Finite Group Databasebeta

← #15846215

Subgroups — #15846215

Order 3328.

658 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13125774C211yesmaximal, characteristic
2C2×#11718749C211yesmaximal, characteristic
3#13123771C211yesmaximal, characteristic, complemented
4#13114406C214yesmaximal, complemented
5#13114282C214yesmaximal, complemented
6#13113582C211yesmaximal, characteristic, complemented
7#13113521C211yesmaximal, characteristic, complemented
8#13113503C211yesmaximal, characteristic
9C13×#3209C211yesmaximal, characteristic, complemented
10C13×#533C2211yescharacteristic
11C2×C13×C4⋊Q8C2211yescharacteristic
12C2×#88128C2211yescharacteristic
13C13×#494—21no—
14C13×#487—21no—
15C13×#462C2214yescomplemented
16C13×#451—22no—
17C13×#431C2211yescharacteristic
18#11718749—21no—
19#11718749—21no—
20#11718749—21no—
21#11718749C2212yes—
22#11718749C2212yes—
23#11718746—21no—
24#11718707—22no—
25#11718688C2214yes—
26#11718685—21no—
27#11718685C2212yescomplemented
28Q8×#1055—21no—
29Q8×#1055C2212yescomplemented
30#11718660—21no—
31#11718658—21no—
32C2×#88049—21no—
33C2×#88049—21no—
34#11718586C2211yescharacteristic
35#11718585C2211yescharacteristic
36C2×C4×#1055C2211yescharacteristic
37#11718139C2214yescomplemented
38#11718077C2214yes—
39#11718072C2212yescomplemented
40#11718071C2212yescomplemented
41C22×Q8×C13—21no—
42C22×D52—21no—
43C22×D52—21no—
44C22×D52—21no—
45C2×C13×Q16—24no—
46C13×C4⋊Q8—21no—
47C13×C4⋊Q8—21no—
48C13×C4⋊Q8—21no—
49C13×C4⋊Q8D411yescharacteristic
50C13×C4⋊Q8D411yescharacteristic
51C13×C4⋊Q8C2312yes—
52C13×#125—21no—
53C13×C22⋊Q8—21no—
54C13×C22⋊Q8D412yes—
55C4×Q8×C13—21no—
56C4×Q8×C13C2312yes—
57C13×#117—21no—
58C2×C13×C4⋊C4—21no—
59C2×C13×C4⋊C4—21no—
60C2×C42×C13C2311yescharacteristic
61C2×#6106—24no—
62C2×#6099—21no—
63C22×#1055D411yescharacteristic
64C22×#1055D411yescharacteristic
65#88128—21no—
66#88128—21no—
67#88128—21no—
68#88128C2312yes—
69C13×C4⋊C8C2312yes—
70C13×#103—24no—
71C13×C22⋊C8—21no—
72C4×C8×C13—21no—
73C22⋊#1055—21no—
74#88049—24no—
75#88049—22no—
76#88049—22no—
77#88049—22no—
78#88049—22no—
79C4⋊#1055—21no—
80C4⋊#1055—22no—
81C4⋊#1055—22no—
82C4⋊#1055C2312yes—
83C4⋊#1055C2312yes—
84#88043—22no—
85C4×#1055—21no—
86C4×#1055—22no—
87C4×#1055D411yescharacteristic
88C4×#1055D411yescharacteristic
89C4×#1055C2311yescharacteristic
90C4×#1055C2311yescharacteristic
91#19214—131nomaximal, Sylow
92C23×D13—41no—
93C2×Q8×C13—21no—
94C2×Q8×C13—22no—
95C2×Q8×C13—41no—
96C2×Q8×C13—82no—
97C2×Q8×C13C2×D412yes—
98C22×C4×C13—41no—
99C22×C4×C13C2×D411yescharacteristic
100C22×C4×C13C2×D411yescharacteristic
101C22×C4×C13#4111yescharacteristic
102C2×D52—22no—
103C2×D52—22no—
104C2×D52—22no—
105C2×D52—42no—
106C2×D52—42no—
107C2×D52—41no—
108C2×D52—41no—
109C2×D52—41no—
110C2×D52—82no—
111C2×D52—82no—
112C13×Q16—84no—
113C2×C8×C13—22no—
114C13×C4⋊C4—21no—
115C13×C4⋊C4—21no—
116C13×C4⋊C4—21no—
117C13×C4⋊C4—22no—
118C13×C4⋊C4—22no—
119C13×C4⋊C4—22no—
120C13×C4⋊C4—22no—
121C13×C4⋊C4—41no—
122C13×C4⋊C4—41no—
123C13×C4⋊C4C2×D412yes—
124C13×C22⋊C4—22no—
