Finite Group Databasebeta

← #10692161

Subgroups — #10692161

Order 768.

398 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 99 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×#3215C211yesmaximal, characteristic
2#83333C211yesmaximal, characteristic, complemented
3#71832C211yesmaximal, characteristic, complemented
4#71720C211yesmaximal, characteristic
5#71719C211yesmaximal, characteristic
6C3⋊#1308C211yesmaximal, characteristic, complemented
7#67757C211yesmaximal, characteristic, complemented
8#9334—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C3×#497—21no—
12C3×#489—21no—
13C3×#464—22no—
14C3×#451C412yes—
15C3×#450C412yes—
16C3×#432C2211yescharacteristic
17#5024—22no—
18#5024—22no—
19#5000—22no—
20#4926C2211yescharacteristic
21#4926C2211yescharacteristic
22#4926C2211yescharacteristic
23#4925C2211yescharacteristic
24#4547—21no—
25#4546—21no—
26#4540—22no—
27#4525—22no—
28#3215D311yescharacteristic
29#2987—31no—
30#1554—31no—
31#1533—31no—
32#1469—31no—
33#1308—31no—
34#1287—31no—
35C22×C3×Q8C2×C411yescharacteristic
36C22×C3×D4C2×C411yescharacteristic
37C2×C3×Q16—22no—
38C2×C3×SD16—22no—
39C3×C4⋊Q8D411yescharacteristic
40C3×C4⋊Q8D411yescharacteristic
41C3×#125—21no—
42C3×#124—21no—
43C3×#124—21no—
44C3×#124—21no—
45C3×#124—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8C2×C412yes—
48C3×C4×Q8—21no—
49C3×C4×Q8D412yes—
50C3×#117—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C42C2311yescharacteristic
54C3×C4⋊C8—21no—
55C3×#103—22no—
56C3×#102—22no—
57C3×C22⋊C8C2×C412yes—
58C3×#97—21no—
59#830—21no—
60#830—21no—
61#830—21no—
62#830D412yes—
63C4⋊(C3⋊C8)—24no—
64C4⋊(C3⋊C8)—21no—
65C4⋊(C3⋊C8)—21no—
66C4⋊(C3⋊C8)—21no—
67C4⋊(C3⋊C8)—21no—
68#803—22no—
69#803—21no—
70#803—21no—
71#803D412yes—
72C4×C3⋊C8—22no—
73C4×C3⋊C8—21no—
74#533C2×D311yescharacteristic
75C2×#124C2×D311yescharacteristic
76#497—21no—
77#489—21no—
78#464—22no—
79#451Dic312yes—
80#450Dic312yes—
81#446—62no—
82#446—62no—
83#433—62no—
84#432—31no—
85#432—31no—
86#432—31no—
87#432C2×D311yescharacteristic
88#431—31no—
89#333—61no—
90#332—61no—
91#329—62no—
92#323—62no—
93C24×C3—21no—
94C2×C3×Q8—21no—
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C2×C3×Q8—81no—
98C2×C3×Q8C22⋊C412yes—
99C2×C3×D4—21no—
100C2×C3×D4—21no—
101C2×C3×D4—81no—
102C22×C3×C4—21no—
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C3×Q16—82no—
107C3×SD16—82no—
108C2×C3×C8—22no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4C22⋊C412yes—
115C3×C22⋊C4—22no—
116C3×C22⋊C4—41no—
117C3×C22⋊C4—41no—
118C3×C22⋊C4—41no—
119C3×C22⋊C4—41no—
120C3×C42—22no—
121C3×C42C2×D411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42#4111yescharacteristic
125C2×C3⋊C8—22no—
126C2×C3⋊C8—22no—
127C2×C3⋊C8—41no—
128C2×C3⋊C8—41no—
129C22×Q8C2×Dic311yescharacteristic
130C22×D4C2×Dic311yescharacteristic
131C2×Q16—22no—
132C2×SD16—22no—
133C4⋊Q8#6711yescharacteristic
134C4⋊Q8#6711yescharacteristic
135#125—21no—
136#124—21no—
137#124—21no—
138#124—21no—
139#124—21no—
140C22⋊Q8—21no—
141C22⋊Q8C2×Dic312yes—
142C4×Q8—21no—
143C4×Q8#6712yes—
144#117—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C42C22×D311yescharacteristic
148C4⋊C8—21no—
149C4⋊C8—64no—
150C4⋊C8—61no—
151C4⋊C8—61no—
152C4⋊C8—61no—
