Finite Group Databasebeta

← #16780704

Subgroups — #16780704

Order 3328.

708 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 156 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
1C4×#88112C212yesmaximal, complemented
2C2×#11718297C212yesmaximal
3#13129714C212yesmaximal, complemented
4#13117400C211yesmaximal, characteristic
5C2×#88194C2211yescharacteristic
6C2×D4×Dic13C411yescharacteristic
7C2×#88147C411yescharacteristic
8C2×#88135—22no—
9C2×#88112C2214yes—
10C2×C4×#1057—22no—
11C13×#390C411yescharacteristic
12#11718637C2211yescharacteristic
13#11718618—22no—
14#11718600C411yescharacteristic
15C42×Dic13C2211yescharacteristic
16#11718297C418yescomplemented
17#11718290—22no—
18C22×D4×C13C2×C411yescharacteristic
19C22×C22⋊D13—22no—
20C23×Dic13—21no—
21C22×D52C2×C411yescharacteristic
22C22×C4×D13—21no—
23C22×#1057—24no—
24C2×C13×C4⋊C4C2×C411yescharacteristic
25C2×C13×C22⋊C4—22no—
26C2×C42×C13C2×C411yescharacteristic
27C2×C22⋊Dic13—21no—
28#88194—22no—
29#88194C2×C414yes—
30D4×Dic13—42no—
31C2×#6103—22no—
32C2×#6103—21no—
33C2×C4⋊Dic13C2×C411yescharacteristic
34C2×#6101—22no—
35C2×C4×Dic13C2×C411yescharacteristic
36C2×C4×Dic13D412yes—
37C2×C4×Dic13D412yes—
38C2×C4×Dic13C2311yescharacteristic
39#88147—42no—
40#88135—44no—
41#88135—44no—
42#88112—24no—
43#88112C2×C418yescomplemented
44C4×#1057—44no—
45C4×#331—131nomaximal, Sylow
46C24×C13—21no—
47C23×D13—21no—
48C2×D4×C13—21no—
49C2×D4×C13—41no—
50C2×D4×C13—41no—
51C2×D4×C13C4212yes—
52C22×C4×C13—21no—
53C22×C4×C13C22⋊C412yes—
54C22×C4×C13C22×C411yescharacteristic
55C2×C22⋊D13—28no—
56C2×C22⋊D13—42no—
57C2×C22⋊D13—42no—
58C2×C22⋊D13—44no—
59C22×Dic13—21no—
60C22×Dic13—42no—
61C22×Dic13—44no—
62C22×Dic13C22⋊C411yescharacteristic
63C22×Dic13C22⋊C411yescharacteristic
64C22×Dic13C2×D412yes—
65C2×D52—21no—
66C2×D52—41no—
67C2×D52—41no—
68C2×D52C4212yes—
69C2×C4×D13—42no—
70C2×C4×D13—44no—
71C2×#1057—28no—
72C2×#1057—44no—
73C2×#1057—44no—
74C13×C4⋊C4—41no—
75C13×C22⋊C4—44no—
76C42×C13—41no—
77C22⋊Dic13—42no—
78#6103—42no—
79#6103—44no—
80C4⋊Dic13—41no—
81#6101—42no—
82#6101—42no—
83C4×Dic13—22no—
84C4×Dic13—24no—
85C4×Dic13D814yes—
86C4×Dic13SD1614yes—
87C4×Dic13C22×C412yes—
88C4×Dic13#4112yes—
89C4×#127—131no—
90C4×C4⋊C8—132no—
91C2×#331—132no—
92#1296—132no—
93C23×C13—22no—
94C23×C13—41no—
95C23×C13—41no—
96C23×C13—42no—
97C23×C13—42no—
98C23×C13C2×C22⋊C411yescharacteristic
99C22×D13—22no—
100C22×D13—41no—
101C22×D13—41no—
102C22×D13—42no—
103C22×D13—42no—
104D4×C13—42no—
105D4×C13—42no—
106C2×C4×C13—21no—
107C2×C4×C13—21no—
108C2×C4×C13—22no—
109C2×C4×C13—22no—
110C2×C4×C13—41no—
111C2×C4×C13—42no—
112C2×C4×C13#10214yes—
113C2×C4×C13C2×C4211yescharacteristic
114C2×C4×C13#11711yescharacteristic
115C22⋊D13—48no—
116C22⋊D13—48no—
117C2×Dic13—21no—
118C2×Dic13—22no—
119C2×Dic13—41no—
120C2×Dic13—42no—
121C2×Dic13—84no—
122C2×Dic13#10012yes—
123C2×Dic13#10214yes—
124C2×Dic13C4×D414yes—
125C2×Dic13C2×D812yes—
126C2×Dic13C2×SD1612yes—
127C2×Dic13#13514yes—
128D52—42no—
129D52—42no—
130C4×D13—84no—
131#1057—48no—
132C2×#127—131no—
133C2×C4×D4—131no—
134C2×C4×D4—131no—
135C2×C4×D4—262no—
136C2×C4⋊C8—134no—
137C2×C4×C8—262no—
138#390—131no—
139#390—262no—
140#390#26111yescharacteristic
141#384—131no—
142C43—131no—
143#336—262no—
144#331—138no—
145C22×C13—42no—
146C22×C13—44no—
147C22×C13—81no—
148C22×C13—82no—
149C22×C13—81no—
150C22×C13C4×C22⋊C411yescharacteristic
151C22×C13C2×#10011yescharacteristic
152C22×C13#41311yescharacteristic
153C22×C13C2×#10212yes—
