Finite Group Databasebeta

← #16784518

Subgroups — #16784518

Order 3328.

618 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 142 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#13130753C211yesmaximal, characteristic, complemented
2C42×#1057C211yesmaximal, characteristic, complemented
3C2×Q8⋊#1057C212yesmaximal, complemented
4C2×#11718243C212yesmaximal
5#13117379C211yesmaximal, characteristic
6C2×#88201C411yescharacteristic
7C2×Q8×Dic13C2211yescharacteristic
8C2×#88147C2211yescharacteristic
9C2×C4⋊#1057C2212yes—
10C2×C4×#1057—22no—
11C2×C4×#1057C2212yes—
12C13×#391C411yescharacteristic
13#11718638C411yescharacteristic
14#11718600C411yescharacteristic
15C42×Dic13C2211yescharacteristic
16#11718331—22no—
17#11718327—22no—
18Q8⋊#1057—24no—
19#11718290—22no—
20#11718243—24no—
21C22×Q8×C13C2×C411yescharacteristic
22C22×D52C2×C411yescharacteristic
23C22×C4×D13—21no—
24C22×#1057—22no—
25C22×#1057—22no—
26C2×C13×C4⋊C4C2×C411yescharacteristic
27C2×C42×C13C2×C411yescharacteristic
28#88201—22no—
29#88201C814yescomplemented
30Q8×Dic13—22no—
31Q8×Dic13—22no—
32C2×#6103—22no—
33C2×#6103—22no—
34C2×#6103—21no—
35C2×C4⋊Dic13—21no—
36C2×C4⋊Dic13C2×C411yescharacteristic
37C2×C4×Dic13—21no—
38C2×C4×Dic13C2×C411yescharacteristic
39C2×C4×Dic13D412yes—
40C2×C4×Dic13D412yes—
41C2×C4×Dic13C2311yescharacteristic
42#88147—22no—
43#88147—22no—
44C4⋊#1057—22no—
45C4⋊#1057—22no—
46C4×#1057—28no—
47C4×#1057—22no—
48C4×#1057D414yes—
49C13×#95—22no—
50#88071—22no—
51#9639—131nomaximal, Sylow
52C23×D13—21no—
53C2×Q8×C13—21no—
54C2×Q8×C13—22no—
55C2×Q8×C13—22no—
56C2×Q8×C13C2×C812yes—
57C22×C4×C13—21no—
58C22×C4×C13—21no—
59C22×C4×C13C22⋊C412yes—
60C22×C4×C13C22×C411yescharacteristic
61C22×Dic13—21no—
62C22×Dic13C22⋊C412yes—
63C22×Dic13C2×D411yescharacteristic
64C22×Dic13C2×D411yescharacteristic
65C2×D52—22no—
66C2×D52—22no—
67C2×D52—21no—
68C2×D52C2×C812yes—
69C2×C4×D13—41no—
70C2×C4×D13—41no—
71C2×C4×D13—44no—
72C2×#1057—28no—
73C2×#1057—24no—
74C2×#1057—42no—
75C2×#1057—44no—
76C13×C4⋊C4—22no—
77C42×C13—22no—
78#6103—28no—
79#6103—44no—
80#6103—42no—
81C4⋊Dic13—22no—
82C4⋊Dic13—42no—
83C4×Dic13—24no—
84C4×Dic13—24no—
85C4×Dic13—41no—
86C4×Dic13—41no—
87C4×Dic13C2×C812yes—
88C4×Dic13M1612yes—
89C4×Dic13C2×D412yes—
90C4×Dic13#4112yes—
91C4×#124—131no—
92#1831—131no—
93C42×C8—131no—
94C2×Q8⋊C8—132no—
95C2×D4⋊C8—132no—
96C23×C13C2×C22⋊C411yescharacteristic
97C22×D13—22no—
98C22×D13—41no—
99C22×D13—41no—
100C22×D13—42no—
101C22×D13—42no—
102Q8×C13—24no—
103Q8×C13—42no—
104C2×C4×C13—22no—
105C2×C4×C13—22no—
106C2×C4×C13—22no—
107C2×C4×C13—41no—
108C2×C4×C13—41no—
109C2×C4×C13—41no—
110C2×C4×C13—42no—
111C2×C4×C13#10414yes—
112C2×C4×C13C4×D412yes—
113C2×C4×C13C22×C811yescharacteristic
114C2×C4×C13C2×M1611yescharacteristic
115C2×C4×C13#13112yes—
116C2×Dic13—22no—
117C2×Dic13—22no—
118C2×Dic13—22no—
119C2×Dic13—41no—
120C2×Dic13—42no—
121C2×Dic13C22⋊C814yes—
122C2×Dic13#12112yes—
123C2×Dic13#12411yescharacteristic
124C2×Dic13#12711yescharacteristic
125C2×Dic13#13514yes—
126D52—24no—
127D52—42no—
128C4×D13—82no—
129C4×D13—82no—
130#1057—28no—
131#1057—82no—
132C2×#124—131no—
133C2×C4×Q8—131no—
134C2×C4×D4—131no—
135C2×C4⋊C8—132no—
136C2×C4×C8—132no—
137C2×C4×C8—262no—
138#391#26111yescharacteristic
139#390—131no—
140#388—262no—
141#382—131no—
142C4×C22⋊C4—262no—
143C43—131no—
144#336—262no—
145Q8⋊C8—264no—
146D4⋊C8—264no—
147C22×C13#39011yescharacteristic
148C22×C13C2×C22⋊C811yescharacteristic
