Finite Group Databasebeta

← #45665356

Subgroups — #45665356

Order 32896.

362 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#22551623C211yesmaximal, characteristic
2#30657311C211yesmaximal, characteristic
3C2×#22551520C211yesmaximal, characteristic
4#30656746C211yesmaximal, characteristic, complemented
5C2×#22551511C211yesmaximal, characteristic, complemented
6#30656497C211yesmaximal, characteristic, complemented
7C2×C4⋊Q8×C257C211yesmaximal, characteristic, complemented
8C2×#16861618C2211yescharacteristic
9#22551623—22no—
10#22551623—22no—
11C2×#16861608—21no—
12C2×#16861605—21no—
13C2×#16861602C2211yescharacteristic
14C2×#16861602C2211yescharacteristic
15#22551590—21no—
16#22551581—21no—
17#22551575—22no—
18#22551568—22no—
19#22551551C2211yescharacteristic
20#22551520—22no—
21#22551516—22no—
22#22551511—22no—
23C22×Q8×C257C2211yescharacteristic
24C22×Q8×C257C2211yescharacteristic
25C4⋊Q8×C257—22no—
26C4⋊Q8×C257—22no—
27C2×C4⋊C4×C257—22no—
28C2×C42×C257C2211yescharacteristic
29#16861618—21no—
30#16861618—21no—
31#16861618—21no—
32#16861618—21no—
33C2×Dic514D411yescharacteristic
34C2×D1028D411yescharacteristic
35C2×C22⋊D257—21no—
36C2×C4×D257—21no—
37#16861608—24no—
38#16861605—24no—
39#16861602—21no—
40#16861602—21no—
41#16861602—21no—
42#16861602D412yes—
43C2×#14523339—21no—
44C2×#14523339—21no—
45C4⋊Dic257D412yes—
46#16861597—24no—
47C22⋊Dic257—21no—
48#16861595—24no—
49C4×Dic257—21no—
50C2×Q8×C257—21no—
51C2×Q8×C257—21no—
52C2×Q8×C257—21no—
53C2×Q8×C257—21no—
54C2×Q8×C257—21no—
55C2×Q8×C257—21no—
56C2×Q8×C257—21no—
57C2×Q8×C257—21no—
58C2×Q8×C257—21no—
59C2×Q8×C257—21no—
60C2×Q8×C257—21no—
61C2×Q8×C257D412yes—
62C22×C4×C257—21no—
63C22×C4×C257—21no—
64C22×C4×C257D412yes—
65C22×C4×C257C2311yescharacteristic
66C4⋊C4×C257—28no—
67C42×C257—21no—
68C42×C257D412yes—
69C22×D257—21no—
70Dic514—21no—
71Dic514—21no—
72D1028—21no—
73D1028—21no—
74C22⋊D257—41no—
75C22⋊D257—41no—
76C2×Dic257—22no—
77C2×Dic257—21no—
78C4×D257—41no—
79C4×D257—41no—
80#14523339—22no—
81#14523339—41no—
82C23×C257C2×D411yescharacteristic
83Q8×C257—22no—
84Q8×C257—22no—
85Q8×C257—41no—
86Q8×C257—41no—
87Q8×C257—41no—
88Q8×C257—41no—
89Q8×C257—41no—
90Q8×C257—41no—
91C2×C4×C257—24no—
92C2×C4×C257—22no—
93C2×C4×C257—22no—
94C2×C4×C257—22no—
95C2×C4×C257—22no—
96C2×C4×C257—22no—
97C2×C4×C257—41no—
98C2×C4×C257—41no—
99C2×C4×C257C2×D411yescharacteristic
100C2×C4×C257C2×D411yescharacteristic
101C2×C4×C257C2×D411yescharacteristic
102C2×C4×C257C2×D411yescharacteristic
103C2×C4×C257C2×D411yescharacteristic
104C2×C4×C257#4111yescharacteristic
105C22×C257—22no—
106C22×C257#12011yescharacteristic
107C22×C257#12011yescharacteristic
108C22×C257#12111yescharacteristic
109C2×D257—41no—
110C2×D257—42no—
111C4×C257—24no—
112C4×C257—42no—
113C4×C257—42no—
114C4×C257#12011yescharacteristic
115C4×C257#12111yescharacteristic
116C4×C257#12111yescharacteristic
117C4×C257#12711yescharacteristic
118Dic257—41no—
119Dic257—42no—
120C2×C257—21no—
