Finite Group Databasebeta

← #61840

Subgroups — #61840

Order 256.

964 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3497C211yesmaximal, characteristic, complemented
2Q8×#41C211yesmaximal, characteristic, complemented
3#3386C213yesmaximal, complemented
4#3384C213yesmaximal, complemented
5Q8×SD16C211yesmaximal, characteristic, complemented
6Q8×D8C211yesmaximal, characteristic, complemented
7#3368C213yesmaximal, complemented
8#3366C213yesmaximal, complemented
9#3362C211yesmaximal, characteristic, complemented
10#3358C211yesmaximal, characteristic, complemented
11#3235C213yesmaximal, complemented
12#3228C213yesmaximal, complemented
13#3115C213yesmaximal, complemented
14#3106C213yesmaximal, complemented
15#2984C211yesmaximal, characteristic, complemented
16C2×2−1+4—21no—
17Q82C2211yescharacteristic
18#557C2213yes—
19#554—23no—
20#554C2213yes—
21#550C2211yescharacteristic
22#550C2213yes—
23D4×Q8—21no—
24D4×Q8C2211yescharacteristic
25D4×Q8C2211yescharacteristic
26D4×Q8C2213yes—
27#548—21no—
28#548—23no—
29#537—23no—
30#533—23no—
31#533C2213yes—
32#532C2213yes—
33#532C2213yes—
34C4×#41C2213yes—
35C2×C4×Q8—21no—
36C2×C4×Q8C2211yescharacteristic
37#501C2216yes—
38#500C2213yes—
39#498C2213yes—
40#490C2216yes—
41#486C2213yes—
42#486C2213yes—
43#482C2213yes—
44#482C2213yes—
45#482C2216yes—
46#478C2216yes—
47#478C2216yes—
48#477C2213yes—
49#476C2213yes—
50#476C2213yes—
51#476C2213yes—
52#475C2213yes—
53#474C2213yes—
54#474C2213yes—
55#474C2213yes—
56#460C2213yes—
57#460C2213yes—
58#459C2213yes—
59#459C2213yes—
60#449C2211yescharacteristic
61#449C2213yes—
62#446C2213yes—
63#446C2213yes—
64C8×Q8C2211yescharacteristic
65C8×Q8C2211yescharacteristic
66#442C2216yes—
67#440C2216yes—
68C4×SD16C2213yes—
69C4×SD16C2213yes—
70C4×D8C2213yes—
71C4×D8C2213yes—
72#432C2213yes—
73#431C2213yes—
742−1+4—48no—
75C2×#41—23no—
76C2×#41—21no—
77C2×#41—23no—
78C2×#41—23no—
79C2×#41C2311yescharacteristic
80C22×Q8—21no—
81C22×Q8—23no—
82C22×Q8—21no—
83C22×Q8D411yescharacteristic
84C2×SD16C2311yescharacteristic
85C2×SD16C2313yes—
86C2×D8C2311yescharacteristic
87C2×D8C2313yes—
88C4⋊Q8—23no—
89C4⋊Q8—23no—
90C4⋊Q8C2313yes—
91C4⋊Q8C2313yes—
92C4⋊Q8C2313yes—
93C4⋊Q8C2313yes—
94#127C2313yes—
95#127C2313yes—
96#125—26no—
97#125C2316yes—
98#124—23no—
99#124—23no—
100C22⋊Q8—23no—
101C22⋊Q8—23no—
102C22⋊Q8—23no—
103C22⋊Q8—23no—
104C22⋊Q8—26no—
105C22⋊Q8—23no—
106C22⋊Q8—26no—
107C22⋊Q8—23no—
108#121—23no—
109#121—23no—
110#121—23no—
111#121—26no—
112#121—23no—
113#121C2311yescharacteristic
114#121C2313yes—
115C4×Q8—21no—
116C4×Q8—23no—
117C4×Q8—23no—
118C4×Q8—23no—
119C4×Q8D411yescharacteristic
120C4×Q8D413yes—
121C4×Q8C2313yes—
122C4×Q8C2313yes—
123C4×Q8C2311yescharacteristic
124C4×Q8C2313yes—
125C4×D4—23no—
126C4×D4—23no—
127C4×D4—23no—
128C4×D4—23no—
129C4×D4C2313yes—
130C4×D4C2313yes—
131C4×D4C2313yes—
132C4×D4C2313yes—
133C4×D4C2316yes—
134#117—23no—
135#117C2313yes—
136C2×C4⋊C4—23no—
137C2×C4⋊C4D413yes—
138C2×C42—23no—
139C2×C42C2313yes—
140#107C2313yes—
141#107C2313yes—
142#107C2316yes—
143#106C2313yes—
144#106C2313yes—
145#106C2316yes—
146C4⋊C8C2313yes—
147C4⋊C8C2313yes—
148C4⋊C8C2313yes—
149C4⋊C8C2313yes—
150#103C2311yescharacteristic
151#103C2313yes—
152#102C2311yescharacteristic
153#102C2313yes—
154#102C2313yes—
155#102C2313yes—
156#102C2313yes—
157#102C2313yes—
158#102C2316yes—
159#102C2316yes—
