Finite Group Databasebeta

← #10692736

Subgroups — #10692736

Order 768.

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

normal subgroups only

This group has 422 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×#3165C211yesmaximal, characteristic
2#83231C211yesmaximal, characteristic
3#71837C211yesmaximal, characteristic
4#71837C211yesmaximal, characteristic
5#71776C211yesmaximal, characteristic
6#67666C211yesmaximal, characteristic
7#67666C211yesmaximal, characteristic
8#9427—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C3×#501—22no—
12C3×#500C412yescomplemented
13C3×#498C412yescomplemented
14C3×#428—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#4538—22no—
25#4538—22no—
26#3165D311yescharacteristic
27#2381—31no—
28#1556—31no—
29#1515—31no—
30#1515—31no—
31#1288—31no—
32#1288—31no—
33C22×C3×Q8—21no—
34C2×C3×M16—21no—
35C3×C4⋊Q8—21no—
36C3×C4⋊Q8—21no—
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×C4⋊Q8C2×C412yes—
40C3×C4⋊Q8D412yes—
41C3×#125—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×#117—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D412yes—
49C2×C3×C42C2311yescharacteristic
50C2×C4⋊Dic3—22no—
51C2×C4×Dic3—22no—
52C3×#107—24no—
53C3×#106—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#533C2×D311yescharacteristic
70C2×C4⋊Q8C2×D311yescharacteristic
71#501—22no—
72#500Dic312yescomplemented
73#498Dic312yescomplemented
74#432—31no—
75#432—31no—
76#428—22no—
77C4×M16C2×D311yescharacteristic
78#391—62no—
79#387—62no—
80#387—62no—
81#383—31no—
82#383—31no—
83#332—62no—
84#328—62no—
85#328—62no—
86C2×C3×Q8—22no—
87C2×C3×Q8—22no—
88C2×C3×Q8—21no—
89C2×C3×Q8—21no—
90C2×C3×Q8—81no—
91C2×C3×Q8—81no—
92C22×C3×C4—22no—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×Dic3—22no—
97C22×Dic3—22no—
98C3×M16—24no—
99C2×C3×C8—22no—
100C3×C4⋊C4—22no—
101C3×C4⋊C4—22no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—21no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4C22⋊C414yes—
110C3×C22⋊C4—41no—
111C3×C42—41no—
112C3×C42C22×C411yescharacteristic
113C3×C42C2×D411yescharacteristic
114C3×C42#4111yescharacteristic
115C3×C42#4111yescharacteristic
116C4⋊Dic3—82no—
117C4×Dic3—82no—
118C2×C3⋊C8—41no—
119C2×C3⋊C8—41no—
120C22×Q8—21no—
121C2×M16—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8C2×Dic311yescharacteristic
125C4⋊Q8C2×Dic311yescharacteristic
126C4⋊Q8C2×Dic312yes—
127C4⋊Q8#6712yes—
128#125—21no—
129C22⋊Q8—21no—
130C22⋊Q8—21no—
131C4×Q8—21no—
132C4×Q8—21no—
133#117—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—62no—
136C2×C4⋊C4#6712yes—
137C2×C42—62no—
138C2×C42C22×D311yescharacteristic
139#107—24no—
140#106—24no—
141C4⋊C8—61no—
142C4⋊C8—61no—
143C22⋊C8—61no—
144C22⋊C8—61no—
145#97—21no—
146#97—61no—
147#97—61no—
148C4×C8C2×Dic312yes—
149#95—32no—
150#95—32no—
151#95—64no—
152#95—61no—
153#95—61no—
154#95—61no—
155#95—61no—
156C23×C3#12011yescharacteristic
157C3×Q8—82no—
158C3×Q8—82no—
159C3×Q8—81no—
160C3×Q8—81no—
161C2×C3×C4—22no—
162C2×C3×C4—24no—
163C2×C3×C4—41no—
164C2×C3×C4—41no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4C2×C22⋊C411yescharacteristic
170C2×C3×C4C4×D411yescharacteristic
171C2×C3×C4C4×D411yescharacteristic
172C2×C3×C4#12111yescharacteristic
173C2×C3×C4#12111yescharacteristic
174C2×C3×C4#12311yescharacteristic
175C2×Dic3—42no—
176C2×Dic3—42no—
177C2×Dic3—42no—
178C2×Dic3—42no—
179C2×Dic3—42no—
180C2×Dic3—42no—
181C3×C8—24no—
182C3⋊C8—81no—
183C3⋊C8—81no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8—81no—
189C2×Q8—81no—
190C22×C4—22no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4D3×D411yescharacteristic
194C22×C4D3×D411yescharacteristic
195C22×C4C2×#6711yescharacteristic
196M16—24no—
197C2×C8—22no—
198C2×C8—121no—
199C2×C8—121no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—242no—
210C4⋊C4#21714yes—
211C22⋊C4—41no—
212C42—41no—
213C42—242no—
214C42#23711yescharacteristic
215C42#23711yescharacteristic
216C42C22×Dic311yescharacteristic
217C42C2×#6711yescharacteristic
218C22×C3—21no—
219C22×C3—21no—
220C22×C3#38611yescharacteristic
221C22×C3#45612yes—
222C3×C4—21no—
223C3×C4—21no—
224C3×C4—21no—
225C3×C4—21no—
226C3×C4—21no—
227C3×C4—21no—
228C3×C4—44no—
229C3×C4—81no—
230C3×C4—81no—
231Dic3—82no—
232Dic3—82no—
233C23#93711yescharacteristic
234Q8—82no—
235Q8—82no—
236Q8—81no—
237Q8—81no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—122no—
247C2×C4—122no—
248C2×C4—122no—
249C2×C4—122no—
250C2×C4—122no—
251C2×C4—122no—
252C2×C4D4×Dic311yescharacteristic
253C2×C4D4×Dic311yescharacteristic
254C2×C4#93511yescharacteristic
255C2×C4#93911yescharacteristic
256C2×C4#93911yescharacteristic
257C2×C4C2×#21711yescharacteristic
258C8—24no—
259C8—241no—
260C8—241no—
261C2×C3—22no—
262C2×C3#188611yescharacteristic
263C2×C3#190411yescharacteristic
264C2×C3#190511yescharacteristic
265C22—21no—
266C22—21no—
267C22#509512yes—
268C22#522011yescharacteristic
269C4—21no—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—21no—
274C4—21no—
275C4—44no—
276C4—81no—
277C4—81no—
278C4—242no—
279C4—242no—
280C3#942711yescharacteristic, complemented, Sylow
281C2—22no—
282C2#7178411yescharacteristic
283C2#7208911yescharacteristic
284C2#7210111yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3165C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#3165D3
hypocenterC3#9427
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
12. C3×#500C4semidirect
13. C3×#498C4semidirect
72. #500Dic3semidirect
73. #498Dic3semidirect
280. C3#9427semidirect

Maximal subgroups

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

subgroupindex
C3×#31652
#832312
#718372
#717762
#676662
#94273