Finite Group Databasebeta

← #11712544

Subgroups — #11712544

Order 768.

697 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 697 conjugacy classes of subgroups — too many to draw. Only its 117 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
1A4×#117C211yesmaximal, characteristic
2C2×#5409C211yesmaximal, characteristic
3C22×A4⋊C8C211yesmaximal, characteristic
4#73044C212yesmaximal, complemented
5C4×A4⋊C8C212yesmaximal, complemented
6#20379—31nomaximal, Sylow
7C22×C4×A4C2211yescharacteristic
8A4×C4⋊C4C412yes—
9A4×C22⋊C4C412yes—
10C42×A4C2212yes—
11#5409C414yescomplemented
12C2×A4⋊C8C414yescomplemented
13C2×A4⋊C8C2211yescharacteristic
14C2×A4⋊C8C2212yes—
15C2×A4⋊C8C2211yescharacteristic
16#4997—41nomaximal
17C22×#117D311yescharacteristic
18C2×#407—31no—
19C22×C22⋊C8—31no—
20C2×#405—31no—
21C2×#405—31no—
22#1750—32no—
23#1750—32no—
24#1749—32no—
25C4×C22⋊C8—32no—
26#1729—31no—
27#1729—31no—
28C23×A4C2×C411yescharacteristic
29C2×C4×A4C2×C412yes—
30C2×C4×A4C2×C412yes—
31C2×C4×A4C2×C414yes—
32C2×C4×A4C2×C411yescharacteristic
33C2×C4×A4C2311yescharacteristic
34C3×#117—41no—
35C2×#208—41no—
36C22×C3⋊C8—41no—
37A4⋊C8C2×C412yes—
38A4⋊C8C2×C412yes—
39A4⋊C8C2×C414yes—
40#803—42no—
41C4×C3⋊C8—42no—
42C24×C4C2×D311yescharacteristic
43C22×M16—31no—
44C22×M16—31no—
45C23×C8—31no—
46C23×C8—31no—
47C2×#117—31no—
48C2×#117—32no—
49C2×#117—32no—
50C2×#117—31no—
51C2×#117—32no—
52C22×C4⋊C4Dic312yes—
53C22×C22⋊C4Dic312yes—
54C22×C42C2×D312yes—
55#407—34no—
56#407—34no—
57C2×C22⋊C8—34no—
58C2×C22⋊C8—31no—
59C2×C22⋊C8—31no—
60C2×C22⋊C8—32no—
61C2×C22⋊C8—34no—
62C2×C22⋊C8—31no—
63C2×C22⋊C8—31no—
64C2×C22⋊C8—32no—
65#405—61no—
66#405—62no—
67#405—62no—
68#405—62no—
69#405—61no—
70C2×#97—32no—
71C2×#97—32no—
72C2×C4×C8—32no—
73C2×C4×C8—32no—
74#336—32no—
75#336—32no—
76#336—32no—
77#336—32no—
78C22×A4—22no—
79C22×A4C4212yes—
80C22×A4C22×C411yescharacteristic
81C22×C3×C4—41no—
82C4×A4—24no—
83C4×A4C4212yes—
84C4×A4C22×C411yescharacteristic
85C4×A4C22×C411yescharacteristic
86C3×C4⋊C4—42no—
87C3×C22⋊C4—42no—
88C3×C42—42no—
89#208—44no—
90C2×C3⋊C8—44no—
91C2×C3⋊C8—41no—
92C2×C3⋊C8—42no—
93C2×C3⋊C8—41no—
94C25C2×Dic311yescharacteristic
95C23×C4—31no—
96C23×C4—31no—
97C23×C4—31no—
98C23×C4—31no—
99C23×C4—32no—
100C23×C4—32no—
101C23×C4C4×D312yes—
102C23×C4C4×D312yes—
103C23×C4C2×Dic314yes—
104C23×C4C2×Dic311yescharacteristic
105C23×C4C22×D311yescharacteristic
106C2×M16—34no—
107C2×M16—34no—
108C2×M16—62no—
109C2×M16—62no—
110C2×M16—61no—
111C2×M16—61no—
112C2×M16—61no—
113C2×M16—61no—
114C22×C8—31no—
115C22×C8—31no—
116C22×C8—31no—
117C22×C8—31no—
118C22×C8—34no—
119C22×C8—34no—
120C22×C8—32no—
121C22×C8—32no—
122C22×C8—62no—
123C22×C8—62no—
124C22×C8—61no—
125C22×C8—61no—
126C22×C8—61no—
127C22×C8—61no—
128#117—31no—
129#117—32no—
130#117—32no—
131#117—32no—
132#117—61no—
133#117—64no—
134#117—61no—
135#117—61no—
136#117S411yescharacteristic
137C2×C4⋊C4—32no—
138C2×C4⋊C4—32no—
139C2×C4⋊C4—34no—
140C2×C22⋊C4—32no—
141C2×C22⋊C4—32no—
142C2×C22⋊C4—34no—
143C2×C42—32no—
144C2×C42—32no—
145C2×C42—34no—
146C22⋊C8—34no—
147C22⋊C8—34no—
148C22⋊C8—32no—
149C22⋊C8—32no—
150C22⋊C8—38no—
151C22⋊C8—34no—
152C22⋊C8—32no—
153C22⋊C8—32no—
154C22⋊C8—34no—
155#97—64no—
156#97—62no—
157#97—62no—
158C4×C8—64no—
159C4×C8—62no—
160C4×C8—62no—
161C23×C3—41no—
162C2×A4—22no—
163C2×A4C2×C4211yescharacteristic
164C2×A4#13112yes—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—41no—
168C2×C3×C4—44no—
169C2×C3×C4—41no—
170C3⋊C8—42no—
171C3⋊C8—42no—
172C3⋊C8—44no—
173C24—22no—