154C22×C13#41612yes—
155C2×D13—42no—
156C2×D13—44no—
157C2×D13—81no—
158C2×D13—82no—
159C2×D13—81no—
160C4×C13—24no—
161C4×C13—21no—
162C4×C13—21no—
163C4×C13—81no—
164Dic13—41no—
165Dic13—81no—
166Dic13#33114yescomplemented
167Dic13C4×D814yes—
168Dic13C4×SD1614yes—
169C22×D4—131no—
170C22×D4—262no—
171C22×D4C2×#26111yescharacteristic
172C23×C4—261no—
173C23×C4—261no—
174C22×C8—264no—
175#127—134no—
176#127—262no—
177C4×D4—524no—
178C4×D4—524no—
179C4×D4—522no—
180C4×D4—522no—
181C2×C4⋊C4—131no—
182C2×C4⋊C4—262no—
183C2×C4⋊C4C2×#26111yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—261no—
186C2×C22⋊C4—261no—
187C2×C22⋊C4—262no—
188C2×C42—132no—
189C2×C42—131no—
190C2×C42—132no—
191C2×C42—131no—
192C2×C42C2×#26111yescharacteristic
193C4⋊C8—138no—
194C4⋊C8—264no—
195C4×C8—524no—
196C2×C13—82no—
197C2×C13C2×#33111yescharacteristic
198C2×C13#154012yes—
199C2×C13#154311yescharacteristic
200C2×C13C4×#10011yescharacteristic
201C2×C13C4×#10212yes—
202D13—82no—
203C24—21no—
204C24—261no—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—132no—
209C2×D4—268no—
210C2×D4—261no—
211C2×D4—521no—
212C2×D4—521no—
213C2×D4—522no—
214C2×D4—522no—
215C2×D4—524no—
216C2×D4C4×#26112yes—
217C22×C4—21no—
218C22×C4—132no—
219C22×C4—131no—
220C22×C4—131no—
221C22×C4—261no—
222C22×C4—524no—
223C22×C4—522no—
224C22×C4—524no—
225C22×C4—522no—
226C22×C4C22⋊#26112yes—
227C22×C4C22×#26111yescharacteristic
228C2×C8—268no—
229C2×C8—524no—
230C2×C8—524no—
231C4⋊C4—41no—
232C4⋊C4—521no—
233C4⋊C4—522no—
234C4⋊C4—522no—
235C22⋊C4—44no—
236C22⋊C4—522no—
237C22⋊C4—522no—
238C22⋊C4—524no—
239C42—41no—
240C42—134no—
241C42—134no—
242C42—132no—
243C42—132no—
244C42—262no—
245C42—264no—
246C13C4×#33111yescharacteristic, complemented, Sylow
247C23—22no—
248C23—42no—
249C23—41no—
250C23—41no—
251C23—42no—
252C23—262no—
253C23—522no—
254C23—521no—
255C23—521no—
256C23—522no—
257C23C2×C22⋊#26111yescharacteristic
258D4—42no—
259D4—42no—
260D4—522no—
261D4—522no—
262D4—528no—
263D4—528no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—134no—
271C2×C4—132no—
272C2×C4—132no—
273C2×C4—134no—
274C2×C4—132no—
275C2×C4—134no—
276C2×C4—261no—
277C2×C4—262no—
278C2×C4—522no—
279C2×C4—521no—
280C2×C4—1044no—
281C2×C4—1044no—
282C2×C4#8811514yes—
283C2×C4C2×C4×#26111yescharacteristic
284C2×C4#8823711yescharacteristic
285C8—528no—
286C22—42no—
287C22—44no—
288C22—81no—
289C22—82no—
290C22—81no—
291C22—522no—
292C22—524no—
293C22—1041no—
294C22—1042no—
295C22—1041no—
296C22C4×C22⋊#26111yescharacteristic
297C22C2×#8811512yes—
298C22#1171914811yescharacteristic
299C22D13⋊#10011yescharacteristic
300C22#1171916512yes—
301C4—21no—
302C4—21no—
303C4—24no—
304C4—81no—
305C4—134no—
306C4—134no—
307C4—134no—
308C4—521no—
309C4—1041no—
310C2—82no—
311C2—1042no—
312C2D13⋊#33111yescharacteristic
313C2#1313033412yes—
314C2#1313034011yescharacteristic
315C2#1313058611yescharacteristic
316C2C4×#8811512yes—
subgroupG/H
centerC23C2×C22⋊#261
commutatorC2×C4×C13C2×C42
FrattiniC22×C4C22×#261
FittingC13×#390C4
socleC23×C13C2×C22⋊C4
prime radicalC2×#88194C22
cosocleC2×C4×Dic13C23
hypercenter#390#261
hypocenterC13C4×#331
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C4×#88112C2semidirect
3. #13129714C2semidirect
16. #11718297C4semidirect
43. #88112C2×C4semidirect
166. Dic13#331semidirect
246. C13C4×#331semidirect

Maximal subgroups

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

subgroupindex
C4×#881122
C2×#117182972
#131297142
#131174002
C4×#33113