Finite Group Databasebeta

← #16783172

Subgroups — #16783172

Order 3328.

498 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 110 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#13131731C211yesmaximal, characteristic
2#13130138C212yesmaximal
3#13130080C211yesmaximal, characteristic
4#13129660C212yesmaximal
5#13117554C211yesmaximal, characteristic
6C2×Q8×Dic13C2211yescharacteristic
7C2×#88198C411yescharacteristic
8C2×#88141C2211yescharacteristic
9#11719125—22no—
10#11719119C2212yes—
11#11719119C2212yes—
12#11719074—24no—
13C13×#391C411yescharacteristic
14#11718638C411yescharacteristic
15#11718593C2211yescharacteristic
16#11718590C411yescharacteristic
17#11718316—22no—
18#11718313—22no—
19Q8⋊#1057—24no—
20#11718286—22no—
21C22×Q8×C13C2×C411yescharacteristic
22C22×Dic26C2×C411yescharacteristic
23C2×C13×C4⋊C4C2×C411yescharacteristic
24C2×C42×C13C2×C411yescharacteristic
25Q8×Dic13—22no—
26Q8×Dic13—22no—
27#88198—22no—
28#88198C814yescomplemented
29C2×C4⋊Dic13—21no—
30C2×C4⋊Dic13C2×C411yescharacteristic
31C2×#6101—22no—
32C2×#6101D412yes—
33C2×C4×Dic13—21no—
34C2×C4×Dic13C2×C411yescharacteristic
35C2×C4×Dic13D412yes—
36C2×C4×Dic13C2311yescharacteristic
37#88141—22no—
38#88141—22no—
39C22⋊#1057—22no—
40C22⋊#1057—22no—
41#88112—24no—
42C4⋊#1057—22no—
43C4⋊#1057—22no—
44C4×#1057—24no—
45C4×#1057—22no—
46C4×#1057D414yes—
47C13×#95—22no—
48#88071—22no—
49#88071—22no—
50#88071—21no—
51#88071—21no—
52#9100—131nomaximal, Sylow
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×Dic13—21no—
63C22×Dic13C22⋊C412yes—
64C22×Dic13C2×D411yescharacteristic
65C22×Dic13C2×D411yescharacteristic
66C2×Dic26—22no—
67C2×Dic26—22no—
68C2×Dic26C2×C812yes—
69C2×Dic26M1612yes—
70C2×#1057—24no—
71C2×#1057—42no—
72C13×C4⋊C4—22no—
73C42×C13—22no—
74C4⋊Dic13—22no—
75C4⋊Dic13—41no—
76C4⋊Dic13—41no—
77#6101—28no—
78#6101—24no—
79C4×Dic13—21no—
80C4×Dic13—22no—
81C4×Dic13—22no—
82C4×Dic13—42no—
83C4×Dic13C2×C812yes—
84C4×Dic13C2×D412yes—
85C4×Dic13#4112yes—
86#2982—131no—
87#2327—131no—
88#1479—132no—
89#1461—131no—
90#1274—132no—
91C23×C13C2×C22⋊C411yescharacteristic
92Q8×C13—24no—
93Q8×C13—42no—
94C2×C4×C13—22no—
95C2×C4×C13—22no—
96C2×C4×C13—41no—
97C2×C4×C13—41no—
98C2×C4×C13—42no—
99C2×C4×C13—41no—
100C2×C4×C13—41no—
101C2×C4×C13—41no—
102C2×C4×C13C4×D412yes—
103C2×C4×C13C22×C811yescharacteristic
104C2×C4×C13C2×M1611yescharacteristic
105C2×C4×C13#13112yes—
106C2×Dic13—22no—
107C2×Dic13—22no—
108C2×Dic13—22no—
109C2×Dic13—41no—
110C2×Dic13—41no—
111C2×Dic13—41no—
112C2×Dic13—41no—
113C2×Dic13—42no—
114C2×Dic13—41no—
115C2×Dic13C22⋊C814yes—
116C2×Dic13#12011yescharacteristic
117C2×Dic13#12112yes—
118C2×Dic13#12311yescharacteristic
119Dic26—24no—
120Dic26—24no—
121#1057—44no—
122#1057—82no—
123C2×C4⋊Q8—131no—
124C2×C4×Q8—131no—
125C2×C4×Q8—131no—
126C8×Q8—264no—
127#431—132no—
128#431—132no—
129#431—262no—
130#391—131no—
131#391#26111yescharacteristic
132#387—262no—
133#387—262no—
134#382—131no—
135#376—131no—
136Q8⋊C8—264no—
137#324—262no—
138C22×C13—21no—
