Finite Group Databasebeta

← #62394

Subgroups — #62394

Order 256.

853 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3522C211yesmaximal, characteristic
2#3502C211yesmaximal, characteristic, complemented
3#3400C213yesmaximal, complemented
4#3392C211yesmaximal, characteristic, complemented
5#3385C213yesmaximal, complemented
6Q8×Q16C211yesmaximal, characteristic, complemented
7#3375C213yesmaximal, complemented
8#3372C213yesmaximal, complemented
9#3363C211yesmaximal, characteristic, complemented
10#3362C211yesmaximal, characteristic, complemented
11#3254C213yesmaximal, complemented
12#3252C213yesmaximal, complemented
13#3182C213yesmaximal, complemented
14#3172C213yesmaximal, complemented
15Q8×M16C211yesmaximal, characteristic, complemented
16#559C2213yes—
17Q82C2211yescharacteristic
18Q82C2211yescharacteristic
19Q82C2213yes—
20#557—21no—
21#557—23no—
22#557C2211yescharacteristic
23#557C2213yes—
24#554C2213yes—
25#550C2211yescharacteristic
26#548—21no—
27#547—23no—
28#544—23no—
29#542—23no—
30#533C2213yes—
31#533C2213yes—
32#532C2213yes—
33#529C2213yes—
34#520—21no—
35#519—21no—
36C2×C4×Q8C2211yescharacteristic
37#501C2213yes—
38#500C2213yes—
39#497C2213yes—
40#492C2213yes—
41#491C2213yes—
42#489C2213yes—
43#487C2213yes—
44#486C2213yes—
45#479C2216yes—
46#477C2213yes—
47#477C2213yes—
48#477C2213yes—
49#476C2213yes—
50#474C2216yes—
51#470C2211yescharacteristic
52#470C2213yes—
53#468C2213yes—
54#468C2213yes—
55#468C2216yes—
56#464C2216yes—
57#462C2213yes—
58#461C2213yes—
59#461C2216yes—
60#460C2213yes—
61#460C2213yes—
62#446C2213yes—
63#446C2213yes—
64C8×Q8C2211yescharacteristic
65C8×Q8C2211yescharacteristic
66#440C2216yes—
67#440C2216yes—
68C4×Q16C2213yes—
69C4×Q16C2213yes—
70C4×SD16C2213yes—
71C4×SD16C2213yes—
72#423C2213yes—
73C4×M16C2213yes—
74C2×#41—21no—
75C2×#41—23no—
76C22×Q8D411yescharacteristic
77C2×Q16C2311yescharacteristic
78C2×Q16C2313yes—
79C2×SD16C2313yes—
80C2×SD16C2313yes—
81C2×SD16C2316yes—
82C2×M16C2311yescharacteristic
83C2×M16C2313yes—
84C4⋊Q8—23no—
85C4⋊Q8—23no—
86C4⋊Q8—23no—
87C4⋊Q8—26no—
88C4⋊Q8C2313yes—
89C4⋊Q8C2313yes—
90C4⋊Q8C2313yes—
91C4⋊Q8C2313yes—
92C4⋊Q8C2313yes—
93#127C2313yes—
94#126—23no—
95#125—23no—
96#125—23no—
97#125—26no—
98#125—23no—
99#125—23no—
100#125C2313yes—
101#124—23no—
102#124—23no—
103#124C2313yes—
104#123—23no—
105#123—26no—
106C22⋊Q8—23no—
107C22⋊Q8—23no—
108C22⋊Q8—23no—
109C22⋊Q8—26no—
110C22⋊Q8C2311yescharacteristic
111C22⋊Q8C2311yescharacteristic
112C22⋊Q8C2313yes—
113#121—23no—
114#121—23no—
115#121C2313yes—
116C4×Q8—23no—
117C4×Q8—21no—
118C4×Q8—23no—
119C4×Q8—21no—
120C4×Q8—21no—
121C4×Q8—23no—
122C4×Q8—23no—
123C4×Q8D411yescharacteristic
124C4×Q8D413yes—
125C4×Q8C2313yes—
126C4×Q8C2313yes—
127C4×Q8C2313yes—
128C4×Q8C2316yes—
129C4×Q8C2311yescharacteristic
130C4×Q8C2313yes—
131C4×D4—23no—
132C4×D4—23no—
133C4×D4—23no—
134C4×D4C2313yes—
135C4×D4C2316yes—
136#117—23no—
137#117—23no—
138#117—23no—
139C2×C4⋊C4—23no—
140C2×C4⋊C4D413yes—
141C2×C42C2313yes—
142#107C2313yes—
143#107C2313yes—
144#107C2316yes—
145#106C2311yescharacteristic
146#106C2313yes—
147C4⋊C8C2313yes—
148C4⋊C8C2313yes—
149C4⋊C8C2316yes—
150#103C2311yescharacteristic
