Finite Group Databasebeta

← #13131568

Subgroups — #13131568

Order 1664.

502 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6122C211yesmaximal, characteristic, complemented
2C2×#88238C211yesmaximal, characteristic, complemented
3#11719291C211yesmaximal, characteristic
4C2×C22⋊#1057C211yesmaximal, characteristic, complemented
5#11719146C211yesmaximal, characteristic, complemented
6#11719145C211yesmaximal, characteristic, complemented
7#11719144C211yesmaximal, characteristic, complemented
8#11719144C211yesmaximal, characteristic, complemented
9D4×#1057C211yesmaximal, characteristic, complemented
10#11719067C211yesmaximal, characteristic, complemented
11#11719066C211yesmaximal, characteristic, complemented
12#11719065C211yesmaximal, characteristic, complemented
13#11719064C211yesmaximal, characteristic, complemented
14#11719063C211yesmaximal, characteristic, complemented
15#11719062C211yesmaximal, characteristic, complemented
16C23×Dic13C2211yescharacteristic
17C2×#6129C2211yescharacteristic
18C2×#6122—24no—
19C2×#6122C2211yescharacteristic
20C2×#6122C2211yescharacteristic
21C2×#6122C2211yescharacteristic
22C2×#6122C2211yescharacteristic
23C2×#6122C2211yescharacteristic, complemented
24C22×#1057C2211yescharacteristic
25C22×#1057C2211yescharacteristic
26C22×#1057C2211yescharacteristic
27C22×#1057C2211yescharacteristic
28C22×#1057C2211yescharacteristic, complemented
29#88238—24no—
30C2×#6118C2211yescharacteristic, complemented
31C2×D13⋊C8C2211yescharacteristic, complemented
32C13×#120C411yescharacteristic
33#88207C411yescharacteristic
34#88193C411yescharacteristic
35#88193C411yescharacteristic
36#88189C411yescharacteristic
37D4×Dic13C2211yescharacteristic
38D4×Dic13C2211yescharacteristic
39#88140C411yescharacteristic
40#88135C2211yescharacteristic
41#88135C2211yescharacteristic
42#88132C411yescharacteristic
43#88132C411yescharacteristic
44#88131C2211yescharacteristic
45C22⋊#1057C2211yescharacteristic
46C22⋊#1057C2211yescharacteristic
47C22⋊#1057C2211yescharacteristic
48C22⋊#1057C2211yescharacteristic
49C22⋊#1057C2211yescharacteristic, complemented
50C22⋊#1057C2211yescharacteristic, complemented
51#88112C2211yescharacteristic, complemented
52#88112C2211yescharacteristic, complemented
53#88111C2211yescharacteristic, complemented
54#88111C2211yescharacteristic, complemented
55#88110C2211yescharacteristic, complemented
56#88110C2211yescharacteristic, complemented
57C4⋊#1057C2211yescharacteristic, complemented
58C4⋊#1057C2211yescharacteristic, complemented
59C4×#1057C2211yescharacteristic, complemented
60C4×#1057C2211yescharacteristic, complemented
61C24×C13C2×C411yescharacteristic
62C2×D4×C13C2×C411yescharacteristic
63C2×D4×C13C2×C411yescharacteristic
64C2×D4×C13C2×C411yescharacteristic