139C22×C13—21no—
140C22×C13#38611yescharacteristic
141C22×C13C2×C22⋊C811yescharacteristic
142C22×C13#40811yescharacteristic
143C4×C13—42no—
144C4×C13—42no—
145C4×C13—81no—
146C4×C13C8×D412yes—
147C4×C13#43512yes—
148Dic13—24no—
149Dic13—22no—
150Dic13—21no—
151Dic13—21no—
152Dic13—42no—
153Dic13—81no—
154C22×Q8—131no—
155C22×Q8C2×#26111yescharacteristic
156C4⋊Q8—134no—
157C4⋊Q8—262no—
158C4×Q8—262no—
159C4×Q8—262no—
160C4×Q8—262no—
161C4×Q8—262no—
162C2×C4⋊C4—132no—
163C2×C4⋊C4—131no—
164C2×C4⋊C4—262no—
165C2×C4⋊C4—261no—
166C2×C4⋊C4C2×#26111yescharacteristic
167C2×C42—131no—
168C2×C42—131no—
169C2×C42—132no—
170C2×C42—261no—
171C2×C42C2×#26111yescharacteristic
172C4⋊C8—262no—
173C4⋊C8—262no—
174C4⋊C8—264no—
175C22⋊C8—262no—
176C22⋊C8—262no—
177C4×C8—134no—
178C4×C8—262no—
179C4×C8—264no—
180#95—22no—
181#95—261no—
182#95—261no—
183#95—262no—
184#95—262no—
185C2×C13—21no—
186C2×C13—21no—
187C2×C13#186911yescharacteristic
188C2×C13#190911yescharacteristic
189C2×C13#191211yescharacteristic
190C2×Q8—21no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—132no—
194C2×Q8—132no—
195C2×Q8—262no—
196C2×Q8—262no—
197C2×Q8D13⋊C812yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—132no—
201C22×C4—131no—
202C22×C4—131no—
203C22×C4—261no—
204C22×C4—261no—
205C22×C4C22⋊#26112yes—
206C22×C4C22×#26111yescharacteristic
207C2×C8—264no—
208C2×C8—522no—
209C4⋊C4—22no—
210C4⋊C4—264no—
211C4⋊C4—268no—
212C4⋊C4—262no—
213C4⋊C4—521no—
214C4⋊C4—521no—
215C42—22no—
216C42—132no—
217C42—132no—
218C42—132no—
219C42—262no—
220C42—262no—
221C42—261no—
222C42—522no—
223C13#910011yescharacteristic, complemented, Sylow
224C23C2×C22⋊#26111yescharacteristic
225Q8—24no—
226Q8—42no—
227Q8—264no—
228Q8—264no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—134no—
238C2×C4—131no—
239C2×C4—131no—
240C2×C4—132no—
241C2×C4—262no—
242C2×C4—262no—
243C2×C4—262no—
244C2×C4—521no—
245C2×C4—521no—
246C2×C4—521no—
247C2×C4—521no—
248C2×C4—522no—
249C2×C4—521no—
250C2×C4C2×D13⋊C811yescharacteristic
251C2×C4D13⋊M1611yescharacteristic
252C2×C4#8823812yes—
253C2×C4D4×#26112yes—
254C8—524no—
255C8—1042no—
256C22—21no—
257C22—21no—
258C22D13⋊(C22⋊C8)11yescharacteristic
259C22#1171914711yescharacteristic
260C22#1171915011yescharacteristic
261C4—42no—
262C4—42no—
263C4—81no—
264C4—264no—
265C4—262no—
266C4—261no—
267C4—261no—
268C4—522no—
269C4—1041no—
270C4#1171906312yes—
271C4#1171906512yes—
272C2—21no—
273C2—21no—
274C2#1313081011yescharacteristic
275C2#1313087711yescharacteristic
276C2#1313088311yescharacteristic
subgroupG/H
centerC22#11719150
commutatorC2×C4×C13C22×C8
FrattiniC22×C4C22×#261
FittingC13×#391C4
socleC22×C13#386
prime radicalC23×C13C2×C22⋊C4
cosocleC2×C4×Dic13C23
hypercenter#391#261
hypocenterC13#9100
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
28. #88198C8semidirect
223. C13#9100semidirect

Maximal subgroups

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

subgroupindex
#131317312
#131301382
#131300802
#131296602
#131175542
#910013