Finite Group Databasebeta

← #18589

Subgroups — #18589

Order 256.

439 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 117 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#3523C211yesmaximal, characteristic
2#3218C211yesmaximal, characteristic
3#3165C211yesmaximal, characteristic, complemented
4C2×#475C211yesmaximal, characteristic, complemented
5#1653C212yesmaximal, complemented
6#1651C212yesmaximal, complemented
7#1650C212yesmaximal, complemented
8#1648C212yesmaximal, complemented
9#1552C211yesmaximal, characteristic, complemented
10#1491C211yesmaximal, characteristic, complemented
11#1483C211yesmaximal, characteristic
12#557—21no—
13#557C2211yescharacteristic
14#557C2211yescharacteristic
15#556—21no—
16#556C2211yescharacteristic
17#552—22no—
18#552—21no—
19D4×Q8—21no—
20D4×Q8C2211yescharacteristic
21#541—22no—
22#541—21no—
23#533—21no—
24#533C2211yescharacteristic
25C2×C4⋊Q8C2211yescharacteristic
26#529—21no—
27#529C2211yescharacteristic
28#519—21no—
29C2×C4×Q8C2211yescharacteristic
30#501—22no—
31#500C2212yes—
32#498C2212yes—
33#497—21no—
34#491—21no—
35#475—21no—
36#475—21no—
37#475C2212yes—
38#475C2212yes—
39#462C2212yes—
40#461C2212yes—
41#451—21no—
42#448—21no—
43#432C2211yescharacteristic
44#428—22no—
45C2×#106—21no—
46C2×C4⋊C8C2211yescharacteristic
47C2×#103—21no—
48C4×M16C2211yescharacteristic
49#332C2212yes—
50#332C2212yes—
51Q8⋊C8C2212yes—
52Q8⋊C8C2212yes—
53Q8⋊C8C2212yes—
54D4⋊C8C2212yes—
55C22×Q8—21no—
56C22×Q8—21no—
57C22×Q8D411yescharacteristic
58C2×Q16—22no—
59C2×SD16—22no—
60C2×M16—21no—
61C22×C8—21no—
62C4⋊Q8—22no—
63C4⋊Q8—21no—
64C4⋊Q8—21no—
65C4⋊Q8—21no—
66C4⋊Q8—21no—
67C4⋊Q8—21no—
68C4⋊Q8—21no—
69C4⋊Q8C2312yes—
70C4⋊Q8C2311yescharacteristic
71C4⋊Q8C2311yescharacteristic
72#126—22no—
73#126—22no—
74#126—22no—
75#126—22no—
76#126—21no—
77#125—22no—
78#125—22no—
79#125—22no—
80#125—21no—
81#125—22no—
82#125—22no—
83#125—21no—
84#125D411yescharacteristic
85#125D412yes—
86#124—21no—
87#124D411yescharacteristic
88#123—22no—
89#123—21no—
90C22⋊Q8—22no—
91C22⋊Q8—22no—
92C22⋊Q8—21no—
93C22⋊Q8—21no—
94C22⋊Q8—21no—
95C22⋊Q8—21no—
96C22⋊Q8—22no—
97C22⋊Q8D411yescharacteristic
98#121D411yescharacteristic
99C4×Q8—22no—
100C4×Q8—22no—
101C4×Q8—21no—
102C4×Q8—21no—
103C4×Q8—21no—
104C4×Q8—21no—
105C4×Q8—21no—
106C4×Q8—21no—
107C4×Q8—21no—
108C4×Q8—21no—
109C4×Q8—21no—
110C4×Q8D411yescharacteristic
111C4×Q8D411yescharacteristic
112C4×Q8C2311yescharacteristic
113C4×Q8C2311yescharacteristic
114C4×Q8C2311yescharacteristic
115C4×D4—22no—
116C4×D4—21no—
117C4×D4—21no—
118C4×D4C2311yescharacteristic
119#117—22no—
120#117—22no—
121#117—22no—
122#117—21no—
123#117—21no—
124#117D412yes—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4D411yescharacteristic
129C2×C42—21no—
130C2×C42C2311yescharacteristic
131#107—24no—
132#106—22no—
133#106—22no—
134#106—24no—
135C4⋊C8—21no—
136C4⋊C8C2312yes—
137C4⋊C8C2312yes—
138#103—22no—
139#103—22no—
140#103—22no—
141#103—22no—
142C22⋊C8—21no—
143#97—21no—
144#97—21no—
145C4×C8C2312yes—
146C2×Q8—22no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—41no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—41no—
156C2×Q8—82no—
157C2×Q8C2×D411yescharacteristic
158C2×Q8C2×D411yescharacteristic
159C2×Q8C2×D411yescharacteristic
160C2×D4—21no—
161C2×D4C2×D411yescharacteristic
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—41no—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170Q16—82no—
171SD16—82no—
172M16—24no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—42no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C4⋊C4C2×D411yescharacteristic
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D412yes—
207C4⋊C4C2×D412yes—
208C22⋊C4—22no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—21no—
212C22⋊C4—22no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C42—22no—
218C42—22no—
219C42—22no—
220C42—21no—
221C42—21no—
222C42—22no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42C2×D411yescharacteristic
227C42C2×D411yescharacteristic
228C42C2×D411yescharacteristic
229C42C2411yescharacteristic
230C23—21no—
231C23#12011yescharacteristic
232Q8—21no—
233Q8—21no—
234Q8—21no—
235Q8—22no—
236Q8—42no—
237Q8—44no—
238Q8—41no—
239Q8—41no—
240Q8—42no—
241Q8—42no—
242Q8—42no—
243Q8—82no—
244Q8#13712yes—
245D4—22no—
246D4—42no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4#12011yescharacteristic
268C2×C4#12011yescharacteristic
269C2×C4#12011yescharacteristic
270C2×C4C22×D411yescharacteristic
271C2×C4C22×D411yescharacteristic
272C2×C4C22×D411yescharacteristic
273C8—24no—
274C8—42no—
275C8—42no—
276C22—21no—
277C22—21no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22#45612yes—
282C22C2×#12011yescharacteristic
283C4—21no—
284C4—22no—
285C4—21no—
286C4—21no—
287C4—21no—
288C4—41no—
289C4—42no—
290C4—41no—
291C4—41no—
292C4—42no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—82no—
297C4C2×#13711yescharacteristic
298C4#57511yescharacteristic
299C4#57712yes—
300C2—22no—
301C2—81no—
302C2#300711yescharacteristic
303C2#301011yescharacteristic
304C2C2×#45611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #3165C2semidirect
4. C2×#475C2semidirect
5. #1653C2semidirect
6. #1651C2semidirect
7. #1650C2semidirect
8. #1648C2semidirect
9. #1552C2semidirect
10. #1491C2semidirect

Maximal subgroups

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

subgroupindex
#35232
#32182
#31652
C2×#4752
#16532
#16512
#16502
#16482
#15522
#14912
#14832