Finite Group Databasebeta

← #35801

Subgroups — #35801

Order 256.

1171 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#548C211yesmaximal, characteristic
2C2×#519C211yesmaximal, characteristic, complemented
3#2624C213yesmaximal, complemented
4#2500C213yesmaximal, complemented
5#2485C213yesmaximal, complemented
6#2448C213yesmaximal, complemented
7#2448C213yesmaximal, complemented
8#2423C213yesmaximal, complemented
9#2421C213yesmaximal, complemented
10#2395C213yesmaximal, complemented
11#2351C211yesmaximal, characteristic, complemented
12Q8×C4⋊C4C211yesmaximal, characteristic, complemented
13#2341C211yesmaximal, characteristic, complemented
14#2341C211yesmaximal, characteristic, complemented
15Q8×C22⋊C4C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18#548—28no—
19C2×#124C2213yes—
20C2×C22⋊Q8C2213yes—
21C2×#121C2213yes—
22#519—28no—
23C2×C4×Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28C2×C4×Q8C2213yes—
29C2×C4×Q8C2213yes—
30C2×C4×D4C2213yes—
31C2×#117C2213yes—
32C2×#117C2213yes—
33#400C2213yes—
34#399C2216yes—
35#399C2213yes—
36#398C2213yes—
37#397C2216yes—
38#397C2211yescharacteristic
39#397C2213yes—
40#397C2213yes—
41#394C2213yes—
42#394C2216yes—
43#394C2213yes—
44#391C2213yes—
45#391C2213yes—
46#390C2213yes—
47#390C2213yes—
48#390C2213yes—
49#390C2213yes—
50#389C2213yes—
51#389C2213yes—
52#389C2213yes—
53#389C2213yes—
54#388C2216yes—
55#388C2216yes—
56#388C2216yes—
57#387C2216yes—
58#384C2213yes—
59#382C2213yes—
60#380C2213yes—
61#380C2213yes—
62#380C2213yes—
63C4×C4⋊C4C2213yes—
64C4×C4⋊C4C2213yes—
65C4×C22⋊C4C2213yes—
66C4×C22⋊C4C2213yes—
67C4×C22⋊C4C2213yes—
68C2×#41—24no—
69C2×#41—24no—
70C2×#41—24no—
71C22×Q8—24no—
72C22×Q8—24no—
73C22×Q8D418yes—
74C22×Q8C2313yes—
75C22×Q8C2311yescharacteristic
76C22×D4C2313yes—
77C22×D4C2313yes—
78C23×C4C2313yes—
79C23×C4C2313yes—
80#124—212no—
81C22⋊Q8—212no—
82#121—212no—
83C4×Q8—212no—
84C4×Q8—212no—
85C4×Q8—24no—
86C4×Q8—24no—
87C4×Q8—24no—
88C4×Q8—24no—
89C4×Q8—24no—
90C4×D4—212no—
91#117—212no—
92#117—212no—
93C2×C4⋊C4C2313yes—
94C2×C4⋊C4C2313yes—
95C2×C4⋊C4C2313yes—
96C2×C4⋊C4C2313yes—
97C2×C4⋊C4C2313yes—
98C2×C4⋊C4C2313yes—
99C2×C4⋊C4C2313yes—
100C2×C4⋊C4C2313yes—
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2313yes—
103C2×C4⋊C4C2313yes—
104C2×C4⋊C4C2313yes—
105C2×C4⋊C4C2316yes—
106C2×C22⋊C4C2311yescharacteristic
107C2×C22⋊C4C2313yes—
108C2×C22⋊C4C2311yescharacteristic
109C2×C22⋊C4C2313yes—
110C2×C22⋊C4C2316yes—
111C2×C22⋊C4C2311yescharacteristic
112C2×C22⋊C4C2313yes—
113C2×C22⋊C4C2313yes—
114C2×C22⋊C4C2313yes—
115C2×C22⋊C4C2313yes—
116C2×C22⋊C4C2313yes—
117C2×C22⋊C4C2316yes—
118C2×C22⋊C4C2316yes—
119C2×C42C2313yes—
120C2×C42C2313yes—
121C2×C42C2316yes—
122C2×C42C2313yes—
123C2×C42C2313yes—
124C2×C42C2313yes—
125#95C2311yescharacteristic
126#95C2311yescharacteristic
127#95C2313yes—
128#95C2313yes—
129#95C2313yes—
130#95C2313yes—
131#95C2311yescharacteristic
132#95C2313yes—
133#95C2313yes—
134#95C2313yes—
135#95C2316yes—
136#95C2316yes—
137C24C2413yes—
138C24C2411yescharacteristic
139#41—416no—
140C2×Q8—232no—
141C2×Q8—26no—
142C2×Q8—26no—
143C2×Q8C2×D4112yes—
144C2×Q8C2×D414yes—
145C2×Q8#4114yes—
146C2×Q8#4114yes—
147C2×D4—26no—
