Finite Group Databasebeta

← #18885

Subgroups — #18885

Order 256.

444 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 129 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#3503C211yesmaximal, characteristic
2#3249C211yesmaximal, characteristic, complemented
3#3230C211yesmaximal, characteristic, complemented
4#3076C211yesmaximal, characteristic
5#1649C212yesmaximal, complemented
6#1648C212yesmaximal, complemented
7#1637C212yesmaximal, complemented
8#1636C212yesmaximal, complemented
9C2×#332C211yesmaximal, characteristic, complemented
10#1481C211yesmaximal, characteristic, complemented
11#1477C211yesmaximal, characteristic
12Q82—21no—
13#557C2212yes—
14#554—21no—
15#554—22no—
16D4×Q8—22no—
17D4×Q8C2212yes—
18#544—22no—
19#544—21no—
20#533—21no—
21#533C2211yescharacteristic
22#533C2211yescharacteristic
23C2×C4⋊Q8—21no—
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#477—21no—
34#477C2211yescharacteristic
35#477C2212yes—
36#476C2211yescharacteristic
37#475C2211yescharacteristic
38#474C2211yescharacteristic
39#467—22no—
40#462—21no—
41#462C2212yes—
42#431C2211yescharacteristic
43#427—21no—
44C2×C4⋊C8C2211yescharacteristic
45C2×C4⋊C8C2211yescharacteristic
46#416—21no—
47#332C2212yes—
48#332C2212yes—
49Q8⋊C8C2212yes—
50Q8⋊C8C2212yes—
51Q8⋊C8C2212yes—
52D4⋊C8C2212yes—
53C2×#41D411yescharacteristic
54C22×Q8—22no—
55C22×Q8—21no—
56C2×Q16—22no—
57C22×C8—21no—
58C22×C8—21no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—24no—
62C4⋊Q8—22no—
63C4⋊Q8—22no—
64C4⋊Q8—21no—
65C4⋊Q8—21no—
66C4⋊Q8D411yescharacteristic
67C4⋊Q8D412yes—
68C4⋊Q8D411yescharacteristic
69C4⋊Q8C2311yescharacteristic
70C4⋊Q8C2311yescharacteristic
71C4⋊Q8C2312yes—
72#125—22no—
73#125—24no—
74#125—22no—
75#125—21no—
76#125—21no—
77C22⋊Q8—22no—
78C22⋊Q8—24no—
79C22⋊Q8—24no—
80C22⋊Q8—21no—
81C22⋊Q8—22no—
82C22⋊Q8—24no—
83C22⋊Q8—21no—
84C22⋊Q8—21no—
85C22⋊Q8D412yes—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×Q8—21no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8D411yescharacteristic
93C4×Q8C2311yescharacteristic
94C4×Q8C2312yes—
95C4×D4—24no—
96C4×D4—22no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4D411yescharacteristic
100C4×D4C2311yescharacteristic
101#117—22no—
102#117—21no—
103#117—21no—
104#117—21no—
105#117D411yescharacteristic
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4D412yes—
110C2×C42—21no—
111C2×C42C2311yescharacteristic
112#107—21no—
113#107—21no—
114#107—22no—
115#107—24no—
116#106—21no—
117#106—21no—
118C4⋊C8—21no—
119C4⋊C8—21no—
120C4⋊C8—21no—
121C4⋊C8C2312yes—
122C4⋊C8C2311yescharacteristic
123C4⋊C8C2311yescharacteristic
124#103—21no—
125#103—21no—
126#103—22no—
127#103—22no—
128#103—24no—
129#102—21no—
130#102—21no—
131C22⋊C8—21no—
132C4×C8C2312yes—
133#41—22no—
134#41Q1614yescomplemented
135C2×Q8—24no—
136C2×Q8—21no—
137C2×Q8—22no—
138C2×Q8—22no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—81no—
142C2×Q8—81no—
143C2×Q8—82no—
144C2×Q8—82no—
145C2×Q8C2×D412yes—
146C2×Q8C2×D411yescharacteristic
147C2×D4—21no—
148C2×D4C2×D411yescharacteristic
149C22×C4—21no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157Q16—82no—
158C2×C8—22no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—22no—
162C2×C8—42no—
163C2×C8—81no—
164C4⋊C4—24no—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4C2×D414yes—
185C4⋊C4C2×D412yes—
186C4⋊C4C2×D411yescharacteristic
187C4⋊C4C2×D411yescharacteristic
188C22⋊C4—24no—
189C22⋊C4—22no—
190C22⋊C4—21no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—41no—
194C42—24no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—22no—
199C42—41no—
200C42C2×D411yescharacteristic
201C42C2×D411yescharacteristic
202C42C2×D411yescharacteristic
203C42C2411yescharacteristic
204C23—21no—
205C23#12011yescharacteristic
206Q8—21no—
207Q8—22no—
208Q8—22no—
209Q8—42no—
210Q8—42no—
211Q8—42no—
212Q8—42no—
213Q8—41no—
214Q8—41no—
215Q8—82no—
216Q8—84no—
217Q8C2×Q1612yes—
218D4—24no—
219D4—21no—
220D4C2×Q1612yes—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—82no—
237C2×C4—84no—
238C2×C4#12011yescharacteristic
239C2×C4#12011yescharacteristic
240C2×C4#12011yescharacteristic
241C2×C4C2×Q1612yes—
242C2×C4#13712yes—
243C2×C4C22×D411yescharacteristic
244C2×C4C22×D411yescharacteristic
245C2×C4C22×D411yescharacteristic
246C8—42no—
247C8—41no—
248C8—41no—
249C8—42no—
250C22—21no—
251C22—21no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22#45112yes—
256C22C2×#12011yescharacteristic
257C4—22no—
258C4—21no—
259C4—21no—
260C4—21no—
261C4—41no—
262C4—42no—
263C4—44no—
264C4—41no—
265C4—42no—
266C4—81no—
267C4—81no—
268C4#45112yes—
269C4C22×Q1611yescharacteristic
270C4C2×#13711yescharacteristic
271C4#57611yescharacteristic
272C4#57711yescharacteristic
273C2—22no—
274C2—42no—
275C2C2×#45111yescharacteristic
276C2#301211yescharacteristic
277C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC2×#41D4
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. #3249C2semidirect
3. #3230C2semidirect
5. #1649C2semidirect
6. #1648C2semidirect
7. #1637C2semidirect
8. #1636C2semidirect
9. C2×#332C2semidirect
10. #1481C2semidirect
134. #41Q16semidirect

Maximal subgroups

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

subgroupindex
#35032
#32492
#32302
#30762
#16492
#16482
#16372
#16362
C2×#3322
#14812
#14772