Finite Group Databasebeta

← #16656

Subgroups — #16656

Order 256.

799 conjugacy classes of subgroups.

The subgroup lattice has 345 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#126C211yesmaximal, characteristic
2C2×#395C211yesmaximal, characteristic, complemented
3C2×#393C211yesmaximal, characteristic
4C2×#391C211yesmaximal, characteristic, complemented
5#1442C212yesmaximal, complemented
6#1441C212yesmaximal, complemented
7#1441C211yesmaximal, characteristic, complemented
8#1440C212yesmaximal
9#1439C212yesmaximal
10#1439C211yesmaximal, characteristic
11#1434C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C2×#126—24no—
14C2×#126—24no—
15C2×#126—24no—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2212yes—
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2212yes—
24C22×C22⋊C4C2211yescharacteristic
25C22×C22⋊C4C2212yes—
26C22×C42C2211yescharacteristic
27#395—24no—
28#393—24no—
29#391—24no—
30C2×#95C2211yescharacteristic
31C2×#95C2212yes—
32C2×#95C2211yescharacteristic
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2212yes—
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C22×Q8—22no—
43C22×Q8—22no—
44C22×Q8—22no—
45C22×Q8—24no—
46C22×Q8—42no—
47C23×C4C2312yes—
48C23×C4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2311yescharacteristic
51C23×C4C2311yescharacteristic
52C23×C4C2311yescharacteristic
53C23×C4C2311yescharacteristic
54C23×C4C2312yes—
55C23×C4C2312yes—
56C23×C4C2311yescharacteristic
57#126—232no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4D414yes—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4D414yes—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42D414yes—
90#95—24no—
91#95—24no—
92#95—24no—
93#95—22no—
94#95—22no—
95#95—24no—
96#95—24no—
97#95—24no—
98#95—42no—
99#95—42no—
100#95—42no—
101C24—21no—
102C24—21no—
103C24—22no—
104C24—22no—
105C24—21no—
106C24—22no—
107C24—21no—
108C24—21no—
109C24—21no—
110C24—22no—
111C24C2×D412yes—
112C24C2411yescharacteristic
113C2×Q8—28no—
114C2×Q8—48no—
115C2×Q8—48no—
116C2×Q8—44no—
117C2×Q8—44no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—21no—
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×C4C2×D412yes—
171C22×C4C2×D412yes—
172C22×C4C2×D412yes—
173C22×C4C2×D412yes—
174C22×C4C2×D412yes—
175C22×C4C2×D412yes—
176C22×C4C2×D412yes—
177C22×C4C2×D412yes—
178C22×C4#4114yes—
179C22×C4#4114yes—
180C4⋊C4—28no—
181C4⋊C4—28no—
182C4⋊C4—28no—
183C4⋊C4—48no—
184C4⋊C4—44no—
185C22⋊C4—28no—
186C22⋊C4—28no—
187C22⋊C4—28no—
188C22⋊C4—48no—
189C22⋊C4—48no—
190C42—28no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—24no—
195C23—24no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23C22×D411yescharacteristic
215C23C22×D411yescharacteristic
216C23C22×D411yescharacteristic
217C23C2×#4112yes—
218C23C2×#4112yes—
219C23H(5,𝔽2)12yes—
220C23H(5,𝔽2)11yescharacteristic
221C232−1+411yescharacteristic
222C232−1+411yescharacteristic
223C232−1+411yescharacteristic
224C232−1+412yes—
225Q8—416no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
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—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#12014yes—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—44no—
292C22—44no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—82no—
298C22—82no—
299C22C2×#12011yescharacteristic
300C22#52511yescharacteristic
301C22#53211yescharacteristic
302C22#53511yescharacteristic
303C22#53611yescharacteristic
304C22#53611yescharacteristic
305C22#53611yescharacteristic
306C22#53611yescharacteristic
307C22#53711yescharacteristic
308C22#53711yescharacteristic
309C22#53711yescharacteristic
310C22#53812yes—
311C22#54112yes—
312C22#54212yes—
313C22#54212yes—
314C22#54612yes—
315C22#54612yes—
316C22#54712yes—
317C22#54812yes—
318C22#55212yes—
319C22#55612yes—
320C22#56312yes—
321C22#56312yes—
322C4—24no—
323C4—42no—
324C4—42no—
325C4—42no—
326C4—44no—
327C4—42no—
328C4—42no—
329C4—44no—
330C4—82no—
331C4—82no—
332C2—82no—
333C2#240411yescharacteristic
334C2#243211yescharacteristic
335C2#260411yescharacteristic
336C2#261611yescharacteristic
337C2#263711yescharacteristic
338C2#266211yescharacteristic
339C2#269812yes—
340C2#270212yes—
341C2#271512yes—
342C2#271612yes—
343C2#282111yescharacteristic
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×#395C2semidirect
4. C2×#391C2semidirect
5. #1442C2semidirect
6. #1441C2semidirect
7. #1441C2semidirect
11. #1434C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1262
C2×#3952
C2×#3932
C2×#3912
#14422
#14412
#14402
#14392
#14342