Finite Group Databasebeta

← #17136

Subgroups — #17136

Order 256.

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

normal subgroups only

This group has 797 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×#399C212yesmaximal, complemented
2C2×#396C212yesmaximal, complemented
3#1441C212yesmaximal, complemented
4#1441C212yesmaximal, 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×C4⋊C4C2212yes—
14C22×C4⋊C4C2212yes—
15C22×C22⋊C4C2211yescharacteristic
16C22×C22⋊C4C2211yescharacteristic
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2211yescharacteristic
23#399—28no—
24#396—28no—
25C2×#95C2212yes—
26C2×#95C2212yes—
27C2×#95C2211yescharacteristic
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2211yescharacteristic
31C2×#95C2212yes—
32C2×#95C2212yes—
33C25C2311yescharacteristic
34C25C2311yescharacteristic
35C22×D4—42no—
36C22×D4—42no—
37C22×D4—41no—
38C22×D4—41no—
39C22×D4Q814yes—
40C23×C4C2312yes—
41C23×C4C2312yes—
42C23×C4C2312yes—
43C23×C4C2312yes—
44C23×C4C2312yes—
45C23×C4C2311yescharacteristic
46C23×C4C2312yes—
47C2×C4⋊C4—24no—
48C2×C4⋊C4—24no—
49C2×C4⋊C4—24no—
50C2×C4⋊C4—24no—
51C2×C4⋊C4—24no—
52C2×C4⋊C4—42no—
53C2×C4⋊C4—42no—
54C2×C4⋊C4—42no—
55C2×C4⋊C4—42no—
56C2×C4⋊C4—42no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4D418yescomplemented
59C2×C22⋊C4—24no—
60C2×C22⋊C4—24no—
61C2×C22⋊C4—24no—
62C2×C22⋊C4—24no—
63C2×C22⋊C4—24no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—41no—
68C2×C22⋊C4—41no—
69C2×C22⋊C4—41no—
70C2×C22⋊C4—42no—
71C2×C22⋊C4—42no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4—42no—
75C2×C22⋊C4—42no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—41no—
78C2×C22⋊C4—42no—
79#95—24no—
80#95—24no—
81#95—41no—
82#95—42no—
83#95—42no—
84#95—42no—
85#95—42no—
86#95—41no—
87C24—22no—
88C24—22no—
89C24—22no—
90C24—22no—
91C24—22no—
92C24—22no—
93C24—22no—
94C24—22no—
95C24—22no—
96C24—22no—
97C24—21no—
98C24—21no—
99C24—21no—
100C24—21no—
101C24—22no—
102C24—22no—
103C24C2×Q812yes—
104C24C2×Q812yes—
105C24C2411yescharacteristic
106C2×D4—44no—
107C2×D4—44no—
108C2×D4—44no—
109C2×D4—48no—
110C2×D4—48no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
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—21no—
125C22×C4—21no—
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—22no—
152C22×C4—22no—
153C22×C4C2×D414yes—
154C22×C4C2×D414yes—
155C22×C4C2×D414yes—
156C22×C4C2×Q812yes—
157C22×C4#4114yes—
158C4⋊C4—48no—
159C4⋊C4—48no—
160C4⋊C4—48no—
161C4⋊C4—48no—
162C22⋊C4—44no—
163C22⋊C4—44no—
164C22⋊C4—48no—
165C22⋊C4—44no—
166C22⋊C4—48no—
167C22⋊C4—48no—
168C22⋊C4—48no—
169C22⋊C4—44no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—24no—
177C23—24no—
178C23—24no—
179C23—24no—
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—41no—
193C23—41no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—41no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23C22×D412yes—
213C23C22×Q811yescharacteristic
214C23C2×#4112yes—
215C23H(5,𝔽2)12yes—
216C23H(5,𝔽2)12yes—
217C23H(5,𝔽2)12yes—
218C23H(5,𝔽2)11yescharacteristic
219C23H(5,𝔽2)11yescharacteristic
220C232−1+412yes—
221D4—416no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—24no—
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—
267C2×C4—42no—
268C2×C4—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—44no—
274C22—44no—
275C22—44no—
276C22—44no—
277C22—42no—
278C22—42no—
279C22—82no—
280C22—82no—
281C22—82no—
282C22—82no—
283C22—82no—
284C22—82no—
285C22—82no—
286C22—82no—
287C22#53412yes—
288C22#53512yes—
289C22#53712yes—
290C22#53912yes—
291C22#54012yes—
292C22#54212yes—
293C22#54311yescharacteristic
294C22#54311yescharacteristic
295C22#54311yescharacteristic
296C22D4211yescharacteristic
297C22#54612yes—
298C22D4×Q812yes—
299C22#55012yes—
300C22#55111yescharacteristic
301C22#55412yes—
302C22#56012yes—
303C22#56012yes—
304C22#56011yescharacteristic
305C22#56212yes—
306C22#56212yes—
307C22#56311yescharacteristic
308C4—42no—
309C4—44no—
310C4—44no—
311C4—44no—
312C4—44no—
313C4—82no—
314C4—82no—
315C2—82no—
316C2—82no—
317C2#267312yes—
318C2#268311yescharacteristic
319C2#269212yes—
320C2#270512yes—
321C2#273212yes—
322C2#274811yescharacteristic
323C2#281312yes—
324C2#283712yes—
325C2#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×#399C2semidirect
2. C2×#396C2semidirect
3. #1441C2semidirect
4. #1441C2semidirect
5. #1439C2semidirect
6. #1439C2semidirect
58. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#3992
C2×#3962
#14412
#14392
#14382