Finite Group Databasebeta

← #16786909

Subgroups — #16786909

Order 3328.

428 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 92 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#13130079C212yesmaximal
2#13129779C212yesmaximal
3#13129767C212yesmaximal
4#13117405C211yesmaximal, characteristic
5C2×Q8×Dic13C2211yescharacteristic
6C2×#88198C411yescharacteristic
7C2×#88141C411yescharacteristic
8#11719119—22no—
9C13×#391C411yescharacteristic
10#11718637C2211yescharacteristic
11C42×Dic13C411yescharacteristic
12#11718590C2211yescharacteristic
13#11718310—22no—
14#11718308—22no—
15#11718285—22no—
16#11718285C2214yes—
17C22×Q8×C13C2×C411yescharacteristic
18C22×Dic26C2×C411yescharacteristic
19C2×C13×C4⋊C4C2×C411yescharacteristic
20C2×C42×C13C2×C411yescharacteristic
21Q8×Dic13—41no—
22Q8×Dic13—41no—
23#88198—22no—
24#88198C814yescomplemented
25C2×C4⋊Dic13D412yes—
26C2×C4⋊Dic13C2311yescharacteristic
27C2×#6101—22no—
28C2×#6101—21no—
29C2×#6101—22no—
30C2×C4×Dic13—21no—
31C2×C4×Dic13—21no—
32C2×C4×Dic13—21no—
33C2×C4×Dic13C2×C411yescharacteristic
34C2×C4×Dic13C2×C411yescharacteristic
35C2×C4×Dic13D412yes—
36#88141—41no—
37#88141—41no—
38C22⋊#1057—24no—
39C4⋊#1057—42no—
40C4×#1057—42no—
41C13×#95—22no—
42#88071—22no—
43#11478—131nomaximal, Sylow
44C2×Q8×C13—41no—
45C2×Q8×C13—41no—
46C2×Q8×C13C2×C812yes—
47C2×Q8×C13M1612yes—
48C22×C4×C13—21no—
49C22×C4×C13—21no—
50C22×C4×C13C22⋊C412yes—
51C22×C4×C13C22×C411yescharacteristic
52C22×Dic13—21no—
53C22×Dic13—21no—
54C22×Dic13C22⋊C411yescharacteristic
55C22×Dic13C22⋊C411yescharacteristic
56C22×Dic13C2×D412yes—
57C2×Dic26—21no—
58C2×Dic26—41no—
59C2×Dic26—41no—
60C2×Dic26C2×C812yes—
61C2×#1057—44no—
62C13×C4⋊C4—41no—
63C42×C13—41no—
64C4⋊Dic13—42no—
65C4⋊Dic13—41no—
66#6101—28no—
67#6101—42no—
68#6101—44no—
69C4×Dic13—24no—
70C4×Dic13—21no—
71C4×Dic13—22no—
72C4×Dic13—22no—
73C4×Dic13—21no—
74C4×Dic13—21no—
75C4×Dic13—42no—
76C4×Dic13—41no—
77C4×Dic13—41no—
78C4×Dic13C2×C812yes—
79C4×C4⋊Q8—131no—
80#1461—132no—
81#1325—132no—
82#1319—132no—
83C23×C13C2×C22⋊C411yescharacteristic
84Q8×C13—42no—
85Q8×C13—42no—
86C2×C4×C13—21no—
87C2×C4×C13—22no—
88C2×C4×C13—21no—
89C2×C4×C13—22no—
90C2×C4×C13—22no—
91C2×C4×C13—41no—
92C2×C4×C13—41no—
93C2×C4×C13—42no—
94C2×C4×C13—41no—
95C2×C4×C13C22⋊C814yes—
96C2×C4×C13C22×C811yescharacteristic
97C2×C4×C13C2×M1611yescharacteristic
98C2×Dic13—22no—
99C2×Dic13—22no—
100C2×Dic13—22no—
101C2×Dic13—22no—
102C2×Dic13—22no—
103C2×Dic13—21no—
104C2×Dic13—22no—
105C2×Dic13—41no—
106C2×Dic13—41no—
107C2×Dic13—41no—
108C2×Dic13—42no—
109C2×Dic13#9911yescharacteristic
110C2×Dic13#9911yescharacteristic
111Dic26—42no—
112Dic26—42no—
113#1057—84no—
114C2×C4⋊Q8—131no—
115C2×C4×Q8—131no—
116C2×C4×Q8—131no—
117#431—262no—
118#391—131no—
119#391#26111yescharacteristic
120#389—262no—
121#384—131no—
122C4×C4⋊C4—262no—
123C43—131no—
124#324—134no—
125#324—262no—
126C22×C13—21no—
127C22×C13—21no—
128C22×C13C2×C22⋊C811yescharacteristic
