Finite Group Databasebeta

← #13128513

Subgroups — #13128513

Order 1664.

1522 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88254C211yesmaximal, characteristic, complemented
2C22×#6131C211yesmaximal, characteristic, complemented
3D4×Q8×C13C211yesmaximal, characteristic, complemented
4#11719522C211yesmaximal, characteristic, complemented
5Q8×C22⋊D13C211yesmaximal, characteristic, complemented
6#11719431C213yesmaximal, complemented
7#11719429C213yesmaximal, complemented
8#11719336C213yesmaximal, complemented
9#11719335C213yesmaximal, complemented
10#11719334C213yesmaximal, complemented
11D13×C22⋊Q8C213yesmaximal, complemented
12#11719281C211yesmaximal, characteristic, complemented
13#11719279C211yesmaximal, characteristic, complemented
14#11719255C213yesmaximal, complemented
15#11719252C213yesmaximal, complemented
16C22×Q8×C13C2211yescharacteristic
17C22×Q8×C13C2211yescharacteristic, complemented
18#88254—28no—
19C2×#6131—28no—
20C2×#6131C2213yes—
21C2×#6131C2213yes—
22C2×#6131C2211yescharacteristic
23C2×#6131C2211yescharacteristic
24C2×#6131C2213yescomplemented
25C2×Q8×D13C2213yescomplemented
26C2×Q8×D13C2211yescharacteristic, complemented
27C2×#6127C2213yescomplemented
28C2×#6127C2213yescomplemented
29C22×D52C2213yes—
30C22×C4×D13C2213yes—
31C13×C4⋊Q8C2213yescomplemented
32C13×C22⋊Q8C2213yes—
33C13×C22⋊Q8C2213yescomplemented
34C4×Q8×C13C2211yescharacteristic, complemented
35C4×D4×C13C2213yes—
36#88201C2213yescomplemented
37#88200C2213yescomplemented
38#88200C2213yescomplemented
39Q8×Dic13C2211yescharacteristic, complemented
40#88198C2213yescomplemented
41#88193C2211yescharacteristic
42#88191C2213yes—
43#88184C2213yescomplemented
44C4×C22⋊D13C2213yes—
45C4×C22⋊D13C2213yescomplemented
46#88178C2213yes—
47#88151C2213yescomplemented
48#88150C2213yescomplemented
49#88150C2216yescomplemented
50#88149C2216yescomplemented
51#88149C2213yescomplemented
52#88148C2216yescomplemented
53#88148C2216yescomplemented
54#88148C2213yescomplemented
55#88147C2213yescomplemented
56C4⋊C4×D13C2216yescomplemented
57C4⋊C4×D13C2213yescomplemented
58#88138C2213yescomplemented
59#88138C2216yescomplemented
60#88137C2213yescomplemented
61#88136C2213yes—
62C22⋊C4×D13C2216yes—
63#88132C2213yes—
64#88129C2213yescomplemented
65C4×D52C2213yescomplemented
66C4×D52C2213yescomplemented
67#88126C2213yescomplemented
68C23×D13C2313yes—
69C2×Q8×C13—24no—
70C2×Q8×C13—24no—
71C2×Q8×C13C2313yes—
72C2×Q8×C13C2313yes—
73C2×Q8×C13C2311yescharacteristic
74C2×D4×C13C2311yescharacteristic
75C22×C4×C13C2313yes—
76C22×C4×C13C2313yes—
77C2×C22⋊D13C2311yescharacteristic
78C2×C22⋊D13C2313yes—
79C2×C22⋊D13C2311yescharacteristic
80C2×C22⋊D13C2313yes—
81C22×Dic13C2311yescharacteristic
82#6131—24no—
83#6131—212no—
