4C4×D411yescharacteristic
197C2×C3×C4#12011yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12311yescharacteristic
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—24no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—42no—
210C2×Dic3—42no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3#10012yes—
214Dic6—42no—
215Dic6—42no—
216C3×C8—42no—
217C3×C8—81no—
218C3⋊C8—42no—
219C3⋊C8—81no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—62no—
225C2×Q8—61no—
226C2×Q8—61no—
227C2×Q8—121no—
228C22×C4—22no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—62no—
234C22×C4C2×C4×D311yescharacteristic
235C22×C4D3×D411yescharacteristic
236C22×C4D3×D411yescharacteristic
237C2×C8—22no—
238C2×C8—41no—
239C2×C8—62no—
240C2×C8—121no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—64no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—124no—
253C4⋊C4—124no—
254C42—61no—
255C42—61no—
256C42—61no—
257C42—61no—
258C42—122no—
259C42#23511yescharacteristic
260C42D3×D411yescharacteristic
261C42#23711yescharacteristic
262C42C2×#6711yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3#38611yescharacteristic
266C22×C3C2×#10011yescharacteristic
267C22×C3#41311yescharacteristic
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—82no—
274C3×C4#45612yes—
275Dic3—22no—
276Dic3—44no—
277Dic3—41no—
278Dic3—41no—
279Dic3—41no—
280C23D3×C22⋊C411yescharacteristic
281Q8—82no—
282Q8—82no—
283Q8—122no—
284Q8—122no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—32no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—64no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—61no—
300C2×C4—61no—
301C2×C4—122no—
302C2×C4—122no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4#67⋊C411yescharacteristic
306C2×C4#88311yescharacteristic
307C2×C4#88411yescharacteristic
308C2×C4C4×#6711yescharacteristic
309C2×C4#93711yescharacteristic
310C2×C4#93811yescharacteristic
311C8—42no—
312C8—81no—
313C8—122no—
314C8—241no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3#188611yescharacteristic
318C2×C3#190011yescharacteristic
319C2×C3#190911yescharacteristic
320C22—21no—
321C22—21no—
322C22D3×#10011yescharacteristic
323C22#474511yescharacteristic
324C22#495611yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—62no—
331C4—82no—
332C4—124no—
333C4—121no—
334C4—121no—
335C4—121no—
336C4#520512yes—
337C3#945211yescharacteristic, complemented, Sylow
338C2—21no—
339C2—21no—
340C2#7008111yescharacteristic
341C2#7034511yescharacteristic
342C2#7035011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1558C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1558D3
hypocenterC3#9452
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
337. C3#9452semidirect

Maximal subgroups

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

subgroupindex
#832392
#793142
C3×#15582
#717772
#708302
#677502
#677492
#94523