Finite Group Databasebeta

← #10692441

Subgroups — #10692441

Order 768.

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

normal subgroups only

This group has 368 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×#3237C211yesmaximal, characteristic
2#83331C211yesmaximal, characteristic
3#71831C211yesmaximal, characteristic
4#71824C211yesmaximal, characteristic
5#71823C211yesmaximal, characteristic
6#67756C211yesmaximal, characteristic
7#67755C211yesmaximal, characteristic
8#9374—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#533C2211yescharacteristic
11C3×#501C412yes—
12C3×#491C412yes—
13C3×#485—22no—
14C3×#477—21no—
15C3×#475—21no—
16C3×#432C2211yescharacteristic
17#5024—22no—
18#5024—21no—
19Q8×C3⋊C8—21no—
20#5000—22no—
21#4926C2211yescharacteristic
22#4926C2211yescharacteristic
23#4925C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4547—21no—
26#4547—21no—
27#4543—22no—
28#4535—21no—
29#4535—21no—
30#3237D311yescharacteristic
31#2985—31no—
32#1552—31no—
33#1537—31no—
34#1481—31no—
35#1290—31no—
36#1287—31no—
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×C4⋊Q8D411yescharacteristic
40C3×C4⋊Q8D411yescharacteristic
41C3×#125—21no—
42C3×#125C2×C412yes—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×#117—21no—
52C3×#117D412yes—
53C2×C3×C42C2311yescharacteristic
54C22×C3⋊C8—21no—
55C3×#107—22no—
56C3×#106—22no—
57C3×C4⋊C8—21no—
58C3×#103—22no—
59C3×#103—22no—
60C3×C22⋊C8—21no—
61C3×#97C2×C412yes—
62#830—21no—
63#830—21no—
64#830—21no—
65C4⋊(C3⋊C8)—22no—
66C4⋊(C3⋊C8)—22no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72#803—22no—
73#803—21no—
74#803D412yes—
75C4×C3⋊C8—22no—
76C4×C3⋊C8D411yescharacteristic
77C4×C3⋊C8D411yescharacteristic
78#533C2×D311yescharacteristic
79#533C2×D311yescharacteristic
80#501Dic312yes—
81#491Dic312yes—
82#485—22no—
83#477—21no—
84#475—21no—
85#446—61no—
86#446—62no—
87C8×Q8—61no—
88#433—62no—
89#432—31no—
90#432—31no—
91#432C2×D311yescharacteristic
92#431—31no—
93C2×C4⋊C8—31no—
94#332—61no—
95#332—61no—
96#330—62no—
97Q8⋊C8—61no—
98Q8⋊C8—61no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C2×C3×C8—22no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4C22⋊C412yes—
118C3×C4⋊C4C22⋊C412yes—
119C3×C22⋊C4—22no—
120C3×C22⋊C4—41no—
121C3×C42—22no—
122C3×C42—41no—
123C3×C42C22×C411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C2×C3⋊C8—21no—
128C2×C3⋊C8—22no—
129C2×C3⋊C8—21no—
130C2×C3⋊C8—41no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8—81no—
133C22×C8—61no—
134C4⋊Q8C2×Dic311yescharacteristic
135C4⋊Q8C2×Dic311yescharacteristic
136C4⋊Q8#6711yescharacteristic
137C4⋊Q8#6711yescharacteristic
138#125—21no—
139#125C2×Dic312yes—
140C22⋊Q8—21no—
141C22⋊Q8—21no—
142C22⋊Q8—21no—
143C22⋊Q8—21no—
144C4×Q8—21no—
145C4×Q8—21no—
146C4×Q8—21no—
147C4×Q8—21no—
148#117—21no—
149#117#6712yes—
150C2×C42C22×D311yescharacteristic
151#107—22no—
152#106—22no—
153C4⋊C8—21no—
154C4⋊C8—62no—
155C4⋊C8—62no—
156C4⋊C8—61no—
157C4⋊C8—61no—
158C4⋊C8—61no—
159C4⋊C8—61no—
160C4⋊C8—61no—
161#103—22no—
162#103—22no—
163C22⋊C8—21no—
164C22⋊C8—61no—
165C22⋊C8—61no—
166C22⋊C8—61no—
167#97—32no—
168#97—62no—
169#97—61no—
170#97C2×Dic312yes—
171C4×C8—31no—
172C4×C8—31no—
173C4×C8—62no—
174C23×C3#12311yescharacteristic
175C3×Q8—42no—
176C3×Q8—42no—
177C3×Q8—41no—
178C3×Q8—41no—
179C3×Q8—81no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4C2×C22⋊C411yescharacteristic
189C2×C3×C4C4×D411yescharacteristic
190C2×C3×C4C4×D411yescharacteristic
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C3×C8—42no—
195C3⋊C8—41no—
196C3⋊C8—42no—
197C3⋊C8—41no—
198C3⋊C8—81no—
199C3⋊C8—81no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—21no—
204C22×C4#23711yescharacteristic
205C22×C4#23711yescharacteristic
206C22×C4C2×#6711yescharacteristic
207C2×C8—22no—
208C2×C8—61no—
209C2×C8—61no—
210C2×C8—62no—
211C2×C8—121no—
212C2×C8—121no—
213C2×C8—241no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4#21712yes—
225C4⋊C4#21712yes—
226C22⋊C4—22no—
227C22⋊C4—41no—
228C42—22no—
229C42—41no—
230C42D3×D411yescharacteristic
231C42D3×D411yescharacteristic
232C42C22×Dic311yescharacteristic
233C42C2×#6711yescharacteristic
234C22×C3—21no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3#38611yescharacteristic
238C3×C4—21no—
239C3×C4—21no—
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—42no—
244C3×C4—42no—
245C3×C4—81no—
246C3×C4—81no—
247C3×C4#47111yescharacteristic
248C3×C4#47311yescharacteristic
249C23#93511yescharacteristic
250Q8—42no—
251Q8—42no—
252Q8—41no—
253Q8—41no—
254Q8—81no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4D4×Dic311yescharacteristic
264C2×C4D4×Dic311yescharacteristic
265C2×C4#93711yescharacteristic
266C2×C4#93911yescharacteristic
267C2×C4#93911yescharacteristic
268C2×C4C2×#21711yescharacteristic
269C8—42no—
270C8—121no—
271C8—121no—
272C8—122no—
273C8—241no—
274C8—241no—
275C2×C3—41no—
276C2×C3#188111yescharacteristic
277C2×C3#190811yescharacteristic
278C2×C3#191211yescharacteristic
279C22—21no—
280C22—21no—
281C22—21no—
282C22#522011yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—21no—
287C4—21no—
288C4—42no—
289C4—42no—
290C4—81no—
291C4—81no—
292C4#520311yescharacteristic
293C4#520411yescharacteristic
294C3#937411yescharacteristic, complemented, Sylow
295C2—41no—
296C2#7186411yescharacteristic
297C2#7210311yescharacteristic
298C2#7211611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3237C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#3237D3
hypocenterC3#9374
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3237

Complements

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

NH ≅ G/Ntype
294. C3#9374semidirect

Maximal subgroups

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

subgroupindex
C3×#32372
#833312
#718312
#718242
#718232
#677562
#677552
#93743