84#6131—212no—
85#6131—212no—
86#6131D418yescomplemented
87Q8×D13—24no—
88Q8×D13—212no—
89#6127—212no—
90#6127—212no—
91C2×D52—212no—
92C2×D52C2313yes—
93C2×D52C2313yes—
94C2×D52C2316yes—
95C2×D52C2313yes—
96C2×D52C2313yes—
97C2×C4×D13—212no—
98C2×C4×D13C2313yes—
99C2×C4×D13C2313yes—
100C2×C4×D13C2316yes—
101C2×C4×D13C2313yes—
102C2×C4×D13C2313yes—
103C2×C4×D13C2316yes—
104C2×C4×D13C2313yes—
105C2×Dic26C2313yes—
106C2×Dic26C2313yes—
107C13×C4⋊C4C2313yes—
108C13×C4⋊C4C2316yes—
109C13×C4⋊C4C2313yes—
110C13×C22⋊C4C2313yes—
111C13×C22⋊C4C2313yes—
112C42×C13C2313yes—
113C22⋊Dic13C2313yes—
114C22⋊Dic13C2311yescharacteristic
115#6103C2313yes—
116#6103C2313yes—
117#6103C2311yescharacteristic
118#6103C2313yes—
119#6103C2316yes—
120#6103C2316yes—
121#6103C2313yes—
122#6103C2313yes—
123C4⋊Dic13C2313yes—
124C4⋊Dic13C2313yes—
125#6101C2311yescharacteristic
126#6101C2313yes—
127#6101C2313yes—
128#6101C2313yes—
129#6101C2316yes—
130C4×Dic13C2313yes—
131#3489—131nomaximal, Sylow
132C23×C13C2411yescharacteristic
133C23×C13C2411yescharacteristic
134C22×D13—26no—
135C22×D13—26no—
136C22×D13—43no—
137C22×D13C2413yes—
138C22×D13C2411yescharacteristic
139C22×D13C2413yes—
140Q8×C13—26no—
141Q8×C13—26no—
142Q8×C13C2×D414yes—
143D4×C13—22no—
144C2×C4×C13—26no—
145C2×C4×C13—26no—
146C2×C4×C13C2413yes—
147C2×C4×C13C2413yes—
148C2×C4×C13C2411yescharacteristic
149C2×C4×C13C2413yes—
150C2×C4×C13C2413yes—
151C22⋊D13—22no—
152C22⋊D13—26no—
153C22⋊D13—22no—
154C22⋊D13—43no—
155C2×Dic13—22no—
156C2×Dic13C2411yescharacteristic
157C2×Dic13C2413yes—
158C2×Dic13C2411yescharacteristic
159C2×Dic13C2413yes—
160C2×Dic13C2411yescharacteristic
161D52—212no—
162D52—26no—
163D52—26no—
164D52—43no—
165D52C2×D4112yes—
166C4×D13—26no—
167C4×D13—212no—
168C4×D13—26no—
169C4×D13—26no—
170C4×D13—26no—
171C4×D13—46no—
172C4×D13C2×D4112yes—
173Dic26—26no—
174Dic26—26no—
175C2×2−1+4—131no—
176C22×#41—131no—
177D4×Q8—131no—
178D4×Q8D1311yescharacteristic, complemented
179#548—131no—
180#548—131no—
181#547—133no—
182#547—133no—
183#546—133no—
184#546—133no—
185#537—133no—
186#536—133no—
187#525—133no—
188C2×C22⋊Q8—133no—
189#520—131no—
190C22×C13—21no—
191C22×C13—21no—
192C22×C13—21no—
193C22×C13—21no—
194C22×C13—21no—
195C22×C132−1+412yescomplemented
196C22×C13C2511yescharacteristic
197C2×D13—23no—
198C2×D13—23no—
199C2×D13—21no—
200C2×D13—23no—
201C2×D13—21no—
202C2×D13—23no—
203C2×D13—21no—
204C2×D13—23no—
205C2×D13—412no—
206C2×D13C22×D416yes—
207C4×C13—23no—
208C4×C13—23no—
209C4×C13—21no—
210C4×C13—23no—
211C4×C13C22×D416yes—
212Dic13—23no—
213Dic13—21no—
214Dic13—21no—
215Dic13—23no—
216Dic13C22×D412yes—
2172−1+4—268no—
218C2×#41—131no—
219C2×#41—133no—
220C2×#41—133no—
221C2×#41—133no—
222C2×#41—131no—
