Finite Group Databasebeta

← #16805

Subgroups — #16805

Order 256.

701 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#397C211yesmaximal, characteristic
2C2×#393C211yesmaximal, characteristic
3C2×#391C211yesmaximal, characteristic, complemented
4C2×#388C211yesmaximal, characteristic
5#1446C211yesmaximal, characteristic, complemented
6#1445C211yesmaximal, characteristic, complemented
7#1443C212yesmaximal, complemented
8#1441C212yesmaximal, complemented
9#1440C212yesmaximal
10#1439C212yesmaximal
11C4⋊#95C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2212yes—
16C22×C22⋊C4C2211yescharacteristic
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2211yescharacteristic
21C22×C42C2211yescharacteristic
22#397—24no—
23#393—24no—
24#391—24no—
25#388—24no—
26C2×#95C2211yescharacteristic
27C2×#95C2212yes—
28C2×#95C2211yescharacteristic
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2211yescharacteristic
34C2×#95C2211yescharacteristic
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C25C2311yescharacteristic
40C22×Q8—22no—
41C22×Q8—22no—
42C22×Q8—24no—
43C22×Q8—22no—
44C22×Q8—42no—
45C23×C4C2312yes—
46C23×C4C2312yes—
47C23×C4C2312yes—
48C23×C4C2311yescharacteristic
49C23×C4C2311yescharacteristic
50C23×C4C2311yescharacteristic
51C23×C4C2311yescharacteristic
52C23×C4C2312yes—
53C23×C4C2311yescharacteristic
54C23×C4C2311yescharacteristic
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—42no—
62C2×C4⋊C4—42no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—42no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4—42no—
75C2×C22⋊C4D414yes—
76C2×C42—42no—
77C2×C42D414yes—
78#95—22no—
79#95—22no—
80#95—24no—
81#95—24no—
82#95—24no—
83#95—22no—
84#95—22no—
85#95—24no—
86#95—24no—
87#95—42no—
88#95—42no—
89#95—42no—
90#95—42no—
91C24—21no—
92C24—21no—
93C24—22no—
94C24—22no—
95C24—21no—
96C24—21no—
97C24—22no—
98C24—21no—
99C24—21no—
100C24—22no—
101C24C2×D412yes—
102C24C2411yescharacteristic
103C2×Q8—28no—
104C2×Q8—48no—
105C2×Q8—48no—
106C2×Q8—44no—
107C2×Q8—44no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
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—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4C2×D412yes—
159C22×C4C2×D412yes—
160C22×C4C2×D412yes—
161C22×C4C2×D412yes—
162C22×C4C2×D412yes—
163C22×C4#4114yes—
164C22×C4#4112yes—
165C22×C4#4114yes—
166C4⋊C4—48no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C22⋊C4—28no—
170C22⋊C4—48no—
171C22⋊C4—44no—
172C22⋊C4—48no—
173C22⋊C4—44no—
174C42—44no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—24no—
179C23—24no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—41no—
190C23—41no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23C22×D411yescharacteristic
199C23C22×D411yescharacteristic
200C23C2×#4111yescharacteristic
201C23C2×#4112yes—
202C23C2×#4112yes—
203C23H(5,𝔽2)12yes—
204C23H(5,𝔽2)11yescharacteristic
205C23H(5,𝔽2)11yescharacteristic
206C232−1+411yescharacteristic
207C232−1+411yescharacteristic
208C232−1+412yes—
209Q8—416no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
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—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4#12114yes—
267C2×C4#12414yes—
268C22—42no—
269C22—44no—
270C22—44no—
271C22—42no—
272C22—42no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—82no—
277C22—82no—
278C22—81no—
279C22C2×#12111yescharacteristic
280C22#52511yescharacteristic
281C22C2×#12411yescharacteristic
282C22#53411yescharacteristic
283C22#53511yescharacteristic
284C22#53611yescharacteristic
285C22#53611yescharacteristic
286C22#53711yescharacteristic
287C22#53711yescharacteristic
288C22#53812yes—
289C22#53912yes—
290C22#54112yes—
291C22#54211yescharacteristic
292C22#54211yescharacteristic
293C22#54212yes—
294C22#54612yes—
295C22#54712yes—
296C22#55112yes—
297C22#55612yes—
298C22#55612yes—
299C22#55912yes—
300C22#56212yes—
301C22#56312yes—
302C4—24no—
303C4—24no—
304C4—44no—
305C4—42no—
306C4—42no—
307C4—81no—
308C4—82no—
309C4—81no—
310C4—82no—
311C4—82no—
312C2—82no—
313C2#253411yescharacteristic
314C2#254711yescharacteristic
315C2#259211yescharacteristic
316C2#262011yescharacteristic
317C2#263711yescharacteristic
318C2#263911yescharacteristic
319C2#268912yes—
320C2#271012yes—
321C2#275912yes—
322C2#279812yes—
323C2#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. C2×#391C2semidirect
5. #1446C2semidirect
6. #1445C2semidirect
7. #1443C2semidirect
8. #1441C2semidirect
11. C4⋊#95C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3972
C2×#3932
C2×#3912
C2×#3882
#14462
#14452
#14432
#14412
#14402
#14392
C4⋊#952