Finite Group Databasebeta

← #15848215

Subgroups — #15848215

Order 3328.

740 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13125751C211yesmaximal, characteristic
2C2×#11718688C211yesmaximal, characteristic, complemented
3#13123917C211yesmaximal, characteristic, complemented
4#13114282C214yesmaximal, complemented
5#13114194C214yesmaximal, complemented
6#13113612C211yesmaximal, characteristic, complemented
7#13113505C211yesmaximal, characteristic, complemented
8#13113485C211yesmaximal, characteristic, complemented
9C13×#3037C211yesmaximal, characteristic, complemented
10C2×C13×#137—21no—
11C13×#532C2211yescharacteristic
12C2×C4×Q8×C13C2211yescharacteristic
13C22×#6106—21no—
14C2×#88128C2211yescharacteristic
15C13×#464—22no—
16C13×#462C2212yescomplemented
17C13×#460C2212yescomplemented
18C13×#423C2211yescharacteristic
19C13×#417—21no—
20#11718749C2212yes—
21#11718734C2212yes—
22#11718697—22no—
23#11718688—22no—
24#11718688C2214yes—
25Q8×#1055—22no—
26Q8×#1055C2214yescomplemented
27#11718676—22no—
28C2×#88049—21no—
29#11718586—22no—
30#11718586C2211yescharacteristic
31#11718586C2211yescharacteristic
32C2×C4⋊#1055C2211yescharacteristic
33#11718139C2212yescomplemented
34#11718136C2212yescomplemented
35#11718077C2214yes—
36#11718071C2214yescomplemented
37C2×C13×#41—21no—
38C22×Q8×C13D411yescharacteristic
39C22×D52—21no—
40C22×D52—21no—
41C22×D52—21no—
42C13×#137—44no—
43C2×C13×Q16—22no—
44C2×C13×SD16—22no—
45C2×C13×M16—21no—
46C13×C4⋊Q8C2311yescharacteristic
47C13×#127C2311yescharacteristic
48C13×#124—21no—
49C13×#121—21no—
50C13×#121—21no—
51C4×Q8×C13—21no—
52C4×Q8×C13—21no—
53C4×Q8×C13D412yes—
54C4×Q8×C13C2312yes—
55C4×D4×C13—21no—
56C4×D4×C13—21no—
57C2×C13×C4⋊C4—21no—
58C2×C13×C4⋊C4D411yescharacteristic
59C2×C42×C13—21no—
60C2×C42×C13C2311yescharacteristic
61C2×#6106—24no—
62C2×#6106—44no—
63C2×#6104—24no—
64C22×#1055—21no—
65#88128—21no—
66#88128—21no—
67#88128—21no—
68#88128C2312yes—
69C13×C4⋊C8—21no—
70C13×C4⋊C8C2312yes—
71C13×#103—22no—
72C13×#102—22no—
73C22⋊#1055—21no—
74C22⋊#1055D412yes—
75#88049—24no—
76#88049—24no—
77C4⋊#1055—21no—
78C4⋊#1055—24no—
79C4⋊#1055D412yes—
80C4⋊#1055C2312yes—
81C4⋊#1055C2312yes—
82C4×#1055—21no—
83C4×#1055—24no—
84C4×#1055C2312yes—
85#19398—131nomaximal, Sylow
86C23×D13—41no—
87C13×#41—42no—
88C13×#41—42no—
89C2×Q8×C13—21no—
90C2×Q8×C13—22no—
91C2×Q8×C13—21no—
92C2×Q8×C13D814yescomplemented
93C2×Q8×C13C2×D412yes—
94C2×D4×C13—21no—
95C2×D4×C13—21no—
96C2×D4×C13—21no—
97C2×D4×C13—41no—
98C22×C4×C13—21no—
99C22×C4×C13—21no—
100C22×C4×C13—41no—
101C22×C4×C13C2×D411yescharacteristic
102C22×C4×C13C2×D411yescharacteristic
103C22×C4×C13#4111yescharacteristic
104C2×D52—22no—
105C2×D52—22no—
106C2×D52—22no—
107C2×D52—42no—
108C2×D52—42no—
109C2×D52—41no—
110C2×D52—41no—
111C2×D52—41no—
112C2×D52—82no—
113C2×D52—82no—
114C13×Q16—44no—
115C13×SD16—44no—
116C13×M16—42no—
117C2×C8×C13—22no—
118C13×C4⋊C4—22no—
119C13×C4⋊C4—21no—
120C13×C4⋊C4—22no—
121C13×C4⋊C4—21no—
122C13×C4⋊C4—21no—
123C13×C4⋊C4C2×D412yes—
124C13×C22⋊C4—41no—
125C13×C22⋊C4—41no—
126C42×C13—22no—
127C42×C13—22no—
128C42×C13C2×D411yescharacteristic
129C42×C13C2×D411yescharacteristic
130C42×C13#4111yescharacteristic
