Finite Group Databasebeta

← #27702

Subgroups — #27702

Order 256.

372 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2500C211yesmaximal, characteristic
2#2441C211yesmaximal, characteristic, complemented
3#2101C211yesmaximal, characteristic, complemented
4#2100C211yesmaximal, characteristic, complemented
5#2071C211yesmaximal, characteristic, complemented
6#2025C211yesmaximal, characteristic, complemented
7#1991C211yesmaximal, characteristic, complemented
8#1934C211yesmaximal, characteristic, complemented
9#1934C211yesmaximal, characteristic, complemented
10#1893C211yesmaximal, characteristic
11#1875C211yesmaximal, characteristic
12#1834C211yesmaximal, characteristic, complemented
13#1825C211yesmaximal, characteristic, complemented
14C8×C22⋊C4C211yesmaximal, characteristic, complemented
15#1735C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C2×#102C2211yescharacteristic
22C2×#102C2211yescharacteristic
23C2×#102C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×C22⋊C8C2211yescharacteristic
26C2×C4×C8C2211yescharacteristic
27C2×C4×C8C2211yescharacteristic
28#399—21no—
29#396—21no—
30#394—21no—
31#394—21no—
32#390C2211yescharacteristic
33#390C2211yescharacteristic
34#389—21no—
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#388—21no—
38#388—21no—
39#387—21no—
40#384C2211yescharacteristic
41#384C2211yescharacteristic
42#382C2211yescharacteristic
43#380C2211yescharacteristic
44C4×C22⋊C4C2211yescharacteristic
45#340C2211yescharacteristic
46#340C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4—21no—
60C22×D4C2311yescharacteristic
61C23×C4C2311yescharacteristic
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—41no—
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69#125—22no—
70#125—41no—
71#121—22no—
72#121—22no—
73#117—22no—
74#117—22no—
75#117—22no—
76#117—41no—
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⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4D411yescharacteristic
101C2×C22⋊C4D411yescharacteristic
102C2×C42—21no—
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106#102—22no—
107#102—22no—
108#102—41no—
109#102—41no—
110C22⋊C8—22no—
111C4×C8—22no—
112C4×C8—41no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95D411yescharacteristic
119#95D411yescharacteristic
120C24—21no—
121C24C2×D411yescharacteristic
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—42no—
129C2×D4—81no—
130C2×D4—81no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
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×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4#4112yes—
148C22×C4#4112yes—
149C22×C4C2411yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
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—44no—
162C2×C8#4112yes—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—81no—
181C4⋊C4—81no—
182C4⋊C4—81no—
183C4⋊C4—81no—
184C4⋊C4#4112yes—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—81no—
193C22⋊C4—81no—
194C22⋊C4—81no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—42no—
200C42—42no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23#12412yes—
215C23C22×D411yescharacteristic
216D4—41no—
217D4—41no—
218D4—41no—
219D4—41no—
220D4—84no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
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—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4#12412yes—
259C2×C4#13512yes—
260C2×C4#13512yes—
261C2×C4C2×#4111yescharacteristic
262C2×C4C2×#4111yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4C2×#4111yescharacteristic
265C2×C4H(5,𝔽2)11yescharacteristic
266C2×C42−1+411yescharacteristic
267C8—22no—
268C8—41no—
269C8—41no—
270C8—41no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22C2×#12411yescharacteristic
286C22#54611yescharacteristic
287C22#54711yescharacteristic
288C22C2×#13511yescharacteristic
289C22C2×#13511yescharacteristic
290C22#57611yescharacteristic
291C22#57711yescharacteristic
292C4—22no—
293C4—22no—
294C4—22no—
295C4—41no—
296C4—42no—
297C4—41no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4#52911yescharacteristic
305C4#53911yescharacteristic
306C4#55111yescharacteristic
307C4#55211yescharacteristic
308C2—42no—
309C2—161no—
310C2#248811yescharacteristic
311C2#313111yescharacteristic
312C2#313611yescharacteristic
313C2#330311yescharacteristic
314C2#330511yescharacteristic
315C2#333711yescharacteristic
316C2#333911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
2. #2441C2semidirect
3. #2101C2semidirect
4. #2100C2semidirect
5. #2071C2semidirect
6. #2025C2semidirect
7. #1991C2semidirect
8. #1934C2semidirect
9. #1934C2semidirect
12. #1834C2semidirect
13. #1825C2semidirect
14. C8×C22⋊C4C2semidirect
15. #1735C2semidirect

Maximal subgroups

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

subgroupindex
#25002
#24412
#21012
#21002
#20712
#20252
#19912
#19342
#18932
#18752
#18342
#18252
C8×C22⋊C42
#17352