Finite Group Databasebeta

← #15848133

Subgroups — #15848133

Order 3328.

674 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13124629C211yesmaximal, characteristic
2C2×#11718744C211yesmaximal, characteristic
3#13123859C211yesmaximal, characteristic
4#13114256C214yesmaximal, complemented
5#13114208C214yesmaximal, complemented
6#13113607C211yesmaximal, characteristic, complemented
7#13113508C211yesmaximal, characteristic, complemented
8#13113493C211yesmaximal, characteristic, complemented
9C13×#3229C211yesmaximal, characteristic, complemented
10C13×#533C2211yescharacteristic
11C2×C13×C4⋊Q8C2211yescharacteristic
12C22×#6106—21no—
13C2×C4×D52C2211yescharacteristic
14C13×#498C2212yescomplemented
15C13×#487C2212yescomplemented
16C13×#483—22no—
17C13×#475—22no—
18C13×#431C2211yescharacteristic
19C2×#88075—21no—
20#11718748C2212yes—
21#11718745C2212yes—
22#11718744—22no—
23#11718744C2214yes—
24#11718727—22no—
25#11718662—21no—
26#11718656—21no—
27#11718586C2211yescharacteristic
28#11718585C2211yescharacteristic
29C2×C4⋊#1055C2211yescharacteristic
30#11718495—22no—
31D13×C22⋊C8—22no—
32C8×D52C2214yes—
33#11718139C2214yescomplemented
34#11718139C2212yescomplemented
35#11718138C2212yescomplemented
36#11718049C2214yescomplemented
37C22×Q8×C13—21no—
38C22×D52D411yescharacteristic
39C22×C4×D13—21no—
40C13×C4⋊Q8—21no—
41C13×C4⋊Q8—21no—
42C13×C4⋊Q8—21no—
43C13×C4⋊Q8C2312yes—
44C13×C4⋊Q8C2311yescharacteristic
45C13×C4⋊Q8C2311yescharacteristic
46C13×#125—21no—
47C13×C22⋊Q8—21no—
48C13×C22⋊Q8—21no—
49C4×Q8×C13—21no—
50C4×Q8×C13—21no—
51C13×#117—21no—
52C2×C13×C4⋊C4—21no—
53C2×C13×C4⋊C4—21no—
54C2×C42×C13C2311yescharacteristic
55C2×#6106—24no—
56C2×#6106—44no—
57C2×#6103—21no—
58C2×C4⋊Dic13D411yescharacteristic
59C2×#6099—21no—
60C22×#1055—21no—
61C2×C8×D13—24no—
62C4×D52—21no—
63C4×D52—21no—
64C4×D52D412yes—
65C4×D52C2312yes—
66C13×#106—24no—
67C13×C4⋊C8D412yes—
68C13×#103—24no—
69C13×C22⋊C8D412yes—
70C4×C8×C13C2312yes—
71#88075—24no—
72C22⋊#1055—21no—
73#88060—24no—
74#88049—22no—
75#88049—22no—
76#88048—22no—
77#88047—22no—
78C4⋊#1055—21no—
79C4⋊#1055—21no—
80C4⋊#1055C2312yes—
81C4⋊#1055C2312yes—
82C4⋊#1055C2312yes—
83C4×#1055—21no—
84#19390—131nomaximal, Sylow
85C23×D13—21no—
86C2×Q8×C13—22no—
87C2×Q8×C13—21no—
88C2×Q8×C13—21no—
89C2×Q8×C13—41no—
90C2×Q8×C13—42no—
91C2×Q8×C13—42no—
92C22×C4×C13—41no—
93C22×C4×C13C2×D411yescharacteristic
94C22×C4×C13C2×D411yescharacteristic
95C22×C4×C13#4111yescharacteristic
96C22×Dic13—21no—
97C2×D52—21no—
98C2×D52—22no—
99C2×D52SD1614yes—
100C2×D52C2×D412yes—
101C2×C4×D13—22no—
102C2×C4×D13—22no—
103C2×C4×D13—82no—
104C2×C8×C13C2×D414yes—
105C13×C4⋊C4—22no—
106C13×C4⋊C4—21no—
107C13×C4⋊C4—21no—
108C13×C4⋊C4—21no—
109C13×C4⋊C4—22no—
110C13×C4⋊C4—21no—
111C13×C4⋊C4—41no—
112C13×C4⋊C4—41no—
113C13×C4⋊C4—41no—
114C13×C4⋊C4—41no—
115C13×C22⋊C4—41no—
116C42×C13—41no—
117C42×C13C2×D411yescharacteristic
118C42×C13C2×D411yescharacteristic
