Finite Group Databasebeta

← #16784524

Subgroups — #16784524

Order 3328.

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

normal subgroups only

This group has 582 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#13130769C211yesmaximal, characteristic, complemented
2#13130550C211yesmaximal, characteristic, complemented
3C2×#11718244C212yesmaximal, complemented
4C2×D4⋊#1057C212yesmaximal
5#13117383C211yesmaximal, characteristic
6C2×#88190C411yescharacteristic
7C2×D4×Dic13C2211yescharacteristic
8C2×#88141C2211yescharacteristic
9C2×#88112—22no—
10C2×C4⋊#1057C2212yes—
11C2×C4×#1057C2212yes—
12C13×#390C411yescharacteristic
13#11718638C411yescharacteristic
14#11718605—22no—
15C22⋊C4×Dic13—22no—
16C42×Dic13C2211yescharacteristic
17#11718590C411yescharacteristic
18D4⋊#1057—24no—
19#11718290—22no—
20#11718244—24no—
21C22×D4×C13C2×C411yescharacteristic
22C23×Dic13—21no—
23C22×Dic26C2×C411yescharacteristic
24C22×#1057—22no—
25C22×#1057—22no—
26C2×C13×C4⋊C4C2×C411yescharacteristic
27C2×C13×C22⋊C4—22no—
28C2×C42×C13C2×C411yescharacteristic
29C2×C22⋊Dic13—22no—
30C2×C22⋊Dic13—21no—
31#88190—22no—
32#88190C814yescomplemented
33D4×Dic13—22no—
34D4×Dic13—22no—
35C2×C4⋊Dic13C2×C411yescharacteristic
36C2×#6101—21no—
37C2×C4×Dic13—21no—
38C2×C4×Dic13C2×C411yescharacteristic
39C2×C4×Dic13D412yes—
40C2×C4×Dic13D412yes—
41C2×C4×Dic13C2311yescharacteristic
42#88141—22no—
43#88141—22no—
44#88112—44no—
45C4⋊#1057—22no—
46C4⋊#1057—22no—
47C4×#1057—22no—
48C4×#1057D414yes—
49#88071—22no—
50#88071—22no—
51#9640—131nomaximal, Sylow
52C24×C13—21no—
53C2×D4×C13—22no—
54C2×D4×C13—21no—
55C2×D4×C13—22no—
56C2×D4×C13C2×C812yes—
57C22×C4×C13—21no—
58C22×C4×C13C22⋊C412yes—
59C22×C4×C13C22×C411yescharacteristic
60C22×Dic13—21no—
61C22×Dic13—21no—
62C22×Dic13—41no—
63C22×Dic13—41no—
64C22×Dic13—42no—
65C22×Dic13—42no—
66C22×Dic13C22⋊C412yes—
67C22×Dic13C2×D411yescharacteristic
68C22×Dic13C2×D411yescharacteristic
69C2×Dic26—22no—
70C2×Dic26—22no—
71C2×Dic26—21no—
72C2×Dic26C2×C812yes—
73C2×#1057—24no—
74C2×#1057—44no—
75C2×#1057—42no—
76C2×#1057—44no—
77C13×C4⋊C4—22no—
78C13×C22⋊C4—44no—
79C42×C13—22no—
80C22⋊Dic13—28no—
81C22⋊Dic13—42no—
82C4⋊Dic13—22no—
83#6101—42no—
84C4×Dic13—22no—
85C4×Dic13—22no—
86C4×Dic13—42no—
87C4×Dic13—41no—
88C4×Dic13—41no—
89C4×Dic13C2×C812yes—
90C4×Dic13M1612yes—
91C4×Dic13C2×D412yes—
92C4×Dic13#4112yes—
93C4×#124—131no—
94#1842—131no—
95#1744—131no—
96C2×Q8⋊C8—132no—
97C2×D4⋊C8—132no—
98C23×C13—22no—
99C23×C13—41no—
100C23×C13—41no—
101C23×C13—42no—
102C23×C13—42no—
103C23×C13C2×C22⋊C411yescharacteristic
104D4×C13—24no—
105D4×C13—42no—
106C2×C4×C13—22no—
107C2×C4×C13—22no—
108C2×C4×C13—22no—
109C2×C4×C13—41no—
110C2×C4×C13—42no—
111C2×C4×C13C4×D412yes—
112C2×C4×C13C22×C811yescharacteristic
113C2×C4×C13C2×M1611yescharacteristic
114C2×C4×C13#13112yes—
115C2×Dic13—22no—
116C2×Dic13—22no—
117C2×Dic13—22no—
118C2×Dic13—22no—
119C2×Dic13—41no—
120C2×Dic13—41no—
121C2×Dic13—41no—
122C2×Dic13—42no—
123C2×Dic13—82no—
124C2×Dic13—81no—
125C2×Dic13—81no—
126C2×Dic13C22⋊C814yes—
127C2×Dic13#12112yes—
128C2×Dic13#12411yescharacteristic
129C2×Dic13#12711yescharacteristic
130C2×Dic13AGL(1,ℤ/8)12yes—
131C2×Dic13#13712yes—
132Dic26—24no—
133Dic26—42no—
134#1057—44no—
135#1057—82no—
136C2×#124—131no—
137C2×C4×Q8—131no—
138C2×C4×D4—131no—
139C2×C4⋊C8—132no—
140C2×C4⋊C8—262no—
141C2×C4×C8—132no—
142#391—131no—
143#390#26111yescharacteristic
144#388—262no—
145#382—131no—
146C4×C22⋊C4—262no—
147C43—131no—
148#336—262no—
149Q8⋊C8—264no—
150D4⋊C8—264no—
151C22×C13—42no—
152C22×C13—44no—
153C22×C13—81no—
