Finite Group Databasebeta

← #25465

Subgroups — #25465

Order 256.

406 conjugacy classes of subgroups.

The subgroup lattice has 307 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#501C211yesmaximal, characteristic, complemented
2C2×#478C211yesmaximal, characteristic
3#2542C211yesmaximal, characteristic
4#2370C211yesmaximal, characteristic, complemented
5#2084C211yesmaximal, characteristic, complemented
6#2083C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic
8#2065C211yesmaximal, characteristic
9#1987C211yesmaximal, characteristic, complemented
10#1948C211yesmaximal, characteristic, complemented
11#1947C211yesmaximal, characteristic, complemented
12#1940C211yesmaximal, characteristic, complemented
13#1926C211yesmaximal, characteristic
14#1925C211yesmaximal, characteristic
15#1762C211yesmaximal, characteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8—21no—
20C2×C4×D4C2211yescharacteristic
21#501—24no—
22#478—24no—
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#97C2211yescharacteristic
37#399—21no—
38#399—21no—
39#390C2211yescharacteristic
40#390C2211yescharacteristic
41#389—21no—
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#387—21no—
45#387—21no—
46#384C2211yescharacteristic
47#384C2211yescharacteristic
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—
72C22⋊Q8—44no—
73C22⋊Q8—44no—
74C4×D4—22no—
75C4×D4—22no—
76C2×C4⋊C4—21no—
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⋊C4—21no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105#107—22no—
106#107—22no—
107#107—22no—
108#106—22no—
109#106—22no—
110#106—22no—
111C4⋊C8—22no—
112C4⋊C8—22no—
113C4⋊C8—22no—
114#102—22no—
115#102—22no—
116#102—22no—
117#102Q814yescomplemented
118#97—22no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95D411yescharacteristic
125#95D411yescharacteristic
126C24—21no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4C2×Q812yes—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2411yescharacteristic
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×C8—21no—
162C2×C8—21no—
163C2×C8C2×Q812yes—
164C2×C8#4112yes—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—21no—
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—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—81no—
187C4⋊C4C2×D412yes—
188C4⋊C4C2×D412yes—
189C4⋊C4C2×Q812yes—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—42no—
200C42—81no—
201C42—81no—
202C42C2×D412yes—
203C42#4112yes—
204C23—22no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23C22×D411yescharacteristic
212Q8—84no—
213D4—22no—
214D4—22no—
215D4—41no—
216D4—41no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
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—84no—
253C2×C4#12112yes—
254C2×C4#12112yes—
255C2×C4AGL(1,ℤ/8)12yes—
256C2×C4AGL(1,ℤ/8)12yes—
257C2×C4C22×D411yescharacteristic
258C2×C4C22×Q811yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C4C2×#4111yescharacteristic
261C2×C42−1+411yescharacteristic
262C2×C42−1+411yescharacteristic
263C8—22no—
264C8—22no—
265C8—41no—
266C8—41no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22C2×#12111yescharacteristic
275C22#54711yescharacteristic
276C22D4×Q811yescharacteristic
277C22C2×AGL(1,ℤ/8)11yescharacteristic
278C22C2×AGL(1,ℤ/8)11yescharacteristic
279C22#57711yescharacteristic
280C22#57811yescharacteristic
281C4—22no—
282C4—22no—
283C4—22no—
284C4—22no—
285C4—41no—
286C4—41no—
287C4—42no—
288C4—42no—
289C4—42no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4—81no—
294C4#53311yescharacteristic
295C4#54111yescharacteristic
296C4#54811yescharacteristic
297C4D4×Q811yescharacteristic
298C2—82no—
299C2#242911yescharacteristic
300C2#303811yescharacteristic
301C2#304611yescharacteristic
302C2#334311yescharacteristic
303C2#334511yescharacteristic
304C2#338411yescharacteristic
305C2#338611yescharacteristic
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. #2084C2semidirect
6. #2083C2semidirect
9. #1987C2semidirect
10. #1948C2semidirect
11. #1947C2semidirect
12. #1940C2semidirect
117. #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
#25422
#23702
#20842
#20832
#20672
#20652
#19872
#19482
#19472
#19402
#19262
#19252
#17622