Finite Group Databasebeta

← #11714117

Subgroups — #11714117

Order 768.

431 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 431 conjugacy classes of subgroups — too many to draw. Only its 65 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#85284C211yesmaximal, characteristic
2#85283C211yesmaximal, characteristic
3A4×#97C211yesmaximal, characteristic
4#73111C212yesmaximal
5#73110C212yesmaximal
6#33217—31nomaximal, Sylow
7C2×A4⋊Q8—21no—
8C2×C8×A4C2212yes—
9C42×A4C2211yescharacteristic
10C4⋊SL(2,ℤ/4)—21no—
11C4⋊SL(2,ℤ/4)C2212yes—
12C4⋊SL(2,ℤ/4)C2211yescharacteristic
13C4⋊SL(2,ℤ/4)C2211yescharacteristic
14SL(2,ℤ/4)⋊C4—21no—
15SL(2,ℤ/4)⋊C4—21no—
16#4702—41nomaximal
17C2×#501—31no—
18C22×#97D311yescharacteristic
19#2857—31no—
20#2851—31no—
21#1993—31no—
22#1992—31no—
23#1944—32no—
24#1943—32no—
25#1847—31no—
26#1808—32no—
27#1807—32no—
28C2×C4×A4D411yescharacteristic
29C2×C4×A4D411yescharacteristic
30C2×C4×A4C2311yescharacteristic
31C2×SL(2,ℤ/4)—22no—
32C2×SL(2,ℤ/4)—21no—
33C2×SL(2,ℤ/4)—21no—
34A4⋊Q8—42no—
35#870—41no—
36#869—41no—
37C8×A4Q814yescomplemented
38C3×#97—41no—
39#818—42no—
40#817—42no—
41C23×C8C2×D312yes—
42C2×C4⋊Q8—31no—
43C2×#125—31no—
44C2×C22⋊Q8—61no—
45C22×C42C2×D311yescharacteristic
46#501—62no—
47#501—61no—
48#501—61no—
49C2×#107—32no—
50C2×#107—32no—
51C2×#106—32no—
52C2×#106—32no—
53C2×#97—31no—
54C2×#97—32no—
55C2×#97—31no—
56#391—61no—
57#389—31no—
58#389—31no—
59#389—32no—
60#389—61no—
61#387—61no—
62#387—61no—
63#385—61no—
64#385—61no—
65#384—31no—
66#382—31no—
67#380—31no—
68#380—32no—
69#380—31no—
70#380—61no—
71#340—32no—
72#340—32no—
73#340—32no—
74#340—32no—
75C22×A4C2×D411yescharacteristic
76C2×Dic6—81no—
77C4×A4—21no—
78C4×A4—21no—
79C4×A4C2×Q812yes—
80SL(2,ℤ/4)—42no—
81SL(2,ℤ/4)—41no—
82SL(2,ℤ/4)—41no—
83C2×C3×C8—42no—
84C3×C42—41no—
85C4⋊Dic3—41no—
86C4⋊Dic3—42no—
87C4⋊Dic3—41no—
88C4⋊Dic3—81no—
89Dic3⋊C4—81no—
90Dic3⋊C4—81no—
91C22×Q8—61no—
92C23×C4D1211yescharacteristic
93C23×C4D1211yescharacteristic
94C23×C4C22×D311yescharacteristic
95C22×C8—32no—
96C22×C8—34no—
97C22×C8—32no—
98C22×C8Dic614yescomplemented
99C4⋊Q8—61no—
100C4⋊Q8—61no—
101C4⋊Q8—61no—
102C4⋊Q8—61no—
103#125—61no—
104#125—61no—
105#125—61no—
106#125—61no—
107C22⋊Q8—122no—
108C22⋊Q8—122no—
109C2×C4⋊C4—31no—
110C2×C4⋊C4—31no—
111C2×C4⋊C4—32no—
112C2×C4⋊C4—31no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—32no—
115C2×C4⋊C4—61no—
116C2×C4⋊C4—62no—
117C2×C4⋊C4—61no—
118C2×C4⋊C4—61no—
119C2×C4⋊C4—61no—
120C2×C4⋊C4—61no—
121C2×C4⋊C4—61no—
122C2×C4⋊C4—61no—
123C2×C22⋊C4—62no—
124C2×C22⋊C4—61no—
125C2×C22⋊C4—61no—
126C2×C42—31no—
127C2×C42—31no—
128C2×C42—31no—
129C2×C42—31no—
130#107—64no—
131#107—62no—
132#107—62no—
133#106—64no—
134#106—62no—
135#106—62no—
136#97—32no—
137#97—31no—
138#97—61no—
139#97S411yescharacteristic
140#95—61no—
141#95—61no—
142#95—62no—
143#95—62no—
144#95—61no—
145#95—61no—
146#95—61no—
147#95—61no—
148C2×A4C4⋊Q811yescharacteristic
149C2×A4AGL(1,ℤ/8)11yescharacteristic
150C2×A4#13711yescharacteristic
151C2×C3×C4—41no—
152C2×C3×C4—41no—
153C2×C3×C4—41no—
154C2×Dic3—81no—
