Finite Group Databasebeta

← #24323

Subgroups — #24323

Order 256.

502 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3503C211yesmaximal, characteristic
2#3078C211yesmaximal, characteristic, complemented
3C4×#137C211yesmaximal, characteristic
4#2567C211yesmaximal, characteristic, complemented
5#2060C212yesmaximal, complemented
6#2028C212yesmaximal, complemented
7#1997C211yesmaximal, characteristic, complemented
8#1986C211yesmaximal, characteristic, complemented
9#1882C212yesmaximal
10#1859C212yesmaximal
11#1806C211yesmaximal, characteristic
12C2×#137C2211yescharacteristic
13Q82—21no—
14#557—21no—
15#554—21no—
16#554—22no—
17D4×Q8—22no—
18D4×Q8—21no—
19#544—22no—
20#544C2212yes—
21#533—21no—
22#533C2211yescharacteristic
23C2×C4⋊Q8—21no—
24C2×C4⋊Q8C2211yescharacteristic
25#520C2212yes—
26C4×#41C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28#479—22no—
29#477—21no—
30#475—21no—
31#441—21no—
32#440—21no—
33C4×Q16—21no—
34C4×SD16—21no—
35#428C2212yes—
36#428C2211yescharacteristic
37#423C2211yescharacteristic
38C2×#104C2212yes—
39#419C2212yes—
40#419C2211yescharacteristic
41#417C2211yescharacteristic
42C4×M16C2211yescharacteristic
43#395—22no—
44#391C2212yes—
45#389—22no—
46#384C2212yes—
47C4×C4⋊C4C2211yescharacteristic
48#343C2212yes—
49#343C2212yes—
50#339C2212yes—
51#339C2212yes—
52#339C2212yes—
53C2×#41C2311yescharacteristic
54C22×Q8—22no—
55C22×Q8D412yes—
56C22×Q8C2311yescharacteristic
57#137—22no—
58#137Q814yescomplemented
59C2×Q16—21no—
60C2×SD16—21no—
61C2×M16C2311yescharacteristic
62C2×M16C2312yes—
63C2×M16C2311yescharacteristic
64C4⋊Q8—22no—
65C4⋊Q8—22no—
66C4⋊Q8—24no—
67C4⋊Q8—22no—
68C4⋊Q8—22no—
69C4⋊Q8—21no—
70C4⋊Q8—21no—
71C4⋊Q8—21no—
72C4⋊Q8—21no—
73C4⋊Q8D411yescharacteristic
74C4⋊Q8D411yescharacteristic
75#125—24no—
76#125—22no—
77#125—22no—
78#125D412yes—
79C22⋊Q8—22no—
80C22⋊Q8—22no—
81C22⋊Q8—24no—
82C22⋊Q8—24no—
83C22⋊Q8—21no—
84C22⋊Q8—21no—
85C22⋊Q8—24no—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×Q8—21no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8D411yescharacteristic
96C4×Q8D411yescharacteristic
97C4×D4—22no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—24no—
101C4×D4D411yescharacteristic
102C4×D4D411yescharacteristic
103#117—22no—
104#117—21no—
105#117C2311yescharacteristic
106#117C2312yes—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4D412yes—
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2312yes—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42C2312yes—
118C2×C42C2311yescharacteristic
119#107—21no—
120#107—22no—
121#106—21no—
122#106—22no—
123C4⋊C8—21no—
124C4⋊C8—21no—
125#104—24no—
126#103—21no—
127#103—21no—
128#103—21no—
129#103—22no—
130#103—22no—
131#102—21no—
132#97—21no—
133C4×C8—21no—
134#95—21no—
135#95—24no—
136#41—21no—
137#41—21no—
138#41C2×Q812yes—
139#41#4112yes—
140C2×Q8—24no—
141C2×Q8—21no—
142C2×Q8—22no—
143C2×Q8—21no—
144C2×Q8—22no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—82no—
151C2×Q8—84no—
152C2×Q8C2×D411yescharacteristic
153C2×Q8C2×Q812yescomplemented
154C2×D4—21no—
155C2×D4C2×D411yescharacteristic
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D414yes—
165C22×C4C2×D411yescharacteristic
166C22×C4C2411yescharacteristic
167Q16—24no—
168SD16—24no—
169M16—21no—
170M16—22no—
171M16—21no—
172M16—21no—
173M16—22no—
174M16C2×Q812yescomplemented
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—22no—
178C4⋊C4—24no—
179C4⋊C4—24no—
180C4⋊C4—24no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—21no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—84no—
198C4⋊C4C2×D412yes—
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D412yes—
201C4⋊C4C2×D411yescharacteristic
202C22⋊C4—24no—
203C22⋊C4—22no—
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—21no—
207C42—24no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—22no—
212C42—22no—
213C42—22no—
214C42—41no—
215C42—41no—
216C42C2×D411yescharacteristic
217C42C2×D411yescharacteristic
218C42C2×D411yescharacteristic
219C42C2×D411yescharacteristic
220C23—21no—
221C23C22×D411yescharacteristic
222Q8—21no—
223Q8—22no—
224Q8—22no—
225Q8—44no—
226Q8—42no—
227Q8—41no—
228Q8—41no—
229Q8—42no—
230Q8—82no—
231Q8—84no—
232Q8C22⋊Q812yes—
233D4—22no—
234D4—22no—
235D4—21no—
236D4C22⋊Q812yes—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—84no—
259C2×C4#12011yescharacteristic
260C2×C4#12011yescharacteristic
261C2×C4#12011yescharacteristic
262C2×C4#12011yescharacteristic
263C2×C4C22×D411yescharacteristic
264C2×C4C22×D411yescharacteristic
265C2×C4C22×Q811yescharacteristic
266C2×C4C2×#4111yescharacteristic
267C2×C42−1+411yescharacteristic
268C2×C42−1+411yescharacteristic
269C8—22no—
270C8—41no—
271C8—42no—
272C22—21no—
273C22—21no—
274C22—22no—
275C22—41no—
276C22—41no—
277C22C2×#12011yescharacteristic
278C22#54811yescharacteristic
279C22D4×Q811yescharacteristic
280C4—22no—
281C4—21no—
282C4—21no—
283C4—21no—
284C4—44no—
285C4—42no—
286C4—41no—
287C4—42no—
288C4—41no—
289C4—44no—
290C4—42no—
291C4—81no—
292C4#45612yes—
293C4C2×C22⋊Q811yescharacteristic
294C4#53611yescharacteristic
295C4#54811yescharacteristic
296C4D4×Q811yescharacteristic
297C2—21no—
298C2—21no—
299C2—42no—
300C2#240911yescharacteristic
301C2C2×#45611yescharacteristic
302C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3078C2semidirect
4. #2567C2semidirect
5. #2060C2semidirect
6. #2028C2semidirect
7. #1997C2semidirect
8. #1986C2semidirect
58. #137Q8semidirect
153. C2×Q8C2×Q8semidirect
174. M16C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#35032
#30782
C4×#1372
#25672
#20602
#20282
#19972
#19862
#18822
#18592
#18062