223C2×#41—133no—
224C2×#41—133no—
225C2×#41—268no—
226C22×Q8—133no—
227C22×Q8—131no—
228C22×Q8C2×D1311yescharacteristic
229C22×Q8C2×D1311yescharacteristic, complemented
230C22×D4—133no—
231C23×C4—133no—
232C4⋊Q8—133no—
233C4⋊Q8C2×D1313yescomplemented
234#124—133no—
235#124—133no—
236#123—133no—
237#123—136no—
238#123—136no—
239#123—133no—
240C22⋊Q8—133no—
241C22⋊Q8—133no—
242C22⋊Q8—133no—
243C22⋊Q8—136no—
244C22⋊Q8—133no—
245C22⋊Q8—133no—
246C22⋊Q8—133no—
247C22⋊Q8C2×D1313yes—
248C22⋊Q8C2×D1313yescomplemented
249#121—131no—
250#121—133no—
251#121—136no—
252#121—133no—
253#121—133no—
254#121—136no—
255#120—133no—
256#120—133no—
257C4×Q8—131no—
258C4×Q8C2×D1311yescharacteristic, complemented
259C4×D4—133no—
260C4×D4—133no—
261C4×D4—133no—
262C4×D4—133no—
263C4×D4—133no—
264C4×D4C2×D1313yes—
265#117—133no—
266C2×C4⋊C4—133no—
267C2×C4⋊C4—136no—
268C2×C22⋊C4—136no—
269C2×C13—22no—
270C2×C13—41no—
271C2×C13C23×D411yescharacteristic
272C2×C13C2×2−1+411yescharacteristic
273C2×C13#58511yescharacteristic
274D13—26no—
275D13—41no—
276D13—43no—
277C24—133no—
278#41—138no—
279#41—2612no—
280#41—2612no—
281#41—2612no—
282#41—2612no—
283#41—2612no—
284#41—264no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—133no—
288C2×Q8—133no—
289C2×Q8—2612no—
290C2×Q8—264no—
291C2×Q8C22×D1313yes—
292C2×Q8C22×D1313yes—
293C2×Q8C22×D1311yescharacteristic
294C2×D4—131no—
295C2×D4—133no—
296C2×D4—131no—
297C2×D4—133no—
298C2×D4—133no—
299C2×D4—133no—
300C2×D4—136no—
301C2×D4—133no—
302C2×D4—133no—
303C2×D4—2612no—
304C2×D4C22×D1311yescharacteristic
305C22×C4—133no—
306C22×C4—131no—
307C22×C4—136no—
308C22×C4—133no—
309C22×C4—133no—
310C22×C4—133no—
311C22×C4—133no—
312C22×C4—136no—
313C22×C4—2612no—
314C22×C4C22×D1313yes—
315C22×C4C22×D1313yes—
316C4⋊C4—133no—
317C4⋊C4—133no—
318C4⋊C4—133no—
319C4⋊C4—133no—
320C4⋊C4—136no—
321C4⋊C4—131no—
322C4⋊C4—133no—
323C4⋊C4C22×D1313yes—
324C4⋊C4C22×D1316yes—
325C4⋊C4C22×D1313yes—
326C22⋊C4—133no—
327C22⋊C4—131no—
328C22⋊C4—136no—
329C22⋊C4—136no—
330C22⋊C4—133no—
331C22⋊C4—133no—
332C22⋊C4—131no—
333C22⋊C4—133no—
334C22⋊C4—133no—
335C22⋊C4—133no—
336C22⋊C4C22×D1313yes—
337C22⋊C4C22×D1313yes—
338C42—133no—
339C42C22×D1313yes—
340C13#348911yescharacteristic, complemented, Sylow
341C23—133no—
342C23—133no—
343C23—131no—
344C23—266no—
345C23—266no—
346C23—523no—
347C23C23×D1311yescharacteristic
348C23C23×D1311yescharacteristic
349Q8—26no—
350Q8—26no—
351Q8—266no—
352Q8—266no—
353Q8D4×D1314yes—
354D4—22no—
355D4—1312no—
356D4—2612no—
357D4—266no—
358D4—262no—
359D4—266no—
360D4—262no—
361D4—266no—
362D4—523no—
363D4—523no—
364C2×C4—26no—
365C2×C4—26no—
366C2×C4—131no—
367C2×C4—1312no—
368C2×C4—131no—
369C2×C4—133no—
370C2×C4—133no—
371C2×C4—131no—
