Finite Group Databasebeta

← #11713006

Subgroups — #11713006

Order 768.

531 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C2×C8×S4C211yesmaximal, characteristic
2#85307C212yesmaximal
3A4×#107C211yesmaximal, characteristic, complemented
4#73111C211yesmaximal, characteristic, complemented
5#73051C212yesmaximal, complemented
6#22656—31nomaximal, Sylow
7C2×C4×S4C2211yescharacteristic
8C2×C8×A4C2211yescharacteristic
9A4×C4⋊C4C2212yes—
10(C2×S4)⋊C4—22no—
11C4⋊SL(2,ℤ/4)C2212yes—
12C4×SL(2,ℤ/4)—22no—
13C2×A4⋊C8C2211yescharacteristic
14C8×S4C414yescomplemented
15#4880—41nomaximal
16C2×C8×D4—31no—
17C2×#427—31no—
18C22×#107D311yescharacteristic, complemented
19#2352—32no—
20#1949—32no—
21#1854—32no—
22#1853—32no—
23#1846—31no—
24#1808—31no—
25#1808—31no—
26C4×#107—31no—
27C22×S4D411yescharacteristic
28C2×C4×A4—22no—
29C2×C4×A4C2311yescharacteristic
30C2×SL(2,ℤ/4)—22no—
31C2×SL(2,ℤ/4)D411yescharacteristic
32C4×S4C2×C412yes—
33C4×S4Q812yes—
34C2×D3×C8—41no—
35#892—42no—
36C8×A4C2×C412yes—
37A4⋊C8C2×C412yes—
38C3×#107—41no—
39#818—41no—
40#808—42no—
41C23×C8—31no—
42C23×C8C2×D311yescharacteristic
43C2×C4×D4—31no—
44C2×#117—32no—
45C22×C4⋊C4C2×D312yes—
46C8×D4—34no—
47C8×D4—34no—
48#427—61no—
49#427—62no—
50#427—61no—
51C2×#107—31no—
52C2×#107—31no—
53C2×#107—32no—
54C2×#107—31no—
55C2×#107—31no—
56C2×#106—32no—
57C2×#106—32no—
58C2×C4⋊C8—31no—
59C2×C22⋊C8—31no—
60C2×C22⋊C8—31no—
61C2×C4×C8—31no—
62#388—62no—
63#387—62no—
64#386—62no—
65#382—32no—
66#380—32no—
67#380—32no—
68C4×C4⋊C4—32no—
69C4×C22⋊C4—62no—
70#340—32no—
71#340—32no—
72#340—32no—
73C22×A4C2×D411yescharacteristic
74C2×S4—21no—
75C2×S4—21no—
76C2×S4C4⋊C412yes—
77C2×C4×D3—41no—
78C4×A4—42no—
79C4×A4C22×C411yescharacteristic
80C4×A4C2×Q811yescharacteristic
81SL(2,ℤ/4)—42no—
82SL(2,ℤ/4)C4⋊C412yes—
83C2×C3×C8—41no—
84C3×C4⋊C4—42no—
85(C2×D3)⋊C4—82no—
86C4⋊Dic3—42no—
87C4×Dic3—82no—
88C2×C3⋊C8—41no—
89D3×C8—44no—
90C22×D4—31no—
91C23×C4—22no—
92C23×C4—31no—
93C23×C4C22×D311yescharacteristic
94C22×C8—31no—
95C22×C8—31no—
96C22×C8—32no—
97C22×C8—31no—
98C22×C8—34no—
99C22×C8—31no—
100C22×C8—62no—
101C22×C8—61no—
102C22×C8—61no—
103C22×C8C4×D312yes—
104C4×D4—32no—
105C4×D4—32no—
106C4×D4—32no—
107C4×D4—32no—
108#117—62no—
109#117—62no—
110#117—122no—
111C2×C4⋊C4—31no—
112C2×C4⋊C4—32no—
113C2×C4⋊C4—32no—
114C2×C4⋊C4—32no—
115C2×C4⋊C4—32no—
116C2×C4⋊C4—62no—
117C2×C22⋊C4—31no—
118C2×C22⋊C4—31no—
119C2×C22⋊C4—62no—
120C2×C22⋊C4—62no—
121C2×C22⋊C4—62no—
122C2×C42—31no—
123C2×C42—62no—
124C2×C42—62no—
125#107—31no—
126#107—32no—
127#107—61no—
128#107—62no—
129#107—61no—
130#107—61no—
131#107S411yescharacteristic, complemented
132#106—62no—
133#106—62no—
134#106—122no—
135C4⋊C8—34no—
136C22⋊C8—34no—
137C22⋊C8—32no—
138C22⋊C8—32no—
139C4×C8—34no—
140#95—62no—
141#95—62no—
142#95—62no—
143#95—62no—
144#95—62no—
145C22×D3—41no—
146C2×A4C2×C4⋊C411yescharacteristic
147C2×A4#13511yescharacteristic
148C2×A4#13511yescharacteristic
149S4—22no—
150C2×C3×C4—41no—
151C2×C3×C4—82no—
152C2×Dic3—41no—
153C2×Dic3—82no—
154C4×D3—42no—
155C4×D3—42no—
156C3×C8—42no—
157C3⋊C8—42no—
158C24—31no—
159C24D3×D411yescharacteristic
160C2×D4—32no—
161C2×D4—32no—
162C2×D4—61no—
163C2×D4—61no—
164C2×D4—61no—
