Finite Group Databasebeta

← #25767

Subgroups — #25767

Order 256.

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

normal subgroups only

This group has 767 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
1C2×#548C211yesmaximal, characteristic
2C2×#497C211yesmaximal, characteristic, complemented
3C2×#462C211yesmaximal, characteristic, complemented
4C2×#452C212yesmaximal, complemented
5C2×#451C212yesmaximal, complemented
6#2542C211yesmaximal, characteristic, complemented
7#2001C212yesmaximal, complemented
8#2000C212yesmaximal, complemented
9#1983C211yesmaximal, characteristic, complemented
10#1955C211yesmaximal, characteristic, complemented
11#1763C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C23×Q8C2211yescharacteristic
14C22×Q16C2212yes—
15C22×Q16C2212yes—
16C22×SD16C2212yes—
17#548—24no—
18#548—24no—
19C2×C4⋊Q8C2211yescharacteristic
20C2×#124—21no—
21C2×#124C2211yescharacteristic
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2212yes—
25C2×C22⋊Q8C2212yes—
26C2×#121—21no—
27C2×#121C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29C2×C4×D4—21no—
30C2×C4×D4C2211yescharacteristic
31#497—24no—
32#462—24no—
33#452—28no—
34#451—28no—
35C2×C4⋊C8C2211yescharacteristic
36C2×#103C2212yes—
37C2×#103C2212yes—
38C2×#103C2212yes—
39C2×#102C2212yes—
40C2×C22⋊C8C2212yes—
41C2×C22⋊C8C2212yes—
42C2×#97C2211yescharacteristic
43#399—21no—
44#399—21no—
45#390—21no—
46#384—21no—
47C4×C4⋊C4C2211yescharacteristic
48#340C2212yes—
49C2×#41—22no—
50C2×#41—22no—
51C2×#41—24no—
52C2×#41—42no—
53C22×Q8—22no—
54C22×Q8—22no—
55C22×Q8—21no—
56C22×Q8—24no—
57C22×Q8—42no—
58C22×Q8C2312yes—
59C22×Q8C2311yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4D411yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C2×Q16—24no—
70C2×Q16—24no—
71C2×Q16—24no—
72C2×Q16—24no—
73C2×Q16—24no—
74C2×Q16—42no—
75C2×SD16—24no—
76C2×SD16—24no—
77C2×SD16—42no—
78C22×C8C2312yes—
79C22×C8C2312yes—
80C4⋊Q8—22no—
81C4⋊Q8—22no—
82#124—28no—
83#124—22no—
84#124D414yes—
85C22⋊Q8—28no—
86C22⋊Q8—22no—
87C22⋊Q8—22no—
88C22⋊Q8—24no—
89C22⋊Q8—24no—
90C22⋊Q8—42no—
91C22⋊Q8—42no—
92#121—28no—
93#121—22no—
94#121—22no—
95C4×Q8—22no—
96C4×Q8D414yes—
97C4×D4—28no—
98C4×D4—22no—
99C4×D4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4D411yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2312yes—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4D411yescharacteristic
116C2×C22⋊C4D411yescharacteristic
117C2×C42—21no—
118C2×C42—21no—
119C2×C42C2311yescharacteristic
120C4⋊C8—22no—
121#103—24no—
122#103—24no—
123#103D418yescomplemented
124#102—24no—
125C22⋊C8—24no—
126C22⋊C8—24no—
127#97—22no—
128#95—21no—
129C24—21no—
130C24C2×D411yescharacteristic
131C24C2×D411yescharacteristic
132#41—48no—
133#41—44no—
134#41—44no—
135C2×Q8—24no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—44no—
148C2×Q8—48no—
149C2×Q8—44no—
150C2×Q8—44no—
151C2×Q8C2×D412yes—
152C2×Q8C2×D414yes—
153C2×Q8C2×D412yes—
154C2×D4—24no—
155C2×D4—24no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4—41no—
162C2×D4—44no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4C2×D412yes—
166C22×C4—21no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2411yescharacteristic
188Q16—48no—
189Q16—48no—
190SD16—48no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8C2×D414yes—
197C4⋊C4—24no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—21no—
201C4⋊C4—22no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4C2×D414yes—
207C4⋊C4C2×D412yes—
208C22⋊C4—24no—
209C22⋊C4—24no—
210C22⋊C4—24no—
211C22⋊C4—24no—
212C22⋊C4—22no—
213C22⋊C4—22no—
214C22⋊C4—44no—
215C22⋊C4—44no—
216C42—24no—
217C42—42no—
218C42C2×D412yes—
219C42C2×D412yes—
220C23—21no—
221C23—22no—
222C23—21no—
223C23—22no—
224C23—21no—
225C23—22no—
226C23—22no—
227C23—21no—
228C23—21no—
229C23—22no—
230C23—42no—
231C23—41no—
232C23—41no—
233C23—42no—
234C23#13712yes—
235C23C22×D411yescharacteristic
236Q8—22no—
237Q8—24no—
238Q8—24no—
239Q8—48no—
240Q8—41no—
241Q8—41no—
242Q8—42no—
243Q8—42no—
244Q8—42no—
245Q8—42no—
246Q8#12714yes—
247D4—24no—
248D4—44no—
249D4—48no—
250D4—48no—
251D4—41no—
252D4—41no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—48no—
266C2×C4—48no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4#12012yes—
282C2×C4#12012yes—
283C2×C4C22×D411yescharacteristic
284C2×C4C22×D412yes—
285C2×C4C22×D411yescharacteristic
286C2×C4H(5,𝔽2)12yes—
287C8—24no—
288C8—42no—
289C22—22no—
290C22—24no—
291C22—41no—
292C22—41no—
293C22—42no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—82no—
303C22—82no—
304C22C2×#12011yescharacteristic
305C22D4212yes—
306C22C2×#13711yescharacteristic
307C22#57511yescharacteristic
308C22#57812yes—
309C4—24no—
310C4—22no—
311C4—22no—
312C4—41no—
313C4—41no—
314C4—42no—
315C4—42no—
316C4—44no—
317C4—82no—
318C4C2×#12711yescharacteristic
319C4#53511yescharacteristic
320C4D4212yes—
321C2—42no—
322C2—81no—
323C2—82no—
324C2#243311yescharacteristic
325C2#300711yescharacteristic
326C2#301111yescharacteristic
327C2#326912yes—
328C2#327012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#548C2
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
2. C2×#497C2semidirect
3. C2×#462C2semidirect
4. C2×#452C2semidirect
5. C2×#451C2semidirect
6. #2542C2semidirect
7. #2001C2semidirect
8. #2000C2semidirect
9. #1983C2semidirect
10. #1955C2semidirect
11. #1763C2semidirect
123. #103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5482
C2×#4972
C2×#4622
C2×#4522
C2×#4512
#25422
#20012
#20002
#19832
#19552
#17632