Finite Group Databasebeta

← #10693272

Subgroups — #10693272

Order 768.

396 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 89 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
1#82448C211yesmaximal, characteristic
2#79316C211yesmaximal, characteristic
3C3×#1467C211yesmaximal, characteristic
4#71622C211yesmaximal, characteristic
5#70825C211yesmaximal, characteristic
6#67758C211yesmaximal, characteristic
7#67756C211yesmaximal, characteristic
8#9528—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13#5228C412yes—
14#5031—21no—
15#5030—22no—
16#5020—21no—
17#4971C412yes—
18#4926C2211yescharacteristic
19#4925C2211yescharacteristic
20#4924C2211yescharacteristic
21#4831—21no—
22#4830—21no—
23#4720—22no—
24#4701—22no—
25C3×#324—22no—
26#4535—21no—
27#4535—21no—
28#4486—21no—
29#4481—21no—
30#3113—31no—
31#2985—31no—
32#1559—31no—
33#1487—31no—
34#1467D311yescharacteristic
35#1307—31no—
36#1287—31no—
37C22×C3×Q8C2×C411yescharacteristic
38C3×C4×Q8—21no—
39C3×C4×Q8—21no—
40C3×C4×Q8—21no—
41C3×C4×Q8—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46C2×#216—22no—
47#926—21no—
48#924—21no—
49#924C2×C412yes—
50C2×#208—21no—
51#870—21no—
52#869D411yescharacteristic
53#869D411yescharacteristic
54C4×Dic6—21no—
55C4×Dic6D412yes—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C22⋊C8—21no—
59C3×C22⋊C8D412yes—
60C3×#97—21no—
61C3×#97—21no—
62#835—22no—
63#830—21no—
64#830C2×C412yes—
65#815—22no—
66Dic3⋊C8—22no—
67Dic3⋊C8—22no—
68C8×Dic3—22no—
69#810—22no—
70#808—22no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75#803—21no—
76C4×C3⋊C8D411yescharacteristic
77C4×C3⋊C8D411yescharacteristic
78#533—31no—
79C2×C4×Q8C2×D311yescharacteristic
80#483—32no—
81#475—61no—
82#462—61no—
83#451—32no—
84#446—61no—
85#446—62no—
86C8×Q8—61no—
87#441—62no—
88#433—62no—
89#432—31no—
90#432C2×D311yescharacteristic
91#432C2×D311yescharacteristic
92#431—31no—
93#423—31no—
94#333—61no—
95#332—61no—
96Q8⋊C8—61no—
97Q8⋊C8—61no—
98#324—22no—
99C2×C3×Q8—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8—42no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C2×Dic6—21no—
108C2×Dic6C22⋊C412yes—
109C2×C3×C8—22no—
110C2×C3×C8—41no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—41no—
115C3×C42—41no—
116C3×C42—41no—
117C3×C42C2×D411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42#4111yescharacteristic
121#217—22no—
122#216—44no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3C22⋊C412yes—
127Dic3⋊C4—22no—
128Dic3⋊C4—22no—
129C4×Dic3—22no—
130#208—81no—
131C2×C3⋊C8—22no—
132C2×C3⋊C8—21no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—41no—
135C22×Q8C4×D311yescharacteristic
136C2×Q16—62no—
137C2×M16—61no—
138C4⋊Q8—31no—
139C4⋊Q8—31no—
140#125—61no—
141C22⋊Q8—32no—
142C22⋊Q8—61no—
143C4×Q8—21no—
144C4×Q8—21no—
145C4×Q8—21no—
146C4×Q8—21no—
147C4×Q8—32no—
148C4×Q8—61no—
149#117—61no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4C4×D311yescharacteristic
152C2×C42—21no—
153C2×C42C22×D311yescharacteristic
154#106—62no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—62no—
158C4⋊C8—62no—
159C4⋊C8—61no—
160C4⋊C8—61no—
161C4⋊C8—61no—
162C4⋊C8—61no—
163#103—62no—
164#103—62no—
165C22⋊C8—21no—
166C22⋊C8—32no—
167C22⋊C8—61no—
168C22⋊C8#6712yes—
169#97—21no—
170#97—21no—
171#97—62no—
172#97—61no—
173C4×C8—31no—
174C4×C8—31no—
175C4×C8—62no—
176C23×C3C4×D411yescharacteristic
177C3×Q8—22no—
178C3×Q8—22no—
179C3×Q8—41no—
180C3×Q8—41no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—22no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—81no—
190C2×C3×C4C2×C22⋊C411yescharacteristic
191C2×C3×C4C4×D411yescharacteristic
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12311yescharacteristic
196C2×Dic3—22no—
197C2×Dic3—22no—
198Dic6—41no—
199Dic6—41no—
200Dic6—42no—
201C3×C8—42no—
202C3×C8—81no—
203C3⋊C8—41no—
204C3⋊C8—41no—
205C3⋊C8—42no—
206C3⋊C8—81no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—32no—
210C2×Q8—42no—
211C2×Q8—61no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4C2×C4×D311yescharacteristic
215C22×C4#23511yescharacteristic
216C22×C4C2×#6711yescharacteristic
217Q16—124no—
218M16—241no—
219C2×C8—22no—
220C2×C8—41no—
221C2×C8—62no—
222C2×C8—61no—
223C2×C8—61no—
224C2×C8—121no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—32no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—62no—
231C4⋊C4—62no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C22⋊C4—62no—
236C42—41no—
237C42—41no—
238C42—62no—
239C42D3×D411yescharacteristic
240C42D3×D411yescharacteristic
241C42D3×D411yescharacteristic
242C42#23711yescharacteristic
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—81no—
254C3×C4#47111yescharacteristic
255C3×C4#47311yescharacteristic
256Dic3—42no—
257Dic3—42no—
258C23C4×#6711yescharacteristic
259Q8—22no—
260Q8—22no—
261Q8—41no—
262Q8—41no—
263Q8—122no—
264Q8—121no—
265Q8—121no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—62no—
275C2×C4—62no—
276C2×C4—81no—
277C2×C4D3×C22⋊C411yescharacteristic
278C2×C4#67⋊C411yescharacteristic
279C2×C4#88311yescharacteristic
280C2×C4#88411yescharacteristic
281C2×C4#93711yescharacteristic
282C2×C4#93811yescharacteristic
283C8—42no—
284C8—81no—
285C8—122no—
286C8—121no—
287C8—121no—
288C8—241no—
289C2×C3—41no—
290C2×C3#189111yescharacteristic
291C2×C3#190511yescharacteristic
292C2×C3#190811yescharacteristic
293C22—21no—
294C22—21no—
295C22—21no—
296C22#495611yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—42no—
303C4—81no—
304C4—122no—
305C4—122no—
306C4#475411yescharacteristic
307C4#475511yescharacteristic
308C3#952811yescharacteristic, complemented, Sylow
309C2—41no—
310C2#6999211yescharacteristic
311C2#7005211yescharacteristic
312C2#7012311yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1467C2
socleC22×C3#386
prime radicalC23×C3C4×D4
cosocleC2×C3×C42C23
hypercenter#1467D3
hypocenterC3#9528
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

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
#824482
#793162
C3×#14672
#716222
#708252
#677582
#677562
#95283