Finite Group Databasebeta

← #22933

Subgroups — #22933

Order 256.

429 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic
2#2363C211yesmaximal, characteristic
3#2347C211yesmaximal, characteristic, complemented
4#1944C211yesmaximal, characteristic, complemented
5#1936C211yesmaximal, characteristic, complemented
6#1925C211yesmaximal, characteristic, complemented
7#1920C211yesmaximal, characteristic, complemented
8#1918C211yesmaximal, characteristic, complemented
9#1893C211yesmaximal, characteristic
10#1864C211yesmaximal, characteristic, complemented
11#1854C211yesmaximal, characteristic, complemented
12#1776C211yesmaximal, characteristic, complemented
13#1762C211yesmaximal, characteristic, complemented
14#1752C211yesmaximal, characteristic, complemented
15#1735C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C2×#117C2211yescharacteristic
21#482C418yescomplemented
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×#97C2211yescharacteristic
32C2×#97C2211yescharacteristic
33#390—21no—
34#390C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#388—21no—
39#388—21no—
40#388—21no—
41#386—21no—
42#383—21no—
43#380C2211yescharacteristic
44C4×C4⋊C4—21no—
45C4×C4⋊C4C2211yescharacteristic
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C22⋊C4—21no—
49C4×C22⋊C4C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C22×D4—21no—
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C23×C4C2311yescharacteristic
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—41no—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71#121—22no—
72#121C2×C414yes—
73C4×D4—22no—
74C4×D4—41no—
75#117—22no—
76#117—22no—
77#117—41no—
78#117C2×C414yescomplemented
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4D411yescharacteristic
97C2×C22⋊C4D411yescharacteristic
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105#107—41no—
106#107C2×C414yescomplemented
107#106—22no—
108#106C2×C414yescomplemented
109#102—22no—
110#102—41no—
111#102C2×C414yes—
112#102C2×C414yes—
113C22⋊C8C2×C414yescomplemented
114#97—22no—
115#97—41no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95D411yescharacteristic
124#95D411yescharacteristic
125C24—21no—
126C24C2×D411yescharacteristic
127C2×D4—24no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—81no—
134C2×D4—81no—
135C2×D4C22×C412yes—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—81no—
150C22×C4C22×C412yes—
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4#4112yes—
157C22×C4C2411yescharacteristic
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×C8—21no—
167C2×C8—44no—
168C2×C8C22×C412yes—
169C2×C8C22×C412yes—
170C2×C8#4112yes—
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—42no—
182C4⋊C4—42no—
183C4⋊C4—81no—
184C4⋊C4C22×C412yes—
185C4⋊C4C22×C412yes—
186C4⋊C4C22×C412yes—
187C4⋊C4#4112yes—
188C22⋊C4—24no—
189C22⋊C4—24no—
190C22⋊C4—41no—
191C22⋊C4—41no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—81no—
196C22⋊C4—81no—
197C42—24no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—42no—
203C42—81no—
204C42—81no—
205C42—81no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—22no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23C4×D412yes—
220C23C22×D411yescharacteristic
221D4—22no—
222D4—22no—
223D4—41no—
224D4—41no—
225D4—84no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
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—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—84no—
266C2×C4C4×D412yes—
267C2×C4C23×C411yescharacteristic
268C2×C4C2×#4111yescharacteristic
269C2×C4C2×#4111yescharacteristic
270C2×C4C2×#4111yescharacteristic
271C2×C4H(5,𝔽2)11yescharacteristic
272C2×C42−1+411yescharacteristic
273C8—22no—
274C8—22no—
275C8—22no—
276C8—41no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22C2×C4×D411yescharacteristic
292C22#54611yescharacteristic
293C22#54711yescharacteristic
294C22#57511yescharacteristic
295C22#57511yescharacteristic
296C22#57611yescharacteristic
297C22#57711yescharacteristic
298C4—22no—
299C4—41no—
300C4—41no—
301C4—42no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—42no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4C4×#4111yescharacteristic
311C4#52011yescharacteristic
312C4#55111yescharacteristic
313C4#55211yescharacteristic
314C2—42no—
315C2—82no—
316C2#235911yescharacteristic
317C2#294311yescharacteristic
318C2#294611yescharacteristic
319C2#330911yescharacteristic
320C2#331111yescharacteristic
321C2#334711yescharacteristic
322C2#335011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2347C2semidirect
4. #1944C2semidirect
5. #1936C2semidirect
6. #1925C2semidirect
7. #1920C2semidirect
8. #1918C2semidirect
10. #1864C2semidirect
11. #1854C2semidirect
12. #1776C2semidirect
13. #1762C2semidirect
14. #1752C2semidirect
15. #1735C2semidirect
21. #482C4semidirect
78. #117C2×C4semidirect
106. #107C2×C4semidirect
108. #106C2×C4semidirect
113. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
#23632
#23472
#19442
#19362
#19252
#19202
#19182
#18932
#18642
#18542
#17762
#17622
#17522
#17352