Finite Group Databasebeta

← #27893

Subgroups — #27893

Order 256.

390 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2850C211yesmaximal, characteristic
2#2283C211yesmaximal, characteristic, complemented
3#2036C211yesmaximal, characteristic, complemented
4#2035C211yesmaximal, characteristic, complemented
5#2034C211yesmaximal, characteristic
6#2033C211yesmaximal, characteristic
7#1958C211yesmaximal, characteristic, complemented
8#1958C211yesmaximal, characteristic, complemented
9#1928C211yesmaximal, characteristic, complemented
10#1927C211yesmaximal, characteristic, complemented
11#1926C211yesmaximal, characteristic, complemented
12#1925C211yesmaximal, characteristic, complemented
13#1846C211yesmaximal, characteristic, complemented
14#1835C211yesmaximal, characteristic, complemented
15C4×C22⋊C8C211yesmaximal, characteristic, complemented
16C2×#124C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C22×C42C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C4×C8C2211yescharacteristic
35C2×C4×C8C2211yescharacteristic
36#391C2211yescharacteristic
37#390C2211yescharacteristic
38#388—21no—
39#388—21no—
40#386—21no—
41#386—21no—
42#383—21no—
43#382C2211yescharacteristic
44#382C2211yescharacteristic
45#382C2211yescharacteristic
46#381—21no—
47#380C2211yescharacteristic
48#380C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51C4×C22⋊C4—21no—
52#376—21no—
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4—21no—
61C22×D4C2311yescharacteristic
62C23×C4D411yescharacteristic
63C23×C4D411yescharacteristic
64C23×C4C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69#124—22no—
70#124—22no—
71C22⋊Q8—41no—
72C22⋊Q8—41no—
73#121—41no—
74#121—41no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42D411yescharacteristic
95C2×C42D411yescharacteristic
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98#107—41no—
99#107—41no—
100#106—41no—
101#106—41no—
102#103—22no—
103#103—22no—
104#103—41no—
105#103—41no—
106#102—22no—
107#102—22no—
108#102—41no—
109#102—41no—
110C22⋊C8—22no—
111C22⋊C8—22no—
112C4×C8—22no—
113C4×C8—22no—
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#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126C24—21no—
127C24C2×D411yescharacteristic
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—42no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2411yescharacteristic
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8#4112yes—
179C2×C8#4112yes—
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—21no—
191C4⋊C4—21no—
192C4⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C42—22no—
202C42—22no—
203C42—21no—
204C42—21no—
205C42—42no—
206C42—42no—
207C42#4112yes—
208C42#4112yes—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23#13512yes—
223C23C22×D411yescharacteristic
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228D4—41no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—84no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—44no—
246C2×C4—44no—
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—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4#12312yes—
273C2×C4#12312yes—
274C2×C4#13512yes—
275C2×C4C2×#4111yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C2×C4C2×#4111yescharacteristic
278C2×C4C2×#4111yescharacteristic
279C2×C4H(5,𝔽2)11yescharacteristic
280C2×C4H(5,𝔽2)11yescharacteristic
281C8—22no—
282C8—22no—
283C8—41no—
284C8—41no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22—81no—
299C22C2×#12311yescharacteristic
300C22#53211yescharacteristic
301C22#53511yescharacteristic
302C22C2×#13511yescharacteristic
303C22C2×#13511yescharacteristic
304C22#57511yescharacteristic
305C22#57511yescharacteristic
306C4—22no—
307C4—22no—
308C4—22no—
309C4—41no—
310C4—41no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4#55311yescharacteristic
319C4#55311yescharacteristic
320C4#55511yescharacteristic
321C4#55511yescharacteristic
322C2—42no—
323C2—161no—
324C2#250011yescharacteristic
325C2#309111yescharacteristic
326C2#309111yescharacteristic
327C2#316811yescharacteristic
328C2#318311yescharacteristic
329C2#321111yescharacteristic
330C2#321211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2283C2semidirect
3. #2036C2semidirect
4. #2035C2semidirect
7. #1958C2semidirect
8. #1958C2semidirect
9. #1928C2semidirect
10. #1927C2semidirect
11. #1926C2semidirect
12. #1925C2semidirect
13. #1846C2semidirect
14. #1835C2semidirect
15. C4×C22⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#28502
#22832
#20362
#20352
#20342
#20332
#19582
#19282
#19272
#19262
#19252
#18462
#18352
C4×C22⋊C82