131C42×C13C2411yescharacteristic
132#6106—48no—
133#6104—84no—
134C2×#1055—22no—
135C2×#1055—22no—
136C2×#1055—42no—
137C2×#1055C2×D414yes—
138#3205—131no—
139#3037D1311yescharacteristic, complemented
140C2×#460—131no—
141#2982—131no—
142#1682—134no—
143#1668—134no—
144#1571—131no—
145#1545—131no—
146#1540—131no—
147C23×C13—41no—
148C23×C13#12111yescharacteristic
149C22×D13—42no—
150C22×D13—82no—
151C22×D13—82no—
152C22×D13—82no—
153Q8×C13—22no—
154Q8×C13—24no—
155Q8×C13—41no—
156Q8×C13—41no—
157Q8×C13C2×D814yes—
158D4×C13—41no—
159D4×C13—41no—
160D4×C13—42no—
161D4×C13—81no—
162D4×C13—82no—
163C2×C4×C13—22no—
164C2×C4×C13—24no—
165C2×C4×C13—22no—
166C2×C4×C13—22no—
167C2×C4×C13—41no—
168C2×C4×C13—41no—
169C2×C4×C13—81no—
170C2×C4×C13—82no—
171C2×C4×C13#12111yescharacteristic
172C2×C4×C13#12111yescharacteristic
173C2×C4×C13#12111yescharacteristic
174C2×C4×C13C2×D812yes—
175C2×C4×C13#13512yes—
176C2×C4×C13C22×D411yescharacteristic
177C2×C4×C13C22×D411yescharacteristic
178C2×C4×C13C2×#4111yescharacteristic
179D52—42no—
180D52—42no—
181D52—44no—
182D52—82no—
183D52—82no—
184D52—82no—
185D52—82no—
186C8×C13—42no—
187#1055—24no—
188#1055—42no—
189#1055—42no—
190C2×#137—21no—
191C22×SD16—261no—
192#532C2×D1311yescharacteristic
193C2×#127—131no—
194C2×C4×Q8C2×D1311yescharacteristic
195#493—132no—
196#486—132no—
197#464—22no—
198#462C2×D1312yescomplemented
199#460—134no—
200#460—262no—
201#460C2×D1312yescomplemented
202#459—262no—
203#447—262no—
204C8×Q8—134no—
205C8×Q8—262no—
206#431—131no—
207#431—131no—
208#431—262no—
209#423C2×D1311yescharacteristic
210C2×C4⋊C8—131no—
211#417—21no—
212C2×#102—261no—
213#335—134no—
214#332—132no—
215#331—132no—
216#331—134no—
217C22×C13—21no—
218C22×C13—21no—
219C22×C13—81no—
220C22×C13—81no—
221C22×C13—81no—
222C22×C13#46612yes—
223C22×C13C2×#12111yescharacteristic
224C2×D13—82no—
225C2×D13—82no—
226C2×D13—82no—
227C2×D13—164no—
228C4×C13—21no—
229C4×C13—24no—
230C4×C13—21no—
231C4×C13—21no—
232C4×C13—42no—
233C4×C13—81no—
234C4×C13#46612yes—
235C4×C13C22×D811yescharacteristic
236C4×C13C2×#13511yescharacteristic
237C4×C13#57712yes—
238C2×#41—21no—
239C22×Q8C22⋊D1311yescharacteristic
240C22×D4—261no—
241C22×D4—261no—
242C22×D4—261no—
243#137—44no—
244C2×Q16—22no—
245C2×SD16—22no—
246C2×SD16—264no—
247C2×SD16—524no—
248C2×D8—264no—
249C2×M16—21no—
250C22×C8—261no—
251C4⋊Q8C22×D1311yescharacteristic
252#127—132no—
253#127—261no—
254#127—261no—
255#127—261no—
256#127C22×D1311yescharacteristic
257#124—21no—
258#121—21no—
259#121—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8C22⋊D1312yes—
263C4×Q8C22×D1312yes—
264C4×D4—21no—
265C4×D4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4C22⋊D1311yescharacteristic
268C2×C42—21no—
269C2×C42C22×D1311yescharacteristic
270C4⋊C8—21no—
271C4⋊C8—132no—
272C4⋊C8—132no—
273C4⋊C8—132no—
274C4⋊C8—264no—
275C4⋊C8—261no—
276C4⋊C8C22×D1312yes—
277#103—22no—
278#102—22no—
279#102—264no—
280#102—264no—
281C22⋊C8—132no—
282C22⋊C8—261no—
283C4×C8—132no—
284C4×C8—264no—
285C4×C8—261no—
286C2×C13—22no—
287C2×C13—161no—
288C2×C13#304311yescharacteristic
289C2×C13C2×#46611yescharacteristic
290C2×C13#306811yescharacteristic
291D13—162no—
