Finite Group Databasebeta

← #10692662

Subgroups — #10692662

Order 768.

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

normal subgroups only

This group has 420 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
1C3×#2980C211yesmaximal, characteristic
2#83294C211yesmaximal, characteristic
3#71813C211yesmaximal, characteristic
4#70837C211yesmaximal, characteristic, complemented
5#70815C211yesmaximal, characteristic, complemented
6#67754C211yesmaximal, characteristic, complemented
7#67752C211yesmaximal, characteristic, complemented
8#9415—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#435—22no—
12C3×#435—22no—
13C3×#432C2211yescharacteristic
14C3×#432C2211yescharacteristic
15C3×#408—22no—
16#5093C411yescharacteristic, complemented
17#5078C411yescharacteristic, complemented
18#5076C411yescharacteristic, complemented
19#5070C411yescharacteristic, complemented
20#5042—22no—
21#5018—22no—
22#4937C2211yescharacteristic
23#4926C2211yescharacteristic
24C4×#208C2211yescharacteristic
25#4537—22no—
26#4467—22no—
27#4465—22no—
28#3182—31no—
29#2980D311yescharacteristic
30#1537—31no—
31#1515—31no—
32#1514—31no—
33#1291—31no—
34#1288—31no—
35C2×C3×#41—21no—
36C2×C3×M16—22no—
37C22×C3×C8—22no—
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×#127C2×C411yescharacteristic
40C3×#124—21no—
41C3×#121—21no—
42C3×#121D412yes—
43C3×C4×D4—21no—
44C3×C4×D4D412yes—
45C2×C3×C42C2311yescharacteristic
46C2×#214—22no—
47#926—21no—
48#924—21no—
49#924—21no—
50C2×#208—21no—
51#870—21no—
52#869C2×C411yescharacteristic
53#869C2×C411yescharacteristic
54C4×Dic6—21no—
55C4×Dic6—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C22⋊C8—24no—
59C3×C22⋊C8—21no—
60C3×C22⋊C8D412yes—
61C3×#97—21no—
62C3×#97D412yes—
63#832—22no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67#830—21no—
68#810—22no—
69#807—22no—
70C4⋊(C3⋊C8)—21no—
71#803—21no—
72#803—21no—
73C4×C3⋊C8C2×C411yescharacteristic
74C4×C3⋊C8C2×C411yescharacteristic
75#533—31no—
76#532C2×D311yescharacteristic
77#500—31no—
78#492—31no—
79#487—31no—
80#486—31no—
81#468—62no—
82#440—62no—
83#435—22no—
84#435—22no—
85#432—31no—
86#432C2×D311yescharacteristic
87#432C2×D311yescharacteristic
88#408—22no—
89C4×M16—31no—
90#383—31no—
91#330—62no—
92#328—62no—
93#327—62no—
94C3×#41—81no—
95C3×#41—81no—
96C2×C3×Q8—21no—
97C2×C3×D4—22no—
98C2×C3×D4—21no—
99C2×C3×D4—21no—
100C2×C3×D4C22⋊C412yes—
101C22×C3×C4—22no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×Dic3—41no—
106C2×Dic6—21no—
107C2×Dic6—21no—
108C3×M16—82no—
109C2×C3×C8—22no—
110C2×C3×C8—22no—
111C2×C3×C8—82no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4C22⋊C412yes—
114C3×C22⋊C4—22no—
115C3×C22⋊C4—22no—
116C3×C42C22×C411yescharacteristic
117C3×C42C2×D411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42#4111yescharacteristic
120#217—41no—
121#214—44no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126Dic3⋊C4—41no—
127Dic3⋊C4—41no—
128C4×Dic3—41no—
129#208—42no—
130C2×C3⋊C8—21no—
131C2×C3⋊C8—21no—
132C2×C3⋊C8—41no—
133C2×#41—21no—
134C2×SD16—62no—
135C2×M16—22no—
136C2×M16—61no—
137C22×C8—22no—
138C4⋊Q8—31no—
139C4⋊Q8—31no—
140C4⋊Q8C4×D311yescharacteristic
141#127C4×D311yescharacteristic
142#125—61no—
143#124—21no—
144C22⋊Q8—61no—
145C22⋊Q8—61no—
146#121—21no—
147#121D1212yes—
148C4×Q8—61no—
149C4×Q8—61no—
150C4×D4—21no—
151C4×D4#6712yes—
152#117—61no—
153C2×C42C22×D311yescharacteristic
154#107—62no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—61no—
