Finite Group Databasebeta

← #16601

Subgroups — #16601

Order 256.

875 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8C211yesmaximal, characteristic
2C2×#400C211yesmaximal, characteristic, complemented
3C2×#395C211yesmaximal, characteristic, complemented
4C2×#385C211yesmaximal, characteristic
5#1446C211yesmaximal, characteristic, complemented
6#1445C211yesmaximal, characteristic, complemented
7#1444C212yesmaximal, complemented
8#1442C212yesmaximal, complemented
9#1440C212yesmaximal
10#1439C212yesmaximal
11C22⋊#95C211yesmaximal, characteristic
12C23×Q8C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C2×C22⋊Q8—24no—
15C2×C22⋊Q8—24no—
16C2×C22⋊Q8—24no—
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2212yes—
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2212yes—
21C22×C4⋊C4C2211yescharacteristic
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2211yescharacteristic
25C22×C22⋊C4C2211yescharacteristic
26#400—24no—
27#395—24no—
28#385—24no—
29C2×#95C2211yescharacteristic
30C2×#95C2212yes—
31C2×#95C2211yescharacteristic
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2211yescharacteristic
37C2×#95C2211yescharacteristic
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C22×Q8—22no—
43C22×Q8—22no—
44C22×Q8—42no—
45C22×Q8—42no—
46C22×Q8D414yes—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—42no—
50C23×C4—42no—
51C23×C4D414yes—
52C23×C4C2311yescharacteristic
53C23×C4C2312yes—
54C23×C4C2311yescharacteristic
55C23×C4C2312yes—
56C23×C4C2312yes—
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C23×C4C2312yes—
61C23×C4C2311yescharacteristic
62C22⋊Q8—232no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4D414yes—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4Q814yescomplemented
88#95—22no—
89#95—22no—
90#95—24no—
91#95—22no—
92#95—24no—
93#95—22no—
94#95—24no—
95#95—24no—
96#95—24no—
97#95—42no—
98#95—42no—
99#95—42no—
100C24—21no—
101C24—21no—
102C24—22no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—21no—
107C24—21no—
108C24C2×D412yes—
109C24C2×Q812yes—
110C24#4112yes—
111C24C2411yescharacteristic
112C2×Q8—28no—
113C2×Q8—48no—
114C2×Q8—44no—
115C2×Q8—44no—
116C2×Q8—48no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
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—21no—
150C22×C4—21no—
151C22×C4—28no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—48no—
172C22×C4—48no—
173C22×C4C2×D414yes—
174C22×C4C2×D414yes—
175C22×C4C2×D412yes—
176C22×C4C2×D412yes—
177C22×C4C2×D412yes—
178C22×C4C2×D412yes—
179C22×C4C2×Q812yes—
180C22×C4C2×Q812yes—
181C22×C4#4114yes—
182C4⋊C4—216no—
183C4⋊C4—28no—
184C4⋊C4—44no—
185C4⋊C4—48no—
186C4⋊C4—44no—
187C22⋊C4—216no—
188C22⋊C4—28no—
189C22⋊C4—48no—
190C22⋊C4—44no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—24no—
198C23—24no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—24no—
204C23—24no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23C22⋊Q814yes—
214C23C22×D411yescharacteristic
215C23C22×D411yescharacteristic
216C23C22×D411yescharacteristic
217C23C22×Q811yescharacteristic
218C23C2×#4111yescharacteristic
219C23C2×#4112yes—
220C23H(5,𝔽2)12yes—
221C23H(5,𝔽2)12yes—
222C232−1+411yescharacteristic
223C232−1+412yes—
224C232−1+411yescharacteristic
225Q8—416no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—416no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4#12014yes—
287C22—24no—
288C22—24no—
289C22—24no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—44no—
295C22—44no—
296C22—44no—
297C22—44no—
298C22C2×#12011yescharacteristic
299C22C2×C22⋊Q811yescharacteristic
300C22#53211yescharacteristic
301C22#53411yescharacteristic
302C22#53411yescharacteristic
303C22#53412yes—
304C22#53512yes—
305C22#53611yescharacteristic
306C22#53611yescharacteristic
307C22#53912yes—
308C22#53912yes—
309C22#54212yes—
310C22#54412yes—
311C22#54712yes—
312C22#54811yescharacteristic
313C22#54811yescharacteristic
314C22#54812yes—
315C22D4×Q811yescharacteristic
316C22D4×Q811yescharacteristic
317C22#55112yes—
318C22#55412yes—
319C22#56312yes—
320C22#56312yes—
321C4—24no—
322C4—42no—
323C4—42no—
324C4—44no—
325C4—42no—
326C4—44no—
327C4—44no—
328C4—81no—
329C4—81no—
330C4—82no—
331C2—44no—
332C2#240911yescharacteristic
333C2#243211yescharacteristic
334C2#256411yescharacteristic
335C2#261311yescharacteristic
336C2#262711yescharacteristic
337C2#266311yescharacteristic
338C2#272012yes—
339C2#272312yes—
340C2#272812yes—
341C2#272912yes—
342C2#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
2. C2×#400C2semidirect
3. C2×#395C2semidirect
5. #1446C2semidirect
6. #1445C2semidirect
7. #1444C2semidirect
8. #1442C2semidirect
87. C2×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C2×#4002
C2×#3952
C2×#3852
#14462
#14452
#14442
#14422
#14402
#14392
C22⋊#952