Finite Group Databasebeta

← #13890

Subgroups — #13890

Order 256.

1165 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D4C211yesmaximal, characteristic
2C2×#387C212yesmaximal, complemented
3C2×#380C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊C4C211yesmaximal, characteristic, complemented
5#1439C212yesmaximal, complemented
6#1438C212yesmaximal
7#1437C212yesmaximal, complemented
8#1434C211yesmaximal, characteristic, complemented
9#1434C211yesmaximal, characteristic, complemented
10C22⋊#95C211yesmaximal, characteristic, complemented
11C22⋊#95C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C2×C4×D4—28no—
16C2×C4×D4C418yescomplemented
17C22×C4⋊C4C2212yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2212yes—
25C22×C42C2212yes—
26C22×C42C2211yescharacteristic
27#387—28no—
28#380—24no—
29C4×C22⋊C4—24no—
30C2×#95C2211yescharacteristic
31C2×#95C2211yescharacteristic
32C2×#95C2211yescharacteristic
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2212yes—
39C2×#95C2211yescharacteristic
40C25C2311yescharacteristic
41C25C2311yescharacteristic
42C22×D4—24no—
43C22×D4—22no—
44C22×D4—22no—
45C22×D4—24no—
46C22×D4C2×C414yes—
47C23×C4—24no—
48C23×C4—24no—
49C23×C4—22no—
50C23×C4—22no—
51C23×C4—22no—
52C23×C4—22no—
53C23×C4—42no—
54C23×C4—42no—
55C23×C4C2×C414yes—
56C23×C4C2×C414yes—
57C23×C4C2312yes—
58C23×C4C2312yes—
59C23×C4C2311yescharacteristic
60C23×C4C2312yes—
61C23×C4C2312yes—
62C23×C4C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C4×D4—232no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4C2×C414yes—
71C2×C4⋊C4D418yes—
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—24no—
80C2×C22⋊C4—24no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4C2×C414yes—
87C2×C22⋊C4C2×C414yes—
88C2×C42—24no—
89C2×C42—24no—
90C2×C42—24no—
91C2×C42—42no—
92C2×C42C2×C414yes—
93#95—22no—
94#95—22no—
95#95—24no—
96#95—24no—
97#95—24no—
98#95—24no—
99#95—24no—
100#95—22no—
101#95—42no—
102#95D414yescomplemented
103C24—21no—
104C24—21no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—22no—
109C24—21no—
110C24—21no—
111C24—22no—
112C24—21no—
113C24—21no—
114C24—21no—
115C24—21no—
116C24—22no—
117C24—22no—
118C24—21no—
119C24—21no—
120C24C22×C412yes—
121C24C22×C412yes—
122C24#4114yes—
123C24#4112yes—
124C24#4112yes—
125C24C2411yescharacteristic
126C2×D4—216no—
127C2×D4—216no—
128C2×D4—28no—
129C2×D4—28no—
130C2×D4—28no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
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—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—28no—
159C22×C4—28no—
160C22×C4—28no—
161C22×C4—28no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—48no—
179C22×C4—48no—
180C22×C4—48no—
181C22×C4—48no—
182C22×C4C22×C412yes—
183C22×C4C22×C412yes—
184C22×C4C22×C412yes—
185C22×C4C22×C412yes—
186C22×C4C22×C412yes—
187C22×C4C2×D414yes—
188C22×C4C2×D414yes—
189C22×C4C2×D414yes—
190C22×C4C2×D414yes—
191C22×C4C2×D412yes—
192C22×C4#4114yes—
193C22×C4#4112yes—
194C4⋊C4—48no—
195C4⋊C4C22⋊C4116yescomplemented
196C22⋊C4—216no—
197C22⋊C4—28no—
198C22⋊C4—28no—
199C22⋊C4—48no—
200C22⋊C4—48no—
201C22⋊C4—48no—
202C42—28no—
203C42—48no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—24no—
217C23—24no—
218C23—24no—
219C23—24no—
220C23—24no—
221C23—24no—
222C23—24no—
223C23—24no—
224C23—22no—
225C23—22no—
226C23—24no—
227C23—22no—
228C23—22no—
229C23—24no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—41no—
239C23—42no—
240C23—42no—
241C23—41no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—41no—
246C23—41no—
247C23—42no—
248C23#11714yes—
249C23#11714yes—
250C23C23×C411yescharacteristic
251C23C22×D412yes—
252C23C22×D411yescharacteristic
253C23C2×#4112yes—
254C23C2×#4111yescharacteristic
255C23C2×#4111yescharacteristic
256C23C2×#4111yescharacteristic
257C23C2×#4112yes—
258C23H(5,𝔽2)12yes—
259C23H(5,𝔽2)11yescharacteristic
260C232−1+411yescharacteristic
261D4—232no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—416no—
288C2×C4—416no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4C2×C22⋊C414yes—
314C2×C4C2×C22⋊C414yes—
315C2×C4C2×C22⋊C414yes—
316C2×C4C4×D414yes—
317C22—24no—
318C22—24no—
319C22—24no—
320C22—24no—
321C22—24no—
322C22—24no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—44no—
331C22—44no—
332C22—44no—
333C22—44no—
334C22—44no—
335C22—44no—
336C22—44no—
337C22—44no—
338C22—42no—
339C22C22×C22⋊C411yescharacteristic
340C22C2×#11711yescharacteristic
341C22C2×#11711yescharacteristic
342C22C2×C4×D411yescharacteristic
343C22C4×#4111yescharacteristic
344C22#51811yescharacteristic
345C22#52011yescharacteristic
346C22#52512yes—
347C22#52512yes—
348C22#52512yes—
349C22#52912yes—
350C22#53812yes—
351C22#53912yes—
352C22D4212yes—
353C22#54612yes—
354C22#54611yescharacteristic
355C22#54812yes—
356C22#54811yescharacteristic
357C22#55012yes—
358C22#55112yes—
359C22#55211yescharacteristic
360C22#55311yescharacteristic
361C22#55312yes—
362C4—24no—
363C4—24no—
364C4—24no—
365C4—24no—
366C4—42no—
367C4—44no—
368C4—44no—
369C4—82no—
370C4—82no—
371C2—44no—
372C2—44no—
373C2#228011yescharacteristic
374C2#229011yescharacteristic
375C2#231811yescharacteristic
376C2D4×C22⋊C411yescharacteristic
377C2#234111yescharacteristic
378C2#235211yescharacteristic
379C2#235311yescharacteristic
380C2#245612yes—
381C2#246012yes—
382C2#246312yes—
383C2#246712yes—
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC23×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#387C2semidirect
3. C2×#380C2semidirect
4. C2×C4×C22⋊C4C2semidirect
5. #1439C2semidirect
7. #1437C2semidirect
8. #1434C2semidirect
9. #1434C2semidirect
10. C22⋊#95C2semidirect
11. C22⋊#95C2semidirect
16. C2×C4×D4C4semidirect
102. #95D4semidirect
195. C4⋊C4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D42
C2×#3872
C2×#3802
C2×C4×C22⋊C42
#14392
#14382
#14372
#14342
C22⋊#952