Finite Group Databasebeta

← #31522

Subgroups — #31522

Order 256.

383 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2567C211yesmaximal, characteristic, complemented
2#2452C211yesmaximal, characteristic
3#2094C211yesmaximal, characteristic
4#2084C211yesmaximal, characteristic, complemented
5#2066C211yesmaximal, characteristic, complemented
6#2058C211yesmaximal, characteristic, complemented
7#2024C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic
9#1990C211yesmaximal, characteristic, complemented
10#1968C211yesmaximal, characteristic
11#1916C211yesmaximal, characteristic, complemented
12#1880C211yesmaximal, characteristic
13#1877C211yesmaximal, characteristic
14#1865C211yesmaximal, characteristic, complemented
15#1802C211yesmaximal, characteristic
16C22×SD16C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×C22⋊Q8—21no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×C4×C8C2211yescharacteristic
33#396—21no—
34#395—21no—
35#395—21no—
36#393—21no—
37#391C2211yescharacteristic
38#391C2211yescharacteristic
39#390C2211yescharacteristic
40#390C2211yescharacteristic
41#389—21no—
42#389—21no—
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#385—21no—
46#384C2211yescharacteristic
47#384C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×Q8—21no—
61C22×Q8C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C2×SD16—41no—
65C2×SD16—41no—
66C2×SD16Q814yescomplemented
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C4⋊Q8—22no—
72C4⋊Q8—41no—
73C4⋊Q8—41no—
74C4⋊Q8—41no—
75C22⋊Q8—44no—
76C4×Q8—41no—
77C4×Q8—41no—
78C4×D4—22no—
79C4×D4—41no—
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⋊C4D411yescharacteristic
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C42—21no—
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106#106—41no—
107#106—41no—
108#106—41no—
109C4⋊C8—41no—
110#103—22no—
111#103—41no—
112#103—41no—
113#102—41no—
114#102—41no—
115#102—41no—
116C4×C8—41no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95D411yescharacteristic
124#95D411yescharacteristic
125C24—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—81no—
135C2×Q8—81no—
136C2×Q8—81no—
137C2×Q8C2×Q812yes—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4C2×Q812yes—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—81no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2411yescharacteristic
161SD16—44no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8C2×Q812yes—
174C4⋊C4—22no—
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—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4SD1614yescomplemented
196C4⋊C4#4112yes—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—81no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—42no—
209C23—22no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23C22×D411yescharacteristic
217Q8—22no—
218Q8—22no—
219Q8—41no—
220Q8—41no—
221Q8—84no—
222Q8—84no—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—41no—
227C2×C4—21no—
228C2×C4—21no—
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—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
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—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—84no—
263C2×C4C2×SD1612yes—
264C2×C4C2×SD1612yes—
265C2×C4C2×SD1612yes—
266C2×C4#13712yes—
267C2×C4C22×Q811yescharacteristic
268C2×C4C2×#4111yescharacteristic
269C2×C4H(5,𝔽2)11yescharacteristic
270C2×C4H(5,𝔽2)11yescharacteristic
271C2×C4H(5,𝔽2)11yescharacteristic
272C2×C42−1+411yescharacteristic
273C8—22no—
274C8—41no—
275C8—41no—
276C8—41no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22#53511yescharacteristic
285C22#54611yescharacteristic
286C22D4×Q811yescharacteristic
287C22C22×SD1611yescharacteristic
288C22C2×#13711yescharacteristic
289C22#57611yescharacteristic
290C22#57711yescharacteristic
291C4—22no—
292C4—22no—
293C4—22no—
294C4—22no—
295C4—42no—
296C4—41no—
297C4—42no—
298C4—41no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4#54211yescharacteristic
305C4#54311yescharacteristic
306C4#55411yescharacteristic
307C4#56211yescharacteristic
308C2—82no—
309C2#269211yescharacteristic
310C2#320611yescharacteristic
311C2#322411yescharacteristic
312C2#329511yescharacteristic
313C2#331611yescharacteristic
314C2Q8×SD1611yescharacteristic
315C2#338511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2567C2semidirect
4. #2084C2semidirect
5. #2066C2semidirect
6. #2058C2semidirect
7. #2024C2semidirect
9. #1990C2semidirect
11. #1916C2semidirect
14. #1865C2semidirect
66. C2×SD16Q8semidirect
195. C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#25672
#24522
#20942
#20842
#20662
#20582
#20242
#20232
#19902
#19682
#19162
#18802
#18772
#18652
#18022