Finite Group Databasebeta

← #24237

Subgroups — #24237

Order 256.

761 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 761 conjugacy classes of subgroups — too many to draw. Only its 151 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×#547C211yesmaximal, characteristic, complemented
2C2×#461C211yesmaximal, characteristic, complemented
3C2×#449C212yesmaximal, complemented
4C2×C4×D8C211yesmaximal, characteristic, complemented
5#2537C211yesmaximal, characteristic, complemented
6#2036C212yesmaximal, complemented
7#2000C212yesmaximal, complemented
8#1956C211yesmaximal, characteristic, complemented
9#1954C211yesmaximal, characteristic, complemented
10#1802C211yesmaximal, characteristic, complemented
11#1788C212yesmaximal, complemented
12C22×#41C2212yes—
13C22×SD16C2212yes—
14C22×D8C2211yescharacteristic
15#547—24no—
16#547—24no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#123—22no—
19C2×C22⋊Q8C2212yes—
20C2×#121—21no—
21C2×#121—21no—
22C2×#121C2212yes—
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C22×C4⋊C4—21no—
27#461—24no—
28#449—28no—
29C4×D8—24no—
30C2×#107C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×#102C2212yes—
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2212yes—
39C2×C22⋊C8C2212yes—
40C2×C4×C8C2211yescharacteristic
41#394—22no—
42#384C2211yescharacteristic
43#380—21no—
44#380C2212yes—
45#340C2212yes—
46C2×#41—24no—
47C2×#41—24no—
48C2×#41—24no—
49C2×#41—24no—
50C2×#41—42no—
51C2×#41—42no—
52C22×Q8C2312yes—
53C22×D4—22no—
54C22×D4—22no—
55C22×D4D411yescharacteristic
56C22×D4C2311yescharacteristic
57C22×D4C2311yescharacteristic
58C23×C4—21no—
59C23×C4—22no—
60C23×C4—21no—
61C23×C4C2312yes—
62C2×SD16—24no—
63C2×SD16—24no—
64C2×SD16—42no—
65C2×D8—22no—
66C2×D8—22no—
67C2×D8D414yescomplemented
68C22×C8C2312yes—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C4⋊Q8—22no—
72C4⋊Q8D414yescomplemented
73#123—216no—
74C22⋊Q8—24no—
75C22⋊Q8—24no—
76#121—28no—
77#121—24no—
78#121—42no—
79#121—44no—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4D414yescomplemented
86C2×C4⋊C4—28no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2312yes—
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—22no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4D412yes—
102C2×C42C2311yescharacteristic
103#107—22no—
104C4⋊C8—22no—
105#103—24no—
106#103—24no—
107#103—42no—
108#102—22no—
109#102—24no—
110#102—22no—
111C22⋊C8—24no—
112C22⋊C8—42no—
113C4×C8—22no—
114#95—22no—
115C24—21no—
116C24—21no—
117C24C2×D412yes—
118#41—48no—
119#41—48no—
120#41—48no—
121#41—48no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8C2×D414yes—
128C2×D4—28no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—44no—
141C2×D4—44no—
142C2×D4—44no—
143C2×D4—44no—
144C2×D4—82no—
145C2×D4C2×D412yes—
146C2×D4C2×D412yes—
147C2×D4C2×D412yes—
148C2×D4#4112yes—
149C22×C4—28no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4C2×D412yes—
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2411yescharacteristic
172SD16—48no—
173D8—28no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—21no—
180C2×C8—22no—
181C2×C8—21no—
182C2×C8C2×D412yes—
183C4⋊C4—28no—
184C4⋊C4—22no—
185C4⋊C4—24no—
186C4⋊C4—24no—
187C4⋊C4—21no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—42no—
193C4⋊C4—81no—
194C4⋊C4C2×D412yes—
195C22⋊C4—28no—
196C22⋊C4—24no—
197C22⋊C4—24no—
198C22⋊C4—44no—
199C22⋊C4—44no—
200C22⋊C4—42no—
201C42C2×D412yes—
202C42C2×D412yes—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23C22×D411yescharacteristic
225Q8—24no—
226Q8—24no—
227Q8—42no—
228Q8—42no—
229D4—22no—
230D4—22no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—416no—
235D4—41no—
236D4—41no—
237D4—44no—
238D4—88no—
239D4#12114yescomplemented
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—416no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—48no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—84no—
264C2×C4#12012yes—
265C2×C4#12012yes—
266C2×C4#13512yes—
267C2×C4AGL(1,ℤ/8)12yes—
268C2×C4C22×D411yescharacteristic
269C2×C4C22×D411yescharacteristic
270C2×C4C22×D411yescharacteristic
271C2×C4C2×#4111yescharacteristic
272C2×C4H(5,𝔽2)11yescharacteristic
273C2×C4H(5,𝔽2)11yescharacteristic
274C8—22no—
275C8—41no—
276C8—42no—
277C22—24no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22C2×#12011yescharacteristic
299C22D4211yescharacteristic
300C22#54611yescharacteristic
301C22C2×#13511yescharacteristic
302C22C2×AGL(1,ℤ/8)11yescharacteristic
303C22#57711yescharacteristic
304C22#57811yescharacteristic
305C4—22no—
306C4—42no—
307C4—44no—
308C4—42no—
309C4—81no—
310C4—82no—
311C4#44914yescomplemented
312C4C2×#12111yescharacteristic
313C4#53511yescharacteristic
314C4D4211yescharacteristic
315C4#54611yescharacteristic
316C2—44no—
317C2—82no—
318C2—82no—
319C2#240811yescharacteristic
320C2C2×#44911yescharacteristic
321C2#301311yescharacteristic
322C2#327211yescharacteristic
323C2#328311yescharacteristic
324C2#330311yescharacteristic
325C2#333011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#547C2semidirect
2. C2×#461C2semidirect
3. C2×#449C2semidirect
4. C2×C4×D8C2semidirect
5. #2537C2semidirect
6. #2036C2semidirect
7. #2000C2semidirect
8. #1956C2semidirect
9. #1954C2semidirect
10. #1802C2semidirect
11. #1788C2semidirect
67. C2×D8D4semidirect
72. C4⋊Q8D4semidirect
85. C4×D4D4semidirect
239. D4#121semidirect
311. C4#449semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4612
C2×#4492
C2×C4×D82
#25372
#20362
#20002
#19562
#19542
#18022
#17882