Finite Group Databasebeta

← #24613

Subgroups — #24613

Order 256.

378 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×#485C211yesmaximal, characteristic
2C2×#433C211yesmaximal, characteristic
3#2625C211yesmaximal, characteristic
4#2316C211yesmaximal, characteristic
5#2088C211yesmaximal, characteristic, complemented
6#2086C211yesmaximal, characteristic, complemented
7#2068C211yesmaximal, characteristic, complemented
8#2068C211yesmaximal, characteristic, complemented
9#1943C211yesmaximal, characteristic, complemented
10#1942C211yesmaximal, characteristic, complemented
11#1935C211yesmaximal, characteristic, complemented
12#1933C211yesmaximal, characteristic, complemented
13#1872C211yesmaximal, characteristic
14#1852C211yesmaximal, characteristic
15#1825C211yesmaximal, characteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×#117C2211yescharacteristic
18C2×#117C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20#485—24no—
21#433—24no—
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×#97C2211yescharacteristic
35C2×C4×C8C2211yescharacteristic
36#400—21no—
37#398—21no—
38#397—21no—
39#397—21no—
40#391C2211yescharacteristic
41#391C2211yescharacteristic
42#389—21no—
43#389—21no—
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#380—21no—
51#380—21no—
52#380C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C23×C4—21no—
63C23×C4C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22⋊Q8—22no—
69C22⋊Q8—22no—
70#117—22no—
71#117—22no—
72#117—22no—
73#117—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4D411yescharacteristic
97C2×C22⋊C4D411yescharacteristic
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#107—22no—
102#107—22no—
103#107—22no—
104#106—22no—
105C4⋊C8—22no—
106C4⋊C8—22no—
107#103—22no—
108#103—22no—
109#103—22no—
110#103—22no—
111C22⋊C8—22no—
112C22⋊C8D414yes—
113#97—22no—
114C4×C8—22no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126C24C2×D411yescharacteristic
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—24no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—81no—
147C22×C4—81no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D412yes—
154C22×C4#4112yes—
155C22×C4C2411yescharacteristic
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8C2×D412yes—
167C2×C8C2×D412yes—
168C4⋊C4—24no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—42no—
184C4⋊C4—81no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C4⋊C4#4112yes—
191C4⋊C4#4112yes—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—81no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—42no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23#12112yes—
209C23C22×D411yescharacteristic
210Q8—41no—
211Q8—41no—
212Q8—41no—
213Q8—41no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
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—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—84no—
257C2×C4#12112yes—
258C2×C4C22×D411yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C4C2×#4111yescharacteristic
261C2×C4C2×#4111yescharacteristic
262C2×C42−1+411yescharacteristic
263C2×C42−1+411yescharacteristic
264C8—22no—
265C8—22no—
266C8—41no—
267C8—41no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22C2×#12111yescharacteristic
276C22#52511yescharacteristic
277C22#53611yescharacteristic
278C22#57611yescharacteristic
279C22#57711yescharacteristic
280C22#57811yescharacteristic
281C22#57811yescharacteristic
282C4—22no—
283C4—42no—
284C4—42no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4#54711yescharacteristic
296C4#54711yescharacteristic
297C4#55211yescharacteristic
298C4#55211yescharacteristic
299C2—42no—
300C2#241311yescharacteristic
301C2#306111yescharacteristic
302C2#306311yescharacteristic
303C2#311811yescharacteristic
304C2#312711yescharacteristic
305C2#324211yescharacteristic
306C2#324311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #2088C2semidirect
6. #2086C2semidirect
7. #2068C2semidirect
8. #2068C2semidirect
9. #1943C2semidirect
10. #1942C2semidirect
11. #1935C2semidirect
12. #1933C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
C2×#4332
#26252
#23162
#20882
#20862
#20682
#19432
#19422
#19352
#19332
#18722
#18522
#18252