Finite Group Databasebeta

← #10692753

Subgroups — #10692753

Order 768.

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

normal subgroups only

This group has 444 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×#3141C211yesmaximal, characteristic
2#83231C211yesmaximal, characteristic, complemented
3#71813C211yesmaximal, characteristic
4#71813C211yesmaximal, characteristic
5#71776C211yesmaximal, characteristic, complemented
6#67666C211yesmaximal, characteristic, complemented
7#67666C211yesmaximal, characteristic, complemented
8#9430—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C3×#489—22no—
12C3×#487C412yescomplemented
13C3×#486C412yescomplemented
14C3×#417—22no—
15C3×C4×M16C2211yescharacteristic
16#5229—22no—
17#4980—22no—
18#4980—22no—
19#4937C2211yescharacteristic
20#4937C2211yescharacteristic
21#4926C2211yescharacteristic
22#4926C2211yescharacteristic
23#4547—22no—
24#4537—22no—
25#4537—22no—
26#3141D311yescharacteristic
27#2381—31no—
28#1556—31no—
29#1514—31no—
30#1514—31no—
31#1288—31no—
32#1288—31no—
33C2×C3×#41—21no—
34C22×C3×Q8—21no—
35C2×C3×M16—21no—
36C3×C4⋊Q8—21no—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×C4⋊Q8C2×C412yes—
40C3×C4⋊Q8D412yes—
41C3×#127C2×C411yescharacteristic
42C3×#124—21no—
43C3×#121—21no—
44C3×#121—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D412yes—
49C2×C3×C42C2311yescharacteristic
50C2×C4⋊Dic3—22no—
51C2×C4×Dic3—22no—
52C3×#103—24no—
53C3×#102—24no—
54C3×#97—21no—
55C3×C4×C8C2×C412yes—
56C3⋊#95—24no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95D412yes—
62C3⋊#95D412yes—
63#830—21no—
64#830—21no—
65C4⋊(C3⋊C8)—21no—
66C4⋊(C3⋊C8)—21no—
67#803—21no—
68#803—21no—
69#532C2×D311yescharacteristic
70C2×C4⋊Q8C2×D311yescharacteristic
71#489—22no—
72#487Dic312yescomplemented
73#486Dic312yescomplemented
74#432—31no—
75#432—31no—
76#417—22no—
77C4×M16C2×D311yescharacteristic
78#391—62no—
79#387—62no—
80#387—62no—
81#383—31no—
82#383—31no—
83#332—62no—
84#327—62no—
85#327—62no—
86C3×#41—44no—
87C2×C3×Q8—22no—
88C2×C3×Q8—22no—
89C2×C3×Q8—21no—
90C2×C3×Q8—81no—
91C2×C3×Q8—81no—
92C2×C3×D4—21no—
93C2×C3×D4—21no—
94C2×C3×D4—21no—
95C2×C3×D4—41no—
96C22×C3×C4—22no—
97C22×C3×C4—41no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×Dic3—22no—
102C22×Dic3—22no—
103C3×M16—42no—
104C2×C3×C8—22no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—22no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4C22⋊C414yes—
111C3×C22⋊C4—41no—
112C3×C22⋊C4—41no—
113C3×C42C22×C411yescharacteristic
114C3×C42C2×D411yescharacteristic
115C3×C42#4111yescharacteristic
116C3×C42#4111yescharacteristic
117C4⋊Dic3—82no—
118C4×Dic3—82no—
119C2×C3⋊C8—41no—
120C2×C3⋊C8—41no—
121C2×#41—21no—
122C22×Q8—21no—
123C2×M16—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8C2×Dic311yescharacteristic
127C4⋊Q8C2×Dic312yes—
128C4⋊Q8#6712yes—
129#127C2×Dic311yescharacteristic
130#124—21no—
131#121—21no—
132#121—21no—
133C4×D4—21no—
134C4×D4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—62no—
137C2×C4⋊C4#6712yes—
138C2×C42—62no—
139C2×C42C22×D311yescharacteristic
140C4⋊C8—61no—
141C4⋊C8—61no—
142#103—24no—
143#102—24no—
144C22⋊C8—61no—
145C22⋊C8—61no—
146#97—21no—
147#97—61no—
148#97—61no—
149C4×C8C2×Dic312yes—
150#95—32no—
151#95—32no—
152#95—64no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
