Finite Group Databasebeta

← #10691176

Subgroups — #10691176

Order 768.

478 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2632C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic, complemented
3#71718C212yesmaximal
4C4×#836C211yesmaximal, characteristic, complemented
5#67633C212yesmaximal, complemented
6#9188—31nomaximal, Sylow
7C2×C3×#124C2211yescharacteristic
8C2×C3×C4×Q8C2211yescharacteristic
9C3×#399—21no—
10C3×#395C411yescharacteristic
11C3×#395C411yescharacteristic
12C3×#394C411yescharacteristic
13C3×#394C411yescharacteristic
14C3×#393—21no—
15C3×#388—21no—
16C3×#388—21no—
17C3×#383C2211yescharacteristic
18C2×#836—21no—
19#5024—24no—
20#4926—21no—
21#4926C2212yes—
22#4925—21no—
23C4×#208C2212yes—
24#4540—24no—
25#4530—21no—
26#2983—31no—
27#2632D311yescharacteristic
28C4×#101—31no—
29#1527—32no—
30#1288—32no—
31C22×C3×Q8C2×C411yescharacteristic
32C22×C3×Q8D411yescharacteristic
33C22×C3×D4C2×C411yescharacteristic
34C3×#124—22no—
35C3×#124—22no—
36C3×C4×Q8—22no—
37C3×C4×Q8—22no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C42D411yescharacteristic
50C2×C3×C42D411yescharacteristic
51C2×C3×C42C2311yescharacteristic
52C2×#208—22no—
53C3×#95—21no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95C2×C411yescharacteristic
58C3×#95C2×C411yescharacteristic
59#836—24no—
60#830—22no—
61C4⋊(C3⋊C8)—22no—
62C4⋊(C3⋊C8)—22no—
63C4⋊(C3⋊C8)—22no—
64#803—22no—
65#803—22no—
66#803D414yes—
67C4×C3⋊C8—22no—
68C4×C3⋊C8—22no—
69C2×#124C2×D311yescharacteristic
70C2×C4×Q8C2×D311yescharacteristic
71#446—64no—
72#432—32no—
73#432—61no—
74#431—61no—
75C2×#101—61no—
76C4×M16—32no—
77#399—21no—
78#395Dic311yescharacteristic
79#395Dic311yescharacteristic
80#394Dic311yescharacteristic
81#394Dic311yescharacteristic
82#393—21no—
83#388—21no—
84#388—21no—
85#383C2×D311yescharacteristic
86#343—61no—
87#329—64no—
88C24×C3—21no—
89C2×C3×Q8—22no—
90C2×C3×Q8—22no—
91C2×C3×Q8—22no—
92C2×C3×Q8—41no—
93C2×C3×Q8—81no—
94C2×C3×D4—22no—
95C2×C3×D4—81no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C3×C4⋊C4—22no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—81no—
110C3×C4⋊C4—81no—
111C3×C22⋊C4—42no—
112C3×C22⋊C4—42no—
113C3×C22⋊C4—41no—
114C3×C22⋊C4—42no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—81no—
117C3×C22⋊C4—81no—
118C3×C42—21no—
119C3×C42—21no—
120C3×C42—21no—
121C3×C42—21no—
122C3×C42C2×D412yes—
123C3×C42#4112yes—
124#208—24no—
125#208—42no—
126C2×C3⋊C8—24no—
127C2×C3⋊C8—42no—
128C22×Q8C2×Dic311yescharacteristic
129C22×Q8#6711yescharacteristic
130C22×D4C2×Dic311yescharacteristic
131C2×M16—62no—
132#124—22no—
133#124—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4C2×Dic311yescharacteristic
139C2×C4⋊C4C2×Dic311yescharacteristic
140C2×C4⋊C4#6711yescharacteristic
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C42#6711yescharacteristic
148C2×C42#6711yescharacteristic
149C2×C42C22×D311yescharacteristic
150C4⋊C8—62no—
151C4⋊C8—62no—
152C4⋊C8—62no—
153#101—64no—
154C22⋊C8—62no—
155#97—34no—
156#97—62no—
157#97—62no—
158C4×C8—62no—
159C4×C8—62no—
160#95—21no—
161#95—21no—
162#95—21no—
163#95—21no—
164#95C2×Dic311yescharacteristic
165#95C2×Dic311yescharacteristic
166C23×C3—41no—
167C23×C3—41no—
168C23×C3—41no—
169C23×C3—41no—
170C23×C3—41no—
171C23×C3—41no—
172C23×C3—41no—
173C23×C3C2×C22⋊C411yescharacteristic
174C3×Q8—44no—
175C3×Q8—82no—
176C3×D4—82no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4#10412yes—
200C2×C3×C4C4×D412yes—
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12112yes—
203C2×C3×C4#12311yescharacteristic
204C3⋊C8—44no—
205C3⋊C8—44no—
206C24—21no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—41no—
211C2×Q8—81no—
212C2×D4—22no—
213C2×D4—81no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4#21711yescharacteristic
218C22×C4#21711yescharacteristic
219C22×C4#21711yescharacteristic
220C22×C4#21711yescharacteristic
221C22×C4C22×Dic311yescharacteristic
222C22×C4C2×#6711yescharacteristic
223C22×C4C2×#6711yescharacteristic
224M16—64no—
225M16—122no—
226C2×C8—64no—
227C2×C8—122no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—42no—
238C22⋊C4—81no—
239C22⋊C4—81no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42D3×D412yes—
245C42#23712yes—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—81no—
249C22×C3—81no—
250C22×C3—81no—
251C22×C3—81no—
252C22×C3—81no—
253C22×C3—81no—
254C22×C3—81no—
255C22×C3#38611yescharacteristic
256C22×C3C2×#10411yescharacteristic
257C22×C3#42111yescharacteristic
258C3×C4—22no—
259C3×C4—22no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—22no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—81no—
267C3×C4—81no—
268C3×C4—81no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23C2×#21711yescharacteristic
277Q8—44no—
278Q8—82no—
279D4—82no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
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—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4#83712yes—
303C2×C4D4×Dic312yes—
304C2×C4#93511yescharacteristic
305C2×C4#93711yescharacteristic
306C2×C4#93912yes—
307C8—124no—
308C8—124no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—161no—
312C2×C3#187111yescharacteristic
313C2×C3#191211yescharacteristic
314C2×C3#191411yescharacteristic
315C22—21no—
316C22—21no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22#522011yescharacteristic
325C22C2×#83711yescharacteristic
326C22#523611yescharacteristic
327C4—22no—
328C4—22no—
329C4—22no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—81no—
336C4—81no—
337C4—81no—
338C3#918811yescharacteristic, complemented, Sylow
339C2—21no—
340C2—21no—
341C2—161no—
342C2#7171311yescharacteristic
343C2#7185511yescharacteristic
344C2#7186411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2632C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenter#2632D3
hypocenterC3#9188
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2632

Complements

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

NH ≅ G/Ntype
2. #82436C2semidirect
4. C4×#836C2semidirect
5. #67633C2semidirect
338. C3#9188semidirect

Maximal subgroups

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

subgroupindex
C3×#26322
#824362
#717182
C4×#8362
#676332
#91883