Finite Group Databasebeta

← #16780715

Subgroups — #16780715

Order 3328.

516 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 134 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#13130621C212yesmaximal
2C2×#11718303C212yesmaximal
3#13129714C212yesmaximal
4C4×#88198C211yesmaximal, characteristic
5C2×C4×Q8×C13C411yescharacteristic
6C2×#88198C2211yescharacteristic
7C2×C4×Dic26C411yescharacteristic
8C2×#88112C2214yes—
9C2×#88110—22no—
10#11718891C411yescharacteristic
11#11718637C2211yescharacteristic
12#11718635C411yescharacteristic
13#11718633—22no—
14C4×#6101—22no—
15C42×Dic13C2211yescharacteristic
16#11718303C418yescomplemented
17#11718290—22no—
18C22×Q8×C13C2×C411yescharacteristic
19C22×Dic26C2×C411yescharacteristic
20C22×#1057—24no—
21C4×Q8×C13—42no—
22C2×C13×C4⋊C4—21no—
23C2×C13×C4⋊C4C2×C411yescharacteristic
24C2×C42×C13—21no—
25C2×C42×C13C2×C411yescharacteristic
26#88198—22no—
27#88198C2×C414yes—
28C2×C4⋊Dic13C2×C411yescharacteristic
29C2×#6101—22no—
30C2×#6101—21no—
31C2×#6101—22no—
32C2×C4×Dic13—21no—
33C2×C4×Dic13C2×C411yescharacteristic
34C2×C4×Dic13D412yes—
35C2×C4×Dic13D412yes—
36C2×C4×Dic13C2311yescharacteristic
37C4×Dic26—42no—
38#88112—24no—
39#88112C2×C418yes—
40#88110—44no—
41#88071—22no—
42#88071—22no—
43#8232—131nomaximal, Sylow
44C2×Q8×C13—21no—
45C2×Q8×C13—41no—
46C2×Q8×C13—41no—
47C2×Q8×C13C4212yes—
48C22×C4×C13—21no—
49C22×C4×C13—21no—
50C22×C4×C13C22⋊C411yescharacteristic
51C22×C4×C13C22⋊C411yescharacteristic
52C22×C4×C13C22×C411yescharacteristic
53C22×Dic13—21no—
54C22×Dic13—21no—
55C22×Dic13C22⋊C412yes—
56C22×Dic13C2×D412yes—
57C2×Dic26—21no—
58C2×Dic26—41no—
59C2×Dic26—41no—
60C2×Dic26C4212yes—
61C2×#1057—28no—
62C2×#1057—44no—
63C2×#1057—44no—
64C13×C4⋊C4—41no—
65C13×C4⋊C4—42no—
66C42×C13—22no—
67C42×C13—42no—
68C4⋊Dic13—41no—
69#6101—28no—
70#6101—42no—
71#6101—42no—
72#6101—42no—
73C4×Dic13—22no—
74C4×Dic13—22no—
75C4×Dic13—24no—
76C4×Dic13—41no—
77C4×Dic13—42no—
78C4×Dic13C22×C412yes—
79C4×Dic13#4112yes—
80C4×C4⋊Q8—131no—
81#1767—132no—
82C2×#333—132no—
83#1296—132no—
84C23×C13C2×C22⋊C411yescharacteristic
85Q8×C13—42no—
86Q8×C13—42no—
87C2×C4×C13—21no—
88C2×C4×C13—21no—
89C2×C4×C13—22no—
90C2×C4×C13—22no—
91C2×C4×C13—21no—
92C2×C4×C13—22no—
93C2×C4×C13—41no—
94C2×C4×C13—41no—
95C2×C4×C13—41no—
96C2×C4×C13—42no—
97C2×C4×C13#10012yes—
98C2×C4×C13C2×C4211yescharacteristic
99C2×C4×C13#11711yescharacteristic
100C2×Dic13—22no—
101C2×Dic13—22no—
102C2×Dic13—22no—
103C2×Dic13—22no—
104C2×Dic13—41no—
105C2×Dic13—41no—
106C2×Dic13—41no—
107C2×Dic13—42no—
108C2×Dic13C4×D414yes—
109C2×Dic13AGL(1,ℤ/8)12yes—
110C2×Dic13#13714yes—
111C2×Dic13#13712yes—
112Dic26—42no—
113Dic26—42no—
114#1057—48no—
115C2×C4⋊Q8—131no—
116C2×C4×Q8—131no—
117C2×C4×Q8#26111yescharacteristic
118C2×C4⋊C8—134no—
119C2×#97—262no—
120#391—131no—
121#391—131no—
122#389—262no—
123#384—131no—
124C4×C4⋊C4—262no—
125C43—131no—
126#336—262no—
127#333—138no—
128C22×C13C4×C22⋊C411yescharacteristic
