Finite Group Databasebeta

← #18207

Subgroups — #18207

Order 256.

504 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#3505C211yesmaximal, characteristic
2C2×#484C211yesmaximal, characteristic, complemented
3C2×#481C211yesmaximal, characteristic, complemented
4#2487C211yesmaximal, characteristic, complemented
5#1618C214yesmaximal, complemented
6#1615C214yesmaximal, complemented
7#1520C211yesmaximal, characteristic, complemented
8#1519C211yesmaximal, characteristic, complemented
9#1459C211yesmaximal, characteristic, complemented
10#553—22no—
11#553—21no—
12#552—21no—
13#551—22no—
14#548—21no—
15#546—21no—
16#539C2212yes—
17#538C2212yes—
18#534—21no—
19#529—21no—
20#525—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C2211yescharacteristic
23#518—21no—
24C2×C4×D4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26#484—22no—
27#484C2214yescomplemented
28#481—22no—
29#481C2214yes—
30C2×#107—21no—
31C2×#106—21no—
32C2×#103—21no—
33C2×#102—21no—
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36#400C2212yescomplemented
37#399C2212yescomplemented
38#389—21no—
39#386—21no—
40#385—21no—
41#380—21no—
42#380C2211yescharacteristic
43#328C2214yescomplemented
44#327C2214yes—
45#324C2212yescomplemented
46#323C2212yescomplemented
47C2×#41—21no—
48C22×Q8D411yescharacteristic
49C22×D4—21no—
50C22×D4D411yescharacteristic
51C22×D4D411yescharacteristic
52C23×C4—21no—
53C23×C4D411yescharacteristic
54C23×C4C2311yescharacteristic
55C22×C8—21no—
56C22×C8—21no—
57#126—22no—
58#126—22no—
59#126—22no—
60#125—21no—
61#125—22no—
62#124—21no—
63#124—22no—
64#123—22no—
65#123—22no—
66#123—22no—
67#123—22no—
68#123—22no—
69C22⋊Q8—21no—
70C22⋊Q8—22no—
71C22⋊Q8—21no—
72C22⋊Q8—21no—
73C22⋊Q8D412yes—
74C22⋊Q8C2312yes—
75#121—21no—
76#121—22no—
77#121—22no—
78#121—22no—
79#121—21no—
80#121—21no—
81#121D412yes—
82#121C2312yes—
83#120—22no—
84#120—22no—
85C4×Q8—21no—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—22no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4—21no—
92C4×D4—21no—
93C4×D4—21no—
94C4×D4—22no—
95C4×D4—22no—
96#117—22no—
97#117—21no—
98#117—22no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4D411yescharacteristic
108C2×C4⋊C4D411yescharacteristic
109C2×C4⋊C4C2312yes—
110C2×C4⋊C4C2311yescharacteristic
111C2×C22⋊C4—22no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4D411yescharacteristic
117C2×C22⋊C4C2311yescharacteristic
118C2×C42D411yescharacteristic
119#107—24no—
120#106—24no—
121#103—24no—
122#102—24no—
123C22⋊C8—21no—
124C22⋊C8—21no—
125C22⋊C8C2312yes—
126C22⋊C8C2312yes—
127#95—22no—
128#95—22no—
129#95C2312yes—
130C24—21no—
131C24C2×D411yescharacteristic
132C24C2×D411yescharacteristic
133#41—82no—
134C2×Q8—21no—
135C2×Q8—42no—
136C2×Q8C2×D412yes—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—21no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4C2×D412yes—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—82no—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2411yescharacteristic
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—42no—
176C2×C8—42no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—42no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—82no—
193C4⋊C4C2×D412yes—
194C4⋊C4C2×D412yes—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—21no—
202C22⋊C4—22no—
203C22⋊C4—21no—
204C22⋊C4—22no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—21no—
208C22⋊C4—21no—
209C22⋊C4—42no—
210C22⋊C4—82no—
211C42—22no—
212C42—22no—
213C42—21no—
214C42—21no—
215C23—21no—
216C23—22no—
217C23—21no—
218C23—21no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—42no—
224C23—81no—
225C23—81no—
226C23#12011yescharacteristic
227C23#12011yescharacteristic
228C23#13512yes—
229C23#13712yes—
230C23C22×D411yescharacteristic
231Q8—42no—
232Q8—42no—
233D4—42no—
234D4—42no—
235D4—42no—
236D4—42no—
237D4—42no—
238D4—44no—
239D4—42no—
240D4—82no—
241D4—82no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4#12011yescharacteristic
274C2×C4#12011yescharacteristic
275C2×C4C22×D411yescharacteristic
276C2×C4C22×D411yescharacteristic
277C8—42no—
278C8—42no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—42no—
294C22—81no—
295C22—162no—
296C22#45212yes—
297C22#45212yes—
298C22C2×#12011yescharacteristic
299C22C2×#13511yescharacteristic
300C22C2×#13711yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—41no—
305C4—42no—
306C4—41no—
307C4—42no—
308C4—41no—
309C4—42no—
310C4—82no—
311C4—82no—
312C4#45812yes—
313C2—21no—
314C2—21no—
315C2—22no—
316C2—22no—
317C2—81no—
318C2—82no—
319C2C2×#45211yescharacteristic
320C2#302111yescharacteristic
321C2C2×#45811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3505C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#484C2semidirect
3. C2×#481C2semidirect
4. #2487C2semidirect
5. #1618C2semidirect
6. #1615C2semidirect
7. #1520C2semidirect
8. #1519C2semidirect
9. #1459C2semidirect
27. #484C22semidirect
36. #400C22semidirect
37. #399C22semidirect
43. #328C22semidirect
45. #324C22semidirect
46. #323C22semidirect

Maximal subgroups

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

subgroupindex
#35052
C2×#4842
C2×#4812
#24872
#16182
#16152
#15202
#15192
#14592