Finite Group Databasebeta

← #10693271

Subgroups — #10693271

Order 768.

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

normal subgroups only

This group has 396 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×#3113C211yesmaximal, characteristic
2#83331C211yesmaximal, characteristic
3#71833C211yesmaximal, characteristic
4#71625C211yesmaximal, characteristic
5#71622C211yesmaximal, characteristic
6#67758C211yesmaximal, characteristic
7#67757C211yesmaximal, characteristic
8#9528—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C3×#483C412yes—
12C3×#475—21no—
13C3×#462—21no—
14C3×#451C412yes—
15C3×#441—22no—
16C3×#432C2211yescharacteristic
17#5024—22no—
18#5024—21no—
19Q8×C3⋊C8—21no—
20#5000—22no—
21#4926C2211yescharacteristic
22#4926C2211yescharacteristic
23#4925C2211yescharacteristic
24#4924C2211yescharacteristic
25#4547—21no—
26#4546—21no—
27#4535—21no—
28#4535—21no—
29#4528—22no—
30#3113D311yescharacteristic
31#2985—31no—
32#1559—31no—
33#1487—31no—
34#1467—31no—
35#1307—31no—
36#1287—31no—
37C22×C3×Q8C2×C411yescharacteristic
38C2×C3×Q16—22no—
39C3×C4⋊Q8D411yescharacteristic
40C3×C4⋊Q8D411yescharacteristic
41C3×#125—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8C2×C412yes—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8D412yes—
50C3×#117—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4C2×C411yescharacteristic
53C2×C3×C42—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#208—21no—
56C3×#106—22no—
57C3×C4⋊C8—21no—
58C3×#103—22no—
59C3×#103—22no—
60C3×C22⋊C8C2×C412yes—
61C3×#97—21no—
62#830—21no—
63#830—21no—
64#830D412yes—
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#803—21no—
75C4×C3⋊C8—22no—
76C4×C3⋊C8D411yescharacteristic
77C4×C3⋊C8D411yescharacteristic
78#533C2×D311yescharacteristic
79C2×C4×Q8C2×D311yescharacteristic
80#483Dic312yes—
81#475—21no—
82#462—21no—
83#451Dic312yes—
84#446—61no—
85#446—62no—
86C8×Q8—61no—
87#441—22no—
88#433—62no—
89#432—31no—
90#432—31no—
91#432C2×D311yescharacteristic
92#431—31no—
93#423—31no—
94#333—61no—
95#332—61no—
96Q8⋊C8—61no—
97Q8⋊C8—61no—
98#324—62no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—42no—
103C2×C3×Q8C22⋊C412yes—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C3×Q16—44no—
110C2×C3×C8—22no—
111C3×C4⋊C4—22no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4C22⋊C412yes—
121C3×C22⋊C4—22no—
122C3×C42—22no—
123C3×C42—41no—
124C3×C42—41no—
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129#208—81no—
130C2×C3⋊C8—21no—
131C2×C3⋊C8—22no—
132C2×C3⋊C8—21no—
133C2×C3⋊C8—41no—
134C2×C3⋊C8—41no—
135C22×Q8C2×Dic311yescharacteristic
136C2×Q16—22no—
137C2×M16—61no—
138C4⋊Q8#6711yescharacteristic
139C4⋊Q8#6711yescharacteristic
140#125—21no—
141C22⋊Q8—21no—
142C22⋊Q8C2×Dic312yes—
143C4×Q8—21no—
144C4×Q8—21no—
145C4×Q8—21no—
146C4×Q8—21no—
147C4×Q8—21no—
148C4×Q8#6712yes—
149#117—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4C2×Dic311yescharacteristic
152C2×C42—21no—
153C2×C42C22×D311yescharacteristic
154#106—22no—
155C4⋊C8—21no—
156C4⋊C8—62no—
157C4⋊C8—62no—
158C4⋊C8—61no—
159C4⋊C8—61no—
160C4⋊C8—61no—
161C4⋊C8—61no—
162C4⋊C8—61no—
163#103—22no—
164#103—22no—
165C22⋊C8—32no—
166C22⋊C8—61no—
167C22⋊C8—61no—
168C22⋊C8C2×Dic312yes—
169#97—21no—
170#97—62no—
171#97—61no—
172#97—61no—
173C4×C8—31no—
174C4×C8—31no—
175C4×C8—62no—
176C23×C3C4×D411yescharacteristic
177C3×Q8—22no—
178C3×Q8—22no—
179C3×Q8—42no—
180C3×Q8—41no—
181C3×Q8—41no—
182C3×Q8—41no—
183C3×Q8—41no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—81no—
195C2×C3×C4C2×C22⋊C411yescharacteristic
196C2×C3×C4C4×D411yescharacteristic
197C2×C3×C4#12011yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12311yescharacteristic
201C3×C8—42no—
202C3⋊C8—41no—
203C3⋊C8—42no—
204C3⋊C8—41no—
205C3⋊C8—81no—
206C3⋊C8—81no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—21no—
210C2×Q8—42no—
211C2×Q8#21712yes—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4D3×D411yescharacteristic
215C22×C4#23711yescharacteristic
216C22×C4C22×Dic311yescharacteristic
217Q16—44no—
218M16—241no—
219C2×C8—22no—
220C2×C8—61no—
221C2×C8—61no—
222C2×C8—62no—
223C2×C8—121no—
224C2×C8—121no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4#21712yes—
235C22⋊C4—22no—
236C42—22no—
237C42—41no—
238C42—41no—
239C42D3×D411yescharacteristic
240C42#23711yescharacteristic
241C42C2×#6711yescharacteristic
242C42C2×#6711yescharacteristic
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3#38611yescharacteristic
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—42no—
253C3×C4—42no—
254C3×C4—42no—
255C3×C4—81no—
256C3×C4#47111yescharacteristic
257C3×C4#47311yescharacteristic
258C23D4×Dic311yescharacteristic
259Q8—22no—
260Q8—22no—
261Q8—42no—
262Q8—41no—
263Q8—41no—
264Q8—41no—
265Q8—41no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4D4×Dic311yescharacteristic
278C2×C4#93511yescharacteristic
279C2×C4#93711yescharacteristic
280C2×C4#93911yescharacteristic
281C2×C4#93911yescharacteristic
282C2×C4C2×#21711yescharacteristic
283C8—42no—
284C8—121no—
285C8—121no—
286C8—122no—
287C8—241no—
288C8—241no—
289C2×C3—41no—
290C2×C3#189111yescharacteristic
291C2×C3#190511yescharacteristic
292C2×C3#190811yescharacteristic
293C22—21no—
294C22—21no—
295C22—21no—
296C22#522011yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—42no—
303C4—42no—
304C4—42no—
305C4—81no—
306C4#520311yescharacteristic
307C4#520411yescharacteristic
308C3#952811yescharacteristic, complemented, Sylow
309C2—41no—
310C2#7171411yescharacteristic
311C2#7210111yescharacteristic
312C2#7210311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3113C2
socleC22×C3#386
prime radicalC23×C3C4×D4
cosocleC2×C3×C42C23
hypercenter#3113D3
hypocenterC3#9528
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
308. C3#9528semidirect

Maximal subgroups

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

subgroupindex
C3×#31132
#833312
#718332
#716252
#716222
#677582
#677572
#95283