Finite Group Databasebeta

← #23865

Subgroups — #23865

Order 256.

877 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 877 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#137C211yesmaximal, characteristic, complemented
2C2×#547C211yesmaximal, characteristic
3C2×#452C212yesmaximal, complemented
4C2×#451C211yesmaximal, characteristic, complemented
5#2409C211yesmaximal, characteristic, complemented
6#2016C212yesmaximal, complemented
7#2001C212yesmaximal, complemented
8#1894C211yesmaximal, characteristic, complemented
9#1877C212yesmaximal, complemented
10#1870C211yesmaximal, characteristic, complemented
11#1788C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×#41C2211yescharacteristic
14C23×Q8C2211yescharacteristic
15C2×#137—24no—
16C2×#137—28no—
17C22×Q16C2212yes—
18C22×Q16C2212yes—
19C22×SD16C2212yes—
20C22×M16C2211yescharacteristic
21#547—24no—
22#547—24no—
23C2×C4⋊Q8C2211yescharacteristic
24C2×#123—21no—
25C2×#123—21no—
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2212yes—
28C2×C22⋊Q8C2211yescharacteristic
29C2×#121—21no—
30C2×C4×D4C2211yescharacteristic
31C2×C4×D4C2211yescharacteristic
32C22×C4⋊C4—21no—
33#452—28no—
34#451—24no—
35C2×#103C2212yes—
36C2×#103C2212yes—
37C2×#103C2212yes—
38C2×#102C2212yes—
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2212yes—
41#395—21no—
42#393—21no—
43#391C2212yes—
44#386—21no—
45#385—21no—
46#380C2211yescharacteristic
47#340C2212yes—
48#336C2212yes—
49C2×#41—22no—
50C2×#41—22no—
51C2×#41—22no—
52C2×#41—24no—
53C2×#41—24no—
54C2×#41—42no—
55C2×#41—42no—
56C2×#41D414yes—
57C22×Q8—21no—
58C22×Q8—24no—
59C22×Q8—22no—
60C22×Q8—22no—
61C22×Q8—42no—
62C22×Q8C2311yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×Q8C2312yes—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4D411yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75#137—232no—
76C2×Q16—24no—
77C2×Q16—24no—
78C2×Q16—24no—
79C2×Q16—24no—
80C2×Q16—24no—
81C2×Q16—42no—
82C2×SD16—24no—
83C2×SD16—24no—
84C2×SD16D418yescomplemented
85C2×M16—22no—
86C2×M16—24no—
87C22×C8C2312yes—
88C22×C8C2312yes—
89C4⋊Q8—22no—
90C4⋊Q8—22no—
91#123—28no—
92#123—28no—
93C22⋊Q8—22no—
94C22⋊Q8—22no—
95C22⋊Q8—22no—
96C22⋊Q8—24no—
97C22⋊Q8—42no—
98C22⋊Q8D414yes—
99#121—28no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—22no—
104C2×C4⋊C4—28no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4D411yescharacteristic
111C2×C4⋊C4C2312yes—
112C2×C4⋊C4C2311yescharacteristic
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4D411yescharacteristic
119C2×C22⋊C4D411yescharacteristic
120C2×C42C2311yescharacteristic
121#103—24no—
122#103—24no—
123#103—42no—
124#102—24no—
125C22⋊C8—22no—
126C22⋊C8—24no—
127#95—21no—
128#95—22no—
129#95—22no—
130C24—21no—
131C24C2×D411yescharacteristic
132C24C2×D411yescharacteristic
133#41—28no—
134#41—44no—
135#41—48no—
136#41—48no—
137#41—44no—
138#41—44no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—22no—
142C2×Q8—28no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—44no—
152C2×Q8—48no—
153C2×Q8—44no—
154C2×Q8—42no—
155C2×Q8C2×D412yes—
156C2×Q8C2×D414yes—
157C2×Q8C2×D412yes—
158C2×D4—24no—
159C2×D4—22no—
160C2×D4—24no—
161C2×D4—22no—
162C2×D4—21no—
163C2×D4—22no—
164C2×D4—41no—
165C2×D4—44no—
166C2×D4—44no—
167C2×D4C2×D414yes—
168C2×D4C2×D412yes—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D412yes—
193C22×C4C2×D412yes—
194C22×C4C2×D411yescharacteristic
195C22×C4C2411yescharacteristic
196Q16—216no—
197Q16—48no—
198SD16—216no—
199M16—28no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8C2×D414yes—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—22no—
210C4⋊C4—24no—
211C4⋊C4—21no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—44no—
215C4⋊C4C2×D412yes—
216C22⋊C4—24no—
217C22⋊C4—24no—
218C22⋊C4—24no—
219C22⋊C4—22no—
220C22⋊C4—22no—
221C22⋊C4—44no—
222C22⋊C4—42no—
223C42—21no—
224C42—21no—
225C23—22no—
226C23—22no—
227C23—24no—
228C23—21no—
229C23—21no—
230C23—22no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—41no—
237C23—41no—
238C23—42no—
239C23#12012yes—
240C23C22×D411yescharacteristic
241Q8—24no—
242Q8—22no—
243Q8—22no—
244Q8—24no—
245Q8—24no—
246Q8—48no—
247Q8—41no—
248Q8—42no—
249Q8—41no—
250Q8—42no—
251Q8—42no—
252D4—22no—
253D4—24no—
254D4—44no—
255D4—48no—
256D4—48no—
257D4#12014yes—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—48no—
273C2×C4—48no—
274C2×C4—48no—
275C2×C4—48no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4#12012yes—
289C2×C4#13712yescomplemented
290C2×C4C22×D411yescharacteristic
291C2×C4C22×D412yes—
292C2×C4C22×D411yescharacteristic
293C2×C4H(5,𝔽2)12yes—
294C8—24no—
295C8—42no—
296C22—22no—
297C22—24no—
298C22—22no—
299C22—22no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—44no—
304C22—42no—
305C22—44no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—42no—
310C22—42no—
311C22C2×#12011yescharacteristic
312C22D4212yes—
313C22C2×#13711yescharacteristic
314C22#57511yescharacteristic
315C22#57812yes—
316C4—22no—
317C4—22no—
318C4—41no—
319C4—42no—
320C4—42no—
321C4—44no—
322C4—42no—
323C4—41no—
324C4—82no—
325C4C2×#12011yescharacteristic
326C4#53411yescharacteristic
327C4D4212yes—
328C2—42no—
329C2—44no—
330C2—81no—
331C2#240311yescharacteristic
332C2#300711yescharacteristic
333C2#301111yescharacteristic
334C2#326912yes—
335C2#327612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#547C2
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. C22×#137C2semidirect
3. C2×#452C2semidirect
4. C2×#451C2semidirect
5. #2409C2semidirect
6. #2016C2semidirect
7. #2001C2semidirect
8. #1894C2semidirect
9. #1877C2semidirect
10. #1870C2semidirect
11. #1788C2semidirect
84. C2×SD16D4semidirect
289. C2×C4#137semidirect

Maximal subgroups

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

subgroupindex
C22×#1372
C2×#5472
C2×#4522
C2×#4512
#24092
#20162
#20012
#18942
#18772
#18702
#17882