Finite Group Databasebeta

← #62362

Subgroups — #62362

Order 256.

873 conjugacy classes of subgroups.

The subgroup lattice has 307 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3523C211yesmaximal, characteristic, complemented
2#3522C211yesmaximal, characteristic, complemented
3#3394C213yesmaximal, complemented
4#3392C211yesmaximal, characteristic, complemented
5#3381C213yesmaximal, complemented
6Q8×SD16C211yesmaximal, characteristic, complemented
7#3365C213yesmaximal, complemented
8#3363C211yesmaximal, characteristic, complemented
9#3360C213yesmaximal, complemented
10#3358C211yesmaximal, characteristic, complemented
11#3248C213yesmaximal, complemented
12#3247C213yesmaximal, complemented
13#3168C213yesmaximal, complemented
14#3167C213yesmaximal, complemented
15C2×C8×Q8C211yesmaximal, characteristic, complemented
16#559C2213yes—
17Q82C2211yescharacteristic
18#557C2211yescharacteristic
19#557C2213yes—
20#556C2213yes—
21#554C2213yes—
22#552—23no—
23#550C2211yescharacteristic
24D4×Q8—21no—
25D4×Q8C2211yescharacteristic
26#548—21no—
27#547—23no—
28#542—23no—
29#541—23no—
30#533C2213yes—
31#532C2213yes—
32#529C2213yes—
33#529C2213yes—
34#520—21no—
35#519—21no—
36C2×C4×Q8C2211yescharacteristic
37#499C2213yes—
38#498C2213yes—
39#495C2213yes—
40#492C2213yes—
41#488C2213yes—
42#488C2213yes—
43#487C2213yes—
44#486C2213yes—
45#479C2216yes—
46#478C2216yes—
47#477C2213yes—
48#476C2213yes—
49#475C2213yes—
50#474C2213yes—
51#467C2213yes—
52#466C2213yes—
53#465C2213yes—
54#465C2216yes—
55#465C2211yescharacteristic
56#464C2216yes—
57#463C2216yes—
58#462C2213yes—
59#461C2213yes—
60#460C2213yes—
61#459C2213yes—
62C8×Q8C2211yescharacteristic
63C8×Q8C2213yes—
64C8×Q8C2211yescharacteristic
65C8×Q8C2213yes—
66C4×Q16C2213yes—
67C4×SD16C2216yes—
68C4×SD16C2213yes—
69C4×SD16C2213yes—
70C4×SD16C2216yes—
71C4×D8C2213yes—
72C2×C4⋊C8C2213yes—
73C2×C4×C8C2213yes—
74C2×#41—21no—
75C2×#41—23no—
76C22×Q8—21no—
77C22×Q8D411yescharacteristic
78C2×Q16C2313yes—
79C2×SD16C2313yes—
80C2×SD16C2311yescharacteristic
81C2×SD16C2316yes—
82C2×D8C2313yes—
83C22×C8C2311yescharacteristic
84C22×C8C2313yes—
85C4⋊Q8—23no—
86C4⋊Q8—23no—
87C4⋊Q8C2313yes—
88C4⋊Q8C2313yes—
89C4⋊Q8C2313yes—
90#127C2313yes—
91#126—26no—
92#126—23no—
93#126—23no—
94#125—26no—
95#125—23no—
96#125—23no—
97#125C2313yes—
98#125C2313yes—
99#124—23no—
100#124—23no—
101#124C2313yes—
102#124C2313yes—
103#123—23no—
104#123—23no—
105#123—26no—
106C22⋊Q8—23no—
107C22⋊Q8—23no—
108C22⋊Q8—23no—
109C22⋊Q8—23no—
110C22⋊Q8—26no—
111C22⋊Q8C2311yescharacteristic
112C22⋊Q8C2313yes—
113#121—23no—
114#121—23no—
115#121C2311yescharacteristic
116#121C2313yes—
117C4×Q8—21no—
118C4×Q8—21no—
119C4×Q8—23no—
120C4×Q8—23no—
121C4×Q8D411yescharacteristic
122C4×Q8D413yes—
123C4×Q8C2313yes—
124C4×Q8C2313yes—
125C4×Q8C2316yes—
126C4×Q8C2311yescharacteristic
127C4×Q8C2313yes—
128C4×D4—23no—
129C4×D4—23no—
130C4×D4—23no—
131C4×D4—23no—
132C4×D4C2313yes—
133C4×D4C2313yes—
134C4×D4C2316yes—
135#117—23no—
136#117—23no—
137#117—23no—
138#117—23no—
139C2×C4⋊C4—23no—
140C2×C4⋊C4D413yes—
141C2×C42C2313yes—
142#107C2313yes—
143#107C2313yes—
144#106C2313yes—
145#106C2311yescharacteristic
146#106C2316yes—
147C4⋊C8C2313yes—
148C4⋊C8C2313yes—
149C4⋊C8C2316yes—
150#103C2313yes—
