Finite Group Databasebeta

← #10694756

Subgroups — #10694756

Order 768.

358 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 97 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×#5065C211yesmaximal, characteristic
2C3×#1528C211yesmaximal, characteristic
3#71737C211yesmaximal, characteristic
4#70831C211yesmaximal, characteristic
5#70794C211yesmaximal, characteristic
6#69863C211yesmaximal, characteristic
7#69847C211yesmaximal, characteristic
8#9792—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13#5065—21no—
14#5065—21no—
15#5065—21no—
16#5065—21no—
17C2×#808C411yescharacteristic
18C2×#808C411yescharacteristic
19C2×#807C411yescharacteristic
20C2×#807C411yescharacteristic
21#4926C2211yescharacteristic
22#4925C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24C3×#330—22no—
25#4528—22no—
26#4480—21no—
27#4479—21no—
28#4469—22no—
29#4465—22no—
30#4454—21no—
31#4454—21no—
32C2×#499—31no—
33#1568—31no—
34#1528—31no—
35#1528D311yescharacteristic
36#1470—31no—
37#1470—31no—
38#1451—31no—
39C2×C3×M16—21no—
40C3×#125—21no—
41C3×#125—21no—
42C3×#125—21no—
43C3×#125—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4C2×C411yescharacteristic
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C42C2311yescharacteristic
49C2×C4⋊Dic3C2×C411yescharacteristic
50C2×C4⋊Dic3C2×C411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C22×C3⋊C8C2×C411yescharacteristic
54C22×C3⋊C8C2×C411yescharacteristic
55#870—21no—
56#870—21no—
57#870—21no—
58#870—21no—
59C3×C4⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×#97—21no—
62C3×#97D412yes—
63C3×C4×C8—21no—
64#830—21no—
65#830D412yes—
66#808—21no—
67#808—21no—
68#808—21no—
69#808—21no—
70#807—21no—
71#807—21no—
72#807—21no—
73#807—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76#803—21no—
77C4×C3⋊C8—21no—
78C4×C3⋊C8D411yescharacteristic
79C4×C3⋊C8D411yescharacteristic
80C4×C3⋊C8D411yescharacteristic
81C4×C3⋊C8D411yescharacteristic
82C2×#125—31no—
83C2×#125C2×D311yescharacteristic
84#499—61no—
85#499—61no—
86#499—61no—
87#499—61no—
88#432—31no—
89#432C2×D311yescharacteristic
90#431—31no—
91C2×#107—31no—
92C2×#107—31no—
93C2×#106—31no—
94C2×#106—31no—
95C4×M16C2×D311yescharacteristic
96C2×C4×C8—31no—
97#335—61no—
98#334—61no—
99#330—22no—
100#330—62no—
101#324—62no—
102#324—62no—
103#322—61no—
104#322—61no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4C22×C411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×Dic3—21no—
111C22×Dic3—21no—
112C3×M16—42no—
113C2×C3×C8—22no—
114C2×C3×C8—41no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42#4111yescharacteristic
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131Dic3⋊C4—41no—
132Dic3⋊C4—41no—
133Dic3⋊C4—41no—
134Dic3⋊C4—41no—
135C2×C3⋊C8—22no—
136C2×C3⋊C8—22no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8C22⋊C411yescharacteristic
140C2×C3⋊C8C22⋊C411yescharacteristic
141C2×C3⋊C8C22⋊C411yescharacteristic
142C2×C3⋊C8C22⋊C411yescharacteristic
143C2×M16—21no—
144C22×C8—31no—
145C22×C8—31no—
146#125—21no—
147#125—21no—
148#125—21no—
149#125—21no—
150#125—61no—
151#125—61no—
152#125—61no—
153#125—61no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4C4×D311yescharacteristic
161C2×C4⋊C4C4×D311yescharacteristic
162C2×C42C22×D311yescharacteristic
163#107—61no—
164#107—61no—
165#107—61no—
166#107—61no—
167#106—61no—
168#106—61no—
169#106—61no—
170#106—61no—
171C4⋊C8—21no—
172C4⋊C8—61no—
173C4⋊C8—61no—
174C22⋊C8—21no—
175C22⋊C8—32no—
176C22⋊C8—61no—
177#97—21no—
178#97—61no—
179#97#6712yes—
180C4×C8—21no—
181C4×C8—31no—
182C4×C8—31no—
183C4×C8—31no—
184C4×C8—31no—
185C4×C8—61no—
186C23×C3C4×D411yescharacteristic
187C2×C3×C4—22no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—42no—
194C2×C3×C4—81no—
195C2×C3×C4C2×C22⋊C411yescharacteristic
196C2×C3×C4C4×D411yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12411yescharacteristic
200C2×C3×C4#12711yescharacteristic
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—42no—
204C2×Dic3—81no—
205C3×C8—42no—
206C3×C8—81no—
207C3⋊C8—21no—
208C3⋊C8—21no—
209C3⋊C8—42no—
210C3⋊C8—81no—
211C3⋊C8#10112yes—
212C3⋊C8#10112yes—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4C2×C4×D311yescharacteristic
218C22×C4D3×D411yescharacteristic
219C22×C4#23711yescharacteristic
220M16—42no—
221C2×C8—22no—
222C2×C8—31no—
223C2×C8—31no—
224C2×C8—31no—
225C2×C8—31no—
226C2×C8—41no—
227C2×C8—62no—
228C2×C8—62no—
229C2×C8—62no—
230C2×C8—121no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C4⋊C4—121no—
247C42#23511yescharacteristic
248C42D3×D411yescharacteristic
249C42D3×D411yescharacteristic
250C42C2×#6711yescharacteristic
251C22×C3—21no—
252C22×C3—21no—
253C22×C3#39011yescharacteristic
254C22×C3#44412yes—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—81no—
261C3×C4—81no—
262C3×C4C2×#10111yescharacteristic
263C3×C4C2×#10111yescharacteristic
264Dic3—81no—
265Dic3—81no—
266C23#67⋊C411yescharacteristic
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—81no—
275C2×C4—121no—
276C2×C4—121no—
277C2×C4—122no—
278C2×C4—241no—
279C2×C4D3×C22⋊C411yescharacteristic
280C2×C4#88411yescharacteristic
281C2×C4#88511yescharacteristic
282C2×C4C4×#6711yescharacteristic
283C2×C4#93911yescharacteristic
284C2×C4#94011yescharacteristic
285C8—32no—
286C8—32no—
287C8—42no—
288C8—61no—
289C8—61no—
290C8—81no—
291C8—122no—
292C8—241no—
293C2×C3—22no—
294C2×C3#196611yescharacteristic
295C2×C3#197511yescharacteristic
296C2×C3#197611yescharacteristic
297C22—21no—
298C22—21no—
299C22#482312yes—
300C22#495711yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—81no—
307C4—81no—
308C4—241no—
309C4—241no—
310C4D3×#10111yescharacteristic
311C4D3×#10111yescharacteristic
312C3#979211yescharacteristic, complemented, Sylow
313C2—22no—
314C2#7009811yescharacteristic
315C2#7037411yescharacteristic
316C2#7039711yescharacteristic
subgroupG/H
centerC22#4957
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1528C2
socleC22×C3#390
prime radicalC23×C3C4×D4
cosocleC2×C3×C42C23
hypercenter#1528D3
hypocenterC3#9792
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1528

Complements

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

NH ≅ G/Ntype
312. C3#9792semidirect

Maximal subgroups

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

subgroupindex
C2×#50652
C3×#15282
#717372
#708312
#707942
#698632
#698472
#97923