Finite Group Databasebeta

← #18177

Subgroups — #18177

Order 256.

475 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 113 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#3491C211yesmaximal, characteristic
2#3117C212yesmaximal, complemented
3#2396C211yesmaximal, characteristic, complemented
4#1615C212yesmaximal, complemented
5#1614C211yesmaximal, characteristic, complemented
6#1613C211yesmaximal, characteristic
7#1603C212yesmaximal, complemented
8#1602C211yesmaximal, characteristic, complemented
9#1601C211yesmaximal, characteristic
10#1509C212yesmaximal, complemented
11#1464C211yesmaximal, characteristic, complemented
12#552—22no—
13#552—22no—
14#551—22no—
15#551—22no—
16#543C2211yescharacteristic
17#539—21no—
18#539C2212yes—
19#534C2211yescharacteristic
20#529—21no—
21#529C2212yes—
22C2×#123—21no—
23#518—21no—
24C2×C4×Q8C2211yescharacteristic, complemented
25C2×#117C2211yescharacteristic
26#484C2212yescomplemented
27#481C2212yes—
28#479—22no—
29#464—22no—
30#451C2212yescomplemented
31#448C2212yes—
32#441—22no—
33#440—22no—
34#432C2212yescomplemented
35#400C2211yescharacteristic, complemented
36#399C2211yescharacteristic
37#398—21no—
38#396—21no—
39#395C2211yescharacteristic, complemented
40#393C2211yescharacteristic
41C4×C4⋊C4—21no—
42C4×C4⋊C4C2211yescharacteristic, complemented
43C4×C22⋊C4—21no—
44#328C2212yescomplemented
45#328C2212yescomplemented
46#327C2212yes—
47#327C2212yes—
48#324C2212yescomplemented
49#323C2211yescharacteristic
50#323C2211yescharacteristic
51C22×Q8C2311yescharacteristic
52C22×D4—21no—
53C23×C4D411yescharacteristic
54C2×Q16—22no—
55C2×SD16—22no—
56#126—22no—
57#126—22no—
58#126—22no—
59#126—22no—
60#125—21no—
61#125—22no—
62#125—22no—
63#125D412yes—
64#124—21no—
65#124D412yes—
66#123—22no—
67#123—22no—
68#123—22no—
69#123—22no—
70#123—22no—
71#123—22no—
72#123—22no—
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76C22⋊Q8—21no—
77C22⋊Q8—22no—
78C22⋊Q8C2312yes—
79#121—21no—
80#121C2312yes—
81#120—22no—
82C4×Q8—21no—
83C4×Q8—22no—
84C4×Q8—22no—
85C4×Q8D412yes—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—21no—
89C4×D4—22no—
90C4×D4—22no—
91C4×D4D412yes—
92#117—22no—
93#117—22no—
94#117—22no—
95#117—22no—
96#117—22no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4D411yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C22⋊C4—22no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4D411yescharacteristic
113C2×C22⋊C4C2311yescharacteristic
114C2×C42—21no—
115C2×C42—21no—
116C2×C42D411yescharacteristic
117C2×C42C2311yescharacteristic
118#107—22no—
119#106—22no—
120C4⋊C8—22no—
121#103—22no—
122#103—22no—
123#103—22no—
124#102—22no—
125C22⋊C8C2312yes—
126C22⋊C8C2312yes—
127#97—22no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95C2311yescharacteristic
133#95C2311yescharacteristic
134C24—21no—
135C24C2×D411yescharacteristic
136C2×Q8—21no—
137C2×Q8—22no—
138C2×Q8—42no—
139C2×Q8C2×D412yes—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—21no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4C2×D412yes—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2411yescharacteristic
165Q16—44no—
166SD16—44no—
167C2×C8—22no—
168C2×C8—22no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4C2×D412yes—
187C4⋊C4C2×D412yes—
188C22⋊C4—22no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C42—22no—
203C42—21no—
204C42—21no—
205C42—22no—
206C42—42no—
207C42—41no—
208C42—41no—
209C42C2×D412yes—
210C42C2×D412yes—
211C23—21no—
212C23—21no—
213C23—22no—
214C23—41no—
215C23—41no—
216C23—42no—
217C23—81no—
218C23—81no—
219C23#13712yes—
220C23C22×D411yescharacteristic
221Q8—24no—
222Q8—42no—
223Q8—42no—
224Q8—81no—
225D4—42no—
226D4—44no—
227D4—42no—
228D4—42no—
229D4—81no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—82no—
260C2×C4—81no—
261C2×C4—82no—
262C2×C4—81no—
263C2×C4#12011yescharacteristic
264C2×C4#12011yescharacteristic
265C2×C4#12012yes—
266C2×C4C22×D412yes—
267C8—42no—
268C8—42no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—81no—
279C22—162no—
280C22C2×#12011yescharacteristic
281C22C2×#13711yescharacteristic
282C22#57511yescharacteristic
283C4—21no—
284C4—21no—
285C4—22no—
286C4—22no—
287C4—42no—
288C4—41no—
289C4—42no—
290C4—41no—
291C4—42no—
292C4—42no—
293C4—41no—
294C4—42no—
295C4—81no—
296C4—81no—
297C2—21no—
298C2—21no—
299C2—42no—
300C2—82no—
301C2#300711yescharacteristic
302C2#301811yescharacteristic
303C2#302511yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#534C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3117C2semidirect
3. #2396C2semidirect
4. #1615C2semidirect
5. #1614C2semidirect
7. #1603C2semidirect
8. #1602C2semidirect
10. #1509C2semidirect
11. #1464C2semidirect
24. C2×C4×Q8C22semidirect
26. #484C22semidirect
30. #451C22semidirect
34. #432C22semidirect
35. #400C22semidirect
39. #395C22semidirect
42. C4×C4⋊C4C22semidirect
44. #328C22semidirect
45. #328C22semidirect
48. #324C22semidirect

Maximal subgroups

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

subgroupindex
#34912
#31172
#23962
#16152
#16142
#16132
#16032
#16022
#16012
#15092
#14642