Finite Group Databasebeta

← #16698

Subgroups — #16698

Order 256.

681 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#381C211yesmaximal, characteristic
2C2×#380C211yesmaximal, characteristic
3#1445C211yesmaximal, characteristic, complemented
4#1445C211yesmaximal, characteristic, complemented
5#1445C212yesmaximal, complemented
6#1444C211yesmaximal, characteristic, complemented
7#1443C212yesmaximal, complemented
8#1442C211yesmaximal, characteristic, complemented
9#1440C212yesmaximal
10#1439C212yesmaximal
11C22⋊#95C211yesmaximal, characteristic
12C24×C4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2212yes—
15C22×C4⋊C4C2211yescharacteristic
16C22×C22⋊C4C2211yescharacteristic
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2212yes—
20#381—24no—
21#380—24no—
22C2×#95C2211yescharacteristic
23C2×#95C2211yescharacteristic
24C2×#95C2212yes—
25C2×#95C2211yescharacteristic
26C2×#95C2211yescharacteristic
27C2×#95C2212yes—
28C2×#95C2211yescharacteristic
29C2×#95C2211yescharacteristic
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C25C2311yescharacteristic
38C23×C4—24no—
39C23×C4—22no—
40C23×C4—22no—
41C23×C4—42no—
42C23×C4D414yes—
43C23×C4C2311yescharacteristic
44C23×C4C2312yes—
45C23×C4C2312yes—
46C23×C4C2311yescharacteristic
47C23×C4C2311yescharacteristic
48C23×C4C2311yescharacteristic
49C23×C4C2312yes—
50C23×C4C2311yescharacteristic
51C23×C4C2312yes—
52C23×C4C2311yescharacteristic
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—42no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4—42no—
61C2×C22⋊C4—24no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—42no—
68C2×C22⋊C4—42no—
69C2×C22⋊C4—42no—
70C2×C22⋊C4—42no—
71#95—24no—
72#95—22no—
73#95—22no—
74#95—22no—
75#95—22no—
76#95—22no—
77#95—24no—
78#95—22no—
79#95—42no—
80#95—42no—
81#95—42no—
82#95—42no—
83#95—42no—
84#95—42no—
85#95—42no—
86C24—22no—
87C24—21no—
88C24—21no—
89C24—21no—
90C24—21no—
91C24—21no—
92C24—22no—
93C24—22no—
94C24—21no—
95C24C2×D412yes—
96C24#4114yes—
97C24C2411yescharacteristic
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—28no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—28no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
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—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—44no—
154C22×C4—48no—
155C22×C4—48no—
156C22×C4C2×D412yes—
157C22×C4C2×D412yes—
158C22×C4#4114yes—
159C22×C4#4112yes—
160C22×C4#4112yes—
161C4⋊C4—44no—
162C4⋊C4—48no—
163C4⋊C4—44no—
164C22⋊C4—48no—
165C22⋊C4—44no—
166C22⋊C4—48no—
167C22⋊C4—44no—
168C23—22no—
169C23—22no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—24no—
175C23—24no—
176C23—24no—
177C23—24no—
178C23—24no—
179C23—24no—
180C23—42no—
181C23—41no—
182C23—41no—
183C23—42no—
184C23—41no—
185C23—41no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23#12314yes—
191C23C22×D411yescharacteristic
192C23C2×#4112yes—
193C23C2×#4111yescharacteristic
194C23C2×#4111yescharacteristic
195C23C2×#4112yes—
196C23H(5,𝔽2)12yes—
197C23H(5,𝔽2)11yescharacteristic
198C23H(5,𝔽2)11yescharacteristic
199C23H(5,𝔽2)11yescharacteristic
200C232−1+412yes—
201C232−1+411yescharacteristic
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—24no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—416no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4#12314yes—
261C22—24no—
262C22—24no—
263C22—24no—
264C22—42no—
265C22—42no—
266C22—44no—
267C22—44no—
268C22—44no—
269C22—44no—
270C22—42no—
271C22—42no—
272C22C2×#12311yescharacteristic
273C22C2×#12311yescharacteristic
274C22#53211yescharacteristic
275C22#53411yescharacteristic
276C22#53511yescharacteristic
277C22#53611yescharacteristic
278C22#53711yescharacteristic
279C22#53812yes—
280C22#53912yes—
281C22#53912yes—
282C22#54011yescharacteristic
283C22#54011yescharacteristic
284C22#54012yes—
285C22#54211yescharacteristic
286C22#54211yescharacteristic
287C22#55112yes—
288C22#55112yes—
289C22#55112yes—
290C22#55212yes—
291C22#55212yes—
292C22#55612yes—
293C22#56212yes—
294C22#56312yes—
295C4—24no—
296C4—42no—
297C4—44no—
298C4—81no—
299C4—82no—
300C4—81no—
301C4—82no—
302C4—82no—
303C4—81no—
304C4—81no—
305C2—44no—
306C2#248311yescharacteristic
307C2#249811yescharacteristic
308C2#249911yescharacteristic
309C2#262311yescharacteristic
310C2#262411yescharacteristic
311C2#266111yescharacteristic
312C2#273812yes—
313C2#274012yes—
314C2#274312yes—
315C2#274512yes—
316C2#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
3. #1445C2semidirect
4. #1445C2semidirect
5. #1445C2semidirect
6. #1444C2semidirect
7. #1443C2semidirect
8. #1442C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3812
C2×#3802
#14452
#14442
#14432
#14422
#14402
#14392
C22⋊#952