292C24—521no—
293#41—42no—
294#41—42no—
295C2×Q8—21no—
296C2×Q8—22no—
297C2×Q8—21no—
298C2×Q8#610414yescomplemented
299C2×Q8C2×C22⋊D1312yes—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—41no—
304C2×D4—262no—
305C2×D4—262no—
306C2×D4—262no—
307C2×D4—522no—
308C2×D4—521no—
309C2×D4—521no—
310C2×D4—521no—
311C2×D4—522no—
312C2×D4—1042no—
313C2×D4—1042no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—41no—
317C22×C4D4×D1311yescharacteristic
318C22×C4#612911yescharacteristic
319C22×C4C2×C22⋊D1311yescharacteristic
320Q16—44no—
321SD16—44no—
322SD16—528no—
323D8—1044no—
324M16—42no—
325C2×C8—22no—
326C2×C8—134no—
327C2×C8—262no—
328C2×C8—262no—
329C2×C8—522no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4C2×C22⋊D1312yes—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C42—22no—
339C42—22no—
340C42D4×D1311yescharacteristic
341C42#612911yescharacteristic
342C42C2×C22⋊D1311yescharacteristic
343C42C23×D1311yescharacteristic
344C13#1939811yescharacteristic, complemented, Sylow
345C23—41no—
346C23—522no—
347C23—1042no—
348C23—1042no—
349C23—1042no—
350C23#8819311yescharacteristic
351Q8—24no—
352Q8—22no—
353Q8—41no—
354Q8—41no—
355Q8C2×#610414yes—
356D4—42no—
357D4—41no—
358D4—41no—
359D4—81no—
360D4—82no—
361D4—524no—
362D4—522no—
363D4—522no—
364D4—1042no—
365D4—1042no—
366D4—1042no—
367D4—1042no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4C2×#610412yes—
377C2×C4#8819311yescharacteristic
378C2×C4#8819311yescharacteristic
379C2×C4#8819311yescharacteristic
380C2×C4#8820412yes—
381C2×C4C2×D4×D1311yescharacteristic
382C2×C4C2×#612911yescharacteristic
383C2×C4C22×C22⋊D1311yescharacteristic
384C8—42no—
385C8—264no—
386C8—522no—
387C8—522no—
388C22—21no—
389C22—21no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—1042no—
394C22—1042no—
395C22—1042no—
396C22—2084no—
397C22#1171869912yes—
398C22C2×#8819311yescharacteristic
399C4—21no—
400C4—24no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—81no—
405C4#1171869912yes—
406C4#1171948312yes—
407C4C22×#610411yescharacteristic
408C4C2×#8820411yescharacteristic
409C2—22no—
410C2—161no—
411C2—2082no—
412C2#1312435911yescharacteristic
413C2C2×#1171869911yescharacteristic
414C2#1312456611yescharacteristic
subgroupG/H
centerC22C2×#88193
commutator, cosocleC42×C13C24
FrattiniC42C23×D13
FittingC13×#3037C2
socleC22×C13C2×#121
prime radical#13125751C2
hypercenter#3037D13
hypocenterC13#19398
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
2. C2×#11718688C2semidirect
3. #13123917C2semidirect
4. #13114282C2semidirect
5. #13114194C2semidirect
6. #13113612C2semidirect
7. #13113505C2semidirect
8. #13113485C2semidirect
9. C13×#3037C2semidirect
16. C13×#462C22semidirect
17. C13×#460C22semidirect
26. Q8×#1055C22semidirect
33. #11718139C22semidirect
34. #11718136C22semidirect
36. #11718071C22semidirect
92. C2×Q8×C13D8semidirect
139. #3037D13semidirect
198. #462C2×D13semidirect
201. #460C2×D13semidirect
298. C2×Q8#6104semidirect
344. C13#19398semidirect

Maximal subgroups

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

subgroupindex
#131257512
C2×#117186882
#131239172
#131142822
#131141942
#131136122
#131135052
#131134852
C13×#30372
#1939813