Finite Group Databasebeta

← #14935

Subgroups — #14935

Order 256.

951 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#385C211yesmaximal, characteristic
2C2×#385C211yesmaximal, characteristic
3C2×#381C211yesmaximal, characteristic
4C2×#381C211yesmaximal, characteristic
5C2×#380C211yesmaximal, characteristic
6C22×#95C211yesmaximal, characteristic
7#1443C212yesmaximal, complemented
8#1441C212yesmaximal, complemented
9#1441C212yesmaximal, complemented
10#1441C212yesmaximal, complemented
11C22⋊#95C211yesmaximal, characteristic
12C24×C4C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2212yes—
17C22×C4⋊C4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2211yescharacteristic
23#385—24no—
24#385—24no—
25#381—24no—
26#381—24no—
27#380—24no—
28C2×#95—24no—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2211yescharacteristic
40C2×#95C2212yes—
41C25C2311yescharacteristic
42C23×C4—22no—
43C23×C4—22no—
44C23×C4—22no—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—24no—
52C23×C4—24no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—42no—
56C23×C4—42no—
57C23×C4D414yes—
58C23×C4Q814yes—
59C23×C4C2311yescharacteristic
60C23×C4C2312yes—
61C23×C4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C23×C4C2312yes—
64C23×C4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C23×C4C2312yes—
67C23×C4C2312yes—
68C23×C4C2311yescharacteristic
69C2×C4⋊C4—24no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4D414yes—
90C2×C22⋊C4D414yes—
91#95—24no—
92#95—24no—
93#95—24no—
94#95—24no—
95#95—24no—
96#95—24no—
97#95—24no—
98#95—24no—
99#95—22no—
100#95—22no—
101#95—24no—
102#95—42no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24C2×D412yes—
108C24C2×D412yes—
109C24C2×D412yes—
110C24C2×Q812yes—
111C24#4112yes—
112C24#4112yes—
113C24#4112yes—
114C24C2411yescharacteristic
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—28no—
119C22×C4—28no—
120C22×C4—28no—
121C22×C4—28no—
122C22×C4—28no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
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—28no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—44no—
172C22×C4—48no—
173C22×C4—48no—
174C22×C4—48no—
175C22×C4—48no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4C2×D412yes—
183C22×C4C2×D414yes—
184C22×C4C2×D412yes—
185C22×C4C2×D412yes—
186C22×C4C2×D412yes—
187C22×C4C2×Q814yes—
188C22×C4#4114yes—
189C22×C4#4114yes—
190C4⋊C4—48no—
191C4⋊C4—48no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C22⋊C4—28no—
195C22⋊C4—28no—
196C22⋊C4—44no—
197C22⋊C4—44no—
198C23—22no—
199C23—22no—
200C23—24no—
201C23—24no—
202C23—22no—
203C23—22no—
204C23—24no—
205C23—24no—
206C23—24no—
207C23—24no—
208C23—24no—
209C23—24no—
210C23—24no—
211C23—24no—
212C23—24no—
213C23—24no—
214C23#12114yes—
215C23#12114yes—
216C23#12114yes—
217C23C22⋊Q814yes—
218C23C22⋊Q814yes—
219C23C22⋊Q814yes—
220C23#12614yes—
221C23C22×D411yescharacteristic
222C23C22×D411yescharacteristic
223C23C22×D411yescharacteristic
224C23C22×Q811yescharacteristic
225C23C2×#4111yescharacteristic
226C23C2×#4111yescharacteristic
227C23C2×#4111yescharacteristic
228C23C2×#4112yes—
229C23C2×#4112yes—
230C23H(5,𝔽2)12yes—
231C232−1+412yes—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—416no—
253C2×C4—416no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
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—416no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4#12314yes—
293C22—24no—
294C22—24no—
295C22—24no—
296C22—24no—
297C22—28no—
298C22—28no—
299C22—28no—
300C22—24no—
301C22—28no—
302C22—24no—
303C22#39618yes—
304C22C2×#12111yescharacteristic
305C22C2×#12111yescharacteristic
306C22C2×#12111yescharacteristic
307C22C2×C22⋊Q811yescharacteristic
308C22C2×C22⋊Q811yescharacteristic
309C22C2×C22⋊Q811yescharacteristic
310C22C2×#12311yescharacteristic
311C22#52511yescharacteristic
312C22C2×#12611yescharacteristic
313C22#52912yes—
314C22#53312yes—
315C22#53411yescharacteristic
316C22#53611yescharacteristic
317C22#53912yes—
318C22#54412yes—
319C22#54612yes—
320C22#54612yes—
321C22#54712yes—
322C22#54712yes—
323C22#55112yes—
324C22#55112yes—
325C22#55212yes—
326C22#55212yes—
327C4—24no—
328C4—44no—
329C4—42no—
330C4—44no—
331C4—42no—
332C4—42no—
333C4—44no—
334C4—81no—
335C4—81no—
336C4—82no—
337C2—28no—
338C2C2×#39611yescharacteristic
339C2#241211yescharacteristic
340C2#241311yescharacteristic
341C2#242512yes—
342C2#243812yes—
343C2#243812yes—
344C2#248611yescharacteristic
345C2#249312yes—
346C2#253411yescharacteristic
347C2#262711yescharacteristic
348C2#285511yescharacteristic
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
7. #1443C2semidirect
8. #1441C2semidirect
9. #1441C2semidirect
10. #1441C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3852
C2×#3812
C2×#3802
C22×#952
#14432
#14412
C22⋊#952