Finite Group Databasebeta

← #16800258

Subgroups — #16800258

Order 3328.

898 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719080C211yesmaximal, characteristic, complemented
2C2×#11719076C211yesmaximal, characteristic
3#13131006C211yesmaximal, characteristic, complemented
4#13130958C211yesmaximal, characteristic
5#13130908C214yesmaximal, complemented
6#13130751C211yesmaximal, characteristic, complemented
7#13130577C214yesmaximal, complemented
8C42×#1057C211yesmaximal, characteristic, complemented
9#13117375C211yesmaximal, characteristic
10C22×#6118C2212yes—
11C2×#88147C2211yescharacteristic
12C2×#88141C2211yescharacteristic
13C2×#88129C411yescharacteristic
14C2×#88111C2212yes—
15C2×#88111C2214yes—
16C2×#88110C2212yes—
17C2×C4⋊#1057C2211yescharacteristic
18C2×C4⋊#1057C2212yes—
19C2×C4⋊#1057C2211yescharacteristic
20C2×C4×#1057C2212yes—
21C2×C4×#1057C2212yes—
22C2×C4×#1057C2211yescharacteristic
23C2×C4×#1057C2211yescharacteristic
24#11719080—24no—
25#11719076—24no—
26C13×#382C411yescharacteristic
27#11718649C411yescharacteristic
28#11718635C411yescharacteristic
29C42×Dic13C2211yescharacteristic
30#11718330C414yes—
31#11718327C2214yes—
32#11718290C2214yes—
33#11718290C2214yes—
34C22×D52C2×C411yescharacteristic
35C22×C4×D13C2312yes—
36C22×Dic26C2×C411yescharacteristic
37C22×#1057C2312yes—
38C22×#1057C2312yes—
39C22×#1057C2311yescharacteristic
40C22×#1057C2312yes—
41C22×#1057C2311yescharacteristic
42C2×#6118—24no—
43C2×#6118—28no—
44C2×C13×C4⋊C4C2×C411yescharacteristic
45C2×C13×C4⋊C4C2×C411yescharacteristic
46C2×C42×C13C2×C411yescharacteristic
47C2×#6103C2×C414yes—
48C2×#6103C2×C412yes—
49C2×#6103C2×C414yes—
50C2×#6101C2×C412yes—
51C2×C4×Dic13C2×C412yes—
52C2×C4×Dic13C2×C411yescharacteristic
53C2×C4×Dic13C2312yes—
54C2×C4×Dic13C2312yes—
55C2×C4×Dic13C2311yescharacteristic
56#88147—22no—
57#88147—22no—
58#88141—22no—
59#88141—22no—
60#88129—24no—
61#88111—24no—
62#88111—28no—
63#88110—24no—
64C4⋊#1057—24no—
65C4⋊#1057—22no—
66C4⋊#1057—22no—
67C4×#1057—24no—
68C4×#1057—24no—
69C4×#1057—22no—
70C4×#1057D414yes—
71C13×#95C2×C414yes—
72#88071C2×C414yes—
73#21631—131nomaximal, Sylow
74C23×D13C22×C412yes—
75C22×C4×C13C22×C412yes—
76C22×C4×C13C22×C412yes—
77C22×C4×C13C22×C412yes—
78C22×C4×C13C22×C411yescharacteristic
79C22×Dic13C22×C412yes—
80C22×Dic13C22×C411yescharacteristic
81C22×Dic13C22×C412yes—
82C22×Dic13C2411yescharacteristic
83C2×D52—22no—
84C2×D52—22no—
85C2×D52—22no—
86C2×C4×D13—22no—
87C2×C4×D13—24no—
88C2×C4×D13—22no—
89C2×C4×D13—24no—
90C2×Dic26—22no—
91C2×Dic26—22no—
92C2×Dic26—22no—
93C2×#1057—22no—
94C2×#1057—22no—
95C2×#1057—22no—
96C2×#1057—22no—
97C2×#1057—22no—
98C2×#1057—24no—
99C2×#1057—24no—
100C2×#1057C2×D412yes—
101C2×#1057C2×D412yes—
102C2×#1057#4114yes—
103C2×#1057#4114yes—
104#6118—48no—
105C13×C4⋊C4—22no—
