Finite Group Databasebeta

← #27697

Subgroups — #27697

Order 256.

395 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2502C211yesmaximal, characteristic
2#2440C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic, complemented
4#2100C211yesmaximal, characteristic, complemented
5#2072C211yesmaximal, characteristic, complemented
6#2024C211yesmaximal, characteristic, complemented
7#1990C211yesmaximal, characteristic, complemented
8#1933C211yesmaximal, characteristic, complemented
9#1933C211yesmaximal, characteristic, complemented
10#1892C211yesmaximal, characteristic
11#1872C211yesmaximal, characteristic
12#1834C211yesmaximal, characteristic, complemented
13#1824C211yesmaximal, characteristic
14C8×C22⋊C4C211yesmaximal, characteristic
15C2×#340C211yesmaximal, characteristic
16C23×C8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C2×#103C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×C22⋊C8C2211yescharacteristic
27C2×C4×C8C2211yescharacteristic
28C2×C4×C8C2211yescharacteristic
29#398—21no—
30#398—21no—
31#398—21no—
32#395—21no—
33#391C2211yescharacteristic
34#391C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#385—21no—
38#385—21no—
39#385—21no—
40#384C2211yescharacteristic
41#384C2211yescharacteristic
42#384C2211yescharacteristic
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×Q8—21no—
60C22×Q8C2311yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C23×C4C2311yescharacteristic
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—41no—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C4⋊Q8—22no—
72C4⋊Q8—41no—
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75C22⋊Q8—44no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
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⋊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—41no—
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
102C2×C22⋊C4D411yescharacteristic
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105#103—22no—
106#103—22no—
107#103—41no—
108#103—41no—
109C22⋊C8—22no—
110C4×C8—22no—
111C4×C8—41no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95D411yescharacteristic
120#95D411yescharacteristic
121C24C2×D411yescharacteristic
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—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×C4—21no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—81no—
147C22×C4—81no—
148C22×C4—81no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4#4112yes—
155C22×C4#4112yes—
156C22×C4C2411yescharacteristic
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—44no—
169C2×C8#4112yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—81no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4#4112yes—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4SD1614yes—
199C22⋊C4Q1614yes—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23#12412yes—
209C23C2×SD1612yes—
210C23C2×Q1612yes—
211C23C22×D411yescharacteristic
212Q8—41no—
213Q8—41no—
214Q8—41no—
215Q8—41no—
216Q8—84no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
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—22no—
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—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—84no—
256C2×C4—84no—
257C2×C4#12412yes—
258C2×C4C2×SD1612yes—
259C2×C4C2×SD1612yes—
260C2×C4C2×Q1612yes—
261C2×C4C2×Q1612yes—
262C2×C4C2×#4111yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4C2×#4111yescharacteristic
265C2×C4C2×#4111yescharacteristic
266C2×C4H(5,𝔽2)11yescharacteristic
267C2×C42−1+411yescharacteristic
268C8—22no—
269C8—41no—
270C8—41no—
271C8—41no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22#48714yes—
279C22C2×#12411yescharacteristic
280C22#54611yescharacteristic
281C22#54711yescharacteristic
282C22C22×SD1611yescharacteristic
283C22C22×Q1611yescharacteristic
284C22#57611yescharacteristic
285C22#57711yescharacteristic
286C4—22no—
287C4—22no—
288C4—22no—
289C4—22no—
290C4—22no—
291C4—42no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4#52911yescharacteristic
300C4#53911yescharacteristic
301C4#55111yescharacteristic
302C4#55211yescharacteristic
303C2—24no—
304C2#248811yescharacteristic
305C2C2×#48711yescharacteristic
306C2#313611yescharacteristic
307C2#329511yescharacteristic
308C2#329911yescharacteristic
309C2#333511yescharacteristic
310C2#333811yescharacteristic
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. #2100C2semidirect
4. #2100C2semidirect
5. #2072C2semidirect
6. #2024C2semidirect
7. #1990C2semidirect
8. #1933C2semidirect
9. #1933C2semidirect
12. #1834C2semidirect

Maximal subgroups

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

subgroupindex
#25022
#24402
#21002
#20722
#20242
#19902
#19332
#18922
#18722
#18342
#18242
C8×C22⋊C42
C2×#3402