Finite Group Databasebeta

← #24007

Subgroups — #24007

Order 256.

760 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 145 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
2C2×#484C211yesmaximal, characteristic, complemented
3C2×#481C211yesmaximal, characteristic, complemented
4C2×#380C211yesmaximal, characteristic, complemented
5#2035C212yesmaximal, complemented
6#2033C212yesmaximal, complemented
7#1874C211yesmaximal, characteristic, complemented
8#1872C211yesmaximal, characteristic, complemented
9#1854C212yesmaximal, complemented
10#1788C212yesmaximal, complemented
11#1779C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C24×C4C2211yescharacteristic
14#525—24no—
15#525—24no—
16C2×#123—21no—
17C2×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×#120—21no—
22C2×C4×D4—21no—
23C2×C4×D4—21no—
24C2×#117C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27C22×C22⋊C4—21no—
28#484—24no—
29#481—24no—
30C2×#107C2212yescomplemented
31C2×#106C2212yescomplemented
32C2×#103C2212yescomplemented
33C2×#103C2212yescomplemented
34C2×#102C2212yescomplemented
35C2×#102C2212yescomplemented
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2212yes—
39C2×C22⋊C8C2212yes—
40#380—21no—
41#380—21no—
42#380C2212yes—
43#380C2212yes—
44C2×#95—21no—
45#340C2212yescomplemented
46#340C2212yescomplemented
47C25D411yescharacteristic
48C2×#41—22no—
49C2×#41—22no—
50C2×#41—24no—
51C2×#41—42no—
52C22×Q8C2311yescharacteristic
53C22×D4—21no—
54C22×D4—21no—
55C22×D4—21no—
56C22×D4C2311yescharacteristic
57C23×C4—21no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—24no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—42no—
64C23×C4D411yescharacteristic
65C23×C4D411yescharacteristic
66C23×C4D411yescharacteristic
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C22×C8C2312yes—
71C22×C8C2312yes—
72#123—28no—
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76C22⋊Q8D414yes—
77#121—22no—
78#121—22no—
79#121—22no—
80#121D414yes—
81#120—28no—
82C4×D4—28no—
83C4×D4—28no—
84#117—22no—
85#117—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4C2312yes—
94C2×C4⋊C4C2312yes—
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—44no—
105C2×C42—21no—
106#107—24no—
107#106—24no—
108#103—24no—
109#103—42no—
110#102—24no—
111#102—42no—
112C22⋊C8—22no—
113C22⋊C8—22no—
114C22⋊C8—24no—
115C22⋊C8—42no—
116#95—22no—
117#95—22no—
118C24—22no—
119C24—22no—
120C24—22no—
121C24—22no—
122C24—22no—
123C24—21no—
124C24—41no—
125C24—41no—
126C24C2×D411yescharacteristic
127C24C2×D411yescharacteristic
128C24C2×D411yescharacteristic
129#41—48no—
130#41—48no—
131C2×Q8—21no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8C2×D412yes—
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—24no—
138C2×D4—21no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4C2×D412yes—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—48no—
173C22×C4—48no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D412yes—
185C22×C4C2×D412yes—
186C22×C4C2×D411yescharacteristic
187C22×C4#4114yes—
188C22×C4C2411yescharacteristic
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—22no—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4C2×D412yes—
206C4⋊C4C2×D412yes—
207C22⋊C4—24no—
208C22⋊C4—24no—
209C22⋊C4—24no—
210C22⋊C4—24no—
211C22⋊C4—24no—
212C22⋊C4—44no—
213C22⋊C4—44no—
214C42—24no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—28no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—44no—
234C23—44no—
235C23—44no—
236C23—44no—
237C23—44no—
238C23—44no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23#12012yes—
246C23#12312yes—
247C23#13512yes—
248C23AGL(1,ℤ/8)12yes—
249C23#13712yes—
250C23C22×D411yescharacteristic
251Q8—24no—
252Q8—42no—
253D4—24no—
254D4—48no—
255D4—48no—
256D4—48no—
257D4—42no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—48no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—84no—
291C2×C4—84no—
292C2×C4#12012yes—
293C2×C4#12312yes—
294C2×C4C22×D411yescharacteristic
295C2×C4C22×D411yescharacteristic
296C2×C4C2×#4112yes—
297C2×C4H(5,𝔽2)12yes—
298C8—42no—
299C8—42no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—416no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22#45214yes—
329C22C2×#12011yescharacteristic
330C22C2×#12311yescharacteristic
331C22#53411yescharacteristic
332C22C2×#13511yescharacteristic
333C22C2×AGL(1,ℤ/8)11yescharacteristic
334C22C2×#13711yescharacteristic
335C22#57511yescharacteristic
336C4—22no—
337C4—22no—
338C4—41no—
339C4—42no—
340C4—42no—
341C4—42no—
342C4—82no—
343C4—82no—
344C4#54612yes—
345C4#54612yes—
346C2—24no—
347C2—42no—
348C2—42no—
349C2—82no—
350C2#240511yescharacteristic
351C2C2×#45211yescharacteristic
352C2#300611yescharacteristic
353C2#309111yescharacteristic
354C2#309211yescharacteristic
355C2#318811yescharacteristic
356C2#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. C2×#484C2semidirect
3. C2×#481C2semidirect
4. C2×#380C2semidirect
5. #2035C2semidirect
6. #2033C2semidirect
7. #1874C2semidirect
8. #1872C2semidirect
9. #1854C2semidirect
10. #1788C2semidirect
11. #1779C2semidirect
30. C2×#107C22semidirect
31. C2×#106C22semidirect
32. C2×#103C22semidirect
33. C2×#103C22semidirect
34. C2×#102C22semidirect
35. C2×#102C22semidirect
45. #340C22semidirect
46. #340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
C2×#4842
C2×#4812
C2×#3802
#20352
#20332
#18742
#18722
#18542
#17882
#17792