125C42×C13—22no—
126C42×C13C2×D411yescharacteristic
127C42×C13C2×D411yescharacteristic
128C42×C13#4111yescharacteristic
129C42×C13C2411yescharacteristic
130#6106—84no—
131#6099—42no—
132C2×#1055—22no—
133C2×#1055—22no—
134C2×#1055—22no—
135C2×#1055—22no—
136C2×#1055C2×D411yescharacteristic
137C2×#1055C2×D411yescharacteristic
138C2×#1055C2×D411yescharacteristic
139C2×#1055C2×D411yescharacteristic
140#3209D1311yescharacteristic, complemented
141#3208—131no—
142C2×#486—131no—
143#2960—131no—
144#1702—134no—
145#1682—134no—
146#1564—131no—
147#1549—131no—
148#1545—131no—
149C23×C13#12111yescharacteristic
150C22×D13—42no—
151C22×D13—82no—
152C22×D13—82no—
153C22×D13—82no—
154Q8×C13—24no—
155Q8×C13—42no—
156Q8×C13—44no—
157Q8×C13—82no—
158C2×C4×C13—22no—
159C2×C4×C13—22no—
160C2×C4×C13—22no—
161C2×C4×C13—22no—
162C2×C4×C13—42no—
163C2×C4×C13—82no—
164C2×C4×C13#12111yescharacteristic
165C2×C4×C13#12111yescharacteristic
166C2×C4×C13#12111yescharacteristic
167C2×C4×C13#13512yes—
168C2×C4×C13#13712yes—
169C2×C4×C13C22×D411yescharacteristic
170C2×C4×C13C22×D411yescharacteristic
171C2×C4×C13C2×#4111yescharacteristic
172D52—42no—
173D52—42no—
174D52—44no—
175D52—82no—
176D52—82no—
177D52—82no—
178D52—82no—
179C8×C13—42no—
180#1055—21no—
181#1055—21no—
182#1055—21no—
183#1055—21no—
184#1055—42no—
185#1055—42no—
186#533C2×D1311yescharacteristic
187C2×C4⋊Q8C2×D1311yescharacteristic
188C2×#127—131no—
189#494—21no—
190#492—261no—
191#487—21no—
192#486—132no—
193#486—132no—
194#486—261no—
195#486—261no—
196#486—261no—
197#462C2×D1314yescomplemented
198#460—134no—
199#451—22no—
200#450—262no—
201#446—132no—
202#446—261no—
203C8×Q8—132no—
204C8×Q8—261no—
205#431—131no—
206#431C2×D1311yescharacteristic
207#424—261no—
208#423—131no—
209C2×#102—261no—
210C2×#102—261no—
211#405—261no—
212C2×C4×C8—131no—
213#335—132no—
214#334—132no—
215#332—134no—
216#331—134no—
217C22×C13—21no—
218C22×C13—21no—
219C22×C13#46412yes—
220C22×C13C2×#12111yescharacteristic
221C2×D13—82no—
222C2×D13—82no—
223C2×D13—82no—
224C2×D13—164no—
225C4×C13—21no—
226C4×C13—21no—
227C4×C13—42no—
228C4×C13—42no—
229C4×C13—82no—
230C4×C13#46412yes—
231C4×C13#47112yes—
232C4×C13C2×#13511yescharacteristic
233C4×C13C2×#13711yescharacteristic
234C4×C13#57612yes—
235C22×Q8—21no—
236C22×D4—261no—
237C22×D4—261no—
238C22×D4—261no—
239C2×Q16—24no—
240C2×SD16—264no—
241C2×M16—261no—
242C22×C8—131no—
243C22×C8—131no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8C22⋊D1311yescharacteristic
248C4⋊Q8C22⋊D1311yescharacteristic
249C4⋊Q8C22×D1312yes—
250#127—132no—
251#127—261no—
252#127—261no—
253#127—261no—
254#125—21no—
255C22⋊Q8—21no—
256C22⋊Q8C22⋊D1312yes—
257C4×Q8—21no—
258C4×Q8C22×D1312yes—
259#117—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C42C22×D1311yescharacteristic
263C4⋊C8—132no—
264C4⋊C8—132no—
265C4⋊C8—262no—
266C4⋊C8—262no—
267C4⋊C8—261no—
268C4⋊C8C22×D1312yes—
269#103—24no—
270#102—262no—
271#102—262no—
272#102—262no—
273#102—262no—
274#102—264no—
275C22⋊C8—21no—
276C22⋊C8—261no—
277#97—262no—
278C4×C8—21no—
