Finite Group Databasebeta

← #23832

Subgroups — #23832

Order 256.

365 conjugacy classes of subgroups.

The subgroup lattice has 303 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2330C211yesmaximal, characteristic, complemented
2#2326C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic, complemented
4#2099C211yesmaximal, characteristic, complemented
5#2072C211yesmaximal, characteristic, complemented
6#2071C211yesmaximal, characteristic
7#2042C211yesmaximal, characteristic
8#2042C211yesmaximal, characteristic
9#2026C211yesmaximal, characteristic, complemented
10#2025C211yesmaximal, characteristic
11#1763C211yesmaximal, characteristic, complemented
12#1762C211yesmaximal, characteristic
13C4×#103C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic
15#1746C211yesmaximal, characteristic, complemented
16C2×#125C2211yescharacteristic
17C2×#124C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×#103C2211yescharacteristic
21C2×#103C2211yescharacteristic
22C2×#103C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#97C2211yescharacteristic
29C2×C4×C8C2211yescharacteristic
30#391C2211yescharacteristic
31#390C2211yescharacteristic
32#389—21no—
33#389—21no—
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#388—21no—
37#388—21no—
38#387—21no—
39#387—21no—
40#385—21no—
41#385—21no—
42#384C2211yescharacteristic
43#384C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
46#376C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66#125—22no—
67#125—22no—
68#124—22no—
69#124—22no—
70C4×Q8—22no—
71C4×Q8—22no—
72C4×D4—22no—
73C4×D4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
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⋊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×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100#103—22no—
101#103—22no—
102#103—22no—
103#103—22no—
104#102—22no—
105#102—22no—
106#102—22no—
107#102—22no—
108#97—22no—
109C4×C8—22no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95D411yescharacteristic
117#95D411yescharacteristic
118#95D411yescharacteristic
119#95D411yescharacteristic
120C24—21no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8#4112yes—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4#4112yes—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—42no—
141C22×C4—81no—
142C22×C4—81no—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
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—21no—
162C4⋊C4—24no—
163C4⋊C4—24no—
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—42no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C4⋊C4—81no—
181C4⋊C4#4112yes—
182C4⋊C4#4112yes—
183C22⋊C4—24no—
184C22⋊C4—24no—
185C22⋊C4—42no—
186C22⋊C4—81no—
187C22⋊C4—81no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—42no—
193C42—81no—
194C42—81no—
195C42#4112yes—
196C42#4112yes—
197C23—22no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23C22×D411yescharacteristic
205Q8—22no—
206Q8—22no—
207Q8—41no—
208Q8—41no—
209D4—22no—
210D4—22no—
211D4—41no—
212D4—41no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
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—84no—
251C2×C4#12412yes—
252C2×C4#12412yes—
253C2×C4C2×#4111yescharacteristic
254C2×C4C2×#4111yescharacteristic
255C2×C4C2×#4111yescharacteristic
256C2×C4C2×#4111yescharacteristic
257C2×C4C2×#4111yescharacteristic
258C2×C4C2×#4111yescharacteristic
259C8—41no—
260C8—41no—
261C8—41no—
262C8—41no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—81no—
267C22—81no—
268C22—81no—
269C22—81no—
270C22#52511yescharacteristic
271C22#52511yescharacteristic
272C22C2×#12411yescharacteristic
273C22#57611yescharacteristic
274C22#57711yescharacteristic
275C22#57711yescharacteristic
276C22#57811yescharacteristic
277C4—22no—
278C4—22no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—42no—
283C4—41no—
284C4—42no—
285C4—42no—
286C4—81no—
287C4—81no—
288C4—81no—
289C4—81no—
290C4#52911yescharacteristic
291C4#52911yescharacteristic
292C4#52911yescharacteristic
293C4#52911yescharacteristic
294C2—82no—
295C2#239811yescharacteristic
296C2#311911yescharacteristic
297C2#312011yescharacteristic
298C2#312411yescharacteristic
299C2#312511yescharacteristic
300C2#313611yescharacteristic
301C2#313711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2330C2semidirect
3. #2100C2semidirect
4. #2099C2semidirect
5. #2072C2semidirect
9. #2026C2semidirect
11. #1763C2semidirect
13. C4×#103C2semidirect
15. #1746C2semidirect

Maximal subgroups

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

subgroupindex
#23302
#23262
#21002
#20992
#20722
#20712
#20422
#20262
#20252
#17632
#17622
C4×#1032
C4×#1022
#17462