Finite Group Databasebeta

← #857⋊Q8 #11713670

Subgroups — #857⋊Q8 #11713670

Order 768.

482 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 34 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#2868⋊C3C211yesmaximal, characteristic, complemented
2#857⋊C4C212yesmaximal
3#72992C212yesmaximal, complemented
4C4×#857C211yesmaximal, characteristic
5#72980C211yesmaximal, characteristic, complemented
6#25830—31nomaximal, Sylow
7Q8×S4—41nomaximal
8C2×#857C2211yescharacteristic
9#384⋊C3C2212yes—
10C4×C42⋊C3C2211yescharacteristic
11#4623C2212yes—
12#4623C2211yescharacteristic
13#3503—31no—
14#3165—31no—
15#3079—31no—
16#2868D311yescharacteristic, complemented
17#2060—32no—
18#2060—32no—
19#1997—31no—
20#1994—31no—
21#1859—32no—
22#1859—32no—
23C4×#104—31no—
24Q8×A4—41no—
25C4×S4—41no—
26C4×S4—42no—
27A4⋊Q8—41no—
28A4⋊Q8—42no—
29C2×C42⋊C3C2311yescharacteristic
30#857Q812yescomplemented
31Q82—61no—
32#557—61no—
33#554—62no—
34#554—61no—
35D4×Q8—61no—
36D4×Q8—61no—
37D4×Q8—61no—
38#544—32no—
39#544—62no—
40#533—31no—
41#533—61no—
42C2×C4⋊Q8—31no—
43C2×C4⋊Q8—31no—
44C2×C4⋊Q8—61no—
45#520—32no—
46C4×#41—31no—
47#501—62no—
48#500—61no—
49#498—61no—
50#477—61no—
51#476—61no—
52#475—61no—
53#474—61no—
54#428—32no—
55#428—32no—
56#428—32no—
57#423—31no—
58C2×#104—31no—
59C2×#104—31no—
60#417—32no—
61C4×M16—31no—
62#384—32no—
63#384C2×D312yes—
64C43C2×D311yescharacteristic
65#343—32no—
66#343—32no—
67#339—31no—
68#339—32no—
69#339—31no—
70#339—32no—
71C2×S4—41no—
72D3×Q8—161no—
73C4×A4—41no—
74C4×A4—42no—
75SL(2,ℤ/4)—42no—
76SL(2,ℤ/4)—41no—
77C42⋊C3C2×Q811yescharacteristic, complemented
78C2×#41—31no—
79C22×Q8—31no—
80C22×Q8—61no—
81C22×Q8—61no—
82C22×Q8S411yescharacteristic
83C2×M16—31no—
84C2×M16—31no—
85C2×M16—32no—
86C4⋊Q8—31no—
87C4⋊Q8—31no—
88C4⋊Q8—61no—
89C4⋊Q8—61no—
90C4⋊Q8—61no—
91C4⋊Q8—61no—
92C4⋊Q8—61no—
93C4⋊Q8—61no—
94C4⋊Q8—61no—
95C4⋊Q8—61no—
96C4⋊Q8—62no—
97C4⋊Q8—62no—
98C4⋊Q8—61no—
99C4⋊Q8—61no—
100C4⋊Q8—62no—
101C4⋊Q8—62no—
102C4⋊Q8—62no—
103#125—32no—
104#125—62no—
105#125—62no—
106#125—62no—
107#125—62no—
108C22⋊Q8—61no—
109C22⋊Q8—62no—
110C22⋊Q8—61no—
111C22⋊Q8—61no—
112C22⋊Q8—62no—
113C22⋊Q8—62no—
114C22⋊Q8—62no—
115C22⋊Q8—62no—
116C22⋊Q8—62no—
117C22⋊Q8—61no—
118C22⋊Q8—62no—
119C4×Q8—31no—
120C4×Q8—31no—
121C4×Q8—61no—
122C4×Q8—61no—
123C4×Q8—62no—
124C4×Q8—61no—
125C4×Q8—61no—
126C4×Q8—62no—
127C4×D4—31no—
128C4×D4—31no—
129C4×D4—61no—
130C4×D4—62no—
131C4×D4—62no—
132C4×D4—61no—
133C4×D4—62no—
134#117—31no—
135#117—32no—
136#117—61no—
137#117—62no—
138C2×C4⋊C4—32no—
139C2×C4⋊C4—32no—
140C2×C4⋊C4—32no—
141C2×C4⋊C4—62no—
142C2×C4⋊C4—62no—
143C2×C4⋊C4—61no—
144C2×C42—31no—
145C2×C42—31no—
146C2×C42—61no—
147C2×C42C22×D311yescharacteristic
148#107—62no—
149#107—62no—
150#107—62no—
151#106—62no—
152#106—62no—
153#106—62no—
154C4⋊C8—61no—
155C4⋊C8—61no—
156#104—32no—
157#104—34no—
158#104—32no—
159#103—62no—
160#102—62no—
161#97—61no—
162C4×C8—61no—
163C2×A4—41no—
164S4—42no—
165C3×Q8—161no—
166C4×D3—161no—
167C4×D3—162no—
168Dic6—161no—
