Finite Group Databasebeta

← #15429

Subgroups — #15429

Order 256.

1045 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#396C212yesmaximal, complemented
2C2×#386C212yesmaximal
3C2×#385C212yesmaximal, complemented
4C2×#380C212yesmaximal, complemented
5#1444C211yesmaximal, characteristic, complemented
6#1442C211yesmaximal, characteristic, complemented
7#1441C212yesmaximal, complemented
8#1439C212yesmaximal, complemented
9#1438C211yesmaximal, characteristic
10C23×D4C2211yescharacteristic
11C24×C4C2212yes—
12C22×C4⋊C4C2212yes—
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2212yes—
15C22×C4⋊C4C2212yes—
16C22×C4⋊C4C2211yescharacteristic
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22#396—28no—
23#386—28no—
24#385—28no—
25#380—28no—
26C2×#95C2212yes—
27C2×#95C2212yes—
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2211yescharacteristic
32C2×#95C2211yescharacteristic
33C2×#95C2211yescharacteristic
34C2×#95C2211yescharacteristic
35C25C2312yes—
36C22×D4—24no—
37C22×D4—24no—
38C22×D4—41no—
39C22×D4—41no—
40C22×D4Q814yes—
41C23×C4—24no—
42C23×C4—24no—
43C23×C4—24no—
44C23×C4—24no—
45C23×C4—24no—
46C23×C4—42no—
47C23×C4—42no—
48C23×C4C2312yes—
49C23×C4C2312yes—
50C23×C4C2311yescharacteristic
51C23×C4C2312yes—
52C23×C4C2311yescharacteristic
53C23×C4C2312yes—
54C23×C4C2311yescharacteristic
55C23×C4C2312yes—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—24no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—42no—
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—24no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4D418yes—
84#95—24no—
85#95—24no—
86#95—24no—
87#95—24no—
88#95—24no—
89#95—41no—
90#95—41no—
91#95D414yescomplemented
92#95D414yescomplemented
93C24—22no—
94C24—22no—
95C24—22no—
96C24—22no—
97C24—22no—
98C24—22no—
99C24—22no—
100C24—22no—
101C24—22no—
102C24—22no—
103C24—22no—
104C24—22no—
105C24C2×D414yes—
106C24C2×Q814yes—
107C24#4114yes—
108C24C2411yescharacteristic
109C2×D4—216no—
110C2×D4—44no—
111C2×D4—44no—
112C2×D4—48no—
113C2×D4—44no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—216no—
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—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
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—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—48no—
158C22×C4—48no—
159C22×C4—48no—
160C22×C4—48no—
161C22×C4—48no—
162C22×C4—48no—
163C22×C4C2×D412yes—
164C22×C4C2×D414yes—
165C22×C4C2×D414yes—
166C22×C4C2×D412yes—
167C22×C4C2×D414yes—
168C22×C4C2×D414yes—
169C22×C4C2×Q812yes—
170C22×C4#4114yes—
171C4⋊C4—48no—
172C4⋊C4—48no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—48no—
176C22⋊C4—216no—
177C22⋊C4—48no—
178C22⋊C4—48no—
179C22⋊C4—48no—
180C22⋊C4—48no—
181C23—24no—
182C23—24no—
183C23—24no—
184C23—24no—
185C23—24no—
186C23—24no—
187C23—24no—
188C23—24no—
189C23—24no—
190C23—24no—
191C23—24no—
192C23—24no—
193C23—24no—
194C23—24no—
195C23—24no—
196C23—24no—
197C23—24no—
198C23—24no—
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—
215C23C22⋊Q818yes—
216C23C22×D411yescharacteristic
217C23C22×D412yes—
218C23C22×D411yescharacteristic
219C23C22×Q811yescharacteristic
220C23C2×#4112yes—
221C23C2×#4112yes—
222C23H(5,𝔽2)11yescharacteristic
223C23H(5,𝔽2)12yes—
224C23H(5,𝔽2)11yescharacteristic
225C232−1+411yescharacteristic
226C232−1+411yescharacteristic
227D4—416no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—24no—
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—24no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—432no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
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—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4#12118yes—
279C22—28no—
280C22—28no—
281C22—28no—
282C22—44no—
283C22—44no—
284C22—44no—
285C22—44no—
286C22—44no—
287C22—44no—
288C22—44no—
289C22—44no—
290C22—44no—
291C22—44no—
292C22—44no—
293C22—44no—
294C22C2×#12112yes—
295C22C2×C22⋊Q812yes—
296C22#52512yes—
297C22#53711yescharacteristic
298C22#53712yes—
299C22#53711yescharacteristic
300C22#53812yes—
301C22#53912yes—
302C22#54311yescharacteristic
303C22D4212yes—
304C22D4211yescharacteristic
305C22D4211yescharacteristic
306C22#54612yes—
307C22#54612yes—
308C22#54712yes—
309C22D4×Q811yescharacteristic
310C22D4×Q811yescharacteristic
311C22#55111yescharacteristic
312C22#55212yes—
313C22#55412yes—
314C22#55511yescharacteristic
315C22#56011yescharacteristic
316C22#56311yescharacteristic
317C4—28no—
318C4—42no—
319C4—44no—
320C4—42no—
321C4—42no—
322C4—44no—
323C4—82no—
324C4—82no—
325C2—48no—
326C2#241712yes—
327C2#242412yes—
328C2#243711yescharacteristic
329C2#244912yes—
330C2#253711yescharacteristic
331C2#260911yescharacteristic
332C2#267511yescharacteristic
333C2#268311yescharacteristic
334C2#269311yescharacteristic
335C2#269512yes—
336C2#278411yescharacteristic
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
3. C2×#385C2semidirect
4. C2×#380C2semidirect
5. #1444C2semidirect
6. #1442C2semidirect
7. #1441C2semidirect
8. #1439C2semidirect
91. #95D4semidirect
92. #95D4semidirect

Maximal subgroups

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

subgroupindex
C2×#3962
C2×#3862
C2×#3852
C2×#3802
#14442
#14422
#14412
#14392
#14382