65C2×C22⋊D13C2×C411yescharacteristic
66C2×C22⋊D13C2×C411yescharacteristic
67C2×C22⋊D13C2×C411yescharacteristic
68C2×C22⋊D13C2×C411yescharacteristic
69C22×Dic13—22no—
70C22×Dic13—22no—
71C22×Dic13C2×C411yescharacteristic
72C22×Dic13C2×C411yescharacteristic
73C22×Dic13C2×C411yescharacteristic
74C22×Dic13C2311yescharacteristic
75C22×Dic13C2311yescharacteristic
76C22×Dic13C2311yescharacteristic
77C22×Dic13C2311yescharacteristic
78#6129—22no—
79#6129—22no—
80C2×C4×D13C2311yescharacteristic
81C2×Dic26C2×C411yescharacteristic
82#6122—22no—
83#6122—22no—
84#6122—22no—
85#6122—22no—
86#6122D414yescomplemented
87C2×#1057—22no—
88C2×#1057—22no—
89C2×#1057—22no—
90C2×#1057—22no—
91C2×#1057—22no—
92C2×#1057C2311yescharacteristic
93C2×#1057C2311yescharacteristic
94C2×#1057C2311yescharacteristic
95C2×#1057C2311yescharacteristic
96C2×#1057C2311yescharacteristic
97C2×#1057C2311yescharacteristic
98C2×#1057C2311yescharacteristic
99C2×#1057C2311yescharacteristic
100#6118—22no—
101D13⋊C8—22no—
102C13×C22⋊C4C2×C411yescharacteristic
103C13×C22⋊C4C2×C411yescharacteristic
104C13×C22⋊C4C2×C411yescharacteristic
105C22⋊Dic13C2×C411yescharacteristic
106C22⋊Dic13C2×C411yescharacteristic
107C22⋊Dic13C2×C411yescharacteristic
108C22⋊Dic13C2×C411yescharacteristic
109C22⋊Dic13C2×C411yescharacteristic
110#6103C2×C411yescharacteristic
111#6103C2×C411yescharacteristic
112C4⋊Dic13C2×C411yescharacteristic
113C4⋊Dic13C2×C411yescharacteristic
114#6101C2×C411yescharacteristic
115#6101C2×C411yescharacteristic
116#6101C2×C411yescharacteristic
117#6101C2×C411yescharacteristic
118C4×Dic13C2311yescharacteristic
119C4×Dic13C2311yescharacteristic
120#2933—131nomaximal, Sylow
121C23×C13—22no—
122C23×C13—22no—
123C23×C13—41no—
124C23×C13C22×C411yescharacteristic
125C23×C13C22×C411yescharacteristic
126C23×C13C22×C411yescharacteristic
127C23×C13C22×C411yescharacteristic
128C22×D13C22×C411yescharacteristic
129D4×C13—22no—
130D4×C13—22no—
131D4×C13—41no—
132C2×C4×C13C22×C411yescharacteristic
133C2×C4×C13C22×C411yescharacteristic
134C2×C4×C13C22×C411yescharacteristic
135C22⋊D13—22no—
136C22⋊D13—22no—
137C22⋊D13—22no—
138C22⋊D13—41no—
139C2×Dic13—22no—
140C2×Dic13—22no—
141C2×Dic13—22no—
142C2×Dic13—21no—
143C2×Dic13—21no—
144C2×Dic13—21no—
145C2×Dic13—21no—
146C2×Dic13—21no—
147C2×Dic13—21no—
148C2×Dic13C22×C411yescharacteristic
149C2×Dic13C22×C411yescharacteristic
150C2×Dic13C22×C411yescharacteristic
151C2×Dic13C22×C411yescharacteristic
152C2×Dic13C22×C411yescharacteristic
153C2×Dic13C22×C411yescharacteristic
154C2×Dic13C2×D412yes—
155C2×Dic13#4112yes—
156C2×Dic13C2411yescharacteristic
157C4×D13—21no—
158C4×D13—21no—
159Dic26—22no—
160#1057—21no—
161#1057—21no—
162#1057—21no—
163#1057—21no—
164#1057—21no—
165#1057—21no—
166#1057C2×D412yes—
167#1057C2×D412yes—
