Finite Group Databasebeta

← #15848102

Subgroups — #15848102

Order 3328.

654 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13124629C211yesmaximal, characteristic
2C2×#11718748C211yesmaximal, characteristic
3#13123859C211yesmaximal, characteristic
4#13114284C214yesmaximal, complemented
5#13114198C214yesmaximal, complemented
6#13113611C211yesmaximal, characteristic, complemented
7#13113521C211yesmaximal, characteristic, complemented
8#13113493C211yesmaximal, characteristic, complemented
9C13×#3209C211yesmaximal, characteristic, complemented
10C13×#533C2211yescharacteristic
11C2×C13×C4⋊Q8C2211yescharacteristic
12C2×C4×D52C2211yescharacteristic
13C13×#494C2212yescomplemented
14C13×#487C2212yescomplemented
15C13×#462—22no—
16C13×#451—22no—
17C13×#431C2211yescharacteristic
18#11718748—22no—
19#11718748C2212yes—
20#11718748C2214yes—
21#11718745C2212yes—
22#11718727—22no—
23#11718662—21no—
24#11718656—21no—
25C2×#88049—21no—
26C2×#88047—21no—
27#11718586C2211yescharacteristic
28#11718585C2211yescharacteristic
29C2×C4⋊#1055C2211yescharacteristic
30#11718495—22no—
31D13×C22⋊C8—22no—
32C8×D52C2214yes—
33#11718139C2212yescomplemented
34#11718139C2214yescomplemented
35#11718138C2212yescomplemented
36#11718050C2214yescomplemented
37C22×Q8×C13—21no—
38C22×D52D411yescharacteristic
39C22×C4×D13—21no—
40C2×C13×Q16—24no—
41C13×C4⋊Q8—21no—
42C13×C4⋊Q8—21no—
43C13×C4⋊Q8—21no—
44C13×C4⋊Q8C2311yescharacteristic
45C13×C4⋊Q8C2312yes—
46C13×C4⋊Q8C2311yescharacteristic
47C13×#125—21no—
48C13×C22⋊Q8—21no—
49C13×C22⋊Q8—21no—
50C4×Q8×C13—21no—
51C4×Q8×C13—21no—
52C13×#117—21no—
53C2×C13×C4⋊C4—21no—
54C2×C13×C4⋊C4—21no—
55C2×C42×C13C2311yescharacteristic
56C2×#6103—21no—
57C2×C4⋊Dic13D411yescharacteristic
58C2×#6099—21no—
59C22×#1055—21no—
60C2×C8×D13—24no—
61C4×D52—21no—
62C4×D52—21no—
63C4×D52D412yes—
64C4×D52C2312yes—
65C13×C4⋊C8D412yes—
66C13×#103—24no—
67C13×C22⋊C8D412yes—
68C4×C8×C13C2312yes—
69C22⋊#1055—21no—
70#88060—24no—
71#88049—22no—
72#88049—22no—
73#88049—24no—
74#88048—22no—
75#88047—22no—
76#88047—24no—
77C4⋊#1055—21no—
78C4⋊#1055—21no—
79C4⋊#1055C2312yes—
80C4⋊#1055C2312yes—
81C4⋊#1055C2312yes—
82C4×#1055—21no—
83#19387—131nomaximal, Sylow
84C23×D13—21no—
85C2×Q8×C13—22no—
86C2×Q8×C13—21no—
87C2×Q8×C13—21no—
88C2×Q8×C13—41no—
89C2×Q8×C13—82no—
90C22×C4×C13—41no—
91C22×C4×C13C2×D411yescharacteristic
92C22×C4×C13C2×D411yescharacteristic
93C22×C4×C13#4111yescharacteristic
94C22×Dic13—21no—
95C2×D52—21no—
96C2×D52—22no—
97C2×D52Q1614yescomplemented
98C2×D52C2×D412yes—
99C2×C4×D13—22no—
100C2×C4×D13—22no—
101C2×C4×D13—82no—
102C13×Q16—84no—
103C2×C8×C13C2×D414yes—
104C13×C4⋊C4—21no—
105C13×C4⋊C4—22no—
106C13×C4⋊C4—21no—
107C13×C4⋊C4—21no—
108C13×C4⋊C4—22no—
109C13×C4⋊C4—21no—
110C13×C4⋊C4—41no—
111C13×C4⋊C4—41no—
112C13×C4⋊C4—41no—
113C13×C4⋊C4—41no—
114C13×C22⋊C4—41no—
115C42×C13—41no—
116C42×C13C2×D411yescharacteristic
117C42×C13C2×D411yescharacteristic