154C22×C13—81no—
155C22×C13—82no—
156C22×C13#39011yescharacteristic
157C22×C13C2×C22⋊C811yescharacteristic
158C22×C13#40811yescharacteristic
159C22×C13#41712yes—
160C22×C13#42112yes—
161C4×C13—42no—
162C4×C13—81no—
163C4×C13C8×D412yes—
164C4×C13#43612yes—
165Dic13—22no—
166Dic13—24no—
167Dic13—42no—
168Dic13—81no—
169Dic13#48912yes—
170Dic13#49611yescharacteristic
171Dic13#49711yescharacteristic
172C22×Q8—131no—
173C22×D4C2×#26111yescharacteristic
174C23×C4—261no—
175C22×C8—262no—
176C22×C8—262no—
177#124—134no—
178#124—262no—
179C4×Q8—262no—
180C4×Q8—262no—
181C4×D4—262no—
182C4×D4—262no—
183C2×C4⋊C4—131no—
184C2×C4⋊C4—261no—
185C2×C4⋊C4C2×#26111yescharacteristic
186C2×C22⋊C4—22no—
187C2×C22⋊C4—262no—
188C2×C22⋊C4—261no—
189C2×C42—131no—
190C2×C42—131no—
191C2×C42—132no—
192C2×C42—132no—
193C2×C42—261no—
194C2×C42C2×#26111yescharacteristic
195C4⋊C8—262no—
196C4⋊C8—262no—
197C4⋊C8—524no—
198C4×C8—134no—
199C4×C8—262no—
200#95—262no—
201#95—262no—
202C2×C13—82no—
203C2×C13#147712yes—
204C2×C13#149512yes—
205C2×C13#195411yescharacteristic
206C2×C13#195911yescharacteristic
207C2×C13#196311yescharacteristic
208C24—21no—
209C2×Q8—132no—
210C2×Q8—261no—
211C2×Q8—262no—
212C2×Q8—262no—
213C2×D4—21no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4D13⋊C812yes—
217C22×C4—21no—
218C22×C4—131no—
219C22×C4—132no—
220C22×C4—131no—
221C22×C4—261no—
222C22×C4—261no—
223C22×C4—521no—
224C22×C4—521no—
225C22×C4—522no—
226C22×C4—522no—
227C22×C4C22⋊#26112yes—
228C22×C4C22×#26111yescharacteristic
229C2×C8—264no—
230C2×C8—524no—
231C2×C8—522no—
232C2×C8—524no—
233C4⋊C4—22no—
234C4⋊C4—262no—
235C4⋊C4—522no—
236C22⋊C4—44no—
237C22⋊C4—268no—
238C22⋊C4—522no—
239C42—22no—
240C42—132no—
241C42—132no—
242C42—132no—
243C42—132no—
244C42—262no—
245C42—262no—
246C42—522no—
247C42—521no—
248C42—521no—
249C13#964011yescharacteristic, complemented, Sylow
250C23—22no—
251C23—42no—
252C23—41no—
253C23—41no—
254C23—42no—
255C23C2×C22⋊#26111yescharacteristic
256Q8—264no—
257Q8—522no—
258D4—24no—
259D4—42no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—131no—
266C2×C4—131no—
267C2×C4—132no—
268C2×C4—132no—
269C2×C4—134no—
270C2×C4—132no—
271C2×C4—262no—
272C2×C4—262no—
273C2×C4—262no—
274C2×C4—262no—
275C2×C4—521no—
276C2×C4—521no—
277C2×C4—521no—
278C2×C4—522no—
279C2×C4—1042no—
280C2×C4—1041no—
281C2×C4—1041no—
282C2×C4C2×D13⋊C811yescharacteristic
283C2×C4D13⋊M1611yescharacteristic
284C2×C4#8823812yes—
285C2×C4D4×#26112yes—
286C8—524no—
287C8—1042no—
288C22—44no—
289C22—42no—
290C22—82no—
291C22—81no—
292C22—81no—
293C22D13⋊(C22⋊C8)11yescharacteristic
294C22#1171914012yes—
295C22#1171914611yescharacteristic
296C22#1171915111yescharacteristic
297C22#1171916412yes—
298C4—42no—
299C4—81no—
300C4—132no—
301C4—131no—
302C4—131no—
303C4—264no—
304C4—262no—
305C4—522no—
306C4—1041no—
307C4#1171906312yes—
308C4#1171906612yes—
309C2—82no—
310C2#1313013212yes—
311C2#1313019112yes—
312C2#1313095911yescharacteristic
313C2#1313096611yescharacteristic
314C2#1313097411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#261
commutatorC2×C4×C13C22×C8
FrattiniC22×C4C22×#261
FittingC13×#390C4
socleC23×C13C2×C22⋊C4
prime radicalC22×D4×C13C2×C4
cosocleC2×C4×Dic13C23
hypercenter#390#261
hypocenterC13#9640
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #13130769C2semidirect
2. #13130550C2semidirect
3. C2×#11718244C2semidirect
32. #88190C8semidirect
249. C13#9640semidirect

Maximal subgroups

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

subgroupindex
#131307692
#131305502
C2×#117182442
C2×D4⋊#10572
#131173832
#964013