106C13×C4⋊C4—22no—
107C42×C13M1614yescomplemented
108#6103—28no—
109#6103—24no—
110#6103—28no—
111#6101—24no—
112C4×Dic13—22no—
113C4×Dic13—24no—
114C4×Dic13—22no—
115C4×Dic13—22no—
116C4×Dic13—22no—
117C4×Dic13—22no—
118C4×Dic13C2×D412yes—
119C4×Dic13#4112yes—
120C2×#446—131no—
121C2×#436—131no—
122C4×#124—131no—
123#1984—131no—
124#1954—131no—
125#1923—134no—
126#1831—131no—
127#1753—134no—
128C42×C8—131no—
129C23×C13C23×C411yescharacteristic
130C22×D13—22no—
131C22×D13—24no—
132C22×D13—22no—
133C22×D13—24no—
134C22×D13—22no—
135C2×C4×C13—22no—
136C2×C4×C13—22no—
137C2×C4×C13—22no—
138C2×C4×C13—22no—
139C2×C4×C13—24no—
140C2×C4×C13—24no—
141C2×C4×C13C4×D412yes—
142C2×C4×C13C2×M1614yes—
143C2×C4×C13C2×M1612yes—
144C2×C4×C13#13112yes—
145C2×Dic13—22no—
146C2×Dic13—22no—
147C2×Dic13—22no—
148C2×Dic13—22no—
149C2×Dic13—22no—
150C2×Dic13—22no—
151C2×Dic13C4×D412yes—
152C2×Dic13#13114yes—
153C2×Dic13C22×D411yescharacteristic
154C2×Dic13C2×#4112yes—
155C2×Dic13C2×#4111yescharacteristic
156C2×Dic13C2×#4112yes—
157D52—44no—
158C4×D13—42no—
159C4×D13—44no—
160C4×D13—42no—
161Dic26—44no—
162#1057—24no—
163#1057—22no—
164#1057—42no—
165#1057—42no—
166#1057#12414yescomplemented
167C22×M16—132no—
168C2×#124—131no—
169C2×C4×Q8—131no—
170C2×C4×D4—131no—
171#446—264no—
172#436—264no—
173C2×C4⋊C8—131no—
174C2×C4⋊C8—131no—
175C2×C4⋊C8—132no—
176C2×C22⋊C8—132no—
177C2×C22⋊C8—134no—
178C2×#97—132no—
179C2×C4×C8—131no—
180C2×C4×C8—132no—
181C2×C4×C8—131no—
182C2×C4×C8—132no—
183#391—131no—
184#390—131no—
185#388—134no—
186#382#26111yescharacteristic
187C4×C22⋊C4—134no—
188C43—131no—
189#336—134no—
190#336—134no—
191C22×C13C2×C4×D411yescharacteristic
192C22×C13C4×#4112yes—
193C22×C13C22×M1611yescharacteristic
194C22×C13C2×#13111yescharacteristic
195C22×C13C2×#13112yes—
196C2×D13—42no—
197C2×D13—42no—
198C2×D13—44no—
199C2×D13—44no—
200C2×D13—42no—
201C4×C13—24no—
202C4×C13—42no—
203C4×C13—42no—
204C4×C13#43614yes—
205Dic13—22no—
206Dic13—24no—
207Dic13—42no—
208Dic13C2×#12411yescharacteristic
209Dic13#52912yes—
210Dic13#53211yescharacteristic
211C22×Q8—131no—
212C22×D4—131no—
213C23×C4—132no—
214C2×M16—268no—
215C2×M16—264no—
216C22×C8—131no—
217C22×C8—132no—
218C22×C8—131no—
219C22×C8—132no—
220C22×C8—132no—
221#124—264no—
222C4×Q8—262no—
223C4×Q8—262no—
224C4×D4—262no—
225C4×D4—262no—
226C2×C4⋊C4—132no—
227C2×C4⋊C4C2×#26111yescharacteristic
228C2×C4⋊C4C2×#26111yescharacteristic
229C2×C22⋊C4—132no—
230C2×C22⋊C4—134no—
231C2×C22⋊C4—134no—
232C2×C42—131no—
233C2×C42—132no—
234C2×C42—132no—
235C2×C42—132no—
236C2×C42—131no—
237C2×C42C2×#26111yescharacteristic
238C4⋊C8—262no—
239C4⋊C8—262no—
240C4⋊C8—264no—
241C22⋊C8—268no—
242C22⋊C8—264no—
243#97—264no—
244C4×C8—134no—
245C4×C8—262no—
246C4×C8—264no—
247C4×C8—264no—
248#95—134no—
249#95C2×#26114yes—
