Finite Group Databasebeta

← #16556

Subgroups — #16556

Order 256.

891 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#396C212yesmaximal, complemented
2C2×#395C211yesmaximal, characteristic, complemented
3C2×#393C212yesmaximal, complemented
4C2×#390C211yesmaximal, characteristic
5#1441C211yesmaximal, characteristic, complemented
6#1439C212yesmaximal, complemented
7#1439C212yesmaximal, complemented
8#1438C212yesmaximal
9#1436C212yesmaximal, complemented
10C23×Q8C2211yescharacteristic
11C23×D4C2211yescharacteristic
12C22×C4⋊C4C2212yes—
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2212yes—
15C22×C4⋊C4C2211yescharacteristic
16C22×C22⋊C4C2212yes—
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C42C2211yescharacteristic
23#396—28no—
24#395—24no—
25#393—28no—
26#390—24no—
27C2×#95C2212yes—
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2211yescharacteristic
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C25C2312yes—
36C22×Q8—24no—
37C22×Q8—22no—
38C22×Q8—42no—
39C22×Q8—41no—
40C22×Q8—41no—
41C22×D4—22no—
42C22×D4—22no—
43C22×D4—41no—
44C22×D4—41no—
45C22×D4—41no—
46C22×D4—41no—
47C22×D4Q814yescomplemented
48C23×C4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2312yes—
51C23×C4C2311yescharacteristic
52C23×C4C2312yes—
53C23×C4C2312yes—
54C23×C4C2312yes—
55C23×C4C2311yescharacteristic
56C2×C4⋊C4—24no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—24no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—41no—
62C2×C4⋊C4—41no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4D414yes—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C42—22no—
86C2×C42—41no—
87C2×C42—41no—
88#95—24no—
89#95—24no—
90#95—22no—
91#95—24no—
92#95—24no—
93#95—42no—
94#95—41no—
95#95D418yescomplemented
96C24—22no—
97C24—22no—
98C24—22no—
99C24—22no—
100C24—22no—
101C24—22no—
102C24—22no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24C2×Q814yes—
111C24C2411yescharacteristic
112C2×Q8—44no—
113C2×Q8—48no—
114C2×Q8—44no—
115C2×Q8—48no—
116C2×Q8—44no—
117C2×D4—28no—
118C2×D4—44no—
119C2×D4—44no—
120C2×D4—44no—
121C2×D4—44no—
122C2×D4—44no—
123C2×D4—44no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
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—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—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
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×C4C2×D414yes—
168C22×C4C2×D414yes—
169C22×C4C2×D412yes—
170C22×C4C2×D414yes—
171C22×C4C2×D412yes—
172C22×C4C2×D412yes—
173C22×C4C2×Q812yes—
174C22×C4#4114yes—
175C22×C4#4112yes—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—48no—
179C4⋊C4—48no—
180C22⋊C4—48no—
181C22⋊C4—48no—
182C22⋊C4—48no—
183C22⋊C4—48no—
184C22⋊C4—48no—
185C22⋊C4—48no—
186C42—44no—
187C23—24no—
188C23—24no—
189C23—24no—
190C23—24no—
191C23—24no—
192C23—24no—
193C23—24no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23C22×D411yescharacteristic
223C23C22×D412yes—
224C23C22×Q811yescharacteristic
225C23C2×#4111yescharacteristic
226C23C2×#4112yes—
227C23H(5,𝔽2)11yescharacteristic
228C23H(5,𝔽2)11yescharacteristic
229C23H(5,𝔽2)11yescharacteristic
230C23H(5,𝔽2)12yes—
231C23H(5,𝔽2)11yescharacteristic
232C232−1+412yes—
233Q8—416no—
234D4—416no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—24no—
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—42no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
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×C4C22⋊Q814yes—
287C22—44no—
288C22—44no—
289C22—44no—
290C22—44no—
291C22—44no—
292C22—44no—
293C22—44no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22—82no—
301C22—82no—
302C22C2×C22⋊Q811yescharacteristic
303C22#53212yes—
304C22#53411yescharacteristic
305C22#53511yescharacteristic
306C22#53512yes—
307C22#53712yes—
308C22#53811yescharacteristic
309C22#53811yescharacteristic
310C22#53812yes—
311C22#54212yes—
312C22#54212yes—
313C22#54311yescharacteristic
314C22#54311yescharacteristic
315C22D4212yes—
316C22D4211yescharacteristic
317C22#54612yes—
318C22#54812yes—
319C22D4×Q812yes—
320C22#55412yes—
321C22#55511yescharacteristic
322C22#56011yescharacteristic
323C22#56212yes—
324C22#56311yescharacteristic
325C4—24no—
326C4—42no—
327C4—42no—
328C4—44no—
329C4—44no—
330C4—44no—
331C4—44no—
332C4—82no—
333C2—84no—
334C2#245212yes—
335C2#255511yescharacteristic
336C2#262611yescharacteristic
337C2#266212yes—
338C2#266911yescharacteristic
339C2#267612yes—
340C2#268311yescharacteristic
341C2#269212yes—
342C2#273111yescharacteristic
343C2#278411yescharacteristic
344C2#283011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC23×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#396C2semidirect
2. C2×#395C2semidirect
3. C2×#393C2semidirect
5. #1441C2semidirect
6. #1439C2semidirect
7. #1439C2semidirect
9. #1436C2semidirect
47. C22×D4Q8semidirect
95. #95D4semidirect

Maximal subgroups

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

subgroupindex
C2×#3962
C2×#3952
C2×#3932
C2×#3902
#14412
#14392
#14382
#14362