Finite Group Databasebeta

← #25466

Subgroups — #25466

Order 256.

378 conjugacy classes of subgroups.

The subgroup lattice has 301 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#501C211yesmaximal, characteristic, complemented
2C2×#478C211yesmaximal, characteristic
3#2563C211yesmaximal, characteristic
4#2370C211yesmaximal, characteristic, complemented
5#2085C211yesmaximal, characteristic, complemented
6#2085C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic
8#2065C211yesmaximal, characteristic
9#1988C211yesmaximal, characteristic, complemented
10#1949C211yesmaximal, characteristic, complemented
11#1949C211yesmaximal, characteristic, complemented
12#1940C211yesmaximal, characteristic, complemented
13#1926C211yesmaximal, characteristic
14#1925C211yesmaximal, characteristic
15#1762C211yesmaximal, characteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#501—24no—
20#478—24no—
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#97C2211yescharacteristic
35#393—21no—
36#393—21no—
37#390C2211yescharacteristic
38#390C2211yescharacteristic
39#389—21no—
40#389C2211yescharacteristic
41#389C2211yescharacteristic
42#387—21no—
43#387—21no—
44#382C2211yescharacteristic
45#382C2211yescharacteristic
46#380—21no—
47#380—21no—
48C4×C4⋊C4—21no—
49C4×C4⋊C4C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C4⋊Q8—22no—
69C4⋊Q8—22no—
70#125—22no—
71#125D414yes—
72C4×D4—22no—
73C4×D4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4D411yescharacteristic
80C2×C4⋊C4D411yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99#107—22no—
100#107—22no—
101#107—22no—
102#106—22no—
103#106—22no—
104#106—22no—
105C4⋊C8—22no—
106C4⋊C8—22no—
107C4⋊C8—22no—
108#102—22no—
109#102—22no—
110#102—22no—
111#102Q814yescomplemented
112#97—22no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95D411yescharacteristic
123#95D411yescharacteristic
124C24—21no—
125C2×Q8—42no—
126C2×Q8—81no—
127C2×Q8—81no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4C2×Q812yes—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2411yescharacteristic
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8C2×Q812yes—
162C2×C8#4112yes—
163C4⋊C4—24no—
164C4⋊C4—24no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—81no—
181C4⋊C4C2×D412yes—
182C4⋊C4C2×D412yes—
183C4⋊C4C2×Q812yes—
184C22⋊C4—42no—
185C22⋊C4—81no—
186C22⋊C4—81no—
187C22⋊C4—81no—
188C22⋊C4—81no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—42no—
194C42—81no—
195C42—81no—
196C42C2×D412yes—
197C42#4112yes—
198C23—22no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23C22×D411yescharacteristic
206Q8—84no—
207D4—22no—
208D4—22no—
209D4—41no—
210D4—41no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
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—84no—
249C2×C4#12112yes—
250C2×C4#12112yes—
251C2×C4C22×D411yescharacteristic
252C2×C4C22×Q811yescharacteristic
253C2×C4C2×#4111yescharacteristic
254C2×C4C2×#4111yescharacteristic
255C2×C42−1+411yescharacteristic
256C2×C42−1+411yescharacteristic
257C8—22no—
258C8—22no—
259C8—41no—
260C8—41no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—81no—
265C22—81no—
266C22—81no—
267C22—81no—
268C22C2×#12111yescharacteristic
269C22#54711yescharacteristic
270C22D4×Q811yescharacteristic
271C22#57511yescharacteristic
272C22#57511yescharacteristic
273C22#57611yescharacteristic
274C22#57711yescharacteristic
275C4—22no—
276C4—22no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—42no—
282C4—42no—
283C4—42no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4#53311yescharacteristic
289C4#54111yescharacteristic
290C4#54811yescharacteristic
291C4D4×Q811yescharacteristic
292C2—82no—
293C2#242911yescharacteristic
294C2#304011yescharacteristic
295C2#304511yescharacteristic
296C2#334711yescharacteristic
297C2#334811yescharacteristic
298C2#338811yescharacteristic
299C2#338911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#501C2semidirect
4. #2370C2semidirect
5. #2085C2semidirect
6. #2085C2semidirect
9. #1988C2semidirect
10. #1949C2semidirect
11. #1949C2semidirect
12. #1940C2semidirect
111. #102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5012
C2×#4782
#25632
#23702
#20852
#20672
#20652
#19882
#19492
#19402
#19262
#19252
#17622