Finite Group Databasebeta

← #15670

Subgroups — #15670

Order 256.

869 conjugacy classes of subgroups.

The subgroup lattice has 329 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×#397C211yesmaximal, characteristic
4C2×#382C211yesmaximal, characteristic, complemented
5#1446C212yesmaximal, complemented
6#1445C211yesmaximal, characteristic, complemented
7#1442C212yesmaximal, complemented
8#1440C212yesmaximal
9#1437C211yesmaximal, characteristic
10#1436C211yesmaximal, characteristic, complemented
11C22⋊#95C212yesmaximal
12C23×Q8C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C2×C22⋊Q8—24no—
15C2×C22⋊Q8—24no—
16C2×C22⋊Q8—24no—
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2211yescharacteristic
25C22×C42C2211yescharacteristic
26#400—24no—
27#397—24no—
28#382—24no—
29C2×#95C2211yescharacteristic
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2211yescharacteristic
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C22×Q8—22no—
43C22×Q8—22no—
44C22×Q8—22no—
45C22×Q8—24no—
46C22×Q8—42no—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—24no—
51C23×C4—42no—
52C23×C4C2312yes—
53C23×C4C2312yes—
54C23×C4C2312yes—
55C23×C4C2311yescharacteristic
56C23×C4C2311yescharacteristic
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59C23×C4C2312yes—
60C23×C4C2311yescharacteristic
61C23×C4C2311yescharacteristic
62C22⋊Q8—232no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4D414yes—
74C2×C4⋊C4D414yes—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4D414yes—
87C2×C42—22no—
88C2×C42—24no—
89#95—24no—
90#95—24no—
91#95—22no—
92#95—24no—
93#95—22no—
94#95—24no—
95#95—24no—
96#95—24no—
97#95—24no—
98#95—42no—
99#95—42no—
100#95—42no—
101C24—21no—
102C24—21no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24C2×D412yes—
109C24#4114yes—
110C24C2411yescharacteristic
111C2×Q8—28no—
112C2×Q8—48no—
113C2×Q8—48no—
114C2×Q8—44no—
115C2×Q8—44no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—28no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
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×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
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—48no—
166C22×C4—48no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4C2×D412yes—
170C22×C4C2×D412yes—
171C22×C4C2×D412yes—
172C22×C4C2×D412yes—
173C22×C4C2×D412yes—
174C22×C4C2×D412yes—
175C22×C4C2×D412yes—
176C22×C4C2×D412yes—
177C22×C4#4114yes—
178C22×C4#4114yes—
179C4⋊C4—28no—
180C4⋊C4—28no—
181C4⋊C4—28no—
182C4⋊C4—44no—
183C22⋊C4—28no—
184C22⋊C4—28no—
185C22⋊C4—48no—
186C22⋊C4—48no—
187C42—28no—
188C23—22no—
189C23—24no—
190C23—22no—
191C23—24no—
192C23—24no—
193C23—24no—
194C23—24no—
195C23—24no—
196C23—24no—
197C23—24no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23#12414yes—
209C23C22×D411yescharacteristic
210C23C22×D411yescharacteristic
211C23C22×D411yescharacteristic
212C23C2×#4112yes—
213C23C2×#4112yes—
214C23C2×#4112yes—
215C23H(5,𝔽2)12yes—
216C232−1+412yes—
217C232−1+412yes—
218Q8—416no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—22no—
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—24no—
238C2×C4—416no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
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—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4#12014yes—
277C2×C4#12414yes—
278C22—24no—
279C22—28no—
280C22—42no—
281C22—44no—
282C22—42no—
283C22—44no—
284C22—44no—
285C22—44no—
286C22—44no—
287C22C2×#12011yescharacteristic
288C22#52511yescharacteristic
289C22C2×#12411yescharacteristic
290C22C2×#12411yescharacteristic
291C22#52912yes—
292C22#53511yescharacteristic
293C22#53611yescharacteristic
294C22#53611yescharacteristic
295C22#53912yes—
296C22#54112yes—
297C22#54212yes—
298C22#54612yes—
299C22#54612yes—
300C22#54712yes—
301C22#54712yes—
302C22#54812yes—
303C22#54812yes—
304C22#55112yes—
305C22#55212yes—
306C22#55212yes—
307C22#56312yes—
308C4—24no—
309C4—24no—
310C4—42no—
311C4—42no—
312C4—42no—
313C4—42no—
314C4—44no—
315C4—82no—
316C4—82no—
317C4—82no—
318C2—44no—
319C2#240611yescharacteristic
320C2#243511yescharacteristic
321C2#243611yescharacteristic
322C2#247712yes—
323C2#248812yes—
324C2#248912yes—
325C2#269012yes—
326C2#269812yes—
327C2#272312yes—
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
4. C2×#382C2semidirect
5. #1446C2semidirect
6. #1445C2semidirect
7. #1442C2semidirect
10. #1436C2semidirect

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×#3972
C2×#3822
#14462
#14452
#14422
#14402
#14372
#14362
C22⋊#952