Finite Group Databasebeta

← #16790021

Subgroups — #16790021

Order 3328.

500 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C2×#11718295C212yesmaximal
2#13130702C212yesmaximal, complemented
3#13129934C212yesmaximal, complemented
4C4×#88190C211yesmaximal, characteristic
5C2×C4×D4×C13C411yescharacteristic
6C2×#88190C2211yescharacteristic
7C2×C4×Dic26C2211yescharacteristic
8C2×#88113C2214yes—
9#11719129—22no—
10#11718882C411yescharacteristic
11C4×C22⋊Dic13—22no—
12#11718638C411yescharacteristic
13#11718635C2211yescharacteristic
14#11718606—22no—
15C42×Dic13C411yescharacteristic
16#11718295—24no—
17#11718291—22no—
18C22×D4×C13C2×C411yescharacteristic
19C23×C4×C13—21no—
20C22×Dic26C2311yescharacteristic
21C2×#6122—24no—
22C4×D4×C13—41no—
23C4×D4×C13—41no—
24C2×C13×C4⋊C4C2×C411yescharacteristic
25C2×C13×C22⋊C4—21no—
26C2×C42×C13C2×C411yescharacteristic
27C2×C22⋊Dic13—22no—
28C2×C22⋊Dic13—22no—
29#88190—22no—
30#88190D412yes—
31#88190Q812yes—
32C2×C4⋊Dic13C2×C411yescharacteristic
33C2×#6101D412yes—
34C2×C4×Dic13—21no—
35C2×C4×Dic13—21no—
36C2×C4×Dic13C2×C411yescharacteristic
37C2×C4×Dic13C2×C411yescharacteristic
38C2×C4×Dic13D412yes—
39C4×Dic26—41no—
40C4×Dic26—41no—
41#88113—24no—
42#88113—24no—
43#88112—42no—
44C4⋊#1057—42no—
45#88071—22no—
46#88071—22no—
47#12801—131nomaximal, Sylow
48C24×C13—21no—
49C2×D4×C13—21no—
50C2×D4×C13—41no—
51C2×D4×C13—41no—
52C2×D4×C13C4⋊C412yes—
53C22×C4×C13—21no—
54C22×C4×C13—44no—
55C22×C4×C13—41no—
56C22×C4×C13—41no—
57C22×C4×C13C22⋊C411yescharacteristic
58C22×C4×C13C22⋊C411yescharacteristic
59C22×C4×C13C22×C411yescharacteristic
60C22×Dic13—21no—
61C22×Dic13—21no—
62C22×Dic13C22⋊C412yes—
63C22×Dic13C2×D412yes—
64C2×Dic26—41no—
65C2×Dic26—41no—
66C2×Dic26C2×D411yescharacteristic
67C2×Dic26C2×Q811yescharacteristic
68C2×Dic26#4112yes—
69#6122—44no—
70#6122—44no—
71C2×#1057—44no—
72C13×C4⋊C4—41no—
73C13×C22⋊C4—42no—
74C42×C13—21no—
75C42×C13—21no—
76C22⋊Dic13—24no—
77C22⋊Dic13—24no—
78C22⋊Dic13—44no—
79C4⋊Dic13—41no—
80#6101—42no—
81C4×Dic13—22no—
82C4×Dic13—22no—
83C4×Dic13—21no—
84C4×Dic13—42no—
85C4×Dic13—42no—
86C4×Dic13—41no—
87C4×Dic13C4⋊C412yes—
88C4×#124—131no—
89C2×#355—132no—
90#1801—132no—
91#1394—132no—
92C23×C13—22no—
93C23×C13—42no—
94C23×C13—42no—
95C23×C13—41no—
96C23×C13—41no—
97C23×C13C2×C22⋊C411yescharacteristic
98D4×C13—42no—
99D4×C13—42no—
100C2×C4×C13—21no—
101C2×C4×C13—21no—
102C2×C4×C13—22no—
103C2×C4×C13—21no—
104C2×C4×C13—41no—
105C2×C4×C13—42no—
106C2×C4×C13—41no—
107C2×C4×C13—82no—
108C2×C4×C13—82no—
109C2×C4×C13#9911yescharacteristic
110C2×C4×C13#9911yescharacteristic
111C2×C4×C13C2×C4⋊C411yescharacteristic
112C2×C4×C13#11711yescharacteristic
113C2×Dic13—22no—
114C2×Dic13—22no—
115C2×Dic13—22no—
116C2×Dic13—22no—
117C2×Dic13—22no—
118C2×Dic13—41no—
119C2×Dic13—41no—
120C2×Dic13—41no—
121C2×Dic13—41no—
122C2×Dic13—42no—
123C2×Dic13#12112yes—
124C2×Dic13C22⋊Q812yes—
125Dic26—41no—
126Dic26—41no—
127Dic26—42no—
128#1057—84no—
129C2×#124—131no—
130C2×C4×Q8—131no—
131C2×C4×D4#26111yescharacteristic
132#423—262no—
133C2×#101—134no—
134#391—131no—
135#390—131no—
136#388—262no—
137#382—131no—
138C4×C22⋊C4—262no—
139C43—131no—
140#355—264no—
141#343—262no—
142C22×C13—21no—
143C22×C13—21no—
144C22×C13—44no—
145C22×C13—42no—
146C22×C13—82no—
147C22×C13—81no—
148C22×C13—81no—
149C22×C13#38011yescharacteristic
