Finite Group Databasebeta

← #15430

Subgroups — #15430

Order 256.

1005 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8C211yesmaximal, characteristic
2C2×#399C211yesmaximal, characteristic
3C2×#398C211yesmaximal, characteristic, complemented
4C2×#395C212yesmaximal, complemented
5C2×#385C212yesmaximal
6C2×#384C211yesmaximal, characteristic, complemented
7#1445C211yesmaximal, characteristic, complemented
8#1444C212yesmaximal, complemented
9#1439C212yesmaximal
10#1437C211yesmaximal, characteristic
11#1436C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C2×C22⋊Q8—24no—
15C2×C22⋊Q8—24no—
16C2×C22⋊Q8—24no—
17C22×C4⋊C4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2212yes—
21C22×C4⋊C4C2212yes—
22C22×C4⋊C4C2212yes—
23C22×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2212yes—
27C22×C22⋊C4C2211yescharacteristic
28C22×C22⋊C4C2212yes—
29C22×C42C2211yescharacteristic
30#399—24no—
31#398—24no—
32#395—28no—
33#385—28no—
34#384—24no—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2211yescharacteristic
38C2×#95C2212yes—
39C2×#95C2211yescharacteristic
40C2×#95C2212yes—
41C2×#95C2212yes—
42C2×#95C2212yes—
43C25C2311yescharacteristic
44C22×Q8—22no—
45C22×Q8—22no—
46C22×Q8—22no—
47C22×Q8—24no—
48C22×Q8—42no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—24no—
53C23×C4—42no—
54C23×C4C2312yes—
55C23×C4C2311yescharacteristic
56C23×C4C2311yescharacteristic
57C23×C4C2311yescharacteristic
58C23×C4C2312yes—
59C23×C4C2311yescharacteristic
60C23×C4C2312yes—
61C23×C4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C23×C4C2312yes—
64C22⋊Q8—232no—
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—24no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4D414yes—
88C2×C4⋊C4D414yes—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4D414yes—
100C2×C22⋊C4Q818yescomplemented
101C2×C42—22no—
102C2×C42—24no—
103#95—24no—
104#95—24no—
105#95—24no—
106#95—22no—
107#95—24no—
108#95—22no—
109#95—24no—
110#95D418yes—
111C24—21no—
112C24—21no—
113C24—22no—
114C24—22no—
115C24—22no—
116C24—22no—
117C24—22no—
118C24C2×D412yes—
119C24C2×Q814yes—
120C24C2411yescharacteristic
121C2×Q8—28no—
122C2×Q8—48no—
123C2×Q8—48no—
124C2×Q8—44no—
125C2×Q8—44no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—28no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
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—48no—
172C22×C4—48no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4C2×D414yes—
176C22×C4C2×D412yes—
177C22×C4C2×D412yes—
178C22×C4C2×D412yes—
179C22×C4C2×D412yes—
180C22×C4C2×D412yes—
181C22×C4C2×D412yes—
182C22×C4C2×D412yes—
183C22×C4C2×D414yes—
184C22×C4C2×D414yes—
185C22×C4C2×D412yes—
186C22×C4C2×Q814yes—
187C22×C4C2×Q814yes—
188C22×C4#4114yes—
189C4⋊C4—28no—
190C4⋊C4—28no—
191C4⋊C4—28no—
192C4⋊C4—48no—
193C4⋊C4—48no—
194C4⋊C4—48no—
195C4⋊C4—44no—
196C4⋊C4—48no—
197C22⋊C4—28no—
198C22⋊C4—28no—
199C22⋊C4—216no—
200C22⋊C4—48no—
201C42—28no—
202C23—22no—
203C23—24no—
204C23—22no—
205C23—24no—
206C23—24no—
207C23—24no—
208C23—24no—
209C23—24no—
210C23—24no—
211C23—24no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23C4⋊Q814yes—
223C23C22×D411yescharacteristic
224C23C22×D411yescharacteristic
225C23C22×D411yescharacteristic
226C23C22×D412yes—
227C23C22×Q812yes—
228C23C2×#4112yes—
229C23H(5,𝔽2)12yes—
230C232−1+412yes—
231C232−1+412yes—
232Q8—416no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—416no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
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#12014yes—
290C2×C4C22⋊Q818yescomplemented
291C2×C4#12714yes—
292C22—24no—
293C22—28no—
294C22—42no—
295C22—44no—
296C22—44no—
297C22—42no—
298C22—44no—
299C22—44no—
300C22—44no—
301C22C2×#12011yescharacteristic
302C22C2×C22⋊Q812yes—
303C22#52511yescharacteristic
304C22C2×#12711yescharacteristic
305C22C2×C4⋊Q811yescharacteristic
306C22#53511yescharacteristic
307C22#53611yescharacteristic
308C22#53612yes—
309C22#53611yescharacteristic
310C22#53912yes—
311C22#54412yes—
312C22D4212yes—
313C22D4212yes—
314C22#54712yes—
315C22#54712yes—
316C22#54712yes—
317C22D4×Q812yes—
318C22D4×Q812yes—
319C22D4×Q812yes—
320C22#55412yes—
321C22#56312yes—
322C4—24no—
323C4—24no—
324C4—28no—
325C4—42no—
326C4—44no—
327C4—42no—
328C4—42no—
329C4—42no—
330C4—44no—
331C4—82no—
332C2—44no—
333C2#240611yescharacteristic
334C2#243311yescharacteristic
335C2#243411yescharacteristic
336C2#244012yes—
337C2#244912yes—
338C2#254212yes—
339C2#267712yes—
340C2#268012yes—
341C2#272012yes—
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
3. C2×#398C2semidirect
4. C2×#395C2semidirect
6. C2×#384C2semidirect
7. #1445C2semidirect
8. #1444C2semidirect
11. #1436C2semidirect
100. C2×C22⋊C4Q8semidirect
290. C2×C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C2×#3992
C2×#3982
C2×#3952
C2×#3852
C2×#3842
#14452
#14442
#14392
#14372
#14362