Finite Group Databasebeta

← #30347

Subgroups — #30347

Order 256.

343 conjugacy classes of subgroups.

The subgroup lattice has 313 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2863C211yesmaximal, characteristic, complemented
2#2861C211yesmaximal, characteristic
3#2068C211yesmaximal, characteristic, complemented
4#2065C211yesmaximal, characteristic
5#2059C211yesmaximal, characteristic, complemented
6#2056C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic
8#2042C211yesmaximal, characteristic
9#2026C211yesmaximal, characteristic, complemented
10#2015C211yesmaximal, characteristic
11#1973C211yesmaximal, characteristic
12#1971C211yesmaximal, characteristic, complemented
13#1936C211yesmaximal, characteristic
14#1935C211yesmaximal, characteristic, complemented
15#1767C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#97C2211yescharacteristic
33C2×#97C2211yescharacteristic
34#391C2211yescharacteristic
35#391C2211yescharacteristic
36#391C2211yescharacteristic
37#391C2211yescharacteristic
38#391C2211yescharacteristic
39#390C2211yescharacteristic
40#390C2211yescharacteristic
41#389C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#388—21no—
45#388—21no—
46#388—21no—
47#388—21no—
48#387—21no—
49#387—21no—
50#387—21no—
51#383—21no—
52C43C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4C2311yescharacteristic
62C2×Q16—22no—
63C2×Q16—41no—
64C2×Q16—41no—
65C2×SD16—22no—
66C2×SD16—41no—
67C2×SD16—41no—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72#125—41no—
73#125—41no—
74#124—41no—
75#124—41no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C42D411yescharacteristic
93C2×C42D411yescharacteristic
94C2×C42D411yescharacteristic
95C2×C42D411yescharacteristic
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99#107—41no—
100#106—41no—
101C4⋊C8—22no—
102C4⋊C8—22no—
103#103—22no—
104#103—41no—
105#103—41no—
106#103—41no—
107#103—41no—
108#102—22no—
109#102—41no—
110#102—41no—
111#97—41no—
112#97—41no—
113#95—21no—
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—
124#95—21no—
125#95—21no—
126C24—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2411yescharacteristic
159Q16—44no—
160SD16—44no—
161C2×C8—21no—
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#4112yes—
172C2×C8#4112yes—
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—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C42—22no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—41no—
202C42—41no—
203C42—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23C22×D411yescharacteristic
212Q8—41no—
213Q8—41no—
214Q8—41no—
215Q8—41no—
216Q8—41no—
217Q8—41no—
218Q8—41no—
219Q8—41no—
220D4—41no—
221D4—41no—
222D4—41no—
223D4—41no—
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—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
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—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4#13712yes—
263C2×C4C2×#4111yescharacteristic
264C2×C4C2×#4111yescharacteristic
265C2×C4H(5,𝔽2)11yescharacteristic
266C2×C4H(5,𝔽2)11yescharacteristic
267C2×C4H(5,𝔽2)11yescharacteristic
268C2×C42−1+411yescharacteristic
269C8—22no—
270C8—22no—
271C8—41no—
272C8—41no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22#53211yescharacteristic
281C22#53511yescharacteristic
282C22#53711yescharacteristic
283C22C2×#13711yescharacteristic
284C22#57511yescharacteristic
285C22#57511yescharacteristic
286C22#57511yescharacteristic
287C4—22no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4#53811yescharacteristic
301C4#53811yescharacteristic
302C4#54211yescharacteristic
303C4#54211yescharacteristic
304C2—161no—
305C2#266211yescharacteristic
306C2#318111yescharacteristic
307C2#318311yescharacteristic
308C2#321611yescharacteristic
309C2#321911yescharacteristic
310C2#325211yescharacteristic
311C2#325611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2863C2semidirect
3. #2068C2semidirect
5. #2059C2semidirect
6. #2056C2semidirect
9. #2026C2semidirect
12. #1971C2semidirect
14. #1935C2semidirect
15. #1767C2semidirect

Maximal subgroups

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

subgroupindex
#28632
#28612
#20682
#20652
#20592
#20562
#20552
#20422
#20262
#20152
#19732
#19712
#19362
#19352
#17672