168C2×#131—131no—
169C22×M16—131no—
170#525—131no—
171#436—131no—
172#436—131no—
173#435—131no—
174#435—131no—
175#435—131no—
176#435—131no—
177C8×D4—131no—
178C8×D4—131no—
179#424—131no—
180#408—131no—
181C2×C22⋊C8—131no—
182#405—131no—
183C22×C13—21no—
184C22×C13—21no—
185C22×C13—21no—
186C22×C13—21no—
187C22×C13—21no—
188C22×C13—21no—
189C22×C13—21no—
190C22×C13—21no—
191C22×C13—21no—
192C22×C13—21no—
193C22×C13—44no—
194C22×C13C4×D412yes—
195C22×C13#13112yescomplemented
196C22×C13C23×C411yescharacteristic
197C2×D13—21no—
198C2×D13—21no—
199C2×D13—21no—
200C4×C13—21no—
201C4×C13—21no—
202C4×C13—21no—
203Dic13—21no—
204Dic13—21no—
205Dic13—21no—
206Dic13—21no—
207Dic13—21no—
208Dic13C4×D412yes—
209Dic13C22×D411yescharacteristic
210Dic13C2×#4111yescharacteristic
211C2×#41—131no—
212C23×C4—131no—
213#131—264no—
214C2×M16—131no—
215C2×M16—131no—
216C2×M16—131no—
217C2×M16—131no—
218C2×M16—131no—
219C2×M16—131no—
220C2×M16—264no—
221C22×C8—131no—
222C22×C8—131no—
223C22×C8—131no—
224C22×C8—131no—
225C22×C8—131no—
226C22×C8—131no—
227#123—131no—
228#123—131no—
229C22⋊Q8—131no—
230C22⋊Q8—131no—
231#121—131no—
232#121—131no—
233#120—131no—
234#120#26111yescharacteristic
235C4×D4—131no—
236C4×D4—131no—
237C4×D4—131no—
238C4×D4—131no—
239#117—131no—
240C4⋊C8—131no—
241C4⋊C8—131no—
242C4⋊C8—131no—
243C4⋊C8—131no—
244C22⋊C8—131no—
245C22⋊C8—131no—
246C22⋊C8—131no—
247C22⋊C8—131no—
248C22⋊C8—131no—
249C22⋊C8—131no—
250C22⋊C8—131no—
251C22⋊C8—131no—
252#97—131no—
253#97—131no—
254C4×C8—131no—
255C4×C8—131no—
256C2×C13—22no—
257C2×C13—22no—
258C2×C13—41no—
259C2×C13—41no—
260C2×C13C2×C4×D411yescharacteristic
261C2×C13C2×#13111yescharacteristic
262C2×C13#56811yescharacteristic
263D13—41no—
264C24C2×#26111yescharacteristic
265#41—262no—
266#41—262no—
267C2×Q8—131no—
268C2×D4—131no—
269C2×D4—131no—
270C2×D4—131no—
271C2×D4—131no—
272C2×D4C2×#26111yescharacteristic
273C2×D4C2×#26111yescharacteristic
274C2×D4C2×#26111yescharacteristic
275C22×C4—131no—
276C22×C4—131no—
277C22×C4—131no—
278C22×C4—131no—
279C22×C4—131no—
280C22×C4—131no—
281C22×C4—131no—
282C22×C4—131no—
283C22×C4—262no—
284C22×C4—262no—
285M16—134no—
286M16—262no—
287M16—262no—
288M16—262no—
289M16—262no—
290M16—262no—
291C2×C8—131no—
292C2×C8—131no—
293C2×C8—131no—
294C2×C8—131no—
295C2×C8—131no—
296C2×C8—131no—
297C2×C8—131no—
298C2×C8—131no—
299C2×C8—262no—
300C2×C8—262no—
301C2×C8—262no—
302C2×C8—262no—
303C2×C8—262no—
304C2×C8—262no—
305C4⋊C4—131no—
306C4⋊C4—131no—
307C4⋊C4—131no—
308C4⋊C4—131no—
309C4⋊C4—131no—
310C4⋊C4—131no—
311C22⋊C4—131no—
312C22⋊C4—131no—
313C22⋊C4—131no—
314C22⋊C4—131no—
315C22⋊C4—131no—
316C22⋊C4—131no—
317C22⋊C4—131no—
318C22⋊C4C2×#26111yescharacteristic
319C22⋊C4C2×#26111yescharacteristic
