Finite Group Databasebeta

← #31857

Subgroups — #31857

Order 256.

374 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
1#2503C211yesmaximal, characteristic
2#2429C211yesmaximal, characteristic
3#2101C211yesmaximal, characteristic, complemented
4#2074C211yesmaximal, characteristic, complemented
5#2056C211yesmaximal, characteristic, complemented
6#2016C211yesmaximal, characteristic, complemented
7#2001C211yesmaximal, characteristic
8#2001C211yesmaximal, characteristic
9#1960C211yesmaximal, characteristic, complemented
10#1935C211yesmaximal, characteristic, complemented
11#1924C211yesmaximal, characteristic
12#1878C211yesmaximal, characteristic, complemented
13#1872C211yesmaximal, characteristic
14#1872C211yesmaximal, characteristic
15#1805C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×#97C2211yescharacteristic
36#400—21no—
37#400—21no—
38#398—21no—
39#396—21no—
40#391C2211yescharacteristic
41#391C2211yescharacteristic
42#391C2211yescharacteristic
43#389—21no—
44#389C2211yescharacteristic
45#388—21no—
46#382C2211yescharacteristic
47#382C2211yescharacteristic
48#381—21no—
49#381—21no—
50C4×C22⋊C4C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×Q8C2311yescharacteristic
62C23×C4—21no—
63C23×C4C2311yescharacteristic
64C2×Q16—22no—
65C2×Q16—41no—
66C2×Q16—41no—
67C2×Q16—41no—
68C2×Q16—41no—
69C2×Q16—41no—
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C4⋊Q8—22no—
75C4⋊Q8—41no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78C22⋊Q8—41no—
79C22⋊Q8—41no—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82C4×Q8—22no—
83C4×Q8—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4D411yescharacteristic
103C2×C22⋊C4D411yescharacteristic
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107C4⋊C8—41no—
108#103—22no—
109#103—41no—
110#103—41no—
111#103—41no—
112#103—41no—
113#103—41no—
114#103—41no—
115#103—41no—
116C22⋊C8—41no—
117C22⋊C8—41no—
118#97—41no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95D411yescharacteristic
131#95D411yescharacteristic
132C24C2×D411yescharacteristic
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8#4112yes—
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—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2411yescharacteristic
174Q16—44no—
175Q16—44no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4#4112yes—
206C22⋊C4—22no—
207C22⋊C4—41no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—42no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23#13712yes—
223C23C22×D411yescharacteristic
224Q8—22no—
225Q8—22no—
226Q8—41no—
227Q8—41no—
228Q8—41no—
229Q8—41no—
230Q8—41no—
231Q8—41no—
232Q8—41no—
233Q8—41no—
234Q8—41no—
235Q8—41no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
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—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—84no—
280C2×C4C2×#4111yescharacteristic
281C2×C4C2×#4111yescharacteristic
282C2×C4H(5,𝔽2)11yescharacteristic
283C2×C4H(5,𝔽2)11yescharacteristic
284C2×C4H(5,𝔽2)11yescharacteristic
285C2×C4H(5,𝔽2)11yescharacteristic
286C8—41no—
287C8—41no—
288C8—41no—
289C8—41no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22#53511yescharacteristic
298C22#54611yescharacteristic
299C22#54611yescharacteristic
300C22C2×#13711yescharacteristic
301C22#57511yescharacteristic
302C22#57811yescharacteristic
303C22#57811yescharacteristic
304C4—41no—
305C4—42no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—42no—
310C4—41no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4—81no—
317C4#53811yescharacteristic
318C4#54011yescharacteristic
319C4#55311yescharacteristic
320C4#56011yescharacteristic
321C2—42no—
322C2#270311yescharacteristic
323C2#321511yescharacteristic
324C2#322311yescharacteristic
325C2#331211yescharacteristic
326C2#331211yescharacteristic
327C2#331511yescharacteristic
328C2#332711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×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. #2101C2semidirect
4. #2074C2semidirect
5. #2056C2semidirect
6. #2016C2semidirect
9. #1960C2semidirect
10. #1935C2semidirect
12. #1878C2semidirect
15. #1805C2semidirect

Maximal subgroups

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

subgroupindex
#25032
#24292
#21012
#20742
#20562
#20162
#20012
#19602
#19352
#19242
#18782
#18722
#18052