Finite Group Databasebeta

← #18889

Subgroups — #18889

Order 256.

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

normal subgroups only

This group has 455 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#3511C211yesmaximal, characteristic
2#3246C211yesmaximal, characteristic, complemented
3#3230C211yesmaximal, characteristic, complemented
4#3076C211yesmaximal, characteristic, complemented
5#1650C212yesmaximal, complemented
6#1647C212yesmaximal, complemented
7#1638C212yesmaximal, complemented
8#1635C212yesmaximal, complemented
9C2×#332C211yesmaximal, characteristic, complemented
10#1480C211yesmaximal, characteristic, complemented
11#1477C211yesmaximal, characteristic, complemented
12#556C2212yes—
13#555—21no—
14#554—21no—
15#553—22no—
16#548—22no—
17#547C2212yes—
18#542—22no—
19#537—21no—
20#533—21no—
21#533C2211yescharacteristic
22#532—21no—
23#532C2211yescharacteristic
24C2×C4⋊Q8C2211yescharacteristic
25#520—21no—
26C4×#41C2211yescharacteristic
27#500C2212yes—
28#487C2212yes—
29#484—22no—
30#479—21no—
31#478—21no—
32#477—22no—
33#477C2211yescharacteristic
34#476C2211yescharacteristic
35#476C2212yes—
36#475—21no—
37#475C2211yescharacteristic
38#474C2211yescharacteristic
39#465—22no—
40#461C2212yes—
41#460—21no—
42#431C2211yescharacteristic
43#427—21no—
44C2×C4⋊C8C2211yescharacteristic
45C2×C4⋊C8C2211yescharacteristic
46#416—21no—
47#332C2212yes—
48#332C2212yes—
49Q8⋊C8C2212yes—
50D4⋊C8C2212yes—
51D4⋊C8C2212yes—
52D4⋊C8C2212yes—
53C2×#41—22no—
54C2×#41—21no—
55C2×#41D411yescharacteristic
56C22×Q8—21no—
57C2×SD16—22no—
58C22×C8—21no—
59C22×C8—21no—
60C4⋊Q8—21no—
61C4⋊Q8—21no—
62C4⋊Q8D412yes—
63C4⋊Q8D411yescharacteristic
64C4⋊Q8C2311yescharacteristic
65C4⋊Q8C2311yescharacteristic
66C4⋊Q8C2312yes—
67#127D411yescharacteristic
68#126—24no—
69#125—22no—
70#125—21no—
71#124—24no—
72#124—22no—
73#124—21no—
74#123—24no—
75C22⋊Q8—22no—
76C22⋊Q8—24no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81#121—22no—
82#121—22no—
83#121—24no—
84#121—22no—
85#121—21no—
86#121D412yes—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×Q8—21no—
90C4×Q8—21no—
91C4×Q8D411yescharacteristic
92C4×Q8C2311yescharacteristic
93C4×D4—22no—
94C4×D4—24no—
95C4×D4—22no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4D411yescharacteristic
100C4×D4C2311yescharacteristic
101C4×D4C2312yes—
102#117—22no—
103#117—22no—
104#117—21no—
105#117—21no—
106#117D411yescharacteristic
107C2×C4⋊C4—21no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4D412yes—
110C2×C42—21no—
111C2×C42C2311yescharacteristic
112#107—21no—
113#107—21no—
114#107—24no—
115#106—21no—
116#106—21no—
117#106—22no—
118C4⋊C8—21no—
119C4⋊C8—21no—
120C4⋊C8—21no—
121C4⋊C8C2311yescharacteristic
122C4⋊C8C2311yescharacteristic
123C4⋊C8C2312yes—
124#103—21no—
125#103—21no—
126#103—22no—
127#103—24no—
128#102—21no—
129#102—21no—
130#102—22no—
131C22⋊C8—21no—
132C4×C8C2312yes—
133#41—22no—
134#41—22no—
135#41—42no—
136#41—42no—
137#41—82no—
138#41—82no—
139C2×Q8—21no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—81no—
145C2×Q8—81no—
146C2×Q8C2×D411yescharacteristic
147C2×D4—21no—
148C2×D4—22no—
149C2×D4—24no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4C2×D411yescharacteristic
153C2×D4C2×D412yes—
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163SD16—82no—
164C2×C8—22no—
165C2×C8—21no—
166C2×C8—22no—
167C2×C8—21no—
168C2×C8—42no—
169C2×C8—81no—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4C2×D412yes—
185C4⋊C4C2×D414yes—
186C4⋊C4C2×D411yescharacteristic
187C4⋊C4C2×D411yescharacteristic
188C22⋊C4—24no—
189C22⋊C4—24no—
190C22⋊C4—24no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—22no—
194C22⋊C4—21no—
195C22⋊C4—22no—
196C22⋊C4—41no—
197C42—24no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—41no—
202C42C2×D411yescharacteristic
203C42C2×D411yescharacteristic
204C42C2×D411yescharacteristic
205C42C2411yescharacteristic
206C23—21no—
207C23—22no—
208C23#12011yescharacteristic
209Q8—21no—
210Q8—21no—
211Q8—42no—
212Q8—42no—
213Q8—42no—
214Q8—42no—
215Q8—41no—
216Q8—41no—
217Q8—82no—
218D4—21no—
219D4—21no—
220D4—22no—
221D4—22no—
222D4—42no—
223D4—42no—
224D4—44no—
225D4—42no—
226D4—84no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—24no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—82no—
244C2×C4—84no—
245C2×C4#12011yescharacteristic
246C2×C4#12011yescharacteristic
247C2×C4#12011yescharacteristic
248C2×C4#13512yes—
249C2×C4#13712yes—
250C2×C4C22×D411yescharacteristic
251C2×C4C22×D411yescharacteristic
252C2×C4C22×D411yescharacteristic
253C8—42no—
254C8—41no—
255C8—41no—
256C8—42no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—42no—
261C22—41no—
262C22—42no—
263C22—41no—
264C22—42no—
265C22#45212yes—
266C22C2×#12011yescharacteristic
267C4—21no—
268C4—21no—
269C4—21no—
270C4—41no—
271C4—42no—
272C4—41no—
273C4—44no—
274C4—41no—
275C4—81no—
276C4—81no—
277C4#45212yes—
278C4C2×#13511yescharacteristic
279C4C2×#13711yescharacteristic
280C4#57611yescharacteristic
281C4#57711yescharacteristic
282C2—22no—
283C2—81no—
284C2—82no—
285C2C2×#45211yescharacteristic
286C2#301211yescharacteristic
287C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3511C2
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
2. #3246C2semidirect
3. #3230C2semidirect
4. #3076C2semidirect
5. #1650C2semidirect
6. #1647C2semidirect
7. #1638C2semidirect
8. #1635C2semidirect
9. C2×#332C2semidirect
10. #1480C2semidirect
11. #1477C2semidirect

Maximal subgroups

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

subgroupindex
#35112
#32462
#32302
#30762
#16502
#16472
#16382
#16352
C2×#3322
#14802
#14772