118C42×C13#4111yescharacteristic
119C42×C13C2411yescharacteristic
120#6103—22no—
121#6103—22no—
122C4⋊Dic13—21no—
123C4⋊Dic13C2×D412yes—
124#6099—42no—
125C2×#1055—22no—
126C2×#1055—22no—
127C2×#1055—22no—
128C2×#1055—42no—
129C8×D13—84no—
130#3209D1311yescharacteristic, complemented
131#3077—131no—
132C2×#474—131no—
133#2972—131no—
134#1683—134no—
135#1669—134no—
136#1571—131no—
137#1549—131no—
138#1542—131no—
139C23×C13#12111yescharacteristic
140C22×D13—22no—
141C22×D13—24no—
142C22×D13—81no—
143C22×D13—81no—
144Q8×C13—42no—
145Q8×C13—44no—
146Q8×C13—82no—
147Q8×C13—81no—
148C2×C4×C13—22no—
149C2×C4×C13—41no—
150C2×C4×C13—41no—
151C2×C4×C13—41no—
152C2×C4×C13—42no—
153C2×C4×C13—82no—
154C2×C4×C13#12111yescharacteristic
155C2×C4×C13#12111yescharacteristic
156C2×C4×C13#12111yescharacteristic
157C2×C4×C13C2×Q1612yes—
158C2×C4×C13#13512yes—
159C2×C4×C13C22×D411yescharacteristic
160C2×C4×C13C22×D411yescharacteristic
161C2×C4×C13C2×#4111yescharacteristic
162C2×Dic13—22no—
163C2×Dic13—81no—
164D52—22no—
165D52—24no—
166D52C2×Q1614yes—
167C4×D13—44no—
168C4×D13—82no—
169C8×C13—24no—
170#1055—42no—
171#1055—42no—
172#1055—42no—
173#533C2×D1311yescharacteristic
174C2×C4⋊Q8C2×D1311yescharacteristic
175C2×C4×D4—131no—
176#494C2×D1312yescomplemented
177#487C2×D1312yescomplemented
178#478—262no—
179#476—132no—
180#474—132no—
181#474—134no—
182#474—262no—
183#462—22no—
184#451—22no—
185C8×D4—134no—
186C8×D4—262no—
187#431—131no—
188#431C2×D1311yescharacteristic
189#428—261no—
190C2×#107—261no—
191#423—131no—
192C2×C4⋊C8—131no—
193#418—261no—
194C2×#102—261no—
195C2×C22⋊C8—262no—
196#335—134no—
197#333—132no—
198#332—132no—
199#332—134no—
200C22×C13—21no—
201C22×C13—21no—
202C22×C13#46712yes—
203C22×C13C2×#12111yescharacteristic
204C2×D13—24no—
205C2×D13—42no—
206C2×D13—42no—
207C2×D13—84no—
208C4×C13—21no—
209C4×C13—21no—
210C4×C13—21no—
211C4×C13—82no—
212C4×C13—81no—
213C4×C13—81no—
214C4×C13#46712yes—
215C4×C13C22×Q1611yescharacteristic
216C4×C13C2×#13511yescharacteristic
217C4×C13#57612yes—
218Dic13—42no—
219C22×Q8—21no—
220C22×D4—131no—
221C23×C4—261no—
222C2×Q16—24no—
223C2×M16—261no—
224C22×C8—261no—
225C22×C8—264no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8C22×D1312yes—
230C4⋊Q8C22×D1311yescharacteristic
231C4⋊Q8C22×D1311yescharacteristic
232#125—21no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×D4—132no—
238C4×D4—132no—
239C4×D4—261no—
240C4×D4—261no—
241#117—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—131no—
245C2×C22⋊C4—261no—
246C2×C42C22×D1311yescharacteristic
247#107—264no—
248#107—262no—
249#106—262no—
250C4⋊C8—132no—
251C4⋊C8—132no—
252C4⋊C8—132no—
253C4⋊C8—261no—
254C4⋊C8—261no—
255C4⋊C8C22⋊D1312yes—
256#103—24no—
257#102—264no—
258#102—262no—
259#102—262no—
260C22⋊C8—261no—
261C22⋊C8—264no—
262C22⋊C8C22⋊D1312yes—
263C4×C8—261no—
264C4×C8C22×D1312yes—
265C2×C13—22no—
266C2×C13#304111yescharacteristic