148C2×D4—26no—
149C2×D4—26no—
150C2×D4—26no—
151C2×D4—43no—
152C2×D4—43no—
153C22×C4—26no—
154C22×C4—26no—
155C22×C4—26no—
156C22×C4—26no—
157C22×C4—26no—
158C22×C4C2×D4112yes—
159C22×C4C2411yescharacteristic
160C22×C4C2411yescharacteristic
161C22×C4C2413yes—
162C22×C4C2413yes—
163C22×C4C2411yescharacteristic
164C22×C4C2413yes—
165C22×C4C2411yescharacteristic
166C22×C4C2413yes—
167C22×C4C2411yescharacteristic
168C22×C4C2413yes—
169C22×C4C2413yes—
170C22×C4C2411yescharacteristic
171C22×C4C2413yes—
172C4⋊C4—26no—
173C4⋊C4—26no—
174C4⋊C4—26no—
175C4⋊C4—26no—
176C4⋊C4—26no—
177C4⋊C4—26no—
178C4⋊C4—26no—
179C4⋊C4—26no—
180C4⋊C4—22no—
181C4⋊C4—43no—
182C4⋊C4—43no—
183C4⋊C4—43no—
184C4⋊C4—46no—
185C22⋊C4—22no—
186C22⋊C4—26no—
187C22⋊C4—22no—
188C22⋊C4—26no—
189C22⋊C4—212no—
190C22⋊C4—22no—
191C22⋊C4—26no—
192C22⋊C4—43no—
193C22⋊C4—43no—
194C22⋊C4—46no—
195C22⋊C4—43no—
196C22⋊C4—43no—
197C22⋊C4—46no—
198C42—212no—
199C42—26no—
200C42—26no—
201C42—26no—
202C42—26no—
203C42—26no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—23no—
211C23—23no—
212C23—23no—
213C23—23no—
214C23—23no—
215C23—23no—
216C23—23no—
217C23C22×D412yes—
218C23C2511yescharacteristic
219Q8—224no—
220Q8#123116yes—
221D4—412no—
222D4—412no—
223C2×C4—21no—
224C2×C4—23no—
225C2×C4—23no—
226C2×C4—21no—
227C2×C4—224no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—23no—
231C2×C4—23no—
232C2×C4—21no—
233C2×C4—23no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—23no—
237C2×C4—23no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—23no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—23no—
248C2×C4—23no—
249C2×C4—23no—
250C2×C4—23no—
251C2×C4—23no—
252C2×C4—23no—
253C2×C4—23no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—23no—
257C2×C4—23no—
258C2×C4—412no—
259C2×C4C22×D416yes—
260C2×C4C22×D416yes—
261C2×C4C2×#4116yes—
262C2×C4C2×#4116yes—
263C2×C42−1+412yes—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—43no—
272C22—43no—
273C22—43no—
274C22—43no—
275C22—43no—
276C22—43no—
277C22—43no—
278C22C23×D411yescharacteristic
279C22C22×#4111yescharacteristic
280C22C22×#4111yescharacteristic
281C22C2×2−1+411yescharacteristic
282C22#58511yescharacteristic
283C22#58511yescharacteristic
284C22#58511yescharacteristic
285C4—26no—
286C4—22no—
287C4—41no—
288C4—41no—
289C4—43no—
290C4—43no—
291C4—41no—
292C4—43no—
293C4—41no—
294C4—41no—
295C4—43no—
296C4—43no—
297C4C2×#123112yes—
298C2—42no—
299C2—83no—
300C2C22×#12311yescharacteristic
301C2#348711yescharacteristic
302C2#349711yescharacteristic
303C2#351611yescharacteristic
304C2#351811yescharacteristic
305C2#352411yescharacteristic
306C2#352411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC2×#548C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#519C2semidirect
3. #2624C2semidirect
4. #2500C2semidirect
5. #2485C2semidirect
6. #2448C2semidirect
7. #2448C2semidirect
8. #2423C2semidirect
9. #2421C2semidirect
10. #2395C2semidirect
11. #2351C2semidirect
12. Q8×C4⋊C4C2semidirect
13. #2341C2semidirect
14. #2341C2semidirect
15. Q8×C22⋊C4C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5482
C2×#5192
#26242
#25002
#24852
#24482
#24232
#24212
#23952
#23512
Q8×C4⋊C42
#23412
Q8×C22⋊C42