Finite Group Databasebeta

← #10705581

Subgroups — #10705581

Order 768.

420 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 91 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×#2638C211yesmaximal, characteristic
2C4×#833C211yesmaximal, characteristic
3#71600C211yesmaximal, characteristic
4#67650C212yesmaximal, complemented
5#67648C212yesmaximal, complemented
6#11739—31nomaximal, Sylow
7C2×C3×C22⋊Q8C411yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C3×#398C2211yescharacteristic
10C3×#397—21no—
11C3×#397C411yescharacteristic
12C3×#395C2211yescharacteristic
13C3×#394—21no—
14C3×#393C411yescharacteristic
15C3×#387—21no—
16C3×#385—21no—
17C3×#381C411yescharacteristic
18C2×#833—21no—
19C2×#830—21no—
20C4×#208—21no—
21#4530C2212yes—
22#4528C2212yes—
23#4525—21no—
24#2638D311yescharacteristic
25C4×#100—31no—
26#1459—31no—
27#1354—32no—
28#1346—32no—
29C22×C3×Q8C2×C411yescharacteristic
30C22×C3×D4D411yescharacteristic
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C3×C22⋊Q8—41no—
34C3×C22⋊Q8—41no—
35C3×C4×D4—41no—
36C3×C4×D4—41no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C4⋊C4C2311yescharacteristic
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4C2×C411yescharacteristic
47C2×C3×C22⋊C4D411yescharacteristic
48C2×C3×C42D411yescharacteristic
49C2×#208—22no—
50C22×C3⋊C8—41no—
51C3×#95—21no—
52C3×#95—21no—
53C3×#95—21no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95C2×C411yescharacteristic
57C3×#95C2×C411yescharacteristic
58#833—24no—
59#830—22no—
60#830—41no—
61#803—41no—
62C4×C3⋊C8—41no—
63C2×C22⋊Q8Dic311yescharacteristic
64C2×C4×D4C2×D311yescharacteristic
65C2×#100—61no—
66C2×C22⋊C8—61no—
67C4×M16—61no—
68#398C2×D311yescharacteristic
69#397—21no—
70#397Dic311yescharacteristic
71#395C2×D311yescharacteristic
72#394—21no—
73#393Dic311yescharacteristic
74#387—21no—
75#385—21no—
76#381Dic311yescharacteristic
77#343—32no—
78#324—32no—
79#323—61no—
80C24×C3C22⋊C411yescharacteristic
81C24×C3C22⋊C411yescharacteristic
82C2×C3×Q8—41no—
83C2×C3×Q8—81no—
84C2×C3×D4—22no—
85C2×C3×D4—22no—
86C2×C3×D4—41no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—81no—
96C22×C3×C4—81no—
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22×C411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C3×C4⋊C4—41no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4—41no—
105C3×C4⋊C4—41no—
106C3×C4⋊C4—42no—
107C3×C4⋊C4—81no—
108C3×C22⋊C4—41no—
109C3×C22⋊C4—41no—
110C3×C22⋊C4—42no—
111C3×C22⋊C4—42no—
112C3×C22⋊C4—81no—
113C3×C22⋊C4—81no—
114C3×C42—21no—
115C3×C42—21no—
116#208—24no—
117#208—42no—
118C2×C3⋊C8—42no—
119C2×C3⋊C8—42no—
120C2×C3⋊C8—82no—
121C22×Q8C2×Dic311yescharacteristic
122C22×D4#6711yescharacteristic
123C23×C4C2×Dic311yescharacteristic
124C23×C4#6711yescharacteristic
125C2×M16—62no—
126C22×C8—121no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C4×D4—41no—
130C4×D4—41no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4C2×Dic311yescharacteristic
135C2×C4⋊C4C22×D311yescharacteristic
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4C2×Dic311yescharacteristic
141C2×C22⋊C4#6711yescharacteristic
142C2×C42#6711yescharacteristic
143#100—64no—
144C22⋊C8—62no—
145C22⋊C8—121no—
146#97—121no—
147C4×C8—121no—
148#95—21no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95C2×Dic311yescharacteristic
154#95C2×Dic311yescharacteristic
155C23×C3—21no—
156C23×C3—21no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—41no—
161C23×C3—41no—
162C23×C3—41no—
163C23×C3—41no—
164C23×C3#10412yes—
165C23×C3C2×C22⋊C411yescharacteristic
166C3×Q8—82no—
167C3×D4—44no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—41no—
174C2×C3×C4—42no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—81no—
191C2×C3×C4—81no—
192C2×C3×C4—81no—
193C2×C3×C4#9911yescharacteristic
194C2×C3×C4#9911yescharacteristic
195C2×C3×C4#13512yes—
196C3⋊C8—44no—
197C3⋊C8—82no—
198C24#21711yescharacteristic
199C24#21711yescharacteristic
200C2×Q8—41no—
201C2×Q8—81no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—81no—
214C22×C4—81no—
215C22×C4#21711yescharacteristic
216C22×C4#21711yescharacteristic
217C22×C4C22×Dic311yescharacteristic
218C22×C4C2×#6711yescharacteristic
219C22×C4C2×#6711yescharacteristic
220M16—64no—
221M16—122no—
222C2×C8—122no—
223C2×C8—122no—
224C2×C8—242no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—81no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C42—21no—
238C42—21no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—22no—
242C22×C3—42no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—81no—
249C22×C3—81no—
250C22×C3—81no—
251C22×C3C2×#9911yescharacteristic
252C22×C3#41611yescharacteristic
253C22×C3C2×#10411yescharacteristic
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—22no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—81no—
261C3×C4—81no—
262C3×C4—81no—
263C3×C4—81no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23#83712yes—
274C23C2×#21711yescharacteristic
275Q8—82no—
276D4—44no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4#83411yescharacteristic
303C2×C4#83411yescharacteristic
304C2×C4#95012yes—
305C8—124no—
306C8—242no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—42no—
310C2×C3—81no—
311C2×C3#150111yescharacteristic
312C2×C3#212111yescharacteristic
313C2×C3#213511yescharacteristic
314C22—21no—
315C22—21no—
316C22—22no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22C2×#83411yescharacteristic
327C22#523311yescharacteristic
328C22C2×#83711yescharacteristic
329C4—22no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—81no—
336C4—81no—
337C4—81no—
338C4—81no—
339C3#1173911yescharacteristic, complemented, Sylow
340C2—21no—
341C2—21no—
342C2—42no—
343C2—81no—
344C2#7170711yescharacteristic
345C2#7184711yescharacteristic
346C2#7187311yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2638C2
socleC22×C3C2×#99
prime radicalC22×C3×D4D4
cosocleC2×C3×C4⋊C4C23
hypercenter#2638D3
hypocenterC3#11739
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2638

Complements

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

NH ≅ G/Ntype
4. #67650C2semidirect
5. #67648C2semidirect
339. C3#11739semidirect

Maximal subgroups

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

subgroupindex
C3×#26382
C4×#8332
#716002
#676502
#676482
#117393