Finite Group Databasebeta

← #14763

Subgroups — #14763

Order 256.

991 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8C211yesmaximal, characteristic
2C2×#395C211yesmaximal, characteristic, complemented
3C2×#389C212yesmaximal, complemented
4C2×#385C211yesmaximal, characteristic
5C2×C4×C22⋊C4C211yesmaximal, characteristic
6#1446C211yesmaximal, characteristic, complemented
7#1444C211yesmaximal, characteristic, complemented
8#1441C211yesmaximal, characteristic, complemented
9#1439C212yesmaximal
10#1439C212yesmaximal
11#1436C212yesmaximal, complemented
12C23×Q8C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C2×C22⋊Q8—24no—
15C2×C22⋊Q8—28no—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2212yes—
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2212yes—
21C22×C4⋊C4C2212yes—
22C22×C22⋊C4C2212yes—
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2211yescharacteristic
26C22×C42C2212yes—
27#395—24no—
28#389—28no—
29#385—24no—
30C4×C22⋊C4—24no—
31C2×#95C2212yes—
32C2×#95C2211yescharacteristic
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2211yescharacteristic
38C2×#95C2211yescharacteristic
39C2×#95C2212yes—
40C2×#95C2211yescharacteristic
41C25C2311yescharacteristic
42C22×Q8—24no—
43C22×Q8—22no—
44C22×Q8—22no—
45C22×Q8—22no—
46C22×Q8—42no—
47C23×C4—24no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—42no—
51C23×C4D414yes—
52C23×C4C2311yescharacteristic
53C23×C4C2311yescharacteristic
54C23×C4C2312yes—
55C23×C4C2312yes—
56C23×C4C2312yes—
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C23×C4C2312yes—
62C22⋊Q8—232no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4D418yes—
79C2×C22⋊C4—24no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4Q818yescomplemented
90C2×C42—24no—
91C2×C42—24no—
92C2×C42—42no—
93#95—22no—
94#95—22no—
95#95—22no—
96#95—24no—
97#95—24no—
98#95—24no—
99#95—24no—
100#95—24no—
101#95—42no—
102#95D414yes—
103C24—22no—
104C24—21no—
105C24—22no—
106C24—21no—
107C24—21no—
108C24—21no—
109C24—22no—
110C24—21no—
111C24—21no—
112C24C2×D412yes—
113C24C2×Q814yes—
114C24C2411yescharacteristic
115C2×Q8—28no—
116C2×Q8—48no—
117C2×Q8—48no—
118C2×Q8—44no—
119C2×Q8—44no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—21no—
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—22no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—28no—
158C22×C4—28no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—44no—
170C22×C4—48no—
171C22×C4—48no—
172C22×C4C2×D414yes—
173C22×C4C2×D414yes—
174C22×C4C2×D412yes—
175C22×C4C2×D414yes—
176C22×C4C2×D412yes—
177C22×C4C2×D412yes—
178C22×C4C2×D414yes—
179C22×C4C2×Q814yes—
180C22×C4C2×Q814yes—
181C22×C4#4112yes—
182C22×C4#4112yes—
183C22×C4#4112yes—
184C22×C4#4114yes—
185C4⋊C4—216no—
186C4⋊C4—28no—
187C4⋊C4—216no—
188C4⋊C4—48no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C22⋊C4—28no—
192C22⋊C4—216no—
193C22⋊C4—216no—
194C22⋊C4—44no—
195C42—48no—
196C23—22no—
197C23—22no—
198C23—24no—
199C23—22no—
200C23—22no—
201C23—24no—
202C23—22no—
203C23—22no—
204C23—24no—
205C23—24no—
206C23—24no—
207C23—24no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23C4⋊Q814yes—
219C23C22×D411yescharacteristic
220C23C22×D412yes—
221C23C22×D411yescharacteristic
222C23C22×Q812yes—
223C23C2×#4111yescharacteristic
224C23C2×#4112yes—
225C23C2×#4111yescharacteristic
226C23C2×#4111yescharacteristic
227C23H(5,𝔽2)12yes—
228C232−1+411yescharacteristic
229C232−1+411yescharacteristic
230Q8—416no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—416no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4#12014yes—
290C2×C4#12114yes—
291C2×C4C22⋊Q818yescomplemented
292C22—24no—
293C22—24no—
294C22—24no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—44no—
300C22—44no—
301C22—44no—
302C22—44no—
303C22C2×#12011yescharacteristic
304C22C2×#12111yescharacteristic
305C22C2×C22⋊Q812yes—
306C22#52511yescharacteristic
307C22#52512yes—
308C22C2×C4⋊Q811yescharacteristic
309C22#53211yescharacteristic
310C22#53212yes—
311C22#53312yes—
312C22#53411yescharacteristic
313C22#53611yescharacteristic
314C22#53912yes—
315C22#54212yes—
316C22D4212yes—
317C22#54612yes—
318C22#54711yescharacteristic
319C22#54811yescharacteristic
320C22D4×Q812yes—
321C22D4×Q812yes—
322C22#55211yescharacteristic
323C22#55412yes—
324C22#55412yes—
325C22#55611yescharacteristic
326C4—24no—
327C4—24no—
328C4—28no—
329C4—44no—
330C4—42no—
331C4—44no—
332C4—42no—
333C4—44no—
334C4—81no—
335C4—81no—
336C2—44no—
337C2#238811yescharacteristic
338C2#241011yescharacteristic
339C2#241511yescharacteristic
340C2#243411yescharacteristic
341C2#244912yes—
342C2#245212yes—
343C2#246512yes—
344C2#255612yes—
345C2#258311yescharacteristic
346C2#266311yescharacteristic
347C2#285711yescharacteristic
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
3. C2×#389C2semidirect
6. #1446C2semidirect
7. #1444C2semidirect
8. #1441C2semidirect
11. #1436C2semidirect
89. C2×C22⋊C4Q8semidirect
291. C2×C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C2×#3952
C2×#3892
C2×#3852
C2×C4×C22⋊C42
#14462
#14442
#14412
#14392
#14362