372C2×C4—266no—
373C2×C4—266no—
374C2×C4—2612no—
375C2×C4—262no—
376C2×C4—266no—
377C2×C4—266no—
378C2×C4—526no—
379C2×C4C23×D1311yescharacteristic
380C2×C4C23×D1313yes—
381C2×C4C23×D1313yes—
382C2×C4C23×D1313yes—
383C2×C4C23×D1313yes—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—136no—
390C22—263no—
391C22—263no—
392C22—261no—
393C22—263no—
394C22—261no—
395C22—263no—
396C22—261no—
397C22—263no—
398C22—5212no—
399C22#8825412yescomplemented
400C22C24×D1311yescharacteristic
401C4—23no—
402C4—23no—
403C4—21no—
404C4—23no—
405C4—132no—
406C4—261no—
407C4—263no—
408C4—261no—
409C4—263no—
410C4C2×D4×D1316yes—
411C2—22no—
412C2—41no—
413C2—266no—
414C2—523no—
415C2—521no—
416C2C22×D4×D1311yescharacteristic
417C2C2×#8825411yescharacteristic
418C2#1171965211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D13
commutator, socle, cosocleC22×C13C25
FittingD4×Q8×C13C2
hypercenterD4×Q8D13
hypocenterC13#3489
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88254C2semidirect
2. C22×#6131C2semidirect
3. D4×Q8×C13C2semidirect
4. #11719522C2semidirect
5. Q8×C22⋊D13C2semidirect
6. #11719431C2semidirect
7. #11719429C2semidirect
8. #11719336C2semidirect
9. #11719335C2semidirect
10. #11719334C2semidirect
11. D13×C22⋊Q8C2semidirect
12. #11719281C2semidirect
13. #11719279C2semidirect
14. #11719255C2semidirect
15. #11719252C2semidirect
17. C22×Q8×C13C22semidirect
24. C2×#6131C22semidirect
25. C2×Q8×D13C22semidirect
26. C2×Q8×D13C22semidirect
27. C2×#6127C22semidirect
28. C2×#6127C22semidirect
31. C13×C4⋊Q8C22semidirect
33. C13×C22⋊Q8C22semidirect
34. C4×Q8×C13C22semidirect
36. #88201C22semidirect
37. #88200C22semidirect
38. #88200C22semidirect
39. Q8×Dic13C22semidirect
40. #88198C22semidirect
43. #88184C22semidirect
45. C4×C22⋊D13C22semidirect
47. #88151C22semidirect
48. #88150C22semidirect
49. #88150C22semidirect
50. #88149C22semidirect
51. #88149C22semidirect
52. #88148C22semidirect
53. #88148C22semidirect
54. #88148C22semidirect
55. #88147C22semidirect
56. C4⋊C4×D13C22semidirect
57. C4⋊C4×D13C22semidirect
58. #88138C22semidirect
59. #88138C22semidirect
60. #88137C22semidirect
64. #88129C22semidirect
65. C4×D52C22semidirect
66. C4×D52C22semidirect
67. #88126C22semidirect
86. #6131D4semidirect
178. D4×Q8D13semidirect
195. C22×C132−1+4semidirect
229. C22×Q8C2×D13semidirect
233. C4⋊Q8C2×D13semidirect
248. C22⋊Q8C2×D13semidirect
258. C4×Q8C2×D13semidirect
340. C13#3489semidirect
399. C22#88254semidirect

Maximal subgroups

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

subgroupindex
C2×#882542
C22×#61312
D4×Q8×C132
#117195222
Q8×C22⋊D132
#117194312
#117194292
#117193362
#117193352
#117193342
D13×C22⋊Q82
#117192812
#117192792
#117192552
#117192522
#348913