150C22×C13C2×#9911yescharacteristic
151C22×C13#41011yescharacteristic
152C4×C13—21no—
153C4×C13—21no—
154C4×C13—21no—
155C4×C13—41no—
156C4×C13—81no—
157C4×C13#35512yescomplemented
158Dic13—22no—
159Dic13—22no—
160Dic13—42no—
161Dic13—42no—
162Dic13—42no—
163C22×Q8—131no—
164C22×D4C2×#26111yescharacteristic
165C23×C4—21no—
166C2×M16—264no—
167#124—132no—
168#124—132no—
169#124—262no—
170C4×Q8—521no—
171C4×Q8—521no—
172C4×D4—41no—
173C4×D4—41no—
174C2×C4⋊C4—132no—
175C2×C4⋊C4—131no—
176C2×C4⋊C4C2×#26111yescharacteristic
177C2×C22⋊C4—21no—
178C2×C22⋊C4—262no—
179C2×C22⋊C4—262no—
180C2×C42—131no—
181C2×C42—132no—
182C2×C42—131no—
183C2×C42—261no—
184C2×C42—261no—
185C2×C42C2×#26111yescharacteristic
186C4⋊C8—522no—
187C4⋊C8—522no—
188#101—264no—
189#101—264no—
190#95—262no—
191#95—262no—
192C2×C13—21no—
193C2×C13—21no—
194C2×C13—82no—
195C2×C13#179911yescharacteristic
196C2×C13C2×#35511yescharacteristic
197C2×C13#213211yescharacteristic
198C24—21no—
199C2×Q8—131no—
200C2×Q8—131no—
201C2×Q8—132no—
202C2×Q8—521no—
203C2×Q8—521no—
204C2×D4—21no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4C4⋊#26112yes—
208C22×C4—21no—
209C22×C4—44no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—132no—
213C22×C4—132no—
214C22×C4—261no—
215C22×C4—261no—
216C22×C4C22⋊#26111yescharacteristic
217C22×C4C22⋊#26111yescharacteristic
218C22×C4C22×#26111yescharacteristic
219M16—524no—
220M16—524no—
221C2×C8—524no—
222C4⋊C4—41no—
223C4⋊C4—522no—
224C4⋊C4—521no—
225C22⋊C4—42no—
226C22⋊C4—264no—
227C22⋊C4—264no—
228C22⋊C4—524no—
229C42—21no—
230C42—21no—
231C42—132no—
232C42—262no—
233C42—262no—
234C42—261no—
235C42—521no—
236C42—522no—
237C42—522no—
238C13#1280111yescharacteristic, complemented, Sylow
239C23—22no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23C2×C22⋊#26111yescharacteristic
245Q8—521no—
246Q8—521no—
247Q8—522no—
248D4—42no—
249D4—42no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—132no—
260C2×C4—132no—
261C2×C4—262no—
262C2×C4—262no—
263C2×C4—262no—
264C2×C4—262no—
265C2×C4—262no—
266C2×C4—521no—
267C2×C4—521no—
268C2×C4—521no—
269C2×C4—521no—
270C2×C4—522no—
271C2×C4#8811911yescharacteristic
272C2×C4#8811911yescharacteristic
273C2×C4C2×C4⋊#26111yescharacteristic
274C2×C4#8823711yescharacteristic
275C8—1044no—
276C22—21no—
277C22—21no—
278C22—44no—
279C22—42no—
280C22—82no—
281C22—81no—
282C22—81no—
283C22#1171911611yescharacteristic
284C22#1171913611yescharacteristic
285C22C2×#8811911yescharacteristic
286C4—21no—
287C4—21no—
288C4—21no—
289C4—41no—
290C4—81no—
291C4—262no—
292C4—262no—
293C4—522no—
294C4—522no—
295C4—522no—
296C4#1171829512yescomplemented
297C2—21no—
298C2—21no—
299C2—82no—
300C2#1313069611yescharacteristic
301C2C2×#1171829511yescharacteristic
302C2#1313112711yescharacteristic
subgroupG/H
centerC22C2×#88119
commutatorC22×C4×C13C22×C4
FrattiniC22×C4C22×#261
FittingC2×C4×D4×C13C4
socleC22×C13C2×#99
prime radicalC22×D4×C13C2×C4
cosocleC22×Dic26C23
hypercenterC2×C4×D4#261
hypocenterC13#12801
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. #13130702C2semidirect
3. #13129934C2semidirect
157. C4×C13#355semidirect
238. C13#12801semidirect
296. C4#11718295semidirect

Maximal subgroups

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

subgroupindex
C2×#117182952
#131307022
#131299342
C4×#881902
#1280113