Finite Group Databasebeta

← #10692878

Subgroups — #10692878

Order 768.

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

normal subgroups only

This group has 448 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#83243C211yesmaximal, characteristic
2#79314C211yesmaximal, characteristic
3C3×#1556C211yesmaximal, characteristic
4#71776C211yesmaximal, characteristic
5#70830C211yesmaximal, characteristic
6#67750C211yesmaximal, characteristic
7#67749C211yesmaximal, characteristic
8#9454—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13#5230—21no—
14#5230—21no—
15#4975C412yes—
16#4973C412yes—
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×#332—22no—
27#4547—22no—
28#4469—21no—
29#4469—21no—
30#2983—31no—
31#2567—31no—
32#1556—31no—
33#1556D311yescharacteristic
34#1467—31no—
35#1285—31no—
36#1285—31no—
37C22×C3×Q8—21no—
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⋊C4C2×C412yes—
45C2×C3×C42C2311yescharacteristic
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3C2×C411yescharacteristic
48C2×Dic3⋊C4—22no—
49C2×Dic3⋊C4—22no—
50C2×Dic3⋊C4D411yescharacteristic
51C2×Dic3⋊C4D411yescharacteristic
52C2×C4×Dic3C2×C412yes—
53C2×C4×Dic3D411yescharacteristic
54C2×C4×Dic3D411yescharacteristic
55C4×Dic6—21no—
56C4×Dic6—21no—
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8D412yes—
61C3×C22⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×#97—21no—
64C3×#97—21no—
65C3⋊#95—21no—
66C3⋊#95—22no—
67C3⋊#95—22no—
68#830—21no—
69#830—21no—
70#815—22no—
71Dic3⋊C8—22no—
72Dic3⋊C8—22no—
73C8×Dic3—22no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)D412yes—
76#803—21no—
77#803—21no—
78C2×C4⋊Q8C2×D311yescharacteristic
79C2×C4×Q8—31no—
80#446—62no—
81#446—62no—
82#432—31no—
83#432—31no—
84#432—61no—
85#432C2×D311yescharacteristic
86#432C2×D311yescharacteristic
87#431—61no—
88#395—61no—
89#395—61no—
90#391—32no—
91#389—61no—
92#389—61no—
93#384—32no—
94C4×C4⋊C4—31no—
95#332—22no—
96#332—62no—
97#324—61no—
98#324—61no—
99C2×C3×Q8—41no—
100C2×C3×Q8—41no—
101C2×C3×Q8—82no—
102C22×C3×C4—22no—
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×Dic3—22no—
107C22×Dic3C22⋊C411yescharacteristic
108C22×Dic3C22⋊C411yescharacteristic
109C22×Dic3C22⋊C411yescharacteristic
110C22×Dic3C22⋊C411yescharacteristic
111C2×Dic6—21no—
112C2×Dic6—21no—
113C2×Dic6—22no—
114C2×Dic6—41no—
115C2×C3×C8—22no—
116C2×C3×C8—41no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C42C2×D411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42#4111yescharacteristic
124C3×C42#4111yescharacteristic
125C4⋊Dic3—24no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128Dic3⋊C4—21no—
129Dic3⋊C4—21no—
130Dic3⋊C4—21no—
131Dic3⋊C4—21no—
132Dic3⋊C4—82no—
133Dic3⋊C4—82no—
134C4×Dic3—21no—
135C4×Dic3—21no—
136C4×Dic3—21no—
137C4×Dic3—21no—
138C4×Dic3—42no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—41no—
141C22×Q8—21no—
142C22×Q8—31no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4×Q8—61no—
148C4×Q8—61no—
149C4×Q8—61no—
150C4×Q8—61no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4—62no—
157C2×C4⋊C4—62no—
158C2×C4⋊C4C4×D312yes—
159C2×C42—31no—
160C2×C42—31no—
161C2×C42—32no—
162C2×C42C22×D311yescharacteristic
163C4⋊C8—21no—
164C4⋊C8—32no—
165C4⋊C8—62no—
166C4⋊C8—62no—
167C4⋊C8—61no—
168C4⋊C8#6712yes—
169C22⋊C8—21no—
170C22⋊C8—21no—
171C22⋊C8—61no—
172C22⋊C8—61no—
173#97—21no—
174#97—21no—
175#97—62no—
176#97—61no—
177#97—61no—
178C4×C8—62no—
179#95—62no—
180#95—62no—
181#95—61no—
182C23×C3C2×C22⋊C411yescharacteristic
183C3×Q8—82no—
184C3×Q8—82no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—22no—
188C2×C3×C4—41no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4C4×D411yescharacteristic
193C2×C3×C4C4×D411yescharacteristic
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12311yescharacteristic
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—24no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—42no—
207C2×Dic3—42no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3#10112yes—
211Dic6—42no—
212Dic6—42no—
213C3×C8—42no—
214C3×C8—81no—
215C3⋊C8—42no—
216C3⋊C8—81no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—62no—
220C2×Q8—61no—
221C2×Q8—61no—
222C2×Q8—82no—
223C2×Q8—121no—
224C22×C4—22no—
225C22×C4—31no—
226C22×C4—31no—
227C22×C4—31no—
228C22×C4—31no—
229C22×C4—62no—
230C22×C4C2×C4×D311yescharacteristic
231C22×C4D3×D411yescharacteristic
232C22×C4D3×D411yescharacteristic
233C2×C8—22no—
234C2×C8—41no—
235C2×C8—62no—
236C2×C8—121no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—64no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—242no—
249C4⋊C4—242no—
250C42—61no—
251C42—61no—
252C42—61no—
253C42—61no—
254C42—122no—
255C42#23511yescharacteristic
256C42D3×D411yescharacteristic
257C42#23711yescharacteristic
258C42C2×#6711yescharacteristic
259C22×C3—21no—
260C22×C3—21no—
261C22×C3#38611yescharacteristic
262C22×C3C2×#10111yescharacteristic
263C22×C3#41311yescharacteristic
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—82no—
270C3×C4#45612yes—
271Dic3—22no—
272Dic3—44no—
273Dic3—41no—
274Dic3—41no—
275Dic3—41no—
276C23D3×C22⋊C411yescharacteristic
277Q8—82no—
278Q8—82no—
279Q8—122no—
280Q8—122no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—32no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—64no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—122no—
298C2×C4—122no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4#67⋊C411yescharacteristic
302C2×C4#88311yescharacteristic
303C2×C4#88411yescharacteristic
304C2×C4C4×#6711yescharacteristic
305C2×C4#93711yescharacteristic
306C2×C4#93811yescharacteristic
307C8—42no—
308C8—81no—
309C8—122no—
310C8—241no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3#188611yescharacteristic
314C2×C3#190111yescharacteristic
315C2×C3#191011yescharacteristic
316C22—21no—
317C22—21no—
318C22D3×#10111yescharacteristic
319C22#475111yescharacteristic
320C22#495611yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—62no—
327C4—82no—
328C4—124no—
329C4—121no—
330C4—121no—
331C4—121no—
332C4#520512yes—
333C3#945411yescharacteristic, complemented, Sylow
334C2—21no—
335C2—21no—
336C2#7011511yescharacteristic
337C2#7033011yescharacteristic
338C2#7034511yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1556C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1556D3
hypocenterC3#9454
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
333. C3#9454semidirect

Maximal subgroups

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

subgroupindex
#832432
#793142
C3×#15562
#717762
#708302
#677502
#677492
#94543