250C2×C13C4×#12411yescharacteristic
251C2×C13C2×#43611yescharacteristic
252C2×C13#293111yescharacteristic
253C2×C13#296512yes—
254C2×C13#296612yes—
255D13—82no—
256C24—132no—
257C2×Q8—262no—
258C2×Q8—262no—
259C2×Q8—262no—
260C2×D4—262no—
261C2×D4—262no—
262C2×D4—262no—
263C22×C4—131no—
264C22×C4—132no—
265C22×C4—131no—
266C22×C4—132no—
267C22×C4—262no—
268C22×C4—264no—
269C22×C4—262no—
270C22×C4—264no—
271C22×C4C22×#26112yes—
272C22×C4C22×#26112yes—
273C22×C4C22×#26112yes—
274C22×C4C22×#26111yescharacteristic
275M16—528no—
276C2×C8—132no—
277C2×C8—134no—
278C2×C8—132no—
279C2×C8—134no—
280C2×C8—262no—
281C2×C8—262no—
282C2×C8—262no—
283C2×C8—262no—
284C2×C8—262no—
285C2×C8—264no—
286C2×C8—264no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—264no—
290C22⋊C4—264no—
291C22⋊C4—268no—
292C22⋊C4—268no—
293C42—132no—
294C42—132no—
295C42—262no—
296C42—262no—
297C42—262no—
298C42—262no—
299C42—264no—
300C42—262no—
301C42#611814yescomplemented
302C13#2163111yescharacteristic, complemented, Sylow
303C23—262no—
304C23—262no—
305C23—262no—
306C23—264no—
307C23—264no—
308C23C23×#26111yescharacteristic
309Q8—524no—
310D4—524no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—131no—
318C2×C4—131no—
319C2×C4—132no—
320C2×C4—132no—
321C2×C4—132no—
322C2×C4—134no—
323C2×C4—262no—
324C2×C4—262no—
325C2×C4—262no—
326C2×C4—262no—
327C2×C4—262no—
328C2×C4—262no—
329C2×C4—522no—
330C2×C4—524no—
331C2×C4—522no—
332C2×C4C2×#611814yes—
333C2×C4C2×#611812yes—
334C2×C4D4×#26112yes—
335C2×C4#8824012yes—
336C8—134no—
337C8—264no—
338C8—262no—
339C8—522no—
340C8—522no—
341C22—522no—
342C22—522no—
343C22—522no—
344C22—524no—
345C22—524no—
346C22C22×#611811yescharacteristic
347C22C2×D4×#26111yescharacteristic
348C22C2×#8824011yescharacteristic
349C22D13⋊#13112yes—
350C22#41×#26112yes—
351C4—24no—
352C4—42no—
353C4—42no—
354C4—131no—
355C4—132no—
356C4—131no—
357C4—262no—
358C4—264no—
359C4—522no—
360C4#1171907614yes—
361C2—1042no—
362C2#124×#26111yescharacteristic
363C2C2×#1171907611yescharacteristic
364C2#1313156011yescharacteristic
365C2#1313164312yes—
366C2#1313165312yes—
subgroupG/H
center, FrattiniC23C23×#261
commutator, socleC23×C13C23×C4
FittingC13×#382C4
prime radicalC22×D52C2×C4
cosocleC22×Dic13C24
hypercenter#382#261
hypocenterC13#21631
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. C2×#11719080C2semidirect
3. #13131006C2semidirect
5. #13130908C2semidirect
6. #13130751C2semidirect
7. #13130577C2semidirect
8. C42×#1057C2semidirect
107. C42×C13M16semidirect
166. #1057#124semidirect
301. C42#6118semidirect
302. C13#21631semidirect

Maximal subgroups

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

subgroupindex
C2×#117190802
C2×#117190762
#131310062
#131309582
#131309082
#131307512
#131305772
C42×#10572
#131173752
#2163113