279C4×C8—131no—
280C4×C8—131no—
281C4×C8—131no—
282C4×C8—131no—
283C4×C8—262no—
284C4×C8—261no—
285C2×C13—22no—
286C2×C13C2×#46411yescharacteristic
287C2×C13#305811yescharacteristic
288C2×C13C2×#47111yescharacteristic
289D13—162no—
290C24—521no—
291C2×Q8—21no—
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—82no—
295C2×Q8C2×C22⋊D1312yes—
296C2×D4—262no—
297C2×D4—262no—
298C2×D4—262no—
299C2×D4—522no—
300C2×D4—522no—
301C2×D4—521no—
302C2×D4—521no—
303C2×D4—521no—
304C2×D4—1042no—
305C2×D4—1042no—
306C22×C4—41no—
307C22×C4D4×D1311yescharacteristic
308C22×C4#612911yescharacteristic
309C22×C4C2×C22⋊D1311yescharacteristic
310Q16—84no—
311SD16—1044no—
312M16—522no—
313C2×C8—22no—
314C2×C8—131no—
315C2×C8—131no—
316C2×C8—131no—
317C2×C8—131no—
318C2×C8—262no—
319C2×C8—262no—
320C2×C8—262no—
321C2×C8—262no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4C2×C22⋊D1312yes—
332C22⋊C4—22no—
333C42—22no—
334C42D4×D1311yescharacteristic
335C42#612911yescharacteristic
336C42C2×C22⋊D1311yescharacteristic
337C42C23×D1311yescharacteristic
338C13#1921411yescharacteristic, complemented, Sylow
339C23—522no—
340C23—1042no—
341C23—1042no—
342C23—1042no—
343C23#8819311yescharacteristic
344Q8—24no—
345Q8—42no—
346Q8—44no—
347Q8—82no—
348D4—522no—
349D4—522no—
350D4—524no—
351D4—1042no—
352D4—1042no—
353D4—1042no—
354D4—1042no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4#8817212yes—
362C2×C4#8817312yes—
363C2×C4#8819311yescharacteristic
364C2×C4#8819311yescharacteristic
365C2×C4#8819311yescharacteristic
366C2×C4C2×D4×D1311yescharacteristic
367C2×C4C2×#612911yescharacteristic
368C2×C4C22×C22⋊D1311yescharacteristic
369C8—42no—
370C8—261no—
371C8—261no—
372C8—261no—
373C8—261no—
374C8—522no—
375C8—522no—
376C22—21no—
377C22—21no—
378C22—1042no—
379C22—1042no—
380C22—1042no—
381C22—2084no—
382C22#1171884612yes—
383C22C2×#8819311yescharacteristic
384C4—21no—
385C4—21no—
386C4—42no—
387C4—42no—
388C4—82no—
389C4#1171884612yes—
390C4#1171886412yes—
391C4C2×#8817211yescharacteristic
392C4C2×#8817311yescharacteristic
393C4#1171954012yes—
394C2—22no—
395C2—2082no—
396C2C2×#1171884611yescharacteristic
397C2#1312448111yescharacteristic
398C2C2×#1171886411yescharacteristic
subgroupG/H
centerC22C2×#88193
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC13×#3209C2
socleC22×C13C2×#121
prime radicalC2×#88128C22
hypercenter#3209D13
hypocenterC13#19214
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
3. #13123771C2semidirect
4. #13114406C2semidirect
5. #13114282C2semidirect
6. #13113582C2semidirect
7. #13113521C2semidirect
9. C13×#3209C2semidirect
15. C13×#462C22semidirect
27. #11718685C22semidirect
29. Q8×#1055C22semidirect
37. #11718139C22semidirect
39. #11718072C22semidirect
40. #11718071C22semidirect
140. #3209D13semidirect
197. #462C2×D13semidirect
338. C13#19214semidirect

Maximal subgroups

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

subgroupindex
#131257742
C2×#117187492
#131237712
#131144062
#131142822
#131135822
#131135212
#131135032
C13×#32092
#1921413