Finite Group Databasebeta

← #10692870

Subgroups — #10692870

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#79314C211yesmaximal, characteristic
2#73711C211yesmaximal, characteristic
3#70830C211yesmaximal, characteristic
4#70824C211yesmaximal, characteristic
5#70824C211yesmaximal, characteristic
6C3×#1285C211yesmaximal, characteristic
7#67621C211yesmaximal, characteristic
8#9452—31nomaximal, Sylow
9C2×#869C412yes—
10C2×#869C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#4984—21no—
16#4984—21no—
17#4929C412yes—
18#4926C2211yescharacteristic
19#4926C2211yescharacteristic
20#4828—22no—
21#4719—21no—
22#4718—21no—
23#4701—22no—
24#4661—21no—
25#4661—21no—
26#4481—22no—
27#4481—22no—
28#4469—21no—
29#4469—21no—
30#2983—31no—
31#2546—31no—
32#1558—31no—
33#1558—31no—
34#1467—31no—
35#1285—31no—
36#1285D311yescharacteristic
37C22×Dic6C2×C411yescharacteristic
38C22×Dic6C2×C412yes—
39C2×C3×C4⋊C4—21no—
40C2×C3×C42C2×C412yes—
41C2×C3×C42C2311yescharacteristic
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—22no—
45C2×C4⋊Dic3—22no—
46C2×C4⋊Dic3C2×C411yescharacteristic
47C2×Dic3⋊C4D411yescharacteristic
48C2×Dic3⋊C4D411yescharacteristic
49C2×C4×Dic3D411yescharacteristic
50C2×C4×Dic3D411yescharacteristic
51#869—21no—
52#869—21no—
53#869—21no—
54#869—21no—
55#869—44no—
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⋊Q8—31no—
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⋊C4C2×D311yescharacteristic
96#333—62no—
97#333—62no—
98#324—61no—
99#324—61no—
100C22×C3×C4—22no—
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×Dic3—22no—
105C22×Dic3C22⋊C411yescharacteristic
106C22×Dic3C22⋊C411yescharacteristic
107C22×Dic3C22⋊C411yescharacteristic
108C22×Dic3C22⋊C411yescharacteristic
109C2×Dic6—21no—
110C2×Dic6—21no—
111C2×Dic6—22no—
112C2×Dic6—22no—
113C2×Dic6—22no—
114C2×Dic6—41no—
115C2×Dic6—42no—
116C2×Dic6—42no—
117C2×C3×C8—22no—
118C2×C3×C8—41no—
119C3×C4⋊C4—24no—
120C3×C42—42no—
121C3×C42C2×D411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42#4111yescharacteristic
124C3×C42#4111yescharacteristic
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—44no—
128C4⋊Dic3—41no—
129C4⋊Dic3—41no—
130C4⋊Dic3—41no—
131C4⋊Dic3—41no—
132C4⋊Dic3—44no—
133Dic3⋊C4—21no—
134Dic3⋊C4—21no—
135Dic3⋊C4—21no—
136Dic3⋊C4—21no—
137C4×Dic3—21no—
138C4×Dic3—21no—
139C4×Dic3—21no—
140C4×Dic3—21no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—41no—
143C22×Q8—31no—
144C22×Q8—32no—
145C4⋊Q8—61no—
146C4⋊Q8—61no—
147C4⋊Q8—61no—
148C4⋊Q8—61no—
149C4⋊Q8—124no—
150C4×Q8—61no—
151C4×Q8—61no—
152C4×Q8—61no—
153C4×Q8—61no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—62no—
159C2×C4⋊C4—62no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—61no—
162C2×C42—31no—
163C2×C42—31no—
164C2×C42C4×D312yes—
165C2×C42C22×D311yescharacteristic
166C4⋊C8—21no—
167C4⋊C8—32no—
168C4⋊C8—62no—
169C4⋊C8—62no—
170C4⋊C8—61no—
171C4⋊C8D1212yes—
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—21no—
183#95—62no—
184#95—62no—
185C23×C3C2×C22⋊C411yescharacteristic
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—22no—
189C2×C3×C4—24no—
190C2×C3×C4—41no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4C4×D411yescharacteristic
194C2×C3×C4C4×D411yescharacteristic
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12311yescharacteristic
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—42no—
209C2×Dic3—42no—
210C2×Dic3—42no—
211C2×Dic3#10012yes—
212Dic6—42no—
213Dic6—42no—
214Dic6—82no—
215Dic6—82no—
216C3×C8—42no—
217C3×C8—81no—
218C3⋊C8—42no—
219C3⋊C8—81no—
220C2×Q8—62no—
221C2×Q8—61no—
222C2×Q8—61no—
223C2×Q8—62no—
224C2×Q8—62no—
225C2×Q8—121no—
226C2×Q8—122no—
227C2×Q8—122no—
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—24no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—124no—
249C4⋊C4—124no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C4⋊C4—121no—
254C42—42no—
255C42—61no—
256C42—61no—
257C42—61no—
258C42—61no—
259C42C2×D1211yescharacteristic
260C42#23511yescharacteristic
261C42D3×D411yescharacteristic
262C42#23911yescharacteristic
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—44no—
274C3×C4#45612yes—
275Dic3—22no—
276Dic3—41no—
277Dic3—41no—
278Dic3—41no—
279Dic3—82no—
280C23D3×C22⋊C411yescharacteristic
281Q8—122no—
282Q8—122no—
283Q8—242no—
284Q8—242no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—32no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—61no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—122no—
303C2×C4—122no—
304C2×C4—122no—
305C2×C4C4×D1211yescharacteristic
306C2×C4#88211yescharacteristic
307C2×C4#88411yescharacteristic
308C2×C4#89311yescharacteristic
309C2×C4#89411yescharacteristic
310C2×C4#89511yescharacteristic
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—
322C22#466511yescharacteristic
323C22D3×#10011yescharacteristic
324C22#475111yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—44no—
331C4—62no—
332C4—121no—
333C4—121no—
334C4—121no—
335C4—242no—
336C4#482612yes—
337C3#945211yescharacteristic, complemented, Sylow
338C2—21no—
339C2—21no—
340C2#6871211yescharacteristic
341C2#6876511yescharacteristic
342C2#6899211yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1285C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1285D3
hypocenterC3#9452
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

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
#793142
#737112
#708302
#708242
C3×#12852
#676212
#94523