Finite Group Databasebeta

← #24154

Subgroups — #24154

Order 256.

697 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 697 conjugacy classes of subgroups — too many to draw. Only its 147 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×#532C211yesmaximal, characteristic
2C2×#475C211yesmaximal, characteristic, complemented
3C2×#460C211yesmaximal, characteristic, complemented
4C4×C22⋊Q8C211yesmaximal, characteristic, complemented
5#2034C212yesmaximal, complemented
6#2000C212yesmaximal, complemented
7#1988C211yesmaximal, characteristic, complemented
8#1982C211yesmaximal, characteristic, complemented
9#1877C212yesmaximal, complemented
10#1864C212yesmaximal, complemented
11#1797C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13C22×SD16C2212yes—
14#532—24no—
15#532—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#124—21no—
19C2×C22⋊Q8C2211yescharacteristic
20C2×#121—21no—
21C2×#121C2212yes—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4—21no—
24C2×C4×D4C2212yes—
25C22×C42C2211yescharacteristic
26#475—24no—
27#460—24no—
28C2×#106C2212yes—
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2212yes—
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×C22⋊C8C2212yes—
36#391C2211yescharacteristic
37#389C2211yescharacteristic
38#387—21no—
39#385—21no—
40#380C2211yescharacteristic
41C4×C4⋊C4—21no—
42C4×C4⋊C4C2211yescharacteristic
43C4×C22⋊C4—21no—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C2×#41—28no—
48C2×#41—44no—
49C2×#41—44no—
50C22×Q8—21no—
51C22×Q8C2311yescharacteristic
52C22×D4—22no—
53C22×D4—21no—
54C22×D4—21no—
55C22×D4C2312yes—
56C23×C4—22no—
57C23×C4D411yescharacteristic
58C23×C4D411yescharacteristic
59C23×C4C2311yescharacteristic
60C2×SD16—24no—
61C2×SD16—24no—
62C2×SD16—42no—
63C22×C8C2312yes—
64C22×C8C2312yes—
65C4⋊Q8—22no—
66C4⋊Q8D414yes—
67#127—22no—
68#127D414yes—
69#124—28no—
70C22⋊Q8—22no—
71C22⋊Q8—22no—
72#121—28no—
73#121—24no—
74#121—24no—
75C4×Q8—22no—
76C4×Q8—22no—
77C4×D4—28no—
78C4×D4—24no—
79C4×D4—24no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2312yes—
87C2×C4⋊C4C2311yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—21no—
92C2×C42—22no—
93C2×C42—22no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42D411yescharacteristic
97C2×C42D411yescharacteristic
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100#106—24no—
101C4⋊C8—22no—
102C4⋊C8—22no—
103#103—24no—
104#102—24no—
105#102—24no—
106#102—42no—
107C22⋊C8—42no—
108#95—21no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113C24—22no—
114C24C2×D411yescharacteristic
115#41—432no—
116C2×Q8—24no—
117C2×Q8—21no—
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8#4112yescomplemented
125C2×D4—28no—
126C2×D4—24no—
127C2×D4—24no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4—44no—
136C2×D4—44no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4C2×D414yes—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—28no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4SD1614yescomplemented
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2411yescharacteristic
166SD16—48no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4C2×D414yes—
184C4⋊C4#4112yescomplemented
185C22⋊C4—28no—
186C22⋊C4—28no—
187C22⋊C4—24no—
188C22⋊C4—42no—
189C42—22no—
190C42—22no—
191C42—21no—
192C42—21no—
193C42—42no—
194C42—42no—
195C42C2×D412yes—
196C42C2×D412yes—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—24no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23C2×SD1612yes—
211C23C22×D411yescharacteristic
212Q8—22no—
213Q8—22no—
214Q8—41no—
215Q8—41no—
216Q8—48no—
217D4—24no—
218D4—24no—
219D4—48no—
220D4—416no—
221D4—42no—
222D4—42no—
223D4—48no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—28no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—24no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—44no—
239C2×C4—416no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—42no—
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#12012yes—
257C2×C4#12012yes—
258C2×C4C2×SD1612yes—
259C2×C4C2×SD1612yes—
260C2×C4AGL(1,ℤ/8)12yes—
261C2×C4C22×D411yescharacteristic
262C2×C4C22×D411yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4C2×#4111yescharacteristic
265C2×C4H(5,𝔽2)11yescharacteristic
266C2×C42−1+411yescharacteristic
267C8—42no—
268C8—42no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—44no—
277C22—44no—
278C22—44no—
279C22—82no—
280C22—82no—
281C22—82no—
282C22—82no—
283C22C2×#12011yescharacteristic
284C22#52511yescharacteristic
285C22#53711yescharacteristic
286C22C22×SD1611yescharacteristic
287C22C2×AGL(1,ℤ/8)11yescharacteristic
288C22#57511yescharacteristic
289C22#57511yescharacteristic
290C4—22no—
291C4—22no—
292C4—42no—
293C4—41no—
294C4—42no—
295C4—44no—
296C4—41no—
297C4—81no—
298C4—81no—
299C4#45014yes—
300C4#54611yescharacteristic
301C4#54711yescharacteristic
302C4#54811yescharacteristic
303C4#55011yescharacteristic
304C2—42no—
305C2—84no—
306C2#240711yescharacteristic
307C2C2×#45011yescharacteristic
308C2#300511yescharacteristic
309C2#310311yescharacteristic
310C2#311511yescharacteristic
311C2#324611yescharacteristic
312C2#325111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#532C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#475C2semidirect
3. C2×#460C2semidirect
4. C4×C22⋊Q8C2semidirect
5. #2034C2semidirect
6. #2000C2semidirect
7. #1988C2semidirect
8. #1982C2semidirect
9. #1877C2semidirect
10. #1864C2semidirect
11. #1797C2semidirect
124. C2×Q8#41semidirect
159. C22×C4SD16semidirect
184. C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5322
C2×#4752
C2×#4602
C4×C22⋊Q82
#20342
#20002
#19882
#19822
#18772
#18642
#17972