Finite Group Databasebeta

← #22710

Subgroups — #22710

Order 256.

449 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#435C211yesmaximal, characteristic
2#2354C211yesmaximal, characteristic
3#2352C211yesmaximal, characteristic
4#1949C211yesmaximal, characteristic, complemented
5#1948C211yesmaximal, characteristic, complemented
6#1854C211yesmaximal, characteristic, complemented
7#1854C211yesmaximal, characteristic, complemented
8#1854C211yesmaximal, characteristic, complemented
9#1853C211yesmaximal, characteristic, complemented
10#1836C211yesmaximal, characteristic, complemented
11#1809C211yesmaximal, characteristic, complemented
12#1807C211yesmaximal, characteristic, complemented
13#1776C211yesmaximal, characteristic, complemented
14#1736C211yesmaximal, characteristic, complemented
15#1735C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#435C418yescomplemented
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×#97C2211yescharacteristic
34#390—21no—
35#388—21no—
36#388—21no—
37#388—21no—
38#387—21no—
39#386—21no—
40#384C2211yescharacteristic
41#382C2211yescharacteristic
42#380C2211yescharacteristic
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45#380C2211yescharacteristic
46C4×C4⋊C4—21no—
47C4×C4⋊C4C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C22⋊C4—21no—
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60C22×D4D411yescharacteristic
61C23×C4—21no—
62C23×C4C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C2×M16—41no—
65C2×M16—41no—
66C2×M16C2×C414yes—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8C2×C414yescomplemented
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C4×D4C2×C414yes—
75C4×D4Q814yes—
76#117—22no—
77#117—22no—
78#117—22no—
79#117—41no—
80#117—41no—
81#117—41no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4D411yescharacteristic
104C2×C22⋊C4D411yescharacteristic
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42C2311yescharacteristic
111#107—22no—
112#107—41no—
113#106—22no—
114#106—41no—
115#106—41no—
116#106—41no—
117C4⋊C8C2×C414yescomplemented
118C22⋊C8C2×C414yes—
119C22⋊C8C2×C414yescomplemented
120#97C2×C414yescomplemented
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129C24C2×D411yescharacteristic
130C24C2×D411yescharacteristic
131C2×D4—41no—
132C2×D4—41no—
133C2×D4C4⋊C414yes—
134C22×C4—21no—
135C22×C4—21no—
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—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—81no—
153C22×C4C22×C412yes—
154C22×C4C22×C412yes—
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×Q812yes—
160C22×C4C2×Q812yes—
161C22×C4C2411yescharacteristic
162M16—44no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—44no—
172C2×C8C22×C412yes—
173C2×C8C22×C412yes—
174C2×C8C22×C412yes—
175C2×C8C22×C412yes—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—81no—
193C4⋊C4C4⋊C414yes—
194C4⋊C4#4112yes—
195C4⋊C4#4112yes—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—81no—
201C22⋊C4—81no—
202C22⋊C4—81no—
203C22⋊C4C4⋊C414yes—
204C22⋊C4C4⋊C414yes—
205C42—42no—
206C42—42no—
207C42—42no—
208C42—42no—
209C42—81no—
210C42C22×C412yes—
211C42C2×Q812yes—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23C2×C4⋊C412yes—
225C23C2×C4⋊C412yes—
226C23C22×D411yescharacteristic
227D4—44no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
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—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—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—84no—
269C2×C4C2×C4⋊C412yes—
270C2×C4C2×C4⋊C412yes—
271C2×C4C2×C4⋊C412yes—
272C2×C4C2×C4⋊C412yes—
273C2×C4C23×C411yescharacteristic
274C2×C4C22×Q811yescharacteristic
275C2×C4C2×#4111yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C2×C4H(5,𝔽2)11yescharacteristic
278C2×C4H(5,𝔽2)11yescharacteristic
279C8—22no—
280C8—22no—
281C8—22no—
282C8—22no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22C22×C4⋊C411yescharacteristic
298C22#54611yescharacteristic
299C22#54611yescharacteristic
300C22#57511yescharacteristic
301C22#57511yescharacteristic
302C22#57711yescharacteristic
303C22#57711yescharacteristic
304C4—22no—
305C4—22no—
306C4—22no—
307C4—22no—
308C4—42no—
309C4—42no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4C4×#4111yescharacteristic
317C4#51811yescharacteristic
318C4#55411yescharacteristic
319C4#55411yescharacteristic
320C2—42no—
321C2—42no—
322C2#234911yescharacteristic
323C2#291111yescharacteristic
324C2#291211yescharacteristic
325C2#330711yescharacteristic
326C2#331111yescharacteristic
327C2#331711yescharacteristic
328C2#332111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1949C2semidirect
5. #1948C2semidirect
6. #1854C2semidirect
7. #1854C2semidirect
8. #1854C2semidirect
9. #1853C2semidirect
10. #1836C2semidirect
11. #1809C2semidirect
12. #1807C2semidirect
13. #1776C2semidirect
14. #1736C2semidirect
15. #1735C2semidirect
23. #435C4semidirect
69. C22×C8C2×C4semidirect
117. C4⋊C8C2×C4semidirect
119. C22⋊C8C2×C4semidirect
120. #97C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4352
#23542
#23522
#19492
#19482
#18542
#18532
#18362
#18092
#18072
#17762
#17362
#17352