174C24—31no—
175C24—31no—
176C24—32no—
177C24—62no—
178C24C4×Dic312yes—
179C24C22×Dic311yescharacteristic
180C22×C4—24no—
181C22×C4—31no—
182C22×C4—32no—
183C22×C4—32no—
184C22×C4—32no—
185C22×C4—32no—
186C22×C4—31no—
187C22×C4—31no—
188C22×C4—31no—
189C22×C4—31no—
190C22×C4—32no—
191C22×C4—32no—
192C22×C4—31no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—31no—
196C22×C4—32no—
197C22×C4—32no—
198C22×C4—32no—
199C22×C4—32no—
200C22×C4—32no—
201C22×C4—34no—
202C22×C4—34no—
203C22×C4—61no—
204C22×C4—61no—
205C22×C4—62no—
206C22×C4—61no—
207C22×C4—62no—
208C22×C4—64no—
209C22×C4C4×Dic312yes—
210C22×C4C2×C4×D311yescharacteristic
211C22×C4C2×C4×D311yescharacteristic
212C22×C4C2×S411yescharacteristic
213M16—64no—
214M16—64no—
215M16—64no—
216M16—64no—
217C2×C8—32no—
218C2×C8—32no—
219C2×C8—32no—
220C2×C8—32no—
221C2×C8—34no—
222C2×C8—34no—
223C2×C8—61no—
224C2×C8—62no—
225C2×C8—61no—
226C2×C8—64no—
227C2×C8—61no—
228C2×C8—61no—
229C2×C8—64no—
230C2×C8—61no—
231C2×C8—61no—
232C2×C8—64no—
233C2×C8—62no—
234C2×C8—62no—
235C2×C8—64no—
236C2×C8—61no—
237C2×C8—62no—
238C2×C8—61no—
239C4⋊C4—34no—
240C4⋊C4—32no—
241C4⋊C4—62no—
242C4⋊C4SL(2,ℤ/4)12yes—
243C22⋊C4—32no—
244C22⋊C4—34no—
245C22⋊C4—62no—
246C22⋊C4SL(2,ℤ/4)12yes—
247C42—34no—
248C42—32no—
249C42—62no—
250C42C2×S412yes—
251C22×C3—42no—
252C22×C3—41no—
253C22×C3—82no—
254A4#40511yescharacteristic, complemented
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—42no—
258C3×C4—84no—
259C23—22no—
260C23—31no—
261C23—31no—
262C23—32no—
263C23—32no—
264C23—32no—
265C23—32no—
266C23—31no—
267C23—62no—
268C23—62no—
269C23—61no—
270C23—62no—
271C23—64no—
272C23—62no—
273C23—62no—
274C23—122no—
275C23C2×C4×Dic311yescharacteristic
276C23#94812yes—
277C23C2×SL(2,ℤ/4)11yescharacteristic
278C2×C4—32no—
279C2×C4—32no—
280C2×C4—31no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—32no—
287C2×C4—32no—
288C2×C4—32no—
289C2×C4—32no—
290C2×C4—32no—
291C2×C4—32no—
292C2×C4—32no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—34no—
296C2×C4—64no—
297C2×C4—64no—
298C2×C4—61no—
299C2×C4—62no—
300C2×C4—62no—
301C2×C4—61no—
302C2×C4—64no—
303C2×C4—64no—
304C2×C4C4×S412yes—
305C2×C4C4×S412yes—
306C2×C4C2×SL(2,ℤ/4)14yes—
307C2×C4C2×SL(2,ℤ/4)11yescharacteristic
308C2×C4C22×S411yescharacteristic
309C8—62no—
310C8—62no—
311C8—64no—
312C8—62no—
313C8—62no—
314C8—64no—
315C2×C3—42no—
316C2×C3—41no—
317C2×C3—82no—
318C22—22no—
319C22—31no—
320C22—32no—
321C22—32no—
322C22—32no—
323C22—31no—
324C22—31no—
325C22—32no—
326C22—62no—
327C22—62no—
328C22—64no—
329C22—61no—
330C22—62no—
331C22—62no—
332C22—62no—
333C22—122no—
334C22#499711yescharacteristic, complemented
335C22C4×SL(2,ℤ/4)12yes—
336C22C22×SL(2,ℤ/4)11yescharacteristic
337C4—24no—
338C4—31no—
339C4—31no—
340C4—32no—
341C4—64no—
342C4C4×SL(2,ℤ/4)12yes—
343C4C2×C4×S411yescharacteristic
344C4C2×C4×S411yescharacteristic
345C3—41noSylow
346C2—22no—
347C2—31no—
348C2—31no—
349C2—32no—
350C2—62no—
351C2C2×C4×SL(2,ℤ/4)11yescharacteristic
352C2#8551112yes—
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC2×A4C2×C42
FittingC22×#117D3
socleC24C22×Dic3
prime radicalC23×A4C2×C4
cosocleC2×C4×A4C23
hypercenter#117S4
hypocenterA4#405
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
4. #73044C2semidirect
5. C4×A4⋊C8C2semidirect
11. #5409C4semidirect
12. C2×A4⋊C8C4semidirect
254. A4#405semidirect
334. C22#4997semidirect

Maximal subgroups

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

subgroupindex
A4×#1172
C2×#54092
C22×A4⋊C82
#730442
C4×A4⋊C82
#203793
#49974