151#103C2313yes—
152#103C2316yes—
153#103C2313yes—
154#103C2311yescharacteristic
155#102C2311yescharacteristic
156#102C2313yes—
157#102C2316yes—
158#102C2313yes—
159#102C2313yes—
160C4×C8C2313yes—
161C4×C8C2313yes—
162C4×C8C2313yes—
163C4×C8C2313yes—
164#41—44no—
165#41—86no—
166C2×Q8—24no—
167C2×Q8—23no—
168C2×Q8—23no—
169C2×Q8—44no—
170C2×Q8C2×D413yes—
171C2×Q8C2×D411yescharacteristic
172C2×Q8C2411yescharacteristic
173C2×Q8C2413yes—
174C2×D4—21no—
175C2×D4—23no—
176C2×D4—23no—
177C2×D4—23no—
178C2×D4C2411yescharacteristic
179C2×D4C2413yes—
180C22×C4—23no—
181C22×C4—23no—
182C22×C4—26no—
183C22×C4C2×D413yes—
184C22×C4C2×D413yes—
185C22×C4C2411yescharacteristic
186Q16—43no—
187SD16—26no—
188SD16—22no—
189SD16—46no—
190D8—43no—
191C2×C8—22no—
192C2×C8—43no—
193C2×C8C2411yescharacteristic
194C2×C8C2413yes—
195C2×C8C2413yes—
196C2×C8C2411yescharacteristic
197C4⋊C4—23no—
198C4⋊C4—23no—
199C4⋊C4—21no—
200C4⋊C4—23no—
201C4⋊C4—23no—
202C4⋊C4—23no—
203C4⋊C4—23no—
204C4⋊C4—23no—
205C4⋊C4—23no—
206C4⋊C4—23no—
207C4⋊C4—26no—
208C4⋊C4—23no—
209C4⋊C4—23no—
210C4⋊C4C2×D413yes—
211C4⋊C4C2×D413yes—
212C4⋊C4C2×D416yes—
213C4⋊C4C2413yes—
214C4⋊C4C2411yescharacteristic
215C4⋊C4C2413yes—
216C4⋊C4C2411yescharacteristic
217C22⋊C4—23no—
218C22⋊C4—21no—
219C22⋊C4—21no—
220C22⋊C4—23no—
221C22⋊C4—23no—
222C22⋊C4—23no—
223C22⋊C4—23no—
224C22⋊C4—23no—
225C22⋊C4—23no—
226C22⋊C4—26no—
227C22⋊C4—23no—
228C42—26no—
229C42—23no—
230C42—23no—
231C42C2×D413yes—
232C42C2×D413yes—
233C42C2413yes—
234C42C2413yes—
235C23—21no—
236C23—23no—
237C23C22×D411yescharacteristic
238Q8—26no—
239Q8—21no—
240Q8—23no—
241Q8—23no—
242Q8—21no—
243Q8—46no—
244Q8—83no—
245Q8#13514yes—
246D4—21no—
247D4—23no—
248D4—21no—
249D4—23no—
250D4—42no—
251D4—46no—
252D4—46no—
253D4—83no—
254C2×C4—26no—
255C2×C4—26no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—23no—
260C2×C4—21no—
261C2×C4—23no—
262C2×C4—23no—
263C2×C4—21no—
264C2×C4—46no—
265C2×C4—46no—
266C2×C4—86no—
267C2×C4C22×D413yes—
268C2×C4C22×D411yescharacteristic
269C2×C4C22×D413yes—
270C2×C4C22×D413yes—
271C2×C4C22×D413yes—
272C2×C4C2511yescharacteristic
273C8—21no—
274C8—23no—
275C8—23no—
276C82−1+412yes—
277C22—21no—
278C22—21no—
279C22—41no—
280C22—43no—
281C22—41no—
282C22—43no—
283C22—43no—
284C22—41no—
285C22#57712yes—
286C22C23×D411yescharacteristic
287C4—21no—
288C4—23no—
289C4—23no—
290C4—23no—
291C4—41no—
292C4—43no—
293C4—41no—
294C4—43no—
295C4—43no—
296C4—41no—
297C4C2×#13516yes—
298C4C2×2−1+411yescharacteristic
299C4#58511yescharacteristic
300C2—22no—
301C2—81no—
302C2—83no—
303C2#349711yescharacteristic
304C2C22×#13511yescharacteristic
305C2C2×#57711yescharacteristic
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. #3523C2semidirect
2. #3522C2semidirect
3. #3394C2semidirect
4. #3392C2semidirect
5. #3381C2semidirect
6. Q8×SD16C2semidirect
7. #3365C2semidirect
8. #3363C2semidirect
9. #3360C2semidirect
10. #3358C2semidirect
11. #3248C2semidirect
12. #3247C2semidirect
13. #3168C2semidirect
14. #3167C2semidirect
15. C2×C8×Q8C2semidirect

Maximal subgroups

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

subgroupindex
#35232
#35222
#33942
#33922
#33812
Q8×SD162
#33652
#33632
#33602
#33582
#32482
#32472
#31682
#31672
C2×C8×Q82