157C23×C3—41no—
158C23×C3#12011yescharacteristic
159C3×Q8—42no—
160C3×Q8—82no—
161C3×Q8—82no—
162C3×D4—42no—
163C3×D4—81no—
164C3×D4—81no—
165C3×D4—82no—
166C2×C3×C4—22no—
167C2×C3×C4—24no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—41no—
173C2×C3×C4—82no—
174C2×C3×C4C2×C22⋊C411yescharacteristic
175C2×C3×C4C4×D411yescharacteristic
176C2×C3×C4C4×D411yescharacteristic
177C2×C3×C4#12111yescharacteristic
178C2×C3×C4#12111yescharacteristic
179C2×C3×C4#12311yescharacteristic
180C2×Dic3—42no—
181C2×Dic3—42no—
182C2×Dic3—42no—
183C2×Dic3—42no—
184C2×Dic3—42no—
185C2×Dic3—42no—
186C3×C8—42no—
187C3⋊C8—81no—
188C3⋊C8—81no—
189#41—44no—
190C2×Q8—22no—
191C2×Q8—22no—
192C2×Q8—21no—
193C2×Q8—81no—
194C2×Q8—81no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—41no—
199C22×C4—22no—
200C22×C4—41no—
201C22×C4—62no—
202C22×C4—62no—
203C22×C4D3×D411yescharacteristic
204C22×C4D3×D411yescharacteristic
205C22×C4C2×#6711yescharacteristic
206M16—42no—
207C2×C8—22no—
208C2×C8—121no—
209C2×C8—121no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—242no—
216C4⋊C4#21714yes—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C42—242no—
220C42#23711yescharacteristic
221C42#23711yescharacteristic
222C42C22×Dic311yescharacteristic
223C42C2×#6711yescharacteristic
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—81no—
227C22×C3—81no—
228C22×C3—81no—
229C22×C3#38611yescharacteristic
230C22×C3#45412yes—
231C3×C4—21no—
232C3×C4—21no—
233C3×C4—21no—
234C3×C4—21no—
235C3×C4—21no—
236C3×C4—21no—
237C3×C4—44no—
238C3×C4—81no—
239Dic3—82no—
240Dic3—82no—
241C23—41no—
242C23#93711yescharacteristic
243Q8—42no—
244Q8—82no—
245Q8—82no—
246D4—42no—
247D4—81no—
248D4—81no—
249D4—82no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—82no—
258C2×C4—122no—
259C2×C4—122no—
260C2×C4—122no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4D4×Dic311yescharacteristic
265C2×C4D4×Dic311yescharacteristic
266C2×C4#93511yescharacteristic
267C2×C4#93911yescharacteristic
268C2×C4#93911yescharacteristic
269C2×C4C2×#21711yescharacteristic
270C8—42no—
271C8—241no—
272C8—241no—
273C2×C3—22no—
274C2×C3—161no—
275C2×C3#188811yescharacteristic
276C2×C3#190411yescharacteristic
277C2×C3#190511yescharacteristic
278C22—21no—
279C22—21no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22#506212yes—
284C22#522011yescharacteristic
285C4—21no—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—21no—
290C4—21no—
291C4—44no—
292C4—81no—
293C4—242no—
294C4—242no—
295C3#943011yescharacteristic, complemented, Sylow
296C2—22no—
297C2—161no—
298C2#7172411yescharacteristic
299C2#7208911yescharacteristic
300C2#7210111yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3141C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3141D3
hypocenterC3#9430
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3141

Complements

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

NH ≅ G/Ntype
2. #83231C2semidirect
5. #71776C2semidirect
6. #67666C2semidirect
7. #67666C2semidirect
12. C3×#487C4semidirect
13. C3×#486C4semidirect
72. #487Dic3semidirect
73. #486Dic3semidirect
295. C3#9430semidirect

Maximal subgroups

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

subgroupindex
C3×#31412
#832312
#718132
#717762
#676662
#94303