160C22⋊C8C2311yescharacteristic
161C22⋊C8C2313yes—
162#97C2316yes—
163C4×C8C2313yes—
164C4×C8C2313yes—
165#41—22no—
166#41—22no—
167#41—412no—
168#41—412no—
169#41—412no—
170#41—44no—
171C2×Q8—24no—
172C2×Q8—23no—
173C2×Q8—23no—
174C2×Q8—23no—
175C2×Q8—23no—
176C2×Q8—412no—
177C2×Q8—44no—
178C2×Q8—44no—
179C2×Q8C2×D413yes—
180C2×Q8C2×D411yescharacteristic
181C2×Q8C2411yescharacteristic
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—23no—
185C2×D4—23no—
186C2×D4—23no—
187C2×D4—23no—
188C2×D4—23no—
189C2×D4C2411yescharacteristic
190C2×D4C2413yes—
191C2×D4C2411yescharacteristic
192C22×C4—21no—
193C22×C4—23no—
194C22×C4—26no—
195C22×C4—23no—
196C22×C4—23no—
197C22×C4—23no—
198C22×C4C2×D413yes—
199C22×C4C2×D413yes—
200C22×C4C2411yescharacteristic
201SD16—22no—
202SD16—26no—
203D8—22no—
204D8—26no—
205C2×C8C2411yescharacteristic
206C2×C8C2411yescharacteristic
207C2×C8C2413yes—
208C2×C8C2413yes—
209C4⋊C4—26no—
210C4⋊C4—23no—
211C4⋊C4—23no—
212C4⋊C4—23no—
213C4⋊C4—23no—
214C4⋊C4—23no—
215C4⋊C4—23no—
216C4⋊C4—26no—
217C4⋊C4C2×D413yes—
218C4⋊C4C2×D416yes—
219C4⋊C4C2×D413yes—
220C4⋊C4C2413yes—
221C4⋊C4C2413yes—
222C4⋊C4C2411yescharacteristic
223C4⋊C4C2413yes—
224C22⋊C4—23no—
225C22⋊C4—23no—
226C22⋊C4—23no—
227C22⋊C4—21no—
228C22⋊C4—23no—
229C22⋊C4—26no—
230C22⋊C4—23no—
231C22⋊C4—23no—
232C42—23no—
233C42—23no—
234C42—23no—
235C42C2×D413yes—
236C42C2×D413yes—
237C42C2413yes—
238C42C2413yes—
239C23—21no—
240C23—23no—
241C23—21no—
242C23C22×D411yescharacteristic
243Q8—26no—
244Q8—21no—
245Q8—46no—
246Q8—46no—
247Q8—46no—
248Q8—83no—
249Q8AGL(1,ℤ/8)14yes—
250Q82−1+412yescomplemented
251D4—24no—
252D4—21no—
253D4—21no—
254D4—23no—
255D4—21no—
256D4—23no—
257D4—46no—
258D4—46no—
259D4—42no—
260D4—46no—
261D4—83no—
262D4—83no—
263D42−1+412yescomplemented
264C2×C4—24no—
265C2×C4—21no—
266C2×C4—26no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—23no—
270C2×C4—23no—
271C2×C4—23no—
272C2×C4—21no—
273C2×C4—43no—
274C2×C4—412no—
275C2×C4—46no—
276C2×C4—46no—
277C2×C4—46no—
278C2×C4—83no—
279C2×C4C22×D411yescharacteristic
280C2×C4C22×D413yes—
281C2×C4C22×D413yes—
282C2×C4C22×D413yes—
283C2×C4C22×D413yes—
284C2×C4C2511yescharacteristic
285C8—21no—
286C8—23no—
287C8—21no—
288C8—23no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—21no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—43no—
297C22—43no—
298C22—41no—
299C22—43no—
300C22—41no—
301C22C23×D411yescharacteristic
302C4—23no—
303C4—23no—
304C4—21no—
305C4—22no—
306C4—23no—
307C4—43no—
308C4—43no—
309C4—41no—
310C4—43no—
311C4C2×AGL(1,ℤ/8)16yes—
312C4C2×2−1+411yescharacteristic
313C4C2×2−1+411yescharacteristic
314C2—41no—
315C2—42no—
316C2—83no—
317C2—81no—
318C2#348611yescharacteristic
319C2C22×AGL(1,ℤ/8)11yescharacteristic
320C2#358411yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3497C2semidirect
2. Q8×#41C2semidirect
3. #3386C2semidirect
4. #3384C2semidirect
5. Q8×SD16C2semidirect
6. Q8×D8C2semidirect
7. #3368C2semidirect
8. #3366C2semidirect
9. #3362C2semidirect
10. #3358C2semidirect
11. #3235C2semidirect
12. #3228C2semidirect
13. #3115C2semidirect
14. #3106C2semidirect
15. #2984C2semidirect
250. Q82−1+4semidirect
263. D42−1+4semidirect

Maximal subgroups

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

subgroupindex
#34972
Q8×#412
#33862
#33842
Q8×SD162
Q8×D82
#33682
#33662
#33622
#33582
#32352
#32282
#31152
#31062
#29842