Finite Group Databasebeta

← #23494

Subgroups — #23494

Order 256.

681 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 681 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C2×#536C211yesmaximal, characteristic
2#2911C211yesmaximal, characteristic, complemented
3C2×#421C211yesmaximal, characteristic
4#2896C211yesmaximal, characteristic
5C22×#101C211yesmaximal, characteristic, complemented
6#2310C211yesmaximal, characteristic, complemented
7#2038C218yesmaximal, complemented
8#1733C211yesmaximal, characteristic, complemented
9C22×#41C2211yescharacteristic
10C23×Q8C2211yescharacteristic
11C22×M16C2211yescharacteristic
12C22×M16C2211yescharacteristic
13#536—22no—
14#536—22no—
15#536—22no—
16#536C2214yes—
17C2×C4⋊Q8—21no—
18C2×#124—21no—
19C2×#123—22no—
20C2×C22⋊Q8—22no—
21C2×#117C2211yescharacteristic
22C2×#117C2211yescharacteristic
23C2×#117C2211yescharacteristic
24#428—22no—
25#428C2214yescomplemented
26#427—22no—
27#421—24no—
28C2×#104C2214yes—
29#419—21no—
30#418—21no—
31#417C2214yes—
32#416—22no—
33C2×#101—24no—
34C2×#101C2214yescomplemented
35#391—22no—
36#391C2214yescomplemented
37#381—22no—
38#339C2214yescomplemented
39C2×#41—21no—
40C2×#41—21no—
41C2×#41—22no—
42C2×#41—22no—
43C2×#41—22no—
44C2×#41—22no—
45C2×#41D412yes—
46C2×#41C2312yes—
47C22×Q8—21no—
48C22×Q8—21no—
49C22×Q8—21no—
50C22×Q8—21no—
51C22×Q8—22no—
52C22×Q8—22no—
53C22×Q8—22no—
54C22×Q8—22no—
55C22×Q8D411yescharacteristic
56C22×Q8D412yes—
57C22×Q8C2312yes—
58C22×D4—21no—
59C22×D4D411yescharacteristic
60C23×C4—21no—
61C23×C4D412yes—
62C23×C4C2311yescharacteristic
63C2×M16—21no—
64C2×M16—22no—
65C2×M16—21no—
66C2×M16—22no—
67C2×M16—22no—
68C2×M16D414yes—
69C2×M16C2312yes—
70C2×M16C2312yes—
71C22×C8—21no—
72C22×C8—21no—
73C4⋊Q8—24no—
74C4⋊Q8—24no—
75#124—24no—
76#124—24no—
77#123—28no—
78#123—28no—
79C22⋊Q8—28no—
80C22⋊Q8—28no—
81#117—21no—
82#117—21no—
83#117—22no—
84#117—22no—
85#117—21no—
86#117—21no—
87#117—21no—
88#117—21no—
89#117D412yes—
90#117C2312yes—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4D412yes—
94C2×C4⋊C4C2312yes—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42C2312yes—
102#107—24no—
103#106—24no—
104#104—28no—
105#103—24no—
106#102—24no—
107#101—28no—
108#95—28no—
109C24—21no—
110C24C2×D411yescharacteristic
111#41—24no—
112#41—24no—
113#41—22no—
114#41—22no—
115#41—22no—
116#41—22no—
117#41—42no—
118#41—42no—
119#41—42no—
120#41—42no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—21no—
124C2×Q8—24no—
125C2×Q8—24no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8C2×D412yes—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—21no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4C2×D412yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4C2×D414yes—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D412yes—
170C22×C4#4112yes—
171C22×C4#4112yes—
172C22×C4#4112yes—
173C22×C4C2411yescharacteristic
174M16—24no—
175M16—22no—
176M16—22no—
177M16—22no—
178M16—42no—
179M16C2×D414yes—
180C2×C8—24no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—42no—
184C4⋊C4—216no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—42no—
189C4⋊C4C2×D414yes—
190C22⋊C4—216no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—41no—
194C22⋊C4—41no—
195C22⋊C4—42no—
196C42—22no—
197C42—22no—
198C42—22no—
199C42—22no—
200C42—41no—
201C42—41no—
202C23—22no—
203C23—42no—
204C23—44no—
205C23—42no—
206C23#12011yescharacteristic
207C23#12011yescharacteristic
208C23#12112yes—
209C23#12312yes—
210C23C22×D411yescharacteristic
211Q8—22no—
212Q8—22no—
213Q8—42no—
214Q8—42no—
215Q8—42no—
216Q8—42no—
217Q8—44no—
218Q8—44no—
219Q8—42no—
220D4—22no—
221D4—22no—
222D4—42no—
223D4—42no—
224D4—44no—
225D4—42no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4#12011yescharacteristic
243C2×C4#12011yescharacteristic
244C2×C4#12112yes—
245C2×C4#12112yes—
246C2×C4#12112yes—
247C2×C4#12112yes—
248C2×C4#12112yes—
249C2×C4#12312yes—
250C2×C4#12312yes—
251C2×C4#12312yes—
252C2×C4#12412yes—
253C2×C4#12412yes—
254C2×C4C22×D411yescharacteristic
255C2×C4C22×D411yescharacteristic
256C2×C4C22×D411yescharacteristic
257C2×C4C2×#4111yescharacteristic
258C2×C4C2×#4111yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C8—24no—
261C8—42no—
262C22—22no—
263C22—22no—
264C22—22no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—44no—
269C22—84no—
270C22#39412yes—
271C22#39412yes—
272C22C2×#12011yescharacteristic
273C22C2×#12111yescharacteristic
274C22C2×#12311yescharacteristic
275C4—44no—
276C4—42no—
277C4—44no—
278C4—82no—
279C4—82no—
280C4#39412yes—
281C4#39412yes—
282C4C2×#12111yescharacteristic
283C4C2×#12111yescharacteristic
284C4C2×#12311yescharacteristic
285C4C2×#12411yescharacteristic
286C2—21no—
287C2—22no—
288C2—21no—
289C2—22no—
290C2—82no—
291C2C2×#39411yescharacteristic
292C2#302012yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2911C2semidirect
5. C22×#101C2semidirect
6. #2310C2semidirect
7. #2038C2semidirect
8. #1733C2semidirect
25. #428C22semidirect
34. C2×#101C22semidirect
36. #391C22semidirect
38. #339C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5362
#29112
C2×#4212
#28962
C22×#1012
#23102
#20382
#17332