155C2×Dic3—81no—
156C2×Dic3—82no—
157Dic6—162no—
158C3×C8—44no—
159C24C2×D1211yescharacteristic
160C2×Q8—121no—
161C2×Q8—122no—
162C2×Q8—122no—
163C2×Q8—121no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—32no—
169C22×C4—31no—
170C22×C4—31no—
171C22×C4—31no—
172C22×C4—31no—
173C22×C4—61no—
174C22×C4—61no—
175C22×C4—61no—
176C22×C4—61no—
177C22×C4—62no—
178C22×C4—61no—
179C22×C4—61no—
180C22×C4—62no—
181C22×C4C2×Dic612yes—
182C2×C8—32no—
183C2×C8—34no—
184C2×C8—34no—
185C2×C8—34no—
186C2×C8—62no—
187C2×C8C2×S412yes—
188C4⋊C4—62no—
189C4⋊C4—62no—
190C4⋊C4—62no—
191C4⋊C4—62no—
192C4⋊C4—61no—
193C4⋊C4—61no—
194C4⋊C4—62no—
195C4⋊C4—61no—
196C4⋊C4—62no—
197C4⋊C4—61no—
198C4⋊C4—121no—
199C4⋊C4—121no—
200C4⋊C4—121no—
201C4⋊C4—122no—
202C4⋊C4—122no—
203C4⋊C4—122no—
204C4⋊C4—122no—
205C4⋊C4—124no—
206C4⋊C4—121no—
207C4⋊C4—121no—
208C4⋊C4—121no—
209C22⋊C4—121no—
210C22⋊C4—121no—
211C22⋊C4—122no—
212C22⋊C4—122no—
213C22⋊C4—121no—
214C22⋊C4—121no—
215C42—31no—
216C42—31no—
217C42—31no—
218C42—61no—
219C42C2×S411yescharacteristic
220C22×C3—41no—
221A4#50111yescharacteristic, complemented
222C3×C4—42no—
223C3×C4—81no—
224C3×C4—81no—
225Dic3—162no—
226Dic3—161no—
227Dic3—161no—
228C23—31no—
229C23—31no—
230C23—31no—
231C23—31no—
232C23#86911yescharacteristic
233C23#90811yescharacteristic
234C23#90911yescharacteristic
235Q8—242no—
236Q8—242no—
237C2×C4—31no—
238C2×C4—32no—
239C2×C4—32no—
240C2×C4—32no—
241C2×C4—31no—
242C2×C4—31no—
243C2×C4—61no—
244C2×C4—61no—
245C2×C4—61no—
246C2×C4—61no—
247C2×C4—61no—
248C2×C4—61no—
249C2×C4—61no—
250C2×C4—61no—
251C2×C4—61no—
252C2×C4—122no—
253C2×C4—122no—
254C2×C4—121no—
255C2×C4—121no—
256C2×C4—121no—
257C2×C4—121no—
258C2×C4—121no—
259C2×C4—122no—
260C2×C4—121no—
261C2×C4—121no—
262C2×C4—121no—
263C2×C4—122no—
264C2×C4—121no—
265C2×C4—121no—
266C2×C4—122no—
267C2×C4—121no—
268C2×C4—121no—
269C2×C4—122no—
270C2×C4C4⋊S411yescharacteristic
271C2×C4C4⋊S411yescharacteristic
272C2×C4C22×S411yescharacteristic
273C8—34no—
274C8A4⋊Q814yescomplemented
275C2×C3—41no—
276C2×C3—41no—
277C2×C3—41no—
278C22—31no—
279C22—31no—
280C22—31no—
281C22—31no—
282C22—31no—
283C22—31no—
284C22—31no—
285C22—31no—
286C22—31no—
287C22—61no—
288C22#470211yescharacteristic, complemented
289C22C2×C4⋊S411yescharacteristic
290C4—21no—
291C4—21no—
292C4—32no—
293C4—61no—
294C4—61no—
295C4—242no—
296C4—241no—
297C4—241no—
298C4—242no—
299C4—241no—
300C4—241no—
301C4C2×A4⋊Q812yes—
302C3—41noSylow
303C2—31no—
304C2—31no—
305C2—31no—
306C2—31no—
307C2#8528311yescharacteristic
308C2#8544311yescharacteristic
309C2#8544411yescharacteristic
subgroupG/H
centerC22C2×C4⋊S4
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×#97D3
socleC24C2×D12
prime radicalC22×A4C2×D4
hypercenter#97S4
hypocenterA4#501
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
37. C8×A4Q8semidirect
98. C22×C8Dic6semidirect
221. A4#501semidirect
274. C8A4⋊Q8semidirect
288. C22#4702semidirect

Maximal subgroups

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

subgroupindex
#852842
#852832
A4×#972
#731112
#731102
#332173
#47024