158#103—62no—
159#102—62no—
160C22⋊C8—24no—
161C22⋊C8—21no—
162C22⋊C8—61no—
163C22⋊C8#6712yes—
164#97—21no—
165#97—61no—
166#97—61no—
167#97#6712yes—
168C4×C8—31no—
169C4×C8—31no—
170#95—61no—
171#95—61no—
172#95—61no—
173C23×C3—22no—
174C23×C3#12111yescharacteristic
175C3×Q8—81no—
176C3×D4—42no—
177C3×D4—81no—
178C3×D4—81no—
179C3×D4—82no—
180C2×C3×C4—22no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4C2×C22⋊C411yescharacteristic
187C2×C3×C4C4×D411yescharacteristic
188C2×C3×C4C4×D411yescharacteristic
189C2×C3×C4#12011yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12311yescharacteristic
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—81no—
195C2×Dic3—81no—
196C2×Dic3—81no—
197Dic6—42no—
198Dic6—81no—
199C3×C8—42no—
200C3×C8—42no—
201C3⋊C8—22no—
202C3⋊C8—41no—
203C3⋊C8—81no—
204#41—81no—
205#41—81no—
206C2×Q8—21no—
207C2×Q8—61no—
208C2×Q8—61no—
209C2×D4—22no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4(C2×D3)⋊C412yes—
213C22×C4—22no—
214C22×C4—121no—
215C22×C4C2×D1211yescharacteristic
216C22×C4#23511yescharacteristic
217C22×C4C2×#6711yescharacteristic
218SD16—124no—
219M16—82no—
220M16—122no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—61no—
224C2×C8—61no—
225C2×C8—82no—
226C2×C8—121no—
227C4⋊C4—21no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—121no—
233C4⋊C4—121no—
234C4⋊C4(C2×D3)⋊C412yes—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—121no—
238C42—121no—
239C42C2×C4×D311yescharacteristic
240C42#23511yescharacteristic
241C42C2×#6711yescharacteristic
242C42C2×#6711yescharacteristic
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—42no—
247C22×C3—42no—
248C22×C3—42no—
249C22×C3#38611yescharacteristic
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—42no—
257Dic3—81no—
258Dic3—81no—
259Dic3—161no—
260C23—22no—
261C23#93011yescharacteristic
262Q8—81no—
263Q8—122no—
264Q8—241no—
265D4—42no—
266D4—81no—
267D4—81no—
268D4—82no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—121no—
276C2×C4—121no—
277C2×C4—241no—
278C2×C4—241no—
279C2×C4—241no—
280C2×C4C2×(C2×D3)⋊C411yescharacteristic
281C2×C4C4×#6711yescharacteristic
282C2×C4C4×#6711yescharacteristic
283C2×C4#92911yescharacteristic
284C2×C4#93011yescharacteristic
285C2×C4#95311yescharacteristic
286C8—42no—
287C8—42no—
288C8—62no—
289C8—121no—
290C8—241no—
291C2×C3—41no—
292C2×C3—82no—
293C2×C3#188411yescharacteristic
294C2×C3#190211yescharacteristic
295C2×C3#190411yescharacteristic
296C22—21no—
297C22—21no—
298C22—21no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22#521311yescharacteristic
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—42no—
310C4—241no—
311C4—241no—
312C4—481no—
313C3#941511yescharacteristic, complemented, Sylow
314C2—41no—
315C2—82no—
316C2#7165111yescharacteristic
317C2#7202111yescharacteristic
318C2#7202211yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2980C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#2980D3
hypocenterC3#9415
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2980

Complements

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

NH ≅ G/Ntype
4. #70837C2semidirect
5. #70815C2semidirect
6. #67754C2semidirect
7. #67752C2semidirect
16. #5093C4semidirect
17. #5078C4semidirect
18. #5076C4semidirect
19. #5070C4semidirect
313. C3#9415semidirect

Maximal subgroups

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

subgroupindex
C3×#29802
#832942
#718132
#708372
#708152
#677542
#677522
#94153