121C2×C257—21no—
122C2×C257#39211yescharacteristic
123C2×C257#45612yes—
124D257—81no—
125C257#200511yescharacteristic, complemented, Sylow
126#2005—2571nomaximal, Sylow
127C2×#137—2571no—
128#536—2571no—
129C2×C4⋊Q8D25711yescharacteristic, complemented
130C2×#104—2571no—
131#419—2571no—
132C2×#101—2571no—
133#339—2571no—
134C2×#41—2571no—
135C22×Q8C2×D25711yescharacteristic
136C22×Q8C2×D25711yescharacteristic
137#137—5142no—
138#137—5142no—
139C2×Q16—5141no—
140C2×SD16—5141no—
141C2×M16—2571no—
142C2×M16—2571no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—5141no—
146#124—5141no—
147#123—5142no—
148C22⋊Q8—5142no—
149#117—2571no—
150C2×C4⋊C4—22no—
151C2×C42C2×D25711yescharacteristic
152#104—5142no—
153#103—5142no—
154#101—5142no—
155#41—5141no—
156#41—5141no—
157#41—5141no—
158#41—5141no—
159C2×Q8—2571no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C22⋊D25712yes—
172C2×D4—2571no—
173C2×D4—5141no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—5141no—
177C22×C4C22⋊D25712yes—
178C22×C4C22×D25711yescharacteristic
179Q16—5144no—
180SD16—5144no—
181M16—2572no—
182M16—5141no—
183M16—5141no—
184M16—5141no—
185C2×C8—5141no—
186C2×C8—5141no—
187C4⋊C4—2572no—
188C4⋊C4—28no—
189C4⋊C4—5144no—
190C22⋊C4—5144no—
191C22⋊C4—5141no—
192C42—21no—
193C42—5141no—
194C42C22⋊D25712yes—
195C23—5141no—
196C23C2×C22⋊D25711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—5141no—
200Q8—5141no—
201Q8—41no—
202Q8—41no—
203Q8—41no—
204Q8—41no—
205Q8—41no—
206Q8—41no—
207D4—5141no—
208D4—5141no—
209D4—10281no—
210D4—10281no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—5142no—
218C2×C4—5141no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—10281no—
222C2×C4—10281no—
223C2×C4D4×D25711yescharacteristic
224C2×C4D4×D25711yescharacteristic
225C2×C4D4×D25711yescharacteristic
226C2×C4C2×C22⋊D25711yescharacteristic
227C2×C4C2×C22⋊D25711yescharacteristic
228C2×C4#1686161711yescharacteristic
229C8—5142no—
230C8—10281no—
231C22—22no—
232C22—10281no—
233C22—10282no—
234C22#2255156011yescharacteristic
235C22#2255156111yescharacteristic
236C22#2255156411yescharacteristic
237C4—24no—
238C4—42no—
239C4—42no—
240C4—10281no—
241C4—10282no—
242C4#2255156011yescharacteristic
243C4#2255156311yescharacteristic
244C4#2255156411yescharacteristic
245C4#2255158811yescharacteristic
246C2—21no—
247C2—21no—
248C2—20561no—
249C2#3065664511yescharacteristic
250C2#3065691912yes—
subgroupG/H
centerC22#22551560
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
FittingC2×C4⋊Q8×C257C2
socleC22×C257#120
prime radicalC2×#16861618C22
hypercenterC2×C4⋊Q8D257
hypocenterC257#2005
LCSG ▷ C22×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #30656746C2semidirect
5. C2×#22551511C2semidirect
6. #30656497C2semidirect
7. C2×C4⋊Q8×C257C2semidirect
125. C257#2005semidirect
129. C2×C4⋊Q8D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225516232
#306573112
C2×#225515202
#306567462
C2×#225515112
#306564972
C2×C4⋊Q8×C2572
#2005257