Finite Group Databasebeta

← #16799739

Subgroups — #16799739

Order 3328.

1708 conjugacy classes of subgroups; only the 288 normal subgroups are shown.

normal subgroups only

This group has 1708 conjugacy classes of subgroups — too many to draw. Only its 288 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
1C24×#1057C211yesmaximal, characteristic, complemented
2C2×#11719064C213yesmaximal, complemented
3#13130811C213yesmaximal, complemented
4#13130779C213yesmaximal, complemented
5#13130658C211yesmaximal, characteristic, complemented
6#13130573C213yesmaximal, complemented
7#13117335C211yesmaximal, characteristic
8C24×Dic13C2211yescharacteristic
9C23×#1057—212no—
10C23×#1057C2213yes—
11C23×#1057C2213yes—
12C22×#6118C2211yescharacteristic, complemented
13C2×#88207C411yescharacteristic
14C2×#88135C2213yes—
15C2×C22⋊#1057C2213yes—
16C2×C22⋊#1057C2213yes—
17C2×#88111C2213yescomplemented
18C2×#88110C2213yescomplemented
19C2×C4⋊#1057C2213yescomplemented
20#11719064—212no—
21C13×#379C411yescharacteristic
22#11718619C413yes—
23#11718607C413yes—
24C22⋊C4×Dic13C2213yes—
25#11718290C2216yescomplemented
26C25×C13C2×C411yescharacteristic
27C22×C22⋊D13C2×C413yes—
28C23×Dic13—26no—
29C23×Dic13—26no—
30C23×Dic13C2×C413yes—
31C23×Dic13C2313yes—
32C22×C4×D13C2311yescharacteristic
33C22×#1057—232no—
34C22×#1057—26no—
35C22×#1057—212no—
36C22×#1057—212no—
37C22×#1057D4112yescomplemented
38C22×#1057C2313yes—
39C22×#1057C2311yescharacteristic
40C22×#1057C2313yes—
41C22×#1057C2311yescharacteristic
42C2×#6118—26no—
43C2×C13×C22⋊C4C2×C413yes—
44C2×C13×C22⋊C4C2×C413yes—
45#88207—24no—
46C2×C22⋊Dic13C2×C413yes—
47C2×#6103C2×C413yes—
48C2×#6101C2×C413yes—
49C2×C4×Dic13C2313yes—
50#88135—26no—
51#88135—26no—
52C22⋊#1057—26no—
53C22⋊#1057—26no—
54#88111—26no—
55#88110—26no—
56C4⋊#1057—26no—
57#88071C2×C416yes—
58#21440—131nomaximal, Sylow
59C24×C13—26no—
60C24×C13—26no—
61C24×C13M1614yescomplemented
62C24×C13C22×C413yes—
63C23×D13C22×C411yescharacteristic
64C22×C4×C13C22×C413yes—
65C22×C4×C13C22×C411yescharacteristic
66C2×C22⋊D13—26no—
67C2×C22⋊D13—26no—
68C2×C22⋊D13—26no—
69C22×Dic13—28no—
70C22×Dic13—224no—
71C22×Dic13—26no—
72C22×Dic13—212no—
73C22×Dic13—26no—
74C22×Dic13—26no—
75C22×Dic13C22×C413yes—
76C22×Dic13C22×C413yes—
77C22×Dic13C2×D416yes—
78C22×Dic13#4116yes—
79C22×Dic13C2411yescharacteristic
80C2×C4×D13—23no—
81C2×C4×D13—23no—
82C2×#1057—224no—
83C2×#1057—26no—
84C2×#1057—224no—
85C2×#1057—23no—
86C2×#1057—23no—
87C2×#1057—26no—
88C2×#1057C2×D416yes—
89C2×#1057C2×D416yes—
90#6118—44no—
91C13×C22⋊C4—26no—
92C13×C22⋊C4—26no—
93C22⋊Dic13—26no—
94#6103—26no—
95#6101—26no—
96C4×Dic13—23no—
97C4×Dic13—23no—
98C24×C8—131no—
99C2×#435—133no—
100C4×#120—131no—
101#1869—133no—
102#1850—133no—
103#1779—131no—
104#1752—133no—
105C23×C13—28no—
106C23×C13—224no—
107C23×C13—26no—
108C23×C13—26no—
109C23×C13—224no—
110C23×C13C4×D416yes—
111C23×C13C2×M1616yes—
112C23×C13#13116yes—
113C23×C13C23×C411yescharacteristic
114C22×D13—23no—
115C22×D13—21no—
116C22×D13—23no—
117C2×C4×C13—23no—
118C2×C4×C13—23no—
119C2×C4×C13—26no—
120C22⋊D13—412no—
121C2×Dic13—224no—
122C2×Dic13—26no—
123C2×Dic13—26no—
124C2×Dic13—26no—
125C2×Dic13—212no—
126C2×Dic13—23no—
127C2×Dic13—23no—
128C2×Dic13—23no—
129C2×Dic13C4×D416yes—
130C2×Dic13C22×D413yes—
131C2×Dic13C2×#4113yes—
132C4×D13—41no—
133C4×D13—43no—
134#1057—26no—
135#1057—41no—
136#1057—43no—
137#1057#12014yescomplemented
138C24×C4—131no—
139C22×M16—131no—
140C23×C8—133no—
141C23×C8—133no—
142C23×C8—2612no—
143C2×#120—131no—
144C2×C4×D4—133no—
145#435—2612no—
146C2×C4⋊C8—133no—
147C2×C22⋊C8—133no—
148C2×C22⋊C8—133no—
149C2×C22⋊C8—133no—
150C2×#97—133no—
151#386—133no—
152#385—133no—
