Finite Group Databasebeta

← #14877

Subgroups — #14877

Order 256.

935 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Q8C211yesmaximal, characteristic, complemented
2C2×#396C212yesmaximal
3C2×#395C211yesmaximal, characteristic, complemented
4C2×#393C211yesmaximal, characteristic
5C2×#388C211yesmaximal, characteristic
6#1441C211yesmaximal, characteristic, complemented
7#1441C212yesmaximal, complemented
8#1439C211yesmaximal, characteristic
9#1437C212yesmaximal
10#1436C212yesmaximal, complemented
11C4⋊#95C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C2×C4×Q8—28no—
14C2×C4×Q8—24no—
15C22×C4⋊C4C2212yes—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2212yes—
19C22×C4⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2211yescharacteristic
25C22×C42C2212yes—
26C22×C42C2211yescharacteristic
27#396—28no—
28#395—24no—
29#393—24no—
30#388—24no—
31C2×#95C2211yescharacteristic
32C2×#95C2212yes—
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2211yescharacteristic
38C2×#95C2212yes—
39C2×#95C2211yescharacteristic
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C22×Q8—24no—
43C22×Q8—22no—
44C22×Q8—24no—
45C22×Q8—22no—
46C22×Q8D414yes—
47C23×C4C2311yescharacteristic
48C23×C4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2312yes—
51C23×C4C2311yescharacteristic
52C23×C4C2312yes—
53C23×C4C2311yescharacteristic
54C23×C4C2312yes—
55C23×C4C2311yescharacteristic
56C23×C4C2311yescharacteristic
57C4×Q8—232no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4D418yes—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4Q814yescomplemented
86C2×C42—24no—
87C2×C42—24no—
88C2×C42—24no—
89C2×C42—24no—
90C2×C42D414yes—
91#95—22no—
92#95—22no—
93#95—22no—
94#95—24no—
95#95—22no—
96#95—24no—
97#95—24no—
98#95—24no—
99#95—24no—
100#95—42no—
101C24—21no—
102C24—22no—
103C24—21no—
104C24—21no—
105C24—22no—
106C24—22no—
107C24—21no—
108C24—21no—
109C24—22no—
110C24—21no—
111C24C2×Q812yes—
112C24C2411yescharacteristic
113C2×Q8—28no—
114C2×Q8—28no—
115C2×Q8—216no—
116C2×Q8—216no—
117C2×Q8—28no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4C2×D414yes—
165C22×C4C2×D412yes—
166C22×C4C2×D414yes—
167C22×C4C2×D412yes—
168C22×C4C2×D414yes—
169C22×C4C2×D414yes—
170C22×C4C2×D412yes—
171C22×C4C2×D412yes—
172C22×C4C2×Q812yes—
173C22×C4C2×Q812yes—
174C22×C4#4114yes—
175C22×C4#4114yes—
176C22×C4#4112yes—
177C22×C4#4112yes—
178C4⋊C4—28no—
179C4⋊C4—216no—
180C4⋊C4—48no—
181C4⋊C4—44no—
182C4⋊C4—48no—
183C22⋊C4—28no—
184C22⋊C4—48no—
185C22⋊C4—44no—
186C22⋊C4—44no—
187C22⋊C4—48no—
188C42—216no—
189C42—28no—
190C23—22no—
191C23—24no—
192C23—24no—
193C23—22no—
194C23—22no—
195C23—41no—
196C23—42no—
197C23—41no—
198C23—41no—
199C23—42no—
200C23—42no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—42no—
212C23—41no—
213C23C22×D411yescharacteristic
214C23C22×D412yes—
215C23C22×D411yescharacteristic
216C23C22×Q811yescharacteristic
217C23C2×#4112yes—
218C23C2×#4111yescharacteristic
219C23C2×#4111yescharacteristic
220C23C2×#4112yes—
221C23H(5,𝔽2)11yescharacteristic
222C232−1+412yes—
223C232−1+411yescharacteristic
224Q8—232no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4#12014yes—
282C2×C4#12118yes—
283C2×C4#12114yes—
284C2×C4C22⋊Q814yes—
285C22—42no—
286C22—42no—
287C22—44no—
288C22—44no—
289C22—42no—
290C22—81no—
291C22—81no—
292C22—82no—
293C22—81no—
294C22—82no—
295C22—81no—
296C22C2×#12011yescharacteristic
297C22C2×#12112yes—
298C22C2×#12111yescharacteristic
299C22C2×C22⋊Q811yescharacteristic
300C22#52511yescharacteristic
301C22#52512yes—
302C22#52912yes—
303C22#53211yescharacteristic
304C22#53312yes—
305C22#53611yescharacteristic
306C22#53711yescharacteristic
307C22#54112yes—
308C22#54212yes—
309C22#54612yes—
310C22#54712yes—
311C22#54812yes—
312C22#54812yes—
313C22D4×Q812yes—
314C22D4×Q811yescharacteristic
315C22#55011yescharacteristic
316C22#55411yescharacteristic
317C22#55612yes—
318C22#55611yescharacteristic
319C4—28no—
320C4—24no—
321C4—24no—
322C4—24no—
323C4—44no—
324C4—44no—
325C4—42no—
326C4—81no—
327C4—82no—
328C4—81no—
329C2—82no—
330C2#239611yescharacteristic
331C2#240911yescharacteristic
332C2#242612yes—
333C2#242912yes—
334C2#243011yescharacteristic
335C2#246512yes—
336C2#246912yes—
337C2#253811yescharacteristic
338C2#254511yescharacteristic
339C2#263711yescharacteristic
340C2#284711yescharacteristic
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
1. C22×C4×Q8C2semidirect
3. C2×#395C2semidirect
6. #1441C2semidirect
7. #1441C2semidirect
10. #1436C2semidirect
11. C4⋊#95C2semidirect
85. C2×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q82
C2×#3962
C2×#3952
C2×#3932
C2×#3882
#14412
#14392
#14372
#14362
C4⋊#952