Finite Group Databasebeta

← #10693260

Subgroups — #10693260

Order 768.

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

normal subgroups only

This group has 436 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#83316C211yesmaximal, characteristic
2#83280C211yesmaximal, characteristic
3C3×#1557C211yesmaximal, characteristic
4#70830C211yesmaximal, characteristic, complemented
5#70500C211yesmaximal, characteristic, complemented
6#67756C211yesmaximal, characteristic, complemented
7#67716C211yesmaximal, characteristic, complemented
8#9527—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#423C2211yescharacteristic
13#5234—22no—
14#5091—21no—
15#5075—21no—
16#5040C412yes—
17#5033C412yes—
18#5018—22no—
19#5012—21no—
20#5011—22no—
21D4×C3⋊C8—21no—
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24#4925C2211yescharacteristic
25C3×#332—21no—
26C3×#331—21no—
27#4484—21no—
28#4484—21no—
29#4469—22no—
30#3112—31no—
31#2981—31no—
32#1557D311yescharacteristic
33#1487—31no—
34#1467—31no—
35#1307—31no—
36#1287—31no—
37C2×C3×#41—21no—
38C22×Dic6C2×C411yescharacteristic
39C2×C3×M16—21no—
40C3×C4⋊Q8D411yescharacteristic
41C3×#127D411yescharacteristic
42C3×#124—21no—
43C3×#121—21no—
44C3×#121C2×C412yes—
45C3×C4×D4—21no—
46C3×C4×D4D412yes—
47C2×C3×C42C2311yescharacteristic
48C2×#214—22no—
49C2×C4⋊Dic3C2×C411yescharacteristic
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3—21no—
52C2×#208—22no—
53C22×C3⋊C8—22no—
54C4×Dic6—21no—
55C4×Dic6—21no—
56C4×Dic6—21no—
57C4×Dic6—21no—
58C3×C4⋊C8—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×#97—21no—
63#832—22no—
64#830—22no—
65#830—22no—
66#830—21no—
67#830C2×C412yes—
68#830D412yes—
69#810—22no—
70#807—22no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74#803—21no—
75#803—21no—
76C4×C3⋊C8D411yescharacteristic
77C4×C3⋊C8D411yescharacteristic
78#532C2×D311yescharacteristic
79C2×C4×Q8—31no—
80#480—32no—
81#477—61no—
82#460—61no—
83#448—32no—
84#440—62no—
85#436—61no—
86#435—62no—
87C8×D4—61no—
88#432—31no—
89#432—31no—
90#432C2×D311yescharacteristic
91#431—31no—
92#423C2×D311yescharacteristic
93#408—62no—
94#332—21no—
95#331—21no—
96Q8⋊C8—61no—
97Q8⋊C8—61no—
98#324—62no—
99C3×#41—82no—
100C2×C3×Q8—21no—
101C2×C3×D4—22no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4C22⋊C412yes—
105C22×C3×C4—22no—
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×Dic3—21no—
110C22×Dic3—21no—
111C2×Dic6—21no—
112C2×Dic6—21no—
113C2×Dic6—42no—
114C3×M16—81no—
115C2×C3×C8—41no—
116C2×C3×C8—41no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4C22⋊C412yes—
119C3×C22⋊C4—22no—
120C3×C22⋊C4—22no—
121C3×C42C2×D411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42#4111yescharacteristic
125#214—44no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128Dic3⋊C4—41no—
129Dic3⋊C4—41no—
130C4×Dic3—41no—
131C4×Dic3—41no—
132#208—82no—
133C2×C3⋊C8—22no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—21no—
136C2×C3⋊C8—22no—
137C2×C3⋊C8—82no—
138C2×#41—21no—
139C22×Q8—31no—
140C2×SD16—62no—
141C2×M16—21no—
142C2×M16—62no—
143C22×C8—62no—
144C4⋊Q8#6711yescharacteristic
145#127#6711yescharacteristic
146#124—21no—
147#121—21no—
148#121C4×D312yes—
149C4×Q8—61no—
150C4×Q8—61no—
151C4×Q8—61no—
152C4×Q8—61no—
153C4×D4—21no—
154C4×D4D1212yes—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—61no—
157C2×C42—61no—
158C2×C42C22×D311yescharacteristic
159#107—62no—
160C4⋊C8—21no—
161C4⋊C8—21no—
162C4⋊C8—21no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166#103—62no—
