Finite Group Databasebeta

← #10693264

Subgroups — #10693264

Order 768.

436 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C3×#3112C211yesmaximal, characteristic
2#83280C211yesmaximal, characteristic
3#71809C211yesmaximal, characteristic
4#71625C211yesmaximal, characteristic, complemented
5#71622C211yesmaximal, characteristic, complemented
6#67758C211yesmaximal, characteristic, complemented
7#67757C211yesmaximal, characteristic, complemented
8#9527—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C3×#480C412yes—
12C3×#477—21no—
13C3×#460—21no—
14C3×#448C412yes—
15C3×#440—22no—
16C3×#432C2211yescharacteristic
17#5234—22no—
18#5012—21no—
19#5011—22no—
20D4×C3⋊C8—21no—
21#4926C2211yescharacteristic
22#4926C2211yescharacteristic
23#4925C2211yescharacteristic
24#4924C2211yescharacteristic
25#4547—21no—
26#4544—21no—
27#4535—21no—
28#4535—21no—
29#4528—22no—
30#3112D311yescharacteristic
31#2981—31no—
32#1557—31no—
33#1487—31no—
34#1467—31no—
35#1307—31no—
36#1287—31no—
37C2×C3×#41—21no—
38C22×C3×Q8C2×C411yescharacteristic
39C2×C3×SD16—22no—
40C3×C4⋊Q8D411yescharacteristic
41C3×#127D411yescharacteristic
42C3×#124—21no—
43C3×#121—21no—
44C3×#121C2×C412yes—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×D4—21no—
50C3×C4×D4D412yes—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4C2×C411yescharacteristic
53C2×C3×C42—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#208—22no—
56C2×#208—21no—
57C22×C3⋊C8—22no—
58C3×#107—22no—
59C3×C4⋊C8—21no—
60C3×#103—22no—
61C3×#102—22no—
62C3×C22⋊C8C2×C412yes—
63C3×#97—21no—
64#830—22no—
65#830—22no—
66#830—21no—
67#830—21no—
68#830D412yes—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
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×Q8C2×D311yescharacteristic
80#480Dic312yes—
81#477—21no—
82#460—21no—
83#448Dic312yes—
84#440—22no—
85#436—61no—
86#435—62no—
87C8×D4—61no—
88#432—31no—
89#432—31no—
90#432C2×D311yescharacteristic
91#431—31no—
92#423—31no—
93#408—62no—
94#332—61no—
95#331—61no—
96Q8⋊C8—61no—
97Q8⋊C8—61no—
98#324—62no—
99C3×#41—82no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8—42no—
104C2×C3×D4—22no—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4C22⋊C412yes—
108C22×C3×C4—21no—
109C22×C3×C4—22no—
110C22×C3×C4—21no—
111C22×C3×C4C22×C411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C3×SD16—44no—
115C2×C3×C8—22no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4C22⋊C412yes—
122C3×C22⋊C4—22no—
123C3×C22⋊C4—22no—
124C3×C42—41no—
125C3×C42—41no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130#208—82no—
131#208—81no—
132C2×C3⋊C8—21no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—22no—
135C2×C3⋊C8—41no—
136C2×C3⋊C8—41no—
137C2×C3⋊C8—82no—
138C2×#41—21no—
139C22×Q8C2×Dic311yescharacteristic
140C2×SD16—22no—
141C2×M16—62no—
142C2×M16—61no—
143C22×C8—62no—
144C4⋊Q8#6711yescharacteristic
145#127#6711yescharacteristic
146#124—21no—
147#121—21no—
148#121C2×Dic312yes—
149C4×Q8—21no—
150C4×Q8—21no—
151C4×Q8—21no—
152C4×Q8—21no—
153C4×D4—21no—
154C4×D4#6712yes—
155C2×C4⋊C4—21no—
156C2×C4⋊C4C2×Dic311yescharacteristic
157C2×C42—21no—
158C2×C42C22×D311yescharacteristic
159#107—22no—
160C4⋊C8—21no—
161C4⋊C8—61no—
162C4⋊C8—61no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166#103—22no—
167#102—22no—
168C22⋊C8—32no—
169C22⋊C8—62no—
170C22⋊C8—62no—
171C22⋊C8—61no—
172C22⋊C8—61no—
173C22⋊C8C2×Dic312yes—
174#97—21no—
175#97—61no—
176#97—61no—
177C4×C8—31no—
178C4×C8—31no—
179C23×C3—22no—
180C23×C3C4×D411yescharacteristic
181C3×Q8—22no—
182C3×Q8—22no—
183C3×Q8—41no—
184C3×Q8—41no—
185C3×Q8—81no—
186C3×D4—42no—
187C3×D4—41no—
188C3×D4—41no—
189C3×D4—81no—
190C3×D4—82no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—81no—
203C2×C3×C4C2×C22⋊C411yescharacteristic
204C2×C3×C4C4×D411yescharacteristic
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12311yescharacteristic
209C3×C8—42no—
210C3⋊C8—41no—
211C3⋊C8—42no—
212C3⋊C8—41no—
213C3⋊C8—81no—
214C3⋊C8—81no—
215#41—82no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—21no—
219C2×Q8—42no—
220C2×D4—22no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4#21712yes—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4D3×D411yescharacteristic
228C22×C4#23711yescharacteristic
229C22×C4C22×Dic311yescharacteristic
230SD16—44no—
231M16—242no—
232M16—241no—
233C2×C8—22no—
234C2×C8—61no—
235C2×C8—61no—
236C2×C8—62no—
237C2×C8—121no—
238C2×C8—121no—
239C2×C8—242no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4#21712yes—
246C22⋊C4—22no—
247C22⋊C4—22no—
248C42—41no—
249C42—41no—
250C42D3×D411yescharacteristic
251C42#23711yescharacteristic
252C42C2×#6711yescharacteristic
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—42no—
268C3×C4—81no—
269C3×C4#47111yescharacteristic
270C3×C4#47311yescharacteristic
271C23—22no—
272C23D4×Dic311yescharacteristic
273Q8—22no—
274Q8—22no—
275Q8—41no—
276Q8—41no—
277Q8—81no—
278D4—42no—
279D4—41no—
280D4—41no—
281D4—81no—
282D4—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4D4×Dic311yescharacteristic
296C2×C4#93511yescharacteristic
297C2×C4#93711yescharacteristic
298C2×C4#93911yescharacteristic
299C2×C4#93911yescharacteristic
300C2×C4C2×#21711yescharacteristic
301C8—42no—
302C8—121no—
303C8—122no—
304C8—121no—
305C8—241no—
306C8—241no—
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#522011yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—42no—
325C4—42no—
326C4—81no—
327C4#520311yescharacteristic
328C4#520411yescharacteristic
329C3#952711yescharacteristic, complemented, Sylow
330C2—41no—
331C2—82no—
332C2#7171411yescharacteristic
333C2#7208911yescharacteristic
334C2#7210311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3112C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3112D3
hypocenterC3#9527
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3112

Complements

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

NH ≅ G/Ntype
4. #71625C2semidirect
5. #71622C2semidirect
6. #67758C2semidirect
7. #67757C2semidirect
329. C3#9527semidirect

Maximal subgroups

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

subgroupindex
C3×#31122
#832802
#718092
#716252
#716222
#677582
#677572
#95273