Finite Group Databasebeta

← #14666

Subgroups — #14666

Order 256.

801 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 801 conjugacy classes of subgroups — too many to draw. Only its 191 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×#386C211yesmaximal, characteristic
2#1443C212yesmaximal, complemented
3#1442C212yesmaximal, complemented
4#1440C212yesmaximal, complemented
5#1439C212yesmaximal, complemented
6#1439C212yesmaximal, complemented
7#1438C211yesmaximal, characteristic
8#1438C211yesmaximal, characteristic
9C22⋊#95C212yesmaximal, complemented
10C23×D4C2211yescharacteristic
11C24×C4C2211yescharacteristic
12C22×C4⋊C4C2212yes—
13C22×C22⋊C4C2212yes—
14C22×C22⋊C4C2212yes—
15C22×C22⋊C4C2211yescharacteristic
16C22×C22⋊C4C2212yes—
17C22×C22⋊C4C2211yescharacteristic
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2212yes—
20#386C418yescomplemented
21C2×#95C2212yes—
22C2×#95C2212yes—
23C2×#95C2212yes—
24C2×#95C2212yes—
25C2×#95C2211yescharacteristic
26C2×#95C2212yes—
27C2×#95C2211yescharacteristic
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C25C2311yescharacteristic
33C25C2311yescharacteristic
34C22×D4—24no—
35C22×D4—42no—
36C22×D4—41no—
37C22×D4—41no—
38C22×D4C2×C414yes—
39C23×C4—41no—
40C23×C4—41no—
41C23×C4—42no—
42C23×C4—42no—
43C23×C4C2×C414yes—
44C23×C4C2312yes—
45C23×C4C2312yes—
46C23×C4C2312yes—
47C23×C4C2312yes—
48C23×C4C2312yes—
49C23×C4C2312yes—
50C23×C4C2311yescharacteristic
51C2×C4⋊C4—42no—
52C2×C4⋊C4—42no—
53C2×C4⋊C4—42no—
54C2×C22⋊C4—24no—
55C2×C22⋊C4—24no—
56C2×C22⋊C4—42no—
57C2×C22⋊C4—42no—
58C2×C22⋊C4—42no—
59C2×C22⋊C4—42no—
60C2×C22⋊C4—42no—
61C2×C22⋊C4—42no—
62C2×C22⋊C4—42no—
63C2×C22⋊C4—41no—
64C2×C22⋊C4—41no—
65C2×C22⋊C4—42no—
66C2×C22⋊C4—42no—
67C2×C22⋊C4—42no—
68C2×C22⋊C4—42no—
69C2×C22⋊C4—42no—
70C2×C22⋊C4C2×C414yes—
71C2×C22⋊C4C2×C418yes—
72#95—42no—
73#95—42no—
74#95—42no—
75#95—42no—
76#95—42no—
77#95—42no—
78#95—42no—
79#95—41no—
80#95—41no—
81#95—42no—
82#95C2×C418yes—
83C24—22no—
84C24—22no—
85C24—22no—
86C24—22no—
87C24—21no—
88C24—21no—
89C24—22no—
90C24—22no—
91C24—22no—
92C24—22no—
93C24—22no—
94C24—21no—
95C24—22no—
96C24—21no—
97C24—22no—
98C24C22×C412yes—
99C24C22×C412yes—
100C24#4114yes—
101C24C2411yescharacteristic
102C2×D4—28no—
103C2×D4—44no—
104C2×D4—44no—
105C2×D4—48no—
106C2×D4—48no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—48no—
152C22×C4—44no—
153C22×C4—48no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4C22×C412yes—
157C22×C4C22×C414yes—
158C22×C4C22×C414yes—
159C4⋊C4—48no—
160C22⋊C4—48no—
161C22⋊C4—44no—
162C22⋊C4—48no—
163C22⋊C4—48no—
164C22⋊C4—44no—
165C22⋊C4—48no—
166C22⋊C4—44no—
167C23—22no—
168C23—22no—
169C23—24no—
170C23—24no—
171C23—24no—
172C23—24no—
173C23—24no—
174C23—24no—
175C23—24no—
176C23—24no—
177C23—22no—
178C23—24no—
179C23—22no—
180C23—22no—
181C23—24no—
182C23—42no—
183C23—41no—
184C23—42no—
185C23—41no—
186C23—42no—
187C23—41no—
188C23—42no—
189C23—42no—
190C23—42no—
191C23—41no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—41no—
199C23—42no—
200C23—42no—
201C23—41no—
202C23—42no—
203C23—41no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—41no—
208C23#11714yes—
209C23C23×C411yescharacteristic
210C23C2×#4112yes—
211C23H(5,𝔽2)11yescharacteristic
212C23H(5,𝔽2)12yes—
213C23H(5,𝔽2)11yescharacteristic
214C23H(5,𝔽2)12yes—
215C23H(5,𝔽2)12yes—
216C23H(5,𝔽2)12yes—
217C232−1+412yes—
218D4—416no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—24no—
228C2×C4—416no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C22—24no—
268C22—28no—
269C22—44no—
270C22—44no—
271C22—44no—
272C22—44no—
273C22—42no—
274C22—42no—
275C22—44no—
276C22—42no—
277C22—44no—
278C22—42no—
279C22—42no—
280C22—44no—
281C22—82no—
282C22—82no—
283C22—82no—
284C22—82no—
285C22C2×#11711yescharacteristic
286C22#51811yescharacteristic
287C22#51811yescharacteristic
288C22#51812yes—
289C22#52012yes—
290C22#53812yes—
291C22#53812yes—
292C22#53812yes—
293C22#53812yes—
294C22#53912yes—
295C22#53912yes—
296C22#56012yes—
297C22#56011yescharacteristic
298C22#56012yes—
299C22#56012yes—
300C22#56011yescharacteristic
301C22H(3,𝔽4)11yescharacteristic
302C22#56212yes—
303C22#56212yes—
304C22#56212yes—
305C22#56311yescharacteristic
306C4—42no—
307C4—44no—
308C4—44no—
309C4—82no—
310C4—82no—
311C4—82no—
312C4—82no—
313C2—44no—
314C2—82no—
315C2#231211yescharacteristic
316C2#231812yes—
317C2#237511yescharacteristic
318C2#237611yescharacteristic
319C2#237712yes—
320C2#282412yes—
321C2#282512yes—
322C2#282612yes—
323C2#282712yes—
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC23×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1443C2semidirect
3. #1442C2semidirect
4. #1440C2semidirect
5. #1439C2semidirect
6. #1439C2semidirect
9. C22⋊#95C2semidirect
20. #386C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3862
#14432
#14422
#14402
#14392
#14382
C22⋊#952