119C42×C13#4111yescharacteristic
120C42×C13C2411yescharacteristic
121#6106—48no—
122#6103—22no—
123#6103—22no—
124C4⋊Dic13—21no—
125C4⋊Dic13C2×D412yes—
126#6099—42no—
127C2×#1055—22no—
128C2×#1055—22no—
129C2×#1055—22no—
130C2×#1055—42no—
131C8×D13—84no—
132#3229D1311yescharacteristic, complemented
133#3077—131no—
134C2×#461—131no—
135#2972—131no—
136#1679—134no—
137#1671—134no—
138#1570—131no—
139#1546—131no—
140#1542—131no—
141C23×C13#12111yescharacteristic
142C22×D13—22no—
143C22×D13—24no—
144C22×D13—81no—
145C22×D13—81no—
146Q8×C13—42no—
147Q8×C13—42no—
148Q8×C13—42no—
149Q8×C13—81no—
150Q8×C13—82no—
151C2×C4×C13—22no—
152C2×C4×C13—41no—
153C2×C4×C13—41no—
154C2×C4×C13—41no—
155C2×C4×C13—42no—
156C2×C4×C13—82no—
157C2×C4×C13#12111yescharacteristic
158C2×C4×C13#12111yescharacteristic
159C2×C4×C13#12111yescharacteristic
160C2×C4×C13C2×SD1612yes—
161C2×C4×C13#13512yes—
162C2×C4×C13C22×D411yescharacteristic
163C2×C4×C13C22×D411yescharacteristic
164C2×C4×C13C2×#4111yescharacteristic
165C2×Dic13—22no—
166C2×Dic13—81no—
167D52—22no—
168D52—24no—
169D52C2×SD1614yes—
170C4×D13—44no—
171C4×D13—82no—
172C8×C13—24no—
173#1055—42no—
174#1055—42no—
175#1055—42no—
176C22×SD16—261no—
177#533C2×D1311yescharacteristic
178C2×C4⋊Q8C2×D1311yescharacteristic
179C2×C4×D4—131no—
180#498C2×D1312yescomplemented
181#487C2×D1312yescomplemented
182#483—22no—
183#478—262no—
184#476—132no—
185#475—22no—
186#474—132no—
187#461—134no—
188#461—262no—
189C8×D4—134no—
190C8×D4—262no—
191#431—131no—
192#431C2×D1311yescharacteristic
193#428—261no—
194#423—131no—
195C2×C4⋊C8—131no—
196#418—261no—
197C2×#103—261no—
198C2×C22⋊C8—262no—
199#334—134no—
200#333—132no—
201#332—132no—
202#332—134no—
203C22×C13—21no—
204C22×C13—21no—
205C22×C13#46512yes—
206C22×C13C2×#12111yescharacteristic
207C2×D13—24no—
208C2×D13—42no—
209C2×D13—42no—
210C2×D13—84no—
211C4×C13—21no—
212C4×C13—21no—
213C4×C13—21no—
214C4×C13—81no—
215C4×C13—82no—
216C4×C13—81no—
217C4×C13#46512yes—
218C4×C13C22×SD1611yescharacteristic
219C4×C13C2×#13511yescharacteristic
220C4×C13#57712yes—
221Dic13—42no—
222C22×Q8—21no—
223C22×D4—131no—
224C23×C4—261no—
225C2×SD16—264no—
226C2×SD16—524no—
227C2×M16—261no—
228C22×C8—261no—
229C22×C8—264no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8C22×D1312yes—
234C4⋊Q8C22×D1311yescharacteristic
235C4⋊Q8C22×D1311yescharacteristic
236#125—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×D4—132no—
242C4×D4—132no—
243C4×D4—261no—
244C4×D4—261no—
245#117—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—131no—
249C2×C22⋊C4—261no—
250C2×C42C22×D1311yescharacteristic
251#107—262no—
252#106—24no—
253#106—262no—
254C4⋊C8—132no—
255C4⋊C8—132no—
256C4⋊C8—132no—
257C4⋊C8—261no—
258C4⋊C8—261no—
259C4⋊C8C22⋊D1312yes—
260#103—24no—
261#103—264no—
262#102—262no—
263#102—262no—
264C22⋊C8—261no—
265C22⋊C8—264no—
266C22⋊C8C22⋊D1312yes—
267C4×C8—261no—
268C4×C8C22×D1312yes—
