Finite Group Databasebeta

← #17624

Subgroups — #17624

Order 256.

449 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3507C211yesmaximal, characteristic
2#2983C211yesmaximal, characteristic, complemented
3#2356C211yesmaximal, characteristic, complemented
4#1521C212yesmaximal, complemented
5#1515C211yesmaximal, characteristic, complemented
6#1514C211yesmaximal, characteristic
7#1512C212yesmaximal, complemented
8#1509C212yesmaximal, complemented
9#1506C211yesmaximal, characteristic, complemented
10#1505C211yesmaximal, characteristic
11#1500C212yesmaximal, complemented
12#559C412yescomplemented
13#557C412yescomplemented
14#556—22no—
15#556C412yescomplemented
16#554C412yescomplemented
17#552—22no—
18#539—21no—
19#539—21no—
20#536—21no—
21#533C2211yescharacteristic
22#532C2211yescharacteristic
23#529C2212yes—
24#529C2212yes—
25#520—21no—
26C2×C4×Q8C2211yescharacteristic
27#446—22no—
28#446—22no—
29#432C2211yescharacteristic
30#432C2211yescharacteristic
31#432C2212yes—
32#431C2212yes—
33#391—21no—
34#387—21no—
35#387—21no—
36#383C2211yescharacteristic
37#383C2211yescharacteristic
38#382C2211yescharacteristic
39C4×C4⋊C4—21no—
40C4×C4⋊C4C2211yescharacteristic
41#376C2211yescharacteristic
42#376C2211yescharacteristic
43#328C2212yes—
44#328C2212yes—
45#328C2212yes—
46#328C2212yes—
47#327C2212yes—
48#327C2212yes—
49#327C2212yes—
50#327C2212yes—
51C2×#41—21no—
52C22×Q8D411yescharacteristic
53C4⋊Q8—22no—
54C4⋊Q8C2×C411yescharacteristic
55C4⋊Q8C2×C411yescharacteristic
56C4⋊Q8C2×C411yescharacteristic
57#127C2×C411yescharacteristic
58#126—22no—
59#126—22no—
60#126—22no—
61#126—22no—
62#125—22no—
63#125—22no—
64#125—21no—
65#125C2×C412yes—
66#125C2×C412yes—
67#124—22no—
68#124—21no—
69#124C2×C412yes—
70#124C2×C412yes—
71#123—22no—
72#123—22no—
73#123—22no—
74#123—22no—
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77C22⋊Q8—21no—
78C22⋊Q8—22no—
79C22⋊Q8—22no—
80C22⋊Q8C2312yes—
81#121—21no—
82#121C2312yes—
83C4×Q8—22no—
84C4×Q8—21no—
85C4×Q8C2×C412yes—
86C4×Q8C2×C412yes—
87C4×Q8C2×C412yes—
88C4×Q8C2×C412yes—
89C4×Q8C2×C412yes—
90C4×Q8D412yes—
91C4×Q8D412yes—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—21no—
96C4×D4C2×C412yes—
97C4×D4C2×C412yes—
98C4×D4C2×C412yes—
99#117—21no—
100#117—22no—
101#117—22no—
102#117—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4D411yescharacteristic
107C2×C4⋊C4D411yescharacteristic
108C2×C4⋊C4D411yescharacteristic
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42C2311yescharacteristic
113C2×C42C2311yescharacteristic
114C2×C42C2311yescharacteristic
115C4⋊C8—22no—
116C4⋊C8—22no—
117C4⋊C8—21no—
118C4⋊C8—21no—
119C22⋊C8C2312yes—
120C22⋊C8C2312yes—
121#97—22no—
122#97—21no—
123#97—21no—
124C4×C8—22no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95C2311yescharacteristic
133#95C2311yescharacteristic
134#95C2311yescharacteristic
135#95C2311yescharacteristic
136#41—42no—
137#41—42no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—41no—
141C2×Q8—41no—
142C2×Q8C22⋊C412yes—
143C2×Q8C22⋊C412yes—
144C2×Q8C22×C412yes—
145C2×D4—22no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4C22×C412yes—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2411yescharacteristic
163C2×C8—22no—
164C2×C8—22no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4C22⋊C412yes—
180C4⋊C4C22⋊C412yes—
181C4⋊C4C22⋊C412yes—
182C4⋊C4C22⋊C412yes—
183C4⋊C4C22⋊C412yes—
184C4⋊C4C22⋊C412yes—
185C4⋊C4C22×C412yes—
186C4⋊C4C22×C412yes—
187C22⋊C4—22no—
188C22⋊C4—22no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C42—22no—
197C42—22no—
198C42—22no—
199C42—42no—
200C42—42no—
201C42—42no—
202C42C22×C411yescharacteristic
203C42C22×C412yes—
204C42C22×C412yes—
205C42C22×C411yescharacteristic
206C42C2×D412yes—
207C42C2×D412yes—
208C42C2×D411yescharacteristic
209C42C2×D411yescharacteristic
210C23—22no—
211C23C22×D411yescharacteristic
212Q8—42no—
213Q8—42no—
214Q8—42no—
215Q8—42no—
216Q8—81no—
217D4—42no—
218D4—42no—
219D4—42no—
220D4—42no—
221D4—81no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—44no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4C2×C22⋊C411yescharacteristic
251C2×C4C2×C22⋊C412yes—
252C2×C4C2×C22⋊C412yes—
253C2×C4C2×C22⋊C412yes—
254C2×C4C2×C22⋊C412yes—
255C2×C4C2×C22⋊C411yescharacteristic
256C2×C4C2×C22⋊C411yescharacteristic
257C2×C4C2×C22⋊C411yescharacteristic
258C2×C4C23×C411yescharacteristic
259C2×C4C22×D411yescharacteristic
260C8—42no—
261C8—42no—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22C22×C22⋊C411yescharacteristic
268C22#57511yescharacteristic
269C22#57511yescharacteristic
270C4—21no—
271C4—21no—
272C4—21no—
273C4—21no—
274C4—22no—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—21no—
279C4—21no—
280C4—42no—
281C4—42no—
282C4—42no—
283C4—42no—
284C4—81no—
285C4—81no—
286C4—81no—
287C2—21no—
288C2—21no—
289C2—82no—
290C2#288411yescharacteristic
291C2#289611yescharacteristic
292C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#532C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2983C2semidirect
3. #2356C2semidirect
4. #1521C2semidirect
5. #1515C2semidirect
7. #1512C2semidirect
8. #1509C2semidirect
9. #1506C2semidirect
11. #1500C2semidirect
12. #559C4semidirect
13. #557C4semidirect
15. #556C4semidirect
16. #554C4semidirect

Maximal subgroups

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

subgroupindex
#35072
#29832
#23562
#15212
#15152
#15142
#15122
#15092
#15062
#15052
#15002