Finite Group Databasebeta

← #14893

Subgroups — #14893

Order 256.

915 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#117C211yesmaximal, characteristic
2C2×#400C211yesmaximal, characteristic, complemented
3C2×#399C211yesmaximal, characteristic
4C2×#389C211yesmaximal, characteristic, complemented
5#1444C212yesmaximal, complemented
6#1441C211yesmaximal, characteristic, complemented
7#1439C212yesmaximal
8#1439C211yesmaximal, characteristic
9#1436C212yesmaximal, complemented
10#1434C211yesmaximal, characteristic, complemented
11C22⋊#95C212yesmaximal
12C24×C4C2211yescharacteristic
13C2×#117—24no—
14C2×#117—24no—
15C2×#117—24no—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2212yes—
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2212yes—
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2211yescharacteristic
26C22×C42C2211yescharacteristic
27C22×C42C2211yescharacteristic
28#400—24no—
29#399—24no—
30#389—24no—
31C2×#95C2211yescharacteristic
32C2×#95C2212yes—
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C23×C4—22no—
43C23×C4—22no—
44C23×C4—22no—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4Q814yes—
49C23×C4C2312yes—
50C23×C4C2312yes—
51C23×C4C2311yescharacteristic
52C23×C4C2311yescharacteristic
53C23×C4C2311yescharacteristic
54C23×C4C2312yes—
55C23×C4C2312yes—
56C23×C4C2311yescharacteristic
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59#117—232no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4D414yes—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4D414yes—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—22no—
94C2×C42D414yes—
95#95—24no—
96#95—22no—
97#95—22no—
98#95—24no—
99#95—24no—
100#95—24no—
101#95—24no—
102#95—24no—
103#95—24no—
104#95—42no—
105C24—21no—
106C24—21no—
107C24—22no—
108C24—22no—
109C24—21no—
110C24—21no—
111C24—22no—
112C24—22no—
113C24C2×D412yes—
114C24C2×Q812yes—
115C24#4112yes—
116C24C2411yescharacteristic
117C22×C4—28no—
118C22×C4—28no—
119C22×C4—28no—
120C22×C4—28no—
121C22×C4—28no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—28no—
133C22×C4—28no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
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—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4C2×D412yes—
173C22×C4C2×D412yes—
174C22×C4C2×D412yes—
175C22×C4C2×D412yes—
176C22×C4C2×D412yes—
177C22×C4C2×D412yes—
178C22×C4C2×D412yes—
179C22×C4C2×D412yes—
180C22×C4C2×Q812yes—
181C22×C4C2×Q812yes—
182C22×C4#4114yes—
183C22×C4#4114yes—
184C22×C4#4112yes—
185C22×C4#4112yes—
186C4⋊C4—28no—
187C4⋊C4—28no—
188C4⋊C4—28no—
189C4⋊C4—48no—
190C4⋊C4—48no—
191C4⋊C4—44no—
192C22⋊C4—28no—
193C22⋊C4—28no—
194C22⋊C4—48no—
195C22⋊C4—48no—
196C42—28no—
197C42—28no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—24no—
205C23—24no—
206C23—24no—
207C23—24no—
208C23—24no—
209C23—24no—
210C23—41no—
211C23—42no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23C22⋊Q814yes—
221C23C22×D411yescharacteristic
222C23C22×D411yescharacteristic
223C23C22×D411yescharacteristic
224C23C22×Q811yescharacteristic
225C23C2×#4111yescharacteristic
226C23C2×#4111yescharacteristic
227C23C2×#4111yescharacteristic
228C23C2×#4112yes—
229C23C2×#4112yes—
230C23H(5,𝔽2)12yes—
231C232−1+412yes—
232C2×C4—232no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
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—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4#12014yes—
293C2×C4C22⋊Q814yes—
294C2×C4C22⋊Q814yes—
295C22—24no—
296C22—28no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—44no—
304C22—44no—
305C22—44no—
306C22C2×#12011yescharacteristic
307C22C2×C22⋊Q811yescharacteristic
308C22C2×C22⋊Q811yescharacteristic
309C22C2×C22⋊Q811yescharacteristic
310C22#52511yescharacteristic
311C22#52511yescharacteristic
312C22#52511yescharacteristic
313C22#52511yescharacteristic
314C22#52912yes—
315C22#53211yescharacteristic
316C22#53312yes—
317C22#53511yescharacteristic
318C22#53611yescharacteristic
319C22#54212yes—
320C22#54412yes—
321C22#54612yes—
322C22#54712yes—
323C22#54712yes—
324C22#55012yes—
325C22#55112yes—
326C22#55212yes—
327C22#55212yes—
328C22#55312yes—
329C4—24no—
330C4—24no—
331C4—24no—
332C4—44no—
333C4—42no—
334C4—42no—
335C4—42no—
336C4—44no—
337C4—82no—
338C4—82no—
339C2—44no—
340C2#239611yescharacteristic
341C2#240611yescharacteristic
342C2#243111yescharacteristic
343C2#243812yes—
344C2#244112yes—
345C2#247012yes—
346C2#247912yes—
347C2#259811yescharacteristic
348C2#261111yescharacteristic
349C2#262811yescharacteristic
350C2#284811yescharacteristic
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×#389C2semidirect
5. #1444C2semidirect
6. #1441C2semidirect
9. #1436C2semidirect
10. #1434C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1172
C2×#4002
C2×#3992
C2×#3892
#14442
#14412
#14392
#14362
#14342
C22⋊#952