Finite Group Databasebeta

← #14870

Subgroups — #14870

Order 256.

819 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#399C211yesmaximal, characteristic
2C2×#398C212yesmaximal, complemented
3C2×#396C212yesmaximal
4C2×#389C211yesmaximal, characteristic, complemented
5C2×#388C211yesmaximal, characteristic
6C2×C4×C4⋊C4C211yesmaximal, characteristic, complemented
7#1445C211yesmaximal, characteristic, complemented
8#1444C211yesmaximal, characteristic, complemented
9#1439C211yesmaximal, characteristic
10#1437C212yesmaximal
11C4⋊#95C212yesmaximal, complemented
12C22×C4⋊C4C2212yes—
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2212yes—
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2212yes—
18C22×C4⋊C4C2212yes—
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2211yescharacteristic
24C22×C42C2212yes—
25C22×C42C2211yescharacteristic
26#399—24no—
27#398—28no—
28#396—28no—
29#389—24no—
30#388—24no—
31C4×C4⋊C4—24no—
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2211yescharacteristic
37C2×#95C2212yes—
38C2×#95C2211yescharacteristic
39C2×#95C2211yescharacteristic
40C2×#95C2211yescharacteristic
41C25C2311yescharacteristic
42C23×C4C2312yes—
43C23×C4C2311yescharacteristic
44C23×C4C2311yescharacteristic
45C23×C4C2312yes—
46C23×C4C2311yescharacteristic
47C23×C4C2312yes—
48C23×C4C2311yescharacteristic
49C23×C4C2312yes—
50C23×C4C2311yescharacteristic
51C23×C4C2311yescharacteristic
52C2×C4⋊C4—24no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—24no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4D418yes—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4Q814yes—
85C2×C42—22no—
86C2×C42—22no—
87C2×C42—24no—
88C2×C42—24no—
89C2×C42—42no—
90C2×C42D414yes—
91#95—22no—
92#95—22no—
93#95—22no—
94#95—24no—
95#95—24no—
96#95—24no—
97#95—24no—
98#95—24no—
99#95D414yes—
100C24—21no—
101C24—21no—
102C24—22no—
103C24—21no—
104C24—22no—
105C24—22no—
106C24—21no—
107C24—21no—
108C24—22no—
109C24—21no—
110C24C2×Q812yes—
111C24C2411yescharacteristic
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
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—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4C2×D414yes—
159C22×C4C2×D412yes—
160C22×C4C2×D412yes—
161C22×C4C2×D414yes—
162C22×C4C2×D414yes—
163C22×C4C2×D412yes—
164C22×C4C2×D412yes—
165C22×C4C2×D414yes—
166C22×C4C2×Q812yes—
167C22×C4C2×Q812yes—
168C22×C4#4114yes—
169C22×C4#4112yes—
170C22×C4#4114yes—
171C22×C4#4112yes—
172C4⋊C4—28no—
173C4⋊C4—216no—
174C4⋊C4—48no—
175C4⋊C4—44no—
176C4⋊C4—48no—
177C4⋊C4—44no—
178C4⋊C4—48no—
179C22⋊C4—28no—
180C22⋊C4—48no—
181C22⋊C4—44no—
182C22⋊C4—48no—
183C22⋊C4—44no—
184C42—28no—
185C42—48no—
186C23—22no—
187C23—24no—
188C23—22no—
189C23—24no—
190C23—22no—
191C23—41no—
192C23—42no—
193C23—41no—
194C23—42no—
195C23—41no—
196C23—41no—
197C23—42no—
198C23—41no—
199C23—41no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—41no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—41no—
208C23—42no—
209C23C22×D411yescharacteristic
210C23C22×D411yescharacteristic
211C23C22×D412yes—
212C23C22×Q811yescharacteristic
213C23C2×#4112yes—
214C23C2×#4111yescharacteristic
215C23C2×#4111yescharacteristic
216C23C2×#4112yes—
217C23H(5,𝔽2)11yescharacteristic
218C232−1+412yes—
219C232−1+411yescharacteristic
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
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—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4#12114yes—
277C2×C4#12118yes—
278C2×C4C22⋊Q814yes—
279C2×C4#12714yes—
280C22—42no—
281C22—42no—
282C22—44no—
283C22—42no—
284C22—44no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—82no—
289C22—82no—
290C22—81no—
291C22C2×#12111yescharacteristic
292C22C2×#12112yes—
293C22C2×C22⋊Q811yescharacteristic
294C22#52511yescharacteristic
295C22#52912yes—
296C22C2×#12711yescharacteristic
297C22#53211yescharacteristic
298C22#53212yes—
299C22#53312yes—
300C22#53611yescharacteristic
301C22#53711yescharacteristic
302C22#53912yes—
303C22#54112yes—
304C22#54611yescharacteristic
305C22#54612yes—
306C22#54712yes—
307C22#54712yes—
308C22#54812yes—
309C22D4×Q811yescharacteristic
310C22D4×Q812yes—
311C22#55212yes—
312C22#55211yescharacteristic
313C22#55411yescharacteristic
314C4—24no—
315C4—24no—
316C4—24no—
317C4—28no—
318C4—44no—
319C4—44no—
320C4—42no—
321C4—81no—
322C4—82no—
323C4—81no—
324C2—82no—
325C2#239611yescharacteristic
326C2#241311yescharacteristic
327C2#241711yescharacteristic
328C2#242512yes—
329C2#242912yes—
330C2#251912yes—
331C2#254011yescharacteristic
332C2#254211yescharacteristic
333C2#255612yes—
334C2#266411yescharacteristic
335C2#285611yescharacteristic
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×#398C2semidirect
4. C2×#389C2semidirect
6. C2×C4×C4⋊C4C2semidirect
7. #1445C2semidirect
8. #1444C2semidirect
11. C4⋊#95C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3992
C2×#3982
C2×#3962
C2×#3892
C2×#3882
C2×C4×C4⋊C42
#14452
#14442
#14392
#14372
C4⋊#952