129C22×C13C2×#10011yescharacteristic
130C22×C13#41311yescharacteristic
131C22×C13#41712yes—
132C22×C13#41912yes—
133C4×C13—22no—
134C4×C13—21no—
135C4×C13—21no—
136C4×C13—41no—
137C4×C13—42no—
138C4×C13—81no—
139Dic13—42no—
140Dic13—42no—
141Dic13—81no—
142Dic13#44014yes—
143Dic13#44114yes—
144C22×Q8—131no—
145C22×Q8C2×#26111yescharacteristic
146C22×C8—264no—
147C4⋊Q8—134no—
148C4⋊Q8—262no—
149C4×Q8—42no—
150C4×Q8—522no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—131no—
153C2×C4⋊C4—262no—
154C2×C4⋊C4—262no—
155C2×C4⋊C4—261no—
156C2×C4⋊C4C2×#26111yescharacteristic
157C2×C42—21no—
158C2×C42—131no—
159C2×C42—132no—
160C2×C42—132no—
161C2×C42—131no—
162C2×C42—261no—
163C2×C42C2×#26111yescharacteristic
164C4⋊C8—138no—
165C4⋊C8—264no—
166#97—524no—
167#95—262no—
168#95—262no—
169C2×C13#155811yescharacteristic
170C2×C13#155912yes—
171C2×C13#156011yescharacteristic
172C2×C13C4×#10011yescharacteristic
173C2×C13#176312yes—
174C2×Q8—21no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—132no—
178C2×Q8—261no—
179C2×Q8—521no—
180C2×Q8—521no—
181C2×Q8C4×#26112yes—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—132no—
185C22×C4—132no—
186C22×C4—261no—
187C22×C4—261no—
188C22×C4C22⋊#26111yescharacteristic
189C22×C4C22⋊#26111yescharacteristic
190C22×C4C22×#26111yescharacteristic
191C2×C8—268no—
192C2×C8—524no—
193C2×C8—524no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—268no—
197C4⋊C4—521no—
198C4⋊C4—522no—
199C4⋊C4—522no—
200C4⋊C4—522no—
201C42—22no—
202C42—42no—
203C42—132no—
204C42—132no—
205C42—262no—
206C42—264no—
207C42—262no—
208C42—521no—
209C42—522no—
210C13#823211yescharacteristic, complemented, Sylow
211C23C2×C22⋊#26111yescharacteristic
212Q8—42no—
213Q8—42no—
214Q8—522no—
215Q8—522no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—134no—
227C2×C4—134no—
228C2×C4—132no—
229C2×C4—132no—
230C2×C4—262no—
231C2×C4—262no—
232C2×C4—262no—
233C2×C4—262no—
234C2×C4—521no—
235C2×C4—521no—
236C2×C4—521no—
237C2×C4—522no—
238C2×C4#8812112yes—
239C2×C4C2×C4×#26111yescharacteristic
240C2×C4#8823711yescharacteristic
241C8—528no—
242C22#1171913211yescharacteristic
243C22C4×C22⋊#26111yescharacteristic
244C22#1171915312yes—
245C22#1171916412yes—
246C22C2×#8812111yescharacteristic
247C4—21no—
248C4—21no—
249C4—22no—
250C4—42no—
251C4—41no—
252C4—81no—
253C4—134no—
254C4—134no—
255C4—522no—
256C4—522no—
257C4—1041no—
258C2#1313036911yescharacteristic
259C2#1313037112yes—
260C2#1313037311yescharacteristic
261C2C4×#8812111yescharacteristic
262C2#1313060812yes—
subgroupG/H
centerC23C2×C22⋊#261
commutatorC2×C4×C13C2×C42
FrattiniC22×C4C22×#261
FittingC2×C4×Q8×C13C4
socle, prime radicalC23×C13C2×C22⋊C4
cosocleC2×C4×Dic13C23
hypercenterC2×C4×Q8#261
hypocenterC13#8232
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
16. #11718303C4semidirect
210. C13#8232semidirect

Maximal subgroups

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

subgroupindex
#131306212
C2×#117183032
#131297142
C4×#881982
#823213