151#103C2313yes—
152#103C2311yescharacteristic
153#103C2313yes—
154#103C2313yes—
155#103C2313yes—
156#103C2316yes—
157#102C2313yes—
158#102C2313yes—
159#102C2316yes—
160#97C2313yes—
161#97C2313yes—
162C4×C8C2313yes—
163C4×C8C2313yes—
164#41—44no—
165#41—86no—
166C2×Q8—24no—
167C2×Q8—23no—
168C2×Q8—23no—
169C2×Q8—23no—
170C2×Q8C2×D413yes—
171C2×Q8C2×D411yescharacteristic
172C2×Q8C2411yescharacteristic
173C2×Q8C2413yes—
174C2×Q8C2411yescharacteristic
175C2×D4—23no—
176C2×D4—23no—
177C2×D4—23no—
178C2×D4C2413yes—
179C22×C4—23no—
180C22×C4—26no—
181C22×C4C2×D413yes—
182C22×C4C2×D413yes—
183C22×C4C2411yescharacteristic
184Q16—22no—
185Q16—43no—
186SD16—26no—
187SD16—46no—
188SD16—43no—
189M16—22no—
190M16—43no—
191C2×C8C2411yescharacteristic
192C2×C8C2413yes—
193C2×C8C2413yes—
194C2×C8C2411yescharacteristic
195C4⋊C4—23no—
196C4⋊C4—23no—
197C4⋊C4—21no—
198C4⋊C4—23no—
199C4⋊C4—23no—
200C4⋊C4—23no—
201C4⋊C4—23no—
202C4⋊C4—23no—
203C4⋊C4—21no—
204C4⋊C4—23no—
205C4⋊C4—23no—
206C4⋊C4—23no—
207C4⋊C4—26no—
208C4⋊C4—23no—
209C4⋊C4—23no—
210C4⋊C4—23no—
211C4⋊C4C2×D413yes—
212C4⋊C4C2×D413yes—
213C4⋊C4C2×D416yes—
214C4⋊C4C2413yes—
215C4⋊C4C2411yescharacteristic
216C4⋊C4C2411yescharacteristic
217C4⋊C4C2413yes—
218C22⋊C4—21no—
219C22⋊C4—23no—
220C22⋊C4—21no—
221C22⋊C4—23no—
222C22⋊C4—23no—
223C22⋊C4—23no—
224C22⋊C4—23no—
225C22⋊C4—26no—
226C42—23no—
227C42—26no—
228C42—23no—
229C42—23no—
230C42C2×D413yes—
231C42C2×D413yes—
232C42C2413yes—
233C42C2413yes—
234C23—23no—
235C23C22×D411yescharacteristic
236Q8—26no—
237Q8—23no—
238Q8—21no—
239Q8—21no—
240Q8—21no—
241Q8—23no—
242Q8—21no—
243Q8—83no—
244Q8—83no—
245Q8—83no—
246Q8#13714yes—
247D4—23no—
248D4—23no—
249D4—46no—
250D4—46no—
251D4—83no—
252C2×C4—26no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—23no—
257C2×C4—21no—
258C2×C4—23no—
259C2×C4—21no—
260C2×C4—23no—
261C2×C4—21no—
262C2×C4—43no—
263C2×C4—46no—
264C2×C4—86no—
265C2×C4C22×D413yes—
266C2×C4C22×D411yescharacteristic
267C2×C4C22×D413yes—
268C2×C4C22×D413yes—
269C2×C4C22×D413yes—
270C2×C4C2511yescharacteristic
271C8—21no—
272C8—23no—
273C8—23no—
274C82−1+412yes—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—43no—
279C22—43no—
280C22—43no—
281C22C23×D411yescharacteristic
282C4—23no—
283C4—23no—
284C4—21no—
285C4—23no—
286C4—43no—
287C4—41no—
288C4—43no—
289C4—41no—
290C4—43no—
291C4—41no—
292C4—41no—
293C4C2×#13716yes—
294C4C2×2−1+411yescharacteristic
295C4#58511yescharacteristic
296C2—41no—
297C2—83no—
298C2#349711yescharacteristic
299C2C22×#13711yescharacteristic
300C2#358511yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3522C2
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
2. #3502C2semidirect
3. #3400C2semidirect
4. #3392C2semidirect
5. #3385C2semidirect
6. Q8×Q16C2semidirect
7. #3375C2semidirect
8. #3372C2semidirect
9. #3363C2semidirect
10. #3362C2semidirect
11. #3254C2semidirect
12. #3252C2semidirect
13. #3182C2semidirect
14. #3172C2semidirect
15. Q8×M16C2semidirect

Maximal subgroups

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

subgroupindex
#35222
#35022
#34002
#33922
#33852
Q8×Q162
#33752
#33722
#33632
#33622
#32542
#32522
#31822
#31722
Q8×M162