Finite Group Databasebeta

← #32658

Subgroups — #32658

Order 256.

353 conjugacy classes of subgroups.

The subgroup lattice has 311 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2582C211yesmaximal, characteristic, complemented
2#2580C211yesmaximal, characteristic
3#2097C211yesmaximal, characteristic, complemented
4#2094C211yesmaximal, characteristic
5#2085C211yesmaximal, characteristic, complemented
6#2083C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic
8#2066C211yesmaximal, characteristic, complemented
9#1985C211yesmaximal, characteristic, complemented
10#1973C211yesmaximal, characteristic
11#1944C211yesmaximal, characteristic, complemented
12#1880C211yesmaximal, characteristic
13#1877C211yesmaximal, characteristic
14#1863C211yesmaximal, characteristic, complemented
15#1862C211yesmaximal, characteristic
16C22×SD16C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#97C2211yescharacteristic
33#399—21no—
34#398—21no—
35#398—21no—
36#393—21no—
37#391—21no—
38#391C2211yescharacteristic
39#391C2211yescharacteristic
40#390C2211yescharacteristic
41#390C2211yescharacteristic
42#389—21no—
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#385—21no—
48#385—21no—
49#384C2211yescharacteristic
50#382C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8D411yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C2×SD16—41no—
64C2×SD16—41no—
65C2×SD16Q814yescomplemented
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C4×Q8—41no—
71C4×Q8—41no—
72C4×Q8—41no—
73C4×Q8—41no—
74C4×D4—41no—
75C4×D4—41no—
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⋊C4D411yescharacteristic
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4D411yescharacteristic
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×C42—21no—
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#107—41no—
102#107—41no—
103#107—41no—
104#106—22no—
105#106—41no—
106C4⋊C8—41no—
107#103—22no—
108#103—41no—
109#103—41no—
110#102—22no—
111#102—41no—
112#102—41no—
113#97—41no—
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—
124C24—21no—
125C2×Q8—22no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8C2×Q812yes—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4C2×Q812yes—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—42no—
148C22×C4—81no—
149C22×C4—81no—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2411yescharacteristic
157SD16—44no—
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×C8C2×Q812yes—
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—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C4⋊C4—81no—
191C4⋊C4—81no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—81no—
196C22⋊C4—81no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—42no—
204C23—22no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23C22×D411yescharacteristic
212Q8—22no—
213Q8—22no—
214Q8—44no—
215Q8—41no—
216Q8—41no—
217D4—22no—
218D4—22no—
219D4—41no—
220D4—41no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
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—41no—
259C2×C4—84no—
260C2×C4#13712yes—
261C2×C4C22×Q811yescharacteristic
262C2×C4C2×#4111yescharacteristic
263C2×C4H(5,𝔽2)11yescharacteristic
264C2×C4H(5,𝔽2)11yescharacteristic
265C2×C4H(5,𝔽2)11yescharacteristic
266C2×C42−1+411yescharacteristic
267C8—22no—
268C8—22no—
269C8—41no—
270C8—41no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22#53411yescharacteristic
279C22D4×Q811yescharacteristic
280C22#55011yescharacteristic
281C22C2×#13711yescharacteristic
282C22#57511yescharacteristic
283C22#57611yescharacteristic
284C22#57711yescharacteristic
285C4—22no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—42no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4#53911yescharacteristic
299C4#54311yescharacteristic
300C4#55411yescharacteristic
301C4#56211yescharacteristic
302C2—82no—
303C2#273211yescharacteristic
304C2#319611yescharacteristic
305C2#319811yescharacteristic
306C2#338511yescharacteristic
307C2#338811yescharacteristic
308C2#340011yescharacteristic
309C2#340311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2582C2semidirect
3. #2097C2semidirect
5. #2085C2semidirect
6. #2083C2semidirect
8. #2066C2semidirect
9. #1985C2semidirect
11. #1944C2semidirect
14. #1863C2semidirect
65. C2×SD16Q8semidirect

Maximal subgroups

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

subgroupindex
#25822
#25802
#20972
#20942
#20852
#20832
#20672
#20662
#19852
#19732
#19442
#18802
#18772
#18632
#18622