Finite Group Databasebeta

← #16303

Subgroups — #16303

Order 256.

841 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#399C211yesmaximal, characteristic
2C2×#395C212yesmaximal, complemented
3C2×#393C211yesmaximal, characteristic
4C2×#380C212yesmaximal
5#1444C212yesmaximal, complemented
6#1442C212yesmaximal, complemented
7#1441C212yesmaximal, complemented
8#1439C212yesmaximal
9C22⋊#95C211yesmaximal, characteristic
10C23×Q8C2211yescharacteristic
11C24×C4C2211yescharacteristic
12C22×C4⋊C4C2212yes—
13C22×C4⋊C4C2212yes—
14C22×C4⋊C4C2212yes—
15C22×C4⋊C4C2212yes—
16C22×C4⋊C4C2211yescharacteristic
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21#399—24no—
22#395—28no—
23#393—24no—
24#380—28no—
25C2×#95C2211yescharacteristic
26C2×#95C2212yes—
27C2×#95C2212yes—
28C2×#95C2212yes—
29C2×#95C2211yescharacteristic
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C25C2311yescharacteristic
36C22×Q8—22no—
37C22×Q8—24no—
38C22×Q8—41no—
39C22×Q8—42no—
40C22×Q8—41no—
41C23×C4—24no—
42C23×C4—24no—
43C23×C4—41no—
44C23×C4—41no—
45C23×C4Q814yescomplemented
46C23×C4C2312yes—
47C23×C4C2312yes—
48C23×C4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2312yes—
51C23×C4C2311yescharacteristic
52C23×C4C2312yes—
53C23×C4C2312yes—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—24no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—41no—
74C2×C22⋊C4—41no—
75C2×C22⋊C4—41no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4D414yes—
80C2×C22⋊C4Q814yescomplemented
81#95—24no—
82#95—24no—
83#95—22no—
84#95—24no—
85#95—24no—
86#95—22no—
87#95—24no—
88#95—42no—
89#95—42no—
90#95D418yes—
91C24—22no—
92C24—22no—
93C24—21no—
94C24—22no—
95C24—21no—
96C24—21no—
97C24—21no—
98C24—22no—
99C24C2×D412yes—
100C24C2×Q812yes—
101C24C2×Q812yes—
102C24C2411yescharacteristic
103C2×Q8—48no—
104C2×Q8—44no—
105C2×Q8—44no—
106C2×Q8—48no—
107C2×Q8—44no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
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—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
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—216no—
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×C4—44no—
154C22×C4—44no—
155C22×C4—48no—
156C22×C4—44no—
157C22×C4C2×D412yes—
158C22×C4C2×D414yes—
159C22×C4C2×D412yes—
160C22×C4C2×D414yes—
161C22×C4C2×D414yes—
162C22×C4C2×Q814yes—
163C22×C4C2×Q814yes—
164C22×C4#4114yes—
165C4⋊C4—48no—
166C4⋊C4—48no—
167C4⋊C4—48no—
168C4⋊C4—44no—
169C4⋊C4—48no—
170C22⋊C4—44no—
171C22⋊C4—44no—
172C22⋊C4—48no—
173C22⋊C4—48no—
174C23—24no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—24no—
182C23—24no—
183C23—24no—
184C23—24no—
185C23—24no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23—41no—
191C23—42no—
192C23—41no—
193C23—41no—
194C23—42no—
195C23—41no—
196C23C4⋊Q814yes—
197C23C22×D412yes—
198C23C22×D411yescharacteristic
199C23C22×Q811yescharacteristic
200C23C22×Q811yescharacteristic
201C23C2×#4112yes—
202C23H(5,𝔽2)11yescharacteristic
203C23H(5,𝔽2)11yescharacteristic
204C23H(5,𝔽2)12yes—
205C23H(5,𝔽2)11yescharacteristic
206C232−1+411yescharacteristic
207C232−1+412yes—
208Q8—416no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—24no—
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—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
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×C4C22⋊Q818yes—
262C22—24no—
263C22—28no—
264C22—42no—
265C22—44no—
266C22—42no—
267C22—44no—
268C22—42no—
269C22—44no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22C2×C22⋊Q812yes—
274C22C2×C4⋊Q811yescharacteristic
275C22#53411yescharacteristic
276C22#53512yes—
277C22#53612yes—
278C22#53711yescharacteristic
279C22#54212yes—
280C22#54212yes—
281C22#54311yescharacteristic
282C22#54311yescharacteristic
283C22#54412yes—
284C22#54411yescharacteristic
285C22#54411yescharacteristic
286C22D4212yes—
287C22D4211yescharacteristic
288C22#54612yes—
289C22#54712yes—
290C22D4×Q812yes—
291C22#55111yescharacteristic
292C22#55412yes—
293C22#56011yescharacteristic
294C22#56212yes—
295C22#56311yescharacteristic
296C4—28no—
297C4—42no—
298C4—44no—
299C4—44no—
300C4—44no—
301C4—42no—
302C4—82no—
303C4—82no—
304C2—44no—
305C2#243711yescharacteristic
306C2#244012yes—
307C2#262812yes—
308C2#265911yescharacteristic
309C2#266911yescharacteristic
310C2#267212yes—
311C2#267711yescharacteristic
312C2#269212yes—
313C2#271711yescharacteristic
314C2#274211yescharacteristic
315C2#284111yescharacteristic
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
2. C2×#395C2semidirect
5. #1444C2semidirect
6. #1442C2semidirect
7. #1441C2semidirect
45. C23×C4Q8semidirect
80. C2×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#3992
C2×#3952
C2×#3932
C2×#3802
#14442
#14422
#14412
#14392
C22⋊#952