267C2×C13C2×#46711yescharacteristic
268C2×C13#306911yescharacteristic
269D13—44no—
270C24—261no—
271C2×Q8—21no—
272C2×Q8—22no—
273C2×Q8—21no—
274C2×Q8—41no—
275C2×Q8—82no—
276C2×D4—134no—
277C2×D4—132no—
278C2×D4—262no—
279C2×D4—261no—
280C22×C4—41no—
281C22×C4—261no—
282C22×C4—262no—
283C22×C4—262no—
284C22×C4—1042no—
285C22×C4D4×D1311yescharacteristic
286C22×C4#612911yescharacteristic
287C22×C4C2×C22⋊D1311yescharacteristic
288Q16—84no—
289M16—522no—
290C2×C8—262no—
291C2×C8—262no—
292C2×C8—262no—
293C2×C8—522no—
294C2×C8—1044no—
295C2×C8C2×C22⋊D1314yes—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—21no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—132no—
307C4⋊C4—261no—
308C22⋊C4—41no—
309C22⋊C4—262no—
310C22⋊C4—262no—
311C42—41no—
312C42D4×D1311yescharacteristic
313C42#612911yescharacteristic
314C42C2×C22⋊D1311yescharacteristic
315C42C23×D1311yescharacteristic
316C13#1938711yescharacteristic, complemented, Sylow
317C23—262no—
318C23—264no—
319C23—1041no—
320C23—1041no—
321C23#8819211yescharacteristic
322Q8—42no—
323Q8—44no—
324Q8—81no—
325Q8—82no—
326D4—134no—
327D4—262no—
328D4—264no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—82no—
335C2×C4—262no—
336C2×C4—524no—
337C2×C4—1042no—
338C2×C4—1041no—
339C2×C4Q16×D1312yes—
340C2×C4#8817312yes—
341C2×C4#8819211yescharacteristic
342C2×C4#8819211yescharacteristic
343C2×C4#8819211yescharacteristic
344C2×C4C2×D4×D1311yescharacteristic
345C2×C4C2×#612911yescharacteristic
346C2×C4C22×C22⋊D1311yescharacteristic
347C8—24no—
348C8—522no—
349C8—522no—
350C8—522no—
351C22—21no—
352C22—21no—
353C22—264no—
354C22—522no—
355C22—522no—
356C22—1044no—
357C22#1171884912yes—
358C22C2×#8819211yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—81no—
363C4—82no—
364C4—81no—
365C4—522no—
366C4#1171884912yes—
367C4C2×Q16×D1311yescharacteristic
368C4C2×#8817311yescharacteristic
369C4#1171954012yes—
370C2—22no—
371C2—524no—
372C2#1312433511yescharacteristic
373C2C2×#1171884911yescharacteristic
374C2#1312456911yescharacteristic
subgroupG/H
centerC22C2×#88192
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC13×#3209C2
socleC22×C13C2×#121
prime radicalC22×D52D4
hypercenter#3209D13
hypocenterC13#19387
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
4. #13114284C2semidirect
5. #13114198C2semidirect
6. #13113611C2semidirect
7. #13113521C2semidirect
8. #13113493C2semidirect
9. C13×#3209C2semidirect
13. C13×#494C22semidirect
14. C13×#487C22semidirect
33. #11718139C22semidirect
34. #11718139C22semidirect
35. #11718138C22semidirect
36. #11718050C22semidirect
97. C2×D52Q16semidirect
130. #3209D13semidirect
176. #494C2×D13semidirect
177. #487C2×D13semidirect
316. C13#19387semidirect

Maximal subgroups

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

subgroupindex
#131246292
C2×#117187482
#131238592
#131142842
#131141982
#131136112
#131135212
#131134932
C13×#32092
#1938713