320C22⋊C4C2×#26111yescharacteristic
321C42—131no—
322C42—131no—
323C13#293311yescharacteristic, complemented, Sylow
324C23—22no—
325C23—22no—
326C23—41no—
327C23—131no—
328C23C22×#26111yescharacteristic
329C23C22×#26111yescharacteristic
330C23C22×#26111yescharacteristic
331C23C22×#26111yescharacteristic
332Q8—262no—
333D4—22no—
334D4—22no—
335D4—41no—
336D4—262no—
337D4—262no—
338D4—262no—
339D4—521no—
340C2×C4—132no—
341C2×C4—132no—
342C2×C4—131no—
343C2×C4—131no—
344C2×C4—131no—
345C2×C4—131no—
346C2×C4—131no—
347C2×C4—131no—
348C2×C4—131no—
349C2×C4—261no—
350C2×C4—261no—
351C2×C4—261no—
352C2×C4—261no—
353C2×C4—261no—
354C2×C4—262no—
355C2×C4—262no—
356C2×C4—261no—
357C2×C4—261no—
358C2×C4—261no—
359C2×C4—262no—
360C2×C4C22×#26111yescharacteristic
361C2×C4C22×#26111yescharacteristic
362C2×C4C22×#26111yescharacteristic
363C8—132no—
364C8—132no—
365C8—261no—
366C8—261no—
367C8—261no—
368C8—261no—
369C8—261no—
370C8—261no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—44no—
382C22—261no—
383C22—261no—
384C22—261no—
385C22#8823812yescomplemented
386C22D4×#26112yes—
387C22C23×#26111yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—132no—
392C4—131no—
393C4—131no—
394C4—261no—
395C4—261no—
396C4—261no—
397C4—261no—
398C4—261no—
399C2—22no—
400C2—22no—
401C2—41no—
402C2—41no—
403C2—521no—
404C2C2×#8823811yescharacteristic
405C2#1171962111yescharacteristic
406C2C2×D4×#26111yescharacteristic
subgroupG/H
center, FrattiniC22C23×#261
commutator, socleC22×C13C23×C4
FittingC13×#120C4
prime radical#11719291C2
cosocleC2×Dic13C24
hypercenter#120#261
hypocenterC13#2933
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×#6122C2semidirect
2. C2×#88238C2semidirect
4. C2×C22⋊#1057C2semidirect
5. #11719146C2semidirect
6. #11719145C2semidirect
7. #11719144C2semidirect
8. #11719144C2semidirect
9. D4×#1057C2semidirect
10. #11719067C2semidirect
11. #11719066C2semidirect
12. #11719065C2semidirect
13. #11719064C2semidirect
14. #11719063C2semidirect
15. #11719062C2semidirect
23. C2×#6122C22semidirect
28. C22×#1057C22semidirect
30. C2×#6118C22semidirect
31. C2×D13⋊C8C22semidirect
49. C22⋊#1057C22semidirect
50. C22⋊#1057C22semidirect
51. #88112C22semidirect
52. #88112C22semidirect
53. #88111C22semidirect
54. #88111C22semidirect
55. #88110C22semidirect
56. #88110C22semidirect
57. C4⋊#1057C22semidirect
58. C4⋊#1057C22semidirect
59. C4×#1057C22semidirect
60. C4×#1057C22semidirect
86. #6122D4semidirect
195. C22×C13#131semidirect
323. C13#2933semidirect
385. C22#88238semidirect

Maximal subgroups

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

subgroupindex
C22×#61222
C2×#882382
#117192912
C2×C22⋊#10572
#117191462
#117191452
#117191442
D4×#10572
#117190672
#117190662
#117190652
#117190642
#117190632
#117190622
#293313