Finite Group Databasebeta

← #16393

Subgroups — #16393

Order 256.

727 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#398C211yesmaximal, characteristic, complemented
2C2×#385C211yesmaximal, characteristic
3C2×#381C211yesmaximal, characteristic
4C2×#380C211yesmaximal, characteristic
5#1445C211yesmaximal, characteristic, complemented
6#1445C212yesmaximal, complemented
7#1442C211yesmaximal, characteristic, complemented
8#1442C212yesmaximal, complemented
9#1441C211yesmaximal, characteristic, complemented
10#1440C212yesmaximal
11#1439C212yesmaximal
12C24×C4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2212yes—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22#398—24no—
23#385—24no—
24#381—24no—
25#380—24no—
26C2×#95C2212yes—
27C2×#95C2212yes—
28C2×#95C2211yescharacteristic
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2211yescharacteristic
32C2×#95C2211yescharacteristic
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2211yescharacteristic
39C25C2311yescharacteristic
40C23×C4—24no—
41C23×C4—22no—
42C23×C4—22no—
43C23×C4—42no—
44C23×C4D414yes—
45C23×C4C2311yescharacteristic
46C23×C4C2312yes—
47C23×C4C2312yes—
48C23×C4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2312yes—
51C23×C4C2311yescharacteristic
52C23×C4C2311yescharacteristic
53C23×C4C2311yescharacteristic
54C23×C4C2311yescharacteristic
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—24no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—42no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—42no—
79#95—24no—
80#95—24no—
81#95—22no—
82#95—22no—
83#95—22no—
84#95—24no—
85#95—24no—
86#95—42no—
87#95—42no—
88#95—42no—
89#95—42no—
90#95—42no—
91#95D414yes—
92C24—21no—
93C24—21no—
94C24—21no—
95C24—21no—
96C24—21no—
97C24—21no—
98C24—22no—
99C24—22no—
100C24—22no—
101C24C2×D412yes—
102C24#4114yes—
103C24C2411yescharacteristic
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—28no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—28no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
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—44no—
160C22×C4—48no—
161C22×C4—48no—
162C22×C4C2×D412yes—
163C22×C4C2×D414yes—
164C22×C4C2×D412yes—
165C22×C4C2×D412yes—
166C22×C4#4112yes—
167C22×C4#4112yes—
168C22×C4#4112yes—
169C4⋊C4—48no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C22⋊C4—28no—
175C22⋊C4—48no—
176C22⋊C4—48no—
177C22⋊C4—44no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—24no—
185C23—24no—
186C23—24no—
187C23—24no—
188C23—24no—
189C23—24no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—42no—
199C23—42no—
200C23#12314yes—
201C23C22×D411yescharacteristic
202C23C22×D411yescharacteristic
203C23C2×#4111yescharacteristic
204C23C2×#4112yes—
205C23C2×#4111yescharacteristic
206C23C2×#4111yescharacteristic
207C23H(5,𝔽2)12yes—
208C23H(5,𝔽2)11yescharacteristic
209C23H(5,𝔽2)11yescharacteristic
210C232−1+412yes—
211C232−1+412yes—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—416no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4#12114yes—
273C22—24no—
274C22—24no—
275C22—24no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—44no—
281C22—44no—
282C22—44no—
283C22—44no—
284C22C2×#12111yescharacteristic
285C22C2×#12311yescharacteristic
286C22#52511yescharacteristic
287C22#53411yescharacteristic
288C22#53511yescharacteristic
289C22#53611yescharacteristic
290C22#53611yescharacteristic
291C22#53612yes—
292C22#53912yes—
293C22#53912yes—
294C22#53912yes—
295C22#54112yes—
296C22#54212yes—
297C22#54612yes—
298C22#54611yescharacteristic
299C22#55011yescharacteristic
300C22#55112yes—
301C22#55212yes—
302C22#55212yes—
303C22#55311yescharacteristic
304C22#55511yescharacteristic
305C22#56212yes—
306C22#56212yes—
307C4—24no—
308C4—42no—
309C4—42no—
310C4—42no—
311C4—44no—
312C4—82no—
313C4—81no—
314C4—81no—
315C4—82no—
316C4—82no—
317C2—44no—
318C2#241211yescharacteristic
319C2#241411yescharacteristic
320C2#248411yescharacteristic
321C2#262111yescharacteristic
322C2#262311yescharacteristic
323C2#263811yescharacteristic
324C2#268712yes—
325C2#269412yes—
326C2#273912yes—
327C2#274412yes—
328C2#281611yescharacteristic
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. C2×#398C2semidirect
5. #1445C2semidirect
6. #1445C2semidirect
7. #1442C2semidirect
8. #1442C2semidirect
9. #1441C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3982
C2×#3852
C2×#3812
C2×#3802
#14452
#14422
#14412
#14402
#14392