153#379#26111yescharacteristic
154C4×C22⋊C4—133no—
155#336—136no—
156C22×C13—232no—
157C22×C13—212no—
158C22×C13—26no—
159C22×C13—212no—
160C22×C13—26no—
161C22×C13#435112yescomplemented
162C22×C13C2×C4×D413yes—
163C22×C13C22×M1611yescharacteristic
164C22×C13C2×#13113yes—
165C2×D13—43no—
166C2×D13—43no—
167C2×D13—41no—
168C4×C13—43no—
169C4×C13—41no—
170Dic13—26no—
171Dic13—43no—
172Dic13C2×#12011yescharacteristic
173Dic13#52513yes—
174C25C2×#26111yescharacteristic
175C22×D4—133no—
176C23×C4—133no—
177C23×C4—133no—
178C23×C4—131no—
179C23×C4—266no—
180C23×C4—266no—
181C2×M16—266no—
182C22×C8—131no—
183C22×C8—133no—
184C22×C8—1312no—
185C22×C8—131no—
186C22×C8—133no—
187C22×C8—2612no—
188C22×C8—2612no—
189C22×C8—266no—
190C22×C8—2632no—
191#120—264no—
192C4×D4—266no—
193C4×D4—266no—
194C2×C4⋊C4—133no—
195C2×C22⋊C4—133no—
196C2×C22⋊C4—133no—
197C2×C22⋊C4C2×#26113yes—
198C2×C22⋊C4C2×#26113yes—
199C2×C42—133no—
200C4⋊C8—266no—
201C22⋊C8—266no—
202C22⋊C8—266no—
203C22⋊C8—266no—
204#97—266no—
205#95—136no—
206C2×C13—212no—
207C2×C13C4×#12011yescharacteristic
208C2×C13C2×#43513yes—
209C2×C13#292913yes—
210D13—81no—
211C24—26no—
212C24—26no—
213C24—131no—
214C24#611814yescomplemented
215C24C22×#26113yes—
216C2×D4—266no—
217C2×D4—266no—
218C2×D4—266no—
219C22×C4—131no—
220C22×C4—136no—
221C22×C4—136no—
222C22×C4—133no—
223C22×C4—133no—
224C22×C4—263no—
225C22×C4—263no—
226C22×C4—266no—
227C22×C4—266no—
228C22×C4—2612no—
229C22×C4—268no—
230C22×C4—2624no—
231C22×C4—266no—
232C22×C4C22×#26113yes—
233C22×C4C22×#26111yescharacteristic
234M16—524no—
235C2×C8—136no—
236C2×C8—136no—
237C2×C8—2624no—
238C2×C8—266no—
239C2×C8—2624no—
240C2×C8—263no—
241C2×C8—263no—
242C2×C8—266no—
243C4⋊C4—266no—
244C22⋊C4—26no—
245C22⋊C4—26no—
246C22⋊C4—266no—
247C22⋊C4—266no—
248C42—263no—
249C42—263no—
250C13#2144011yescharacteristic, complemented, Sylow
251C23—28no—
252C23—224no—
253C23—224no—
254C23—26no—
255C23—26no—
256C23—261no—
257C23—263no—
258C23—263no—
259C23C2×#611816yes—
260C23#8823816yes—
261C23D4×#26116yes—
262C23C23×#26111yescharacteristic
263D4—5212no—
264C2×C4—26no—
265C2×C4—23no—
266C2×C4—23no—
267C2×C4—133no—
268C2×C4—133no—
269C2×C4—136no—
270C2×C4—266no—
271C2×C4—266no—
272C2×C4—2612no—
273C2×C4—2624no—
274C2×C4—266no—
275C2×C4—263no—
276C2×C4—263no—
277C2×C4—263no—
278C2×C4—521no—
279C2×C4—523no—
280C8—134no—
281C8—266no—
282C8—523no—
283C8—521no—
284C22—232no—
285C22—212no—
286C22—212no—
287C22—26no—
288C22—26no—
289C22—523no—
290C22—523no—
291C22—521no—
292C22#11719064112yescomplemented
293C22C22×#611811yescharacteristic
294C22C2×#8823813yes—
295C22C2×D4×#26113yes—
296C4—43no—
297C4—41no—
298C4—131no—
299C4—133no—
300C4—266no—
301C4—523no—
302C2—212no—
303C2—1041no—
304C2#120×#26111yescharacteristic
305C2C2×#1171906413yes—
306C2#1313155113yes—
subgroupG/H
center, FrattiniC23C23×#261
commutator, socleC23×C13C23×C4
FittingC13×#379C4
prime radicalC2×#88207C4
cosocleC22×Dic13C24
hypercenter#379#261
hypocenterC13#21440
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C24×#1057C2semidirect
2. C2×#11719064C2semidirect
3. #13130811C2semidirect
4. #13130779C2semidirect
5. #13130658C2semidirect
6. #13130573C2semidirect
12. C22×#6118C22semidirect
17. C2×#88111C22semidirect
18. C2×#88110C22semidirect
19. C2×C4⋊#1057C22semidirect
25. #11718290C22semidirect
37. C22×#1057D4semidirect
61. C24×C13M16semidirect
137. #1057#120semidirect
161. C22×C13#435semidirect
214. C24#6118semidirect
250. C13#21440semidirect
292. C22#11719064semidirect

Maximal subgroups

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

subgroupindex
C24×#10572
C2×#117190642
#131308112
#131307792
#131306582
#131305732
#131173352
#2144013