Finite Group Databasebeta

← #16264

Subgroups — #16264

Order 256.

921 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8C211yesmaximal, characteristic
2C2×#398C212yesmaximal, complemented
3C2×#395C211yesmaximal, characteristic, complemented
4C2×#385C211yesmaximal, characteristic
5C2×#380C211yesmaximal, characteristic
6#1446C211yesmaximal, characteristic, complemented
7#1442C211yesmaximal, characteristic, complemented
8#1441C211yesmaximal, characteristic, complemented
9#1441C212yesmaximal, complemented
10#1440C212yesmaximal
11#1439C212yesmaximal
12C23×Q8C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C2×C22⋊Q8—24no—
15C2×C22⋊Q8—28no—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2212yes—
20C22×C4⋊C4C2212yes—
21C22×C4⋊C4C2212yes—
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2211yescharacteristic
25C22×C22⋊C4C2211yescharacteristic
26#398—28no—
27#395—24no—
28#385—24no—
29#380—24no—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2211yescharacteristic
35C2×#95C2211yescharacteristic
36C2×#95C2211yescharacteristic
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2211yescharacteristic
41C25C2311yescharacteristic
42C22×Q8—24no—
43C22×Q8—22no—
44C22×Q8—22no—
45C22×Q8—22no—
46C22×Q8—42no—
47C23×C4—24no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—42no—
51C23×C4D414yes—
52C23×C4C2311yescharacteristic
53C23×C4C2312yes—
54C23×C4C2311yescharacteristic
55C23×C4C2312yes—
56C23×C4C2312yes—
57C23×C4C2312yes—
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C23×C4C2311yescharacteristic
62C22⋊Q8—232no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4D418yes—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4Q814yescomplemented
90#95—22no—
91#95—22no—
92#95—22no—
93#95—24no—
94#95—24no—
95#95—24no—
96#95—24no—
97#95—42no—
98#95—42no—
99#95—42no—
100#95D414yes—
101C24—22no—
102C24—21no—
103C24—21no—
104C24—21no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—21no—
109C24C2×D412yes—
110C24C2×Q812yes—
111C24#4112yes—
112C24C2411yescharacteristic
113C2×Q8—28no—
114C2×Q8—48no—
115C2×Q8—48no—
116C2×Q8—44no—
117C2×Q8—44no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
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—21no—
151C22×C4—28no—
152C22×C4—28no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—44no—
171C22×C4—48no—
172C22×C4—48no—
173C22×C4C2×D414yes—
174C22×C4C2×D414yes—
175C22×C4C2×D414yes—
176C22×C4C2×D412yes—
177C22×C4C2×D414yes—
178C22×C4C2×D412yes—
179C22×C4C2×D412yes—
180C22×C4C2×Q812yes—
181C22×C4C2×Q812yes—
182C22×C4#4112yes—
183C4⋊C4—28no—
184C4⋊C4—216no—
185C4⋊C4—48no—
186C4⋊C4—48no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C22⋊C4—216no—
190C22⋊C4—28no—
191C22⋊C4—48no—
192C22⋊C4—44no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—24no—
200C23—24no—
201C23—22no—
202C23—24no—
203C23—24no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23C22⋊Q814yes—
216C23C22×D411yescharacteristic
217C23C22×D412yes—
218C23C22×D411yescharacteristic
219C23C22×Q811yescharacteristic
220C23C2×#4111yescharacteristic
221C23C2×#4111yescharacteristic
222C23H(5,𝔽2)12yes—
223C23H(5,𝔽2)11yescharacteristic
224C232−1+412yes—
225C232−1+411yescharacteristic
226C232−1+412yes—
227Q8—416no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—416no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
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—42no—
287C2×C4#12014yes—
288C2×C4#12114yes—
289C22—24no—
290C22—24no—
291C22—24no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—44no—
296C22—44no—
297C22—44no—
298C22—44no—
299C22—42no—
300C22C2×#12011yescharacteristic
301C22C2×#12111yescharacteristic
302C22C2×C22⋊Q811yescharacteristic
303C22#53411yescharacteristic
304C22#53611yescharacteristic
305C22#53612yes—
306C22#53611yescharacteristic
307C22#53612yes—
308C22#53711yescharacteristic
309C22#53712yes—
310C22#53912yes—
311C22#54112yes—
312C22#54412yes—
313C22D4212yes—
314C22#54612yes—
315C22#54611yescharacteristic
316C22#54812yes—
317C22D4×Q812yes—
318C22D4×Q811yescharacteristic
319C22#55211yescharacteristic
320C22#55411yescharacteristic
321C22#56212yes—
322C22#56312yes—
323C4—24no—
324C4—24no—
325C4—44no—
326C4—42no—
327C4—44no—
328C4—42no—
329C4—44no—
330C4—81no—
331C4—82no—
332C4—81no—
333C2—44no—
334C2#240911yescharacteristic
335C2#241011yescharacteristic
336C2#241711yescharacteristic
337C2#261611yescharacteristic
338C2#262111yescharacteristic
339C2#262711yescharacteristic
340C2#267712yes—
341C2#268112yes—
342C2#269012yes—
343C2#270412yes—
344C2#281411yescharacteristic
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
3. C2×#395C2semidirect
6. #1446C2semidirect
7. #1442C2semidirect
8. #1441C2semidirect
9. #1441C2semidirect
89. 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×#3982
C2×#3952
C2×#3852
C2×#3802
#14462
#14422
#14412
#14402
#14392