Finite Group Databasebeta

← #17272

Subgroups — #17272

Order 256.

713 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 713 conjugacy classes of subgroups — too many to draw. Only its 151 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#1443C212yesmaximal, complemented
2#1441C212yesmaximal, complemented
3#1439C212yesmaximal, complemented
4#1439C212yesmaximal, complemented
5#1439C212yesmaximal, complemented
6#1439C212yesmaximal, complemented
7#1438C211yesmaximal, characteristic
8#1438C211yesmaximal, characteristic
9#1438C211yesmaximal, characteristic
10C23×D4C2211yescharacteristic
11C22×C4⋊C4C2212yes—
12C22×C4⋊C4C2212yes—
13C22×C22⋊C4C2211yescharacteristic
14C22×C22⋊C4C2212yes—
15C22×C22⋊C4C2212yes—
16C22×C22⋊C4C2212yes—
17C22×C22⋊C4C2211yescharacteristic
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21C2×#95C2212yes—
22C2×#95C2211yescharacteristic
23C2×#95C2212yes—
24C2×#95C2212yes—
25C2×#95C2211yescharacteristic
26C2×#95C2212yes—
27C2×#95C2212yes—
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C25C2311yescharacteristic
32C25C2311yescharacteristic
33C22×D4—41no—
34C22×D4—42no—
35C22×D4—42no—
36C22×D4—41no—
37C22×D4Q814yes—
38C23×C4C2312yes—
39C23×C4C2312yes—
40C23×C4C2312yes—
41C23×C4C2312yes—
42C23×C4C2311yescharacteristic
43C23×C4C2312yes—
44C23×C4C2312yes—
45C2×C4⋊C4—24no—
46C2×C4⋊C4—42no—
47C2×C4⋊C4—42no—
48C2×C4⋊C4—42no—
49C2×C4⋊C4—42no—
50C2×C4⋊C4—42no—
51C2×C22⋊C4—24no—
52C2×C22⋊C4—24no—
53C2×C22⋊C4—24no—
54C2×C22⋊C4—24no—
55C2×C22⋊C4—41no—
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⋊C4—41no—
71#95—24no—
72#95—42no—
73#95—42no—
74#95—42no—
75#95—42no—
76#95—41no—
77#95—41no—
78#95—42no—
79#95—42no—
80#95—42no—
81C24—22no—
82C24—22no—
83C24—22no—
84C24—22no—
85C24—22no—
86C24—22no—
87C24—22no—
88C24—22no—
89C24—22no—
90C24—22no—
91C24—22no—
92C24—21no—
93C24—21no—
94C24—22no—
95C24—21no—
96C24—21no—
97C24C2×Q812yes—
98C24C2×Q812yes—
99C24C2411yescharacteristic
100C2×D4—44no—
101C2×D4—44no—
102C2×D4—44no—
103C2×D4—48no—
104C2×D4—48no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
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×C4C2×Q812yes—
150C22×C4#4114yes—
151C22×C4#4114yes—
152C4⋊C4—48no—
153C4⋊C4—48no—
154C22⋊C4—44no—
155C22⋊C4—48no—
156C22⋊C4—48no—
157C22⋊C4—48no—
158C22⋊C4—44no—
159C22⋊C4—48no—
160C22⋊C4—44no—
161C22⋊C4—44no—
162C23—22no—
163C23—22no—
164C23—22no—
165C23—22no—
166C23—22no—
167C23—22no—
168C23—24no—
169C23—24no—
170C23—24no—
171C23—24no—
172C23—42no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—42no—
182C23—42no—
183C23—42no—
184C23—41no—
185C23—41no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—41no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—41no—
203C23—41no—
204C23C22×Q811yescharacteristic
205C23C2×#4112yes—
206C23C2×#4112yes—
207C23H(5,𝔽2)12yes—
208C23H(5,𝔽2)12yes—
209C23H(5,𝔽2)11yescharacteristic
210C23H(5,𝔽2)12yes—
211C23H(5,𝔽2)12yes—
212C23H(5,𝔽2)11yescharacteristic
213D4—416no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
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—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—44no—
266C22—44no—
267C22—44no—
268C22—44no—
269C22—42no—
270C22—42no—
271C22—82no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22#53812yes—
280C22#53812yes—
281C22#53812yes—
282C22#53812yes—
283C22#54012yes—
284C22#54012yes—
285C22#54311yescharacteristic
286C22#54311yescharacteristic
287C22#54311yescharacteristic
288C22#55111yescharacteristic
289C22#55312yes—
290C22#55412yes—
291C22#55412yes—
292C22#55512yes—
293C22#55911yescharacteristic
294C22#56011yescharacteristic
295C22#56012yes—
296C22#56012yes—
297C22#56012yes—
298C22#56012yes—
299C22H(3,𝔽4)11yescharacteristic
300C4—44no—
301C4—42no—
302C4—44no—
303C4—82no—
304C4—82no—
305C4—82no—
306C4—82no—
307C2—82no—
308C2—82no—
309C2#274811yescharacteristic
310C2#275612yes—
311C2#278012yes—
312C2#278412yes—
313C2#280312yes—
314C2#280411yescharacteristic
315C2#282512yes—
316C2#282812yes—
317C2#284211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC23×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #1443C2semidirect
2. #1441C2semidirect
3. #1439C2semidirect
4. #1439C2semidirect
5. #1439C2semidirect
6. #1439C2semidirect

Maximal subgroups

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

subgroupindex
#14432
#14412
#14392
#14382