169Dic6—162no—
170#41—32no—
171#41—32no—
172#41—61no—
173#41—61no—
174C2×Q8—31no—
175C2×Q8—61no—
176C2×Q8—61no—
177C2×Q8—61no—
178C2×Q8—61no—
179C2×Q8—61no—
180C2×Q8—61no—
181C2×Q8—62no—
182C2×Q8—62no—
183C2×Q8—121no—
184C2×Q8—121no—
185C2×Q8—241no—
186C2×Q8—241no—
187C2×Q8—241no—
188C2×Q8—241no—
189C2×D4—31no—
190C2×D4—61no—
191C22×C4—31no—
192C22×C4—31no—
193C22×C4—32no—
194C22×C4—61no—
195C22×C4—61no—
196C22×C4—61no—
197C22×C4—62no—
198C22×C4—62no—
199C22×C4C2×S411yescharacteristic
200C22×C4C2×S412yes—
201M16—32no—
202M16—32no—
203M16—61no—
204M16—61no—
205M16—62no—
206M16—62no—
207C2×C8—61no—
208C2×C8—61no—
209C2×C8—62no—
210C4⋊C4—32no—
211C4⋊C4—31no—
212C4⋊C4—32no—
213C4⋊C4—31no—
214C4⋊C4—62no—
215C4⋊C4—62no—
216C4⋊C4—62no—
217C4⋊C4—62no—
218C4⋊C4—61no—
219C4⋊C4—61no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—62no—
223C4⋊C4—62no—
224C4⋊C4—62no—
225C4⋊C4—62no—
226C4⋊C4—62no—
227C4⋊C4—62no—
228C4⋊C4—62no—
229C4⋊C4—62no—
230C4⋊C4—62no—
231C4⋊C4—62no—
232C4⋊C4—62no—
233C4⋊C4—62no—
234C4⋊C4—121no—
235C4⋊C4—121no—
236C22⋊C4—61no—
237C22⋊C4—61no—
238C22⋊C4—62no—
239C22⋊C4—62no—
240C22⋊C4—62no—
241C22⋊C4—61no—
242C42—31no—
243C42—31no—
244C42—31no—
245C42—31no—
246C42—31no—
247C42—32no—
248C42—61no—
249C42—61no—
250C42—61no—
251C42—61no—
252C42—62no—
253C42—62no—
254C42—62no—
255C42D3×Q811yescharacteristic, complemented
256C2×D3—161no—
257A4—41no—
258C3×C4—162no—
259C3×C4—161no—
260Dic3—161no—
261Dic3—162no—
262C23—61no—
263C23C22×S411yescharacteristic
264Q8—32no—
265Q8—41no—
266Q8—61no—
267Q8—121no—
268Q8—122no—
269Q8—121no—
270Q8—121no—
271Q8—121no—
272Q8—121no—
273Q8—241no—
274Q8—242no—
275Q8—242no—
276D4—32no—
277D4—61no—
278D4—62no—
279D4—62no—
280C2×C4—31no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—62no—
291C2×C4—62no—
292C2×C4—62no—
293C2×C4—62no—
294C2×C4—61no—
295C2×C4—62no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—122no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—122no—
304C2×C4—242no—
305C2×C4—242no—
306C8—62no—
307C8—62no—
308C8—122no—
309C2×C3—161no—
310D3—162no—
311C22—31no—
312C22—31no—
313C22—62no—
314C22—121no—
315C22—121no—
316C22Q8×S411yescharacteristic
317C4—31no—
318C4—31no—
319C4—31no—
320C4—31no—
321C4—32no—
322C4—31no—
323C4—42no—
324C4—61no—
325C4—62no—
326C4—121no—
327C4—122no—
328C4—121no—
329C4—122no—
330C4—122no—
331C4#857⋊C211yescharacteristic
332C3—161noSylow
333C2—31no—
334C2—31no—
335C2—122no—
336C2C2×#857⋊C211yescharacteristic
subgroupG/H
centerC2C2×#857⋊C2
commutator, cosocleC2×C42⋊C3C23
Frattini, socleC23C22×S4
Fitting#2868D3
prime radicalC2×#857C22
hypercenterC4#857⋊C2
hypocenterC42⋊C3C2×Q8
LCSG ▷ C2×C42⋊C3 ▷ C42⋊C3
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #2868⋊C3C2semidirect
3. #72992C2semidirect
5. #72980C2semidirect
16. #2868D3semidirect
30. #857Q8semidirect
77. C42⋊C3C2×Q8semidirect
255. C42D3×Q8semidirect

Maximal subgroups

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

subgroupindex
#2868⋊C32
#857⋊C42
#729922
C4×#8572
#729802
#258303
Q8×S44