Finite Group Databasebeta

← #16675

Subgroups — #16675

Order 256.

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

normal subgroups only

This group has 727 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×#397C211yesmaximal, characteristic
2C2×#385C212yesmaximal
3C2×#381C211yesmaximal, characteristic
4#1444C212yesmaximal, complemented
5#1443C212yesmaximal, complemented
6#1442C212yesmaximal, complemented
7#1441C212yesmaximal, complemented
8#1440C212yesmaximal
9#1439C211yesmaximal, characteristic
10C24×C4C2211yescharacteristic
11C22×C4⋊C4C2211yescharacteristic
12C22×C4⋊C4C2211yescharacteristic
13C22×C4⋊C4C2212yes—
14C22×C4⋊C4C2212yes—
15C22×C22⋊C4C2212yes—
16C22×C22⋊C4C2212yes—
17C22×C22⋊C4C2211yescharacteristic
18C22×C22⋊C4C2211yescharacteristic
19#397—24no—
20#385—28no—
21#381—24no—
22C2×#95C2212yes—
23C2×#95C2211yescharacteristic
24C2×#95C2212yes—
25C2×#95C2212yes—
26C2×#95C2212yes—
27C2×#95C2212yes—
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2211yescharacteristic
32C2×#95C2212yes—
33C2×#95C2212yes—
34C25C2311yescharacteristic
35C23×C4—22no—
36C23×C4—22no—
37C23×C4—41no—
38C23×C4—41no—
39C23×C4—41no—
40C23×C4—41no—
41C23×C4D414yes—
42C23×C4C2312yes—
43C23×C4C2312yes—
44C23×C4C2312yes—
45C23×C4C2311yescharacteristic
46C23×C4C2311yescharacteristic
47C23×C4C2312yes—
48C23×C4C2312yes—
49C23×C4C2312yes—
50C2×C4⋊C4—24no—
51C2×C4⋊C4—24no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—41no—
55C2×C4⋊C4—42no—
56C2×C4⋊C4—42no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—42no—
61C2×C22⋊C4—24no—
62C2×C22⋊C4—24no—
63C2×C22⋊C4—24no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—41no—
69C2×C22⋊C4—41no—
70C2×C22⋊C4—41no—
71C2×C22⋊C4—41no—
72C2×C22⋊C4D414yes—
73#95—24no—
74#95—24no—
75#95—24no—
76#95—24no—
77#95—22no—
78#95—24no—
79#95—24no—
80#95—24no—
81#95—24no—
82#95—41no—
83#95—42no—
84#95—42no—
85C24—22no—
86C24—22no—
87C24—21no—
88C24—21no—
89C24—21no—
90C24—21no—
91C24—22no—
92C24—22no—
93C24C2×D412yes—
94C24C2×D412yes—
95C24#4112yes—
96C24C2411yescharacteristic
97C22×C4—28no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
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—44no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4C2×D412yes—
152C22×C4C2×D412yes—
153C22×C4C2×D414yes—
154C22×C4#4114yes—
155C22×C4#4114yes—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C4⋊C4—48no—
159C4⋊C4—48no—
160C22⋊C4—216no—
161C22⋊C4—44no—
162C22⋊C4—44no—
163C22⋊C4—48no—
164C23—24no—
165C23—22no—
166C23—22no—
167C23—22no—
168C23—22no—
169C23—24no—
170C23—22no—
171C23—22no—
172C23—24no—
173C23—24no—
174C23—24no—
175C23—24no—
176C23—42no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—41no—
181C23—42no—
182C23—41no—
183C23—41no—
184C23—42no—
185C23—41no—
186C23#12114yes—
187C23C22×D411yescharacteristic
188C23C22×D411yescharacteristic
189C23C2×#4111yescharacteristic
190C23C2×#4112yes—
191C23C2×#4112yes—
192C23H(5,𝔽2)11yescharacteristic
193C23H(5,𝔽2)11yescharacteristic
194C23H(5,𝔽2)11yescharacteristic
195C23H(5,𝔽2)11yescharacteristic
196C232−1+412yes—
197C232−1+412yes—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—24no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—416no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
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—
251C22—24no—
252C22—28no—
253C22—42no—
254C22—42no—
255C22—44no—
256C22—42no—
257C22—44no—
258C22—42no—
259C22—42no—
260C22—44no—
261C22—42no—
262C22C2×#12111yescharacteristic
263C22#52512yes—
264C22#53411yescharacteristic
265C22#53411yescharacteristic
266C22#53411yescharacteristic
267C22#53511yescharacteristic
268C22#53612yes—
269C22#53811yescharacteristic
270C22#53811yescharacteristic
271C22#53912yes—
272C22#53912yes—
273C22#54212yes—
274C22#54212yes—
275C22#54712yes—
276C22#54712yes—
277C22#55111yescharacteristic
278C22#55212yes—
279C22#55212yes—
280C22#55312yes—
281C22#55911yescharacteristic
282C22#56311yescharacteristic
283C22#56312yes—
284C22#56311yescharacteristic
285C4—44no—
286C4—44no—
287C4—44no—
288C4—42no—
289C4—42no—
290C4—82no—
291C4—82no—
292C4—82no—
293C2—44no—
294C2#241312yes—
295C2#260011yescharacteristic
296C2#261711yescharacteristic
297C2#263812yes—
298C2#271711yescharacteristic
299C2#271812yes—
300C2#272411yescharacteristic
301C2#276411yescharacteristic
302C2#276512yes—
303C2#277111yescharacteristic
304C2#281711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC25C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1444C2semidirect
5. #1443C2semidirect
6. #1442C2semidirect
7. #1441C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3972
C2×#3852
C2×#3812
#14442
#14432
#14422
#14412
#14402
#14392