Finite Group Databasebeta

← #18665

Subgroups — #18665

Order 256.

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

normal subgroups only

This group has 463 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#3501C211yesmaximal, characteristic
2#3249C211yesmaximal, characteristic, complemented
3#3170C211yesmaximal, characteristic, complemented
4#3077C211yesmaximal, characteristic
5#1649C212yesmaximal, complemented
6#1644C212yesmaximal, complemented
7#1637C212yesmaximal, complemented
8#1632C212yesmaximal, complemented
9#1550C211yesmaximal, characteristic, complemented
10#1493C211yesmaximal, characteristic, complemented
11C2×D4⋊C8C211yesmaximal, characteristic
12#554—21no—
13#554—22no—
14#554—22no—
15#554C2212yes—
16D4×Q8—21no—
17D4×Q8C2212yes—
18#544—21no—
19#543—22no—
20#533C2211yescharacteristic
21#533C2211yescharacteristic
22#533C2211yescharacteristic
23C2×C22⋊Q8—22no—
24#518—21no—
25C2×C4×D4C2211yescharacteristic
26#500C2211yescharacteristic, complemented
27#494C2211yescharacteristic, complemented
28#487C2211yescharacteristic, complemented
29#487C2211yescharacteristic, complemented
30#478—22no—
31#477—21no—
32#477C2212yescomplemented
33#476C2212yes—
34#474C2212yes—
35#467—22no—
36#467—22no—
37#462—21no—
38#462C2212yescomplemented
39C4×Q16—22no—
40#428—21no—
41#423C2211yescharacteristic
42C2×C4⋊C8C2211yescharacteristic
43#418—21no—
44C2×C4×C8C2211yescharacteristic
45#333C2212yescomplemented
46#332C2212yescomplemented
47Q8⋊C8C2212yescomplemented
48Q8⋊C8C2212yescomplemented
49D4⋊C8C2212yes—
50D4⋊C8C2212yes—
51C22×Q8—22no—
52C22×D4D411yescharacteristic
53C23×C4—21no—
54C2×Q16—22no—
55C2×Q16—22no—
56C2×M16—21no—
57C22×C8—21no—
58C22×C8—21no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8D411yescharacteristic
62C4⋊Q8D411yescharacteristic
63C4⋊Q8C2311yescharacteristic
64C4⋊Q8C2311yescharacteristic
65C4⋊Q8C2311yescharacteristic
66C4⋊Q8C2311yescharacteristic
67#125—22no—
68#125—22no—
69#125—21no—
70#125—21no—
71#125—21no—
72C22⋊Q8—24no—
73C22⋊Q8—24no—
74C22⋊Q8—24no—
75C22⋊Q8—24no—
76C22⋊Q8—22no—
77C22⋊Q8—24no—
78C22⋊Q8—21no—
79C22⋊Q8—21no—
80C22⋊Q8—22no—
81C22⋊Q8—24no—
82C22⋊Q8—21no—
83C22⋊Q8—21no—
84C22⋊Q8D412yes—
85C22⋊Q8D412yes—
86C4×Q8—21no—
87C4×Q8—21no—
88C4×Q8—21no—
89C4×Q8—21no—
90C4×Q8D412yes—
91C4×Q8C2312yes—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4D412yes—
99C4×D4C2312yes—
100#117—21no—
101#117—21no—
102#117—21no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4D411yescharacteristic
106C2×C22⋊C4—24no—
107C2×C22⋊C4—21no—
108C2×C42C2311yescharacteristic
109#107—22no—
110#107—22no—
111#107—22no—
112#106—22no—
113C4⋊C8—21no—
114C4⋊C8—21no—
115C4⋊C8C2312yes—
116C4⋊C8C2312yes—
117#103—22no—
118#103—22no—
119#103—22no—
120#103—22no—
121#102—22no—
122#102—22no—
123C4×C8—21no—
124C4×C8C2311yescharacteristic
125C4×C8C2311yescharacteristic
126C24—21no—
127C2×Q8—24no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—82no—
133C2×Q8—82no—
134C2×Q8C2×D412yes—
135C2×Q8C2×D412yes—
136C2×D4—21no—
137C2×D4—22no—
138C2×D4Q1614yescomplemented
139C2×D4C2×D412yes—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—81no—
146C22×C4—81no—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150Q16—44no—
151Q16—82no—
152M16—42no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—42no—
158C2×C8—81no—
159C4⋊C4—24no—
160C4⋊C4—24no—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4C2×D412yes—
180C4⋊C4C2×D412yes—
181C4⋊C4C2×D412yes—
182C22⋊C4—24no—
183C22⋊C4—24no—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—22no—
188C22⋊C4—41no—
189C42—22no—
190C42—22no—
191C42—41no—
192C42C2×D411yescharacteristic
193C42C2×D411yescharacteristic
194C42C2×D411yescharacteristic
195C42C2411yescharacteristic
196C23—22no—
197C23—24no—
198C23—81no—
199C23—81no—
200C23#12011yescharacteristic
201Q8—22no—
202Q8—22no—
203Q8—42no—
204Q8—42no—
205Q8—41no—
206Q8—41no—
207Q8—41no—
208Q8—41no—
209Q8—81no—
210Q8—84no—
211D4—22no—
212D4—24no—
213D4C2×Q1614yes—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—82no—
227C2×C4—81no—
228C2×C4—84no—
229C2×C4—84no—
230C2×C4#12011yescharacteristic
231C2×C4#12011yescharacteristic
232C2×C4#12011yescharacteristic
233C2×C4C2×Q1612yes—
234C2×C4#13712yes—
235C2×C4C22×D411yescharacteristic
236C2×C4C22×D411yescharacteristic
237C2×C4C22×D411yescharacteristic
238C8—22no—
239C8—41no—
240C8—42no—
241C8—42no—
242C22—21no—
243C22—21no—
244C22—24no—
245C22—42no—
246C22—42no—
247C22—84no—
248C22#45112yes—
249C22C2×#12011yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—42no—
254C4—42no—
255C4—42no—
256C4—42no—
257C4—42no—
258C4—81no—
259C4—81no—
260C4#45112yes—
261C4C22×Q1611yescharacteristic
262C4C2×#13711yescharacteristic
263C4#57611yescharacteristic
264C4#57711yescharacteristic
265C2—22no—
266C2—44no—
267C2C2×#45111yescharacteristic
268C2#301211yescharacteristic
269C2#301811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC22×D4D4
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. #3170C2semidirect
5. #1649C2semidirect
6. #1644C2semidirect
7. #1637C2semidirect
8. #1632C2semidirect
9. #1550C2semidirect
10. #1493C2semidirect
26. #500C22semidirect
27. #494C22semidirect
28. #487C22semidirect
29. #487C22semidirect
32. #477C22semidirect
38. #462C22semidirect
45. #333C22semidirect
46. #332C22semidirect
47. Q8⋊C8C22semidirect
48. Q8⋊C8C22semidirect
138. C2×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#35012
#32492
#31702
#30772
#16492
#16442
#16372
#16322
#15502
#14932
C2×D4⋊C82