149C22×C13#40811yescharacteristic
150C22×C13#41612yes—
151C22×C13C2×#10412yes—
152C2×D13—42no—
153C2×D13—44no—
154C2×D13—81no—
155C2×D13—81no—
156C2×D13—82no—
157C4×C13—24no—
158C4×C13—42no—
159C4×C13—81no—
160C4×C13C8×D412yes—
161C4×C13#43612yes—
162Dic13—22no—
163Dic13—24no—
164Dic13—81no—
165Dic13#48812yes—
166Dic13#49512yes—
167C22×Q8C2×#26111yescharacteristic
168C22×D4—131no—
169C23×C4—261no—
170C22×C8—262no—
171C22×C8—262no—
172#124—134no—
173#124—262no—
174C4×Q8—262no—
175C4×Q8—262no—
176C4×D4—262no—
177C4×D4—262no—
178C2×C4⋊C4—131no—
179C2×C4⋊C4—261no—
180C2×C4⋊C4C2×#26111yescharacteristic
181C2×C22⋊C4—262no—
182C2×C22⋊C4—261no—
183C2×C22⋊C4—262no—
184C2×C42—131no—
185C2×C42—131no—
186C2×C42—132no—
187C2×C42—132no—
188C2×C42—261no—
189C2×C42C2×#26111yescharacteristic
190C4⋊C8—262no—
191C4⋊C8—262no—
192C4×C8—134no—
193C4×C8—262no—
194C4×C8—268no—
195#95—22no—
196#95—262no—
197C2×C13#147612yes—
198C2×C13#149412yes—
199C2×C13#195411yescharacteristic
200C2×C13#195811yescharacteristic
201C2×C13#196211yescharacteristic
202D13—82no—
203C24—261no—
204C2×Q8—21no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8D13⋊C812yes—
208C2×D4—132no—
209C2×D4—261no—
210C2×D4—262no—
211C2×D4—262no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—131no—
215C22×C4—132no—
216C22×C4—131no—
217C22×C4—261no—
218C22×C4—521no—
219C22×C4—521no—
220C22×C4—524no—
221C22×C4C22⋊#26112yes—
222C22×C4C22×#26111yescharacteristic
223C2×C8—264no—
224C2×C8—268no—
225C2×C8—522no—
226C2×C8—524no—
227C4⋊C4—22no—
228C4⋊C4—262no—
229C4⋊C4—522no—
230C22⋊C4—268no—
231C22⋊C4—522no—
232C22⋊C4—524no—
233C42—22no—
234C42—132no—
235C42—132no—
236C42—132no—
237C42—132no—
238C42—264no—
239C42—264no—
240C42—521no—
241C42—521no—
242C13#963911yescharacteristic, complemented, Sylow
243C23—262no—
244C23—521no—
245C23—521no—
246C23—522no—
247C23—522no—
248C23C2×C22⋊#26111yescharacteristic
249Q8—24no—
250Q8—42no—
251D4—264no—
252D4—522no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—131no—
261C2×C4—131no—
262C2×C4—134no—
263C2×C4—134no—
264C2×C4—132no—
265C2×C4—262no—
266C2×C4—262no—
267C2×C4—262no—
268C2×C4—521no—
269C2×C4—522no—
270C2×C4—1042no—
271C2×C4—1042no—
272C2×C4#8811814yes—
273C2×C4C2×D13⋊C811yescharacteristic
274C2×C4D13⋊M1611yescharacteristic
275C2×C4#8823812yes—
276C2×C4D4×#26112yes—
277C8—268no—
278C8—1042no—
279C22—524no—
280C22—522no—
281C22—1042no—
282C22—1041no—
283C22—1041no—
284C22D13⋊(C22⋊C8)11yescharacteristic
285C22#1171914611yescharacteristic
286C22#1171915111yescharacteristic
287C22C2×#8811812yes—
288C22#1171916512yes—
289C4—24no—
290C4—42no—
291C4—81no—
292C4—132no—
293C4—132no—
294C4—264no—
295C4—262no—
296C4—1041no—
297C4#1171906312yes—
298C4#1171906612yes—
299C2—1042no—
300C2#1313013012yes—
301C2#1313018912yes—
302C2#1313095911yescharacteristic
303C2#1313096511yescharacteristic
304C2#1313096911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#261
commutatorC2×C4×C13C22×C8
FrattiniC22×C4C22×#261
FittingC13×#391C4
socleC23×C13C2×C22⋊C4
prime radicalC22×D52C2×C4
cosocleC2×C4×Dic13C23
hypercenter#391#261
hypocenterC13#9639
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #13130753C2semidirect
2. C42×#1057C2semidirect
3. C2×Q8⋊#1057C2semidirect
29. #88201C8semidirect
242. C13#9639semidirect

Maximal subgroups

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

subgroupindex
#131307532
C42×#10572
C2×Q8⋊#10572
C2×#117182432
#131173792
#963913