269C2×C13—22no—
270C2×C13#304211yescharacteristic
271C2×C13C2×#46511yescharacteristic
272C2×C13#306911yescharacteristic
273D13—44no—
274C24—261no—
275C2×Q8—22no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—41no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×D4—134no—
282C2×D4—132no—
283C2×D4—262no—
284C2×D4—261no—
285C22×C4—41no—
286C22×C4—261no—
287C22×C4—262no—
288C22×C4—262no—
289C22×C4—1042no—
290C22×C4D4×D1311yescharacteristic
291C22×C4#612911yescharacteristic
292C22×C4C2×C22⋊D1311yescharacteristic
293SD16—528no—
294M16—522no—
295C2×C8—262no—
296C2×C8—262no—
297C2×C8—262no—
298C2×C8—522no—
299C2×C8—1044no—
300C2×C8C2×C22⋊D1314yes—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—132no—
312C4⋊C4—261no—
313C22⋊C4—41no—
314C22⋊C4—262no—
315C22⋊C4—262no—
316C42—41no—
317C42D4×D1311yescharacteristic
318C42#612911yescharacteristic
319C42C2×C22⋊D1311yescharacteristic
320C42C23×D1311yescharacteristic
321C13#1939011yescharacteristic, complemented, Sylow
322C23—262no—
323C23—264no—
324C23—1041no—
325C23—1041no—
326C23#8819211yescharacteristic
327Q8—42no—
328Q8—42no—
329Q8—42no—
330Q8—81no—
331Q8—82no—
332D4—134no—
333D4—262no—
334D4—264no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—82no—
341C2×C4—262no—
342C2×C4—524no—
343C2×C4—1042no—
344C2×C4—1041no—
345C2×C4SD16×D1312yes—
346C2×C4#8817012yes—
347C2×C4#8819211yescharacteristic
348C2×C4#8819211yescharacteristic
349C2×C4#8819211yescharacteristic
350C2×C4C2×D4×D1311yescharacteristic
351C2×C4C2×#612911yescharacteristic
352C2×C4C22×C22⋊D1311yescharacteristic
353C8—24no—
354C8—522no—
355C8—522no—
356C8—522no—
357C22—21no—
358C22—21no—
359C22—264no—
360C22—522no—
361C22—522no—
362C22—1044no—
363C22#1171883212yes—
364C22C2×#8819211yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—82no—
369C4—81no—
370C4—81no—
371C4—522no—
372C4#1171883212yes—
373C4C2×SD16×D1311yescharacteristic
374C4C2×#8817011yescharacteristic
375C4#1171954212yes—
376C2—22no—
377C2—524no—
378C2#1312434411yescharacteristic
379C2C2×#1171883211yescharacteristic
380C2#1312456911yescharacteristic
subgroupG/H
centerC22C2×#88192
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC13×#3229C2
socleC22×C13C2×#121
prime radicalC22×D52D4
hypercenter#3229D13
hypocenterC13#19390
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
4. #13114256C2semidirect
5. #13114208C2semidirect
6. #13113607C2semidirect
7. #13113508C2semidirect
8. #13113493C2semidirect
9. C13×#3229C2semidirect
14. C13×#498C22semidirect
15. C13×#487C22semidirect
33. #11718139C22semidirect
34. #11718139C22semidirect
35. #11718138C22semidirect
36. #11718049C22semidirect
132. #3229D13semidirect
180. #498C2×D13semidirect
181. #487C2×D13semidirect
321. C13#19390semidirect

Maximal subgroups

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

subgroupindex
#131246292
C2×#117187442
#131238592
#131142562
#131142082
#131136072
#131135082
#131134932
C13×#32292
#1939013