153C4⋊C8—61no—
154#103—22no—
155#102—22no—
156C22⋊C8—32no—
157C22⋊C8—61no—
158C22⋊C8—61no—
159C22⋊C8—61no—
160C22⋊C8C2×Dic312yes—
161#97—21no—
162#97—32no—
163#97—62no—
164#97—61no—
165#97—61no—
166C4×C8—62no—
167C4×C8—61no—
168C23×C3—42no—
169C23×C3—41no—
170C23×C3—81no—
171C23×C3—81no—
172C23×C3C4×D411yescharacteristic
173C3×Q8—42no—
174C3×Q8—42no—
175C3×Q8—81no—
176C3×Q8—81no—
177C3×D4—42no—
178C3×D4—81no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—81no—
186C2×C3×C4C2×C22⋊C411yescharacteristic
187C2×C3×C4C4×D411yescharacteristic
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12111yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12311yescharacteristic
192C3×C8—42no—
193C3⋊C8—42no—
194C3⋊C8—42no—
195C3⋊C8—81no—
196C3⋊C8—81no—
197C24—21no—
198C2×Q8—21no—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×Q8—81no—
202C2×Q8#21712yes—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—81no—
206C22×C4—21no—
207C22×C4D3×D411yescharacteristic
208C22×C4#23711yescharacteristic
209C22×C4C22×Dic311yescharacteristic
210Q16—82no—
211SD16—82no—
212C2×C8—22no—
213C2×C8—62no—
214C2×C8—62no—
215C2×C8—121no—
216C2×C8—121no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4#21712yes—
223C22⋊C4—22no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C42—22no—
229C42D3×D411yescharacteristic
230C42#23711yescharacteristic
231C42C2×#6711yescharacteristic
232C42C2×#6711yescharacteristic
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—21no—
236C22×C3—81no—
237C22×C3—81no—
238C22×C3—81no—
239C22×C3—162no—
240C22×C3#38611yescharacteristic
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—21no—
246C3×C4—21no—
247C3×C4—42no—
248C3×C4—42no—
249C3×C4—81no—
250C23—42no—
251C23—41no—
252C23—81no—
253C23—81no—
254C23D4×Dic311yescharacteristic
255Q8—42no—
256Q8—42no—
257Q8—81no—
258Q8—81no—
259D4—42no—
260D4—81no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—81no—
268C2×C4D4×Dic311yescharacteristic
269C2×C4#93511yescharacteristic
270C2×C4#93711yescharacteristic
271C2×C4#93911yescharacteristic
272C2×C4#93911yescharacteristic
273C2×C4C2×#21711yescharacteristic
274C8—42no—
275C8—122no—
276C8—122no—
277C8—241no—
278C8—241no—
279C2×C3—41no—
280C2×C3—161no—
281C2×C3#188111yescharacteristic
282C2×C3#190511yescharacteristic
283C2×C3#191411yescharacteristic
284C22—21no—
285C22—21no—
286C22—21no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—162no—
291C22#522011yescharacteristic
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—42no—
299C4—42no—
300C4—81no—
301C3#933411yescharacteristic, complemented, Sylow
302C2—41no—
303C2—161no—
304C2#7185511yescharacteristic
305C2#7210111yescharacteristic
306C2#7211611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3215C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenter#3215D3
hypocenterC3#9334
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3215

Complements

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

NH ≅ G/Ntype
2. #83333C2semidirect
3. #71832C2semidirect
6. C3⋊#1308C2semidirect
7. #67757C2semidirect
301. C3#9334semidirect

Maximal subgroups

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

subgroupindex
C3×#32152
#833332
#718322
#717202
#717192
C3⋊#13082
#677572
#93343