165C2×D4—61no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—31no—
171C22×C4—31no—
172C22×C4—31no—
173C22×C4—32no—
174C22×C4—32no—
175C22×C4—42no—
176C22×C4—62no—
177C22×C4—61no—
178C22×C4—61no—
179C22×C4—61no—
180C22×C4—61no—
181C22×C4—62no—
182C22×C4—62no—
183C22×C4—62no—
184C22×C4—122no—
185C22×C4C2×C4×D311yescharacteristic
186C22×C4D3×Q811yescharacteristic
187C2×C8—31no—
188C2×C8—32no—
189C2×C8—32no—
190C2×C8—32no—
191C2×C8—32no—
192C2×C8—32no—
193C2×C8—61no—
194C2×C8—64no—
195C2×C8—61no—
196C2×C8—61no—
197C2×C8—61no—
198C2×C8—61no—
199C2×C8—64no—
200C2×C8C2×S411yescharacteristic
201C4⋊C4—32no—
202C4⋊C4—32no—
203C4⋊C4—32no—
204C4⋊C4—62no—
205C4⋊C4—62no—
206C4⋊C4—62no—
207C4⋊C4—62no—
208C4⋊C4—62no—
209C4⋊C4—62no—
210C4⋊C4C2×S412yes—
211C22⋊C4—32no—
212C22⋊C4—32no—
213C22⋊C4—32no—
214C22⋊C4—32no—
215C22⋊C4—122no—
216C22⋊C4—122no—
217C22⋊C4—122no—
218C22⋊C4—122no—
219C22⋊C4—242no—
220C42—32no—
221C42—32no—
222C42—122no—
223C42—122no—
224C42—242no—
225C22×C3—41no—
226C2×D3—42no—
227C2×D3—81no—
228C2×D3—81no—
229A4#42711yescharacteristic, complemented
230C3×C4—41no—
231C3×C4—41no—
232C3×C4—162no—
233Dic3—42no—
234Dic3—162no—
235C23—31no—
236C23—31no—
237C23—31no—
238C23—31no—
239C23—32no—
240C23—61no—
241C23—61no—
242C23—61no—
243C23—61no—
244C23—61no—
245C23—61no—
246C23D3×C4⋊C411yescharacteristic
247C23#91211yescharacteristic
248C23#91911yescharacteristic
249D4—62no—
250D4—62no—
251D4—62no—
252D4—62no—
253C2×C4—22no—
254C2×C4—31no—
255C2×C4—31no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—32no—
262C2×C4—32no—
263C2×C4—32no—
264C2×C4—61no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—61no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—62no—
274C2×C4—62no—
275C2×C4—62no—
276C2×C4—122no—
277C2×C4—122no—
278C2×C4—122no—
279C2×C4—122no—
280C2×C4—122no—
281C2×C4—122no—
282C2×C4—122no—
283C2×C4—122no—
284C2×C4—122no—
285C2×C4—122no—
286C2×C4C22×S411yescharacteristic
287C8—32no—
288C8—62no—
289C8—62no—
290C8C4×S412yes—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3—41no—
294D3—82no—
295C22—31no—
296C22—31no—
297C22—31no—
298C22—31no—
299C22—31no—
300C22—31no—
301C22—31no—
302C22—31no—
303C22—31no—
304C22—61no—
305C22—62no—
306C22—62no—
307C22—62no—
308C22—121no—
309C22—121no—
310C22—121no—
311C22—121no—
312C22#488011yescharacteristic, complemented
313C22D4×S411yescharacteristic
314C4—31no—
315C4—31no—
316C4—42no—
317C4—62no—
318C4—62no—
319C4—62no—
320C4—122no—
321C4—242no—
322C4—242no—
323C4C2×C4×S411yescharacteristic
324C4Q8×S411yescharacteristic
325C3—41noSylow
326C2—31no—
327C2—31no—
328C2—31no—
329C2—31no—
330C2—122no—
331C2C4⋊C4×S411yescharacteristic
332C2#8545811yescharacteristic
333C2#8546511yescharacteristic
subgroupG/H
centerC22D4×S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC22×#107D3
socleC24D3×D4
prime radicalC22×S4D4
cosocleC2×C4×A4C23
hypercenter#107S4
hypocenterA4#427
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
3. A4×#107C2semidirect
4. #73111C2semidirect
5. #73051C2semidirect
14. C8×S4C4semidirect
18. C22×#107D3semidirect
131. #107S4semidirect
229. A4#427semidirect
312. C22#4880semidirect

Maximal subgroups

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

subgroupindex
C2×C8×S42
#853072
A4×#1072
#731112
#730512
#226563
#48804