Finite Group Databasebeta

← #23922

Subgroups — #23922

Order 256.

964 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 964 conjugacy classes of subgroups — too many to draw. Only its 161 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525C211yesmaximal, characteristic
2C22×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#452C212yesmaximal, complemented
4C2×#451C211yesmaximal, characteristic, complemented
5C2×#448C211yesmaximal, characteristic, complemented
6#2001C212yesmaximal, complemented
7#2000C212yesmaximal, complemented
8#1874C211yesmaximal, characteristic, complemented
9#1873C212yesmaximal, complemented
10#1872C211yesmaximal, characteristic, complemented
11#1779C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C23×Q8C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C22×Q16C2212yescomplemented
16C22×SD16C2212yescomplemented
17#525—24no—
18#525—24no—
19C2×#123—21no—
20C2×C22⋊Q8—21no—
21C2×C22⋊Q8—21no—
22C2×C22⋊Q8—28no—
23C2×C22⋊Q8C2212yes—
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2211yescharacteristic
26C2×C22⋊Q8C2212yes—
27C2×#121C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×#120—21no—
30C2×C4×D4—21no—
31C2×C4×D4—21no—
32C2×#117C2211yescharacteristic
33C22×C4⋊C4—21no—
34C22×C4⋊C4C2211yescharacteristic
35C22×C22⋊C4—21no—
36#452—28no—
37#451—24no—
38#448—24no—
39C2×#103C2212yescomplemented
40C2×#103C2212yescomplemented
41C2×#103C2212yescomplemented
42C2×#102C2212yescomplemented
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2212yes—
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2212yes—
47#340C2212yescomplemented
48#340C2212yescomplemented
49C25D411yescharacteristic
50C2×#41—22no—
51C2×#41—22no—
52C2×#41—24no—
53C2×#41—42no—
54C22×Q8—24no—
55C22×Q8—21no—
56C22×Q8—22no—
57C22×Q8—22no—
58C22×Q8—42no—
59C22×Q8C2312yes—
60C22×Q8C2311yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—24no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—42no—
72C23×C4D411yescharacteristic
73C23×C4D411yescharacteristic
74C23×C4D411yescharacteristic
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C23×C4C2311yescharacteristic
78C2×Q16—24no—
79C2×Q16—24no—
80C2×Q16—42no—
81C2×SD16—24no—
82C2×SD16—24no—
83C2×SD16—42no—
84C22×C8C2312yes—
85C22×C8C2312yes—
86#123—28no—
87C22⋊Q8—28no—
88C22⋊Q8—28no—
89C22⋊Q8—24no—
90C22⋊Q8—22no—
91C22⋊Q8—24no—
92C22⋊Q8—22no—
93C22⋊Q8—24no—
94C22⋊Q8—22no—
95C22⋊Q8D414yes—
96C22⋊Q8D418yescomplemented
97#121—22no—
98#121—22no—
99#121—22no—
100#121D414yes—
101#120—28no—
102C4×D4—28no—
103C4×D4—28no—
104#117—22no—
105#117—22no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4C2312yes—
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C22⋊C4—28no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C42—21no—
125#103—24no—
126#103—24no—
127#103—24no—
128#102—24no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—24no—
132C22⋊C8—42no—
133C24—22no—
134C24—22no—
135C24—22no—
136C24—22no—
137C24—22no—
138C24—21no—
139C24—41no—
140C24—41no—
141C24C2×D411yescharacteristic
142C24C2×D411yescharacteristic
143C24C2×D411yescharacteristic
144#41—48no—
145#41—48no—
146C2×Q8—24no—
147C2×Q8—22no—
148C2×Q8—21no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—48no—
156C2×Q8—48no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8C2×D412yes—
160C2×Q8C2×D414yes—
161C2×D4—24no—
162C2×D4—24no—
163C2×D4—24no—
164C2×D4—22no—
165C2×D4—21no—
166C2×D4—22no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4C2×D412yes—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—48no—
201C22×C4—48no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D412yes—
212C22×C4C2×D412yes—
213C22×C4C2×D414yes—
214C22×C4C2411yescharacteristic
215Q16—48no—
216SD16—48no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—22no—
223C4⋊C4—24no—
224C4⋊C4—28no—
225C4⋊C4—24no—
226C4⋊C4—28no—
227C4⋊C4—24no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4C2×D414yes—
232C4⋊C4C2×D412yes—
233C4⋊C4C2×D412yes—
234C22⋊C4—28no—
235C22⋊C4—24no—
236C22⋊C4—24no—
237C22⋊C4—24no—
238C22⋊C4—28no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C42—24no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—28no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—44no—
261C23—44no—
262C23—44no—
263C23—44no—
264C23—44no—
265C23—44no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23#12012yes—
273C23#12012yes—
274C23#13512yes—
275C23#13712yes—
276C23#13712yes—
277C23C22×D411yescharacteristic
278Q8—24no—
279Q8—24no—
280Q8—24no—
281Q8—48no—
282Q8—42no—
283Q8—42no—
284Q8—42no—
285D4—24no—
286D4—48no—
287D4—48no—
288D4—48no—
289D4—42no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—48no—
310C2×C4—48no—
311C2×C4—48no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4#12012yes—
325C2×C4#12012yes—
326C2×C4C22×D412yes—
327C2×C4C22×D411yescharacteristic
328C2×C4C22×D411yescharacteristic
329C2×C4H(5,𝔽2)12yes—
330C8—42no—
331C8—42no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—416no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22#45214yescomplemented
361C22C2×#12011yescharacteristic
362C22C2×#12011yescharacteristic
363C22#53411yescharacteristic
364C22C2×#13511yescharacteristic
365C22C2×#13711yescharacteristic
366C22C2×#13711yescharacteristic
367C22#57511yescharacteristic
368C4—22no—
369C4—22no—
370C4—41no—
371C4—42no—
372C4—44no—
373C4—42no—
374C4—44no—
375C4—42no—
376C4D4212yes—
377C4D4212yes—
378C2—24no—
379C2—42no—
380C2—42no—
381C2—82no—
382C2#240311yescharacteristic
383C2C2×#45211yescharacteristic
384C2#300211yescharacteristic
385C2#300411yescharacteristic
386C2#300711yescharacteristic
387C2#319011yescharacteristic
388C2#319611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#525C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C22⋊Q8C2semidirect
3. C2×#452C2semidirect
4. C2×#451C2semidirect
5. C2×#448C2semidirect
6. #2001C2semidirect
7. #2000C2semidirect
8. #1874C2semidirect
9. #1873C2semidirect
10. #1872C2semidirect
11. #1779C2semidirect
15. C22×Q16C22semidirect
16. C22×SD16C22semidirect
39. C2×#103C22semidirect
40. C2×#103C22semidirect
41. C2×#103C22semidirect
42. C2×#102C22semidirect
47. #340C22semidirect
48. #340C22semidirect
96. C22⋊Q8D4semidirect
360. C22#452semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
C22×C22⋊Q82
C2×#4522
C2×#4512
C2×#4482
#20012
#20002
#18742
#18732
#18722
#17792