Finite Group Databasebeta

← #17196

Subgroups — #17196

Order 256.

755 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 755 conjugacy classes of subgroups — too many to draw. Only its 159 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×#396C212yesmaximal, complemented
2#1445C212yesmaximal, complemented
3#1445C211yesmaximal, characteristic, complemented
4#1444C211yesmaximal, characteristic, complemented
5#1439C212yesmaximal, complemented
6#1439C212yesmaximal, complemented
7#1439C212yesmaximal, complemented
8#1438C212yesmaximal
9#1438C211yesmaximal, characteristic
10C23×D4C2211yescharacteristic
11C22×C4⋊C4C2212yes—
12C22×C4⋊C4C2212yes—
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C22⋊C4C2212yes—
16C22×C22⋊C4C2212yes—
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21#396—28no—
22C2×#95C2212yes—
23C2×#95C2211yescharacteristic
24C2×#95C2212yes—
25C2×#95C2211yescharacteristic
26C2×#95C2211yescharacteristic
27C2×#95C2212yes—
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2211yescharacteristic
31C2×#95C2212yes—
32C25C2312yes—
33C22×D4—41no—
34C22×D4—42no—
35C22×D4—41no—
36C22×D4—42no—
37C22×D4Q814yes—
38C23×C4C2312yes—
39C23×C4C2312yes—
40C23×C4C2312yes—
41C23×C4C2311yescharacteristic
42C23×C4C2312yes—
43C23×C4C2311yescharacteristic
44C23×C4C2312yes—
45C23×C4C2311yescharacteristic
46C2×C4⋊C4—24no—
47C2×C4⋊C4—24no—
48C2×C4⋊C4—24no—
49C2×C4⋊C4—41no—
50C2×C4⋊C4—41no—
51C2×C4⋊C4—42no—
52C2×C4⋊C4—42no—
53C2×C4⋊C4—42no—
54C2×C4⋊C4—41no—
55C2×C4⋊C4—42no—
56C2×C4⋊C4D414yescomplemented
57C2×C22⋊C4—24no—
58C2×C22⋊C4—24no—
59C2×C22⋊C4—24no—
60C2×C22⋊C4—24no—
61C2×C22⋊C4—24no—
62C2×C22⋊C4—24no—
63C2×C22⋊C4—42no—
64C2×C22⋊C4—42no—
65C2×C22⋊C4—42no—
66C2×C22⋊C4—42no—
67C2×C22⋊C4—42no—
68C2×C22⋊C4—42no—
69C2×C22⋊C4—42no—
70C2×C22⋊C4—42no—
71C2×C22⋊C4—42no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4—42no—
75#95—24no—
76#95—22no—
77#95—42no—
78#95—42no—
79#95—41no—
80#95—42no—
81#95—42no—
82#95—41no—
83#95—41no—
84#95—42no—
85C24—22no—
86C24—22no—
87C24—22no—
88C24—22no—
89C24—22no—
90C24—22no—
91C24—22no—
92C24—22no—
93C24—22no—
94C24—22no—
95C24—22no—
96C24—22no—
97C24—22no—
98C24—22no—
99C24C2×Q814yes—
100C24C2411yescharacteristic
101C2×D4—48no—
102C2×D4—44no—
103C2×D4—44no—
104C2×D4—48no—
105C2×D4—44no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—21no—
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—21no—
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×C4C2×D414yes—
152C22×C4C2×D412yes—
153C22×C4C2×Q812yes—
154C22×C4#4112yes—
155C22×C4#4114yes—
156C4⋊C4—44no—
157C4⋊C4—48no—
158C4⋊C4—48no—
159C4⋊C4—44no—
160C22⋊C4—48no—
161C22⋊C4—48no—
162C22⋊C4—48no—
163C22⋊C4—48no—
164C22⋊C4—48no—
165C22⋊C4—48no—
166C23—24no—
167C23—24no—
168C23—24no—
169C23—24no—
170C23—24no—
171C23—24no—
172C23—24no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—42no—
182C23—42no—
183C23—42no—
184C23—42no—
185C23—42no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23C22×D411yescharacteristic
202C23C22×Q811yescharacteristic
203C23C2×#4111yescharacteristic
204C23C2×#4112yes—
205C23H(5,𝔽2)11yescharacteristic
206C23H(5,𝔽2)11yescharacteristic
207C23H(5,𝔽2)12yes—
208C23H(5,𝔽2)12yes—
209C23H(5,𝔽2)11yescharacteristic
210C23H(5,𝔽2)12yes—
211C232−1+411yescharacteristic
212D4—416no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—41no—
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—
263C22—44no—
264C22—44no—
265C22—44no—
266C22—44no—
267C22—44no—
268C22—44no—
269C22—44no—
270C22—82no—
271C22—82no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22#53411yescharacteristic
279C22#53512yes—
280C22#53812yes—
281C22#53812yes—
282C22#53911yescharacteristic
283C22#54012yes—
284C22#54212yes—
285C22#54312yes—
286C22#54311yescharacteristic
287C22#54612yes—
288C22D4×Q811yescharacteristic
289C22#55011yescharacteristic
290C22#55312yes—
291C22#55412yes—
292C22#55411yescharacteristic
293C22#55511yescharacteristic
294C22#56011yescharacteristic
295C22#56011yescharacteristic
296C22#56012yes—
297C22#56012yes—
298C22#56011yescharacteristic
299C22#56212yes—
300C22#56211yescharacteristic
301C4—42no—
302C4—42no—
303C4—44no—
304C4—44no—
305C4—42no—
306C4—82no—
307C4—82no—
308C4—82no—
309C2—84no—
310C2#269212yes—
311C2#269911yescharacteristic
312C2#271412yes—
313C2#273211yescharacteristic
314C2#278012yes—
315C2#278411yescharacteristic
316C2#279511yescharacteristic
317C2#281511yescharacteristic
318C2#282611yescharacteristic
319C2#282912yes—
320C2#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. C2×#396C2semidirect
2. #1445C2semidirect
3. #1445C2semidirect
4. #1444C2semidirect
5. #1439C2semidirect
6. #1439C2semidirect
7. #1439C2semidirect
56. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#3962
#14452
#14442
#14392
#14382