167#102—62no—
168C22⋊C8—21no—
169C22⋊C8—32no—
170C22⋊C8—32no—
171C22⋊C8—62no—
172C22⋊C8—62no—
173C22⋊C8—61no—
174#97—21no—
175#97—61no—
176#97—61no—
177C4×C8—31no—
178C4×C8—31no—
179C23×C3—22no—
180C23×C3C4×D411yescharacteristic
181C3×Q8—81no—
182C3×D4—42no—
183C3×D4—41no—
184C3×D4—41no—
185C3×D4—81no—
186C3×D4—82no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—22no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4C2×C22⋊C411yescharacteristic
195C2×C3×C4C4×D411yescharacteristic
196C2×C3×C4#12011yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12311yescharacteristic
200C2×Dic3—22no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—81no—
205Dic6—22no—
206Dic6—22no—
207Dic6—41no—
208Dic6—41no—
209C3×C8—81no—
210C3×C8—81no—
211C3⋊C8—41no—
212C3⋊C8—42no—
213C3⋊C8—41no—
214C3⋊C8—42no—
215#41—82no—
216C2×Q8—21no—
217C2×Q8—61no—
218C2×Q8—61no—
219C2×Q8—122no—
220C2×D4—22no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4(C2×D3)⋊C412yes—
224C22×C4—22no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4C2×C4×D311yescharacteristic
228C22×C4D3×D411yescharacteristic
229C22×C4#23711yescharacteristic
230SD16—124no—
231M16—81no—
232M16—242no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—61no—
236C2×C8—61no—
237C2×C8—62no—
238C2×C8—62no—
239C2×C8—242no—
240C4⋊C4—21no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C4⋊C4(C2×D3)⋊C412yes—
246C22⋊C4—22no—
247C22⋊C4—22no—
248C42—121no—
249C42—121no—
250C42C2×D1211yescharacteristic
251C42D3×D411yescharacteristic
252C42#23711yescharacteristic
253C42C2×#6711yescharacteristic
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—42no—
258C22×C3—42no—
259C22×C3—42no—
260C22×C3#38611yescharacteristic
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—42no—
267C3×C4#47111yescharacteristic
268C3×C4#47311yescharacteristic
269Dic3—42no—
270Dic3—81no—
271C23—22no—
272C23#67⋊C411yescharacteristic
273Q8—62no—
274Q8—62no—
275Q8—81no—
276Q8—121no—
277Q8—121no—
278D4—42no—
279D4—41no—
280D4—41no—
281D4—81no—
282D4—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—62no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—241no—
295C2×C4#67⋊C411yescharacteristic
296C2×C4#88211yescharacteristic
297C2×C4#88611yescharacteristic
298C2×C4C2×(C2×D3)⋊C411yescharacteristic
299C2×C4#93911yescharacteristic
300C2×C4#93911yescharacteristic
301C8—81no—
302C8—81no—
303C8—121no—
304C8—122no—
305C8—121no—
306C8—122no—
307C2×C3—41no—
308C2×C3—82no—
309C2×C3#189111yescharacteristic
310C2×C3#190411yescharacteristic
311C2×C3#190811yescharacteristic
312C22—21no—
313C22—21no—
314C22—21no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22#495811yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—42no—
325C4—122no—
326C4—241no—
327C4#520011yescharacteristic
328C4#520411yescharacteristic
329C3#952711yescharacteristic, complemented, Sylow
330C2—41no—
331C2—82no—
332C2#7013611yescharacteristic
333C2#7030411yescharacteristic
334C2#7031411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1557C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1557D3
hypocenterC3#9527
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1557

Complements

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

NH ≅ G/Ntype
4. #70830C2semidirect
5. #70500C2semidirect
6. #67756C2semidirect
7. #67716C2semidirect
329. C3#9527semidirect

Maximal subgroups

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

subgroupindex
#833162
#832802
C3×#15572
#708302
#705002
#677562
#677162
#95273