129C22×C13C2×#9911yescharacteristic
130C22×C13#41311yescharacteristic
131C4×C13—21no—
132C4×C13—21no—
133C4×C13—24no—
134C4×C13—42no—
135C4×C13—81no—
136Dic13—24no—
137Dic13—21no—
138Dic13—42no—
139Dic13—41no—
140Dic13—42no—
141Dic13#35612yes—
142C22×Q8—131no—
143C22×Q8C2×#26111yescharacteristic
144C4⋊Q8—134no—
145C4⋊Q8—262no—
146C4×Q8—521no—
147C4×Q8—521no—
148C4×Q8—521no—
149C4×Q8—521no—
150C2×C4⋊C4—132no—
151C2×C4⋊C4—131no—
152C2×C4⋊C4—262no—
153C2×C4⋊C4—261no—
154C2×C4⋊C4—262no—
155C2×C4⋊C4C2×#26111yescharacteristic
156C2×C42—131no—
157C2×C42—132no—
158C2×C42—131no—
159C2×C42—261no—
160C2×C42—261no—
161C2×C42—261no—
162C2×C42C2×#26111yescharacteristic
163C4⋊C8—522no—
164C22⋊C8—264no—
165C4×C8—522no—
166#95—22no—
167#95—262no—
168C2×C13—21no—
169C2×C13—21no—
170C2×C13#145911yescharacteristic
171C2×C13#212611yescharacteristic
172C2×C13C2×#35611yescharacteristic
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—132no—
176C2×Q8—261no—
177C2×Q8—521no—
178C2×Q8—521no—
179C2×Q8D13⋊C812yes—
180C2×Q8#611812yes—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—131no—
184C22×C4—132no—
185C22×C4—131no—
186C22×C4—261no—
187C22×C4—261no—
188C22×C4C22⋊#26112yes—
189C22×C4C22×#26111yescharacteristic
190C2×C8—524no—
191C4⋊C4—41no—
192C4⋊C4—268no—
193C4⋊C4—521no—
194C4⋊C4—522no—
195C4⋊C4—522no—
196C4⋊C4—524no—
197C42—41no—
198C42—132no—
199C42—262no—
200C42—262no—
201C42—261no—
202C42—261no—
203C42—264no—
204C42—261no—
205C42—521no—
206C42—521no—
207C42—522no—
208C13#1147811yescharacteristic, complemented, Sylow
209C23C2×C22⋊#26111yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—522no—
213Q8—522no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—131no—
224C2×C4—131no—
225C2×C4—262no—
226C2×C4—262no—
227C2×C4—262no—
228C2×C4—262no—
229C2×C4—262no—
230C2×C4—261no—
231C2×C4—262no—
232C2×C4—521no—
233C2×C4—521no—
234C2×C4—521no—
235C2×C4—522no—
236C2×C4#8811114yes—
237C2×C4C2×D13⋊C811yescharacteristic
238C2×C4C2×#611811yescharacteristic
239C8—1044no—
240C22—21no—
241C22—21no—
242C22C2×#8811111yescharacteristic
243C22D13⋊#9911yescharacteristic
244C22#1171914811yescharacteristic
245C4—24no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—81no—
250C4—132no—
251C4—264no—
252C4—261no—
253C4—522no—
254C4—522no—
255C4—521no—
256C2—21no—
257C2—21no—
258C2#1313007411yescharacteristic
259C2#1313110811yescharacteristic
260C2D13⋊#35611yescharacteristic
subgroupG/H
centerC22D13⋊#99
commutatorC2×C4×C13C22×C8
FrattiniC22×C4C22×#261
FittingC13×#391C4
socleC22×C13C2×#99
prime radicalC23×C13C2×C22⋊C4
cosocleC2×C4⋊Dic13C23
hypercenter#391#261
hypocenterC13#11478
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #391

Complements

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

NH ≅ G/Ntype
24. #88198C8semidirect
208. C13#11478semidirect

Maximal subgroups

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

subgroupindex
#131300792
#131297792
#131297672
#131174052
#1147813