Finite Group Databasebeta

← #13119920

Subgroups — #13119920

Order 1664.

492 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88178C211yesmaximal, characteristic
2C2×#88124C211yesmaximal, characteristic, complemented
3C4×C13×C22⋊C4C211yesmaximal, characteristic
4#11718871C211yesmaximal, characteristic
5#11718612C211yesmaximal, characteristic
6#11718612C211yesmaximal, characteristic
7#11718610C211yesmaximal, characteristic
8#11718609C211yesmaximal, characteristic
9#11718594C211yesmaximal, characteristic, complemented
10C4×#6101C211yesmaximal, characteristic, complemented
11#11718589C211yesmaximal, characteristic, complemented
12#11718322C211yesmaximal, characteristic, complemented
13#11718317C211yesmaximal, characteristic, complemented
14#11718308C211yesmaximal, characteristic, complemented
15#11718307C211yesmaximal, characteristic, complemented
16C23×C4×C13C2211yescharacteristic
17C22×Dic26C2211yescharacteristic
18C2×C13×C22⋊C4C2211yescharacteristic
19C2×C13×C22⋊C4C2211yescharacteristic
20C2×C42×C13C2211yescharacteristic
21C2×C42×C13C2211yescharacteristic
22C2×C22⋊Dic13C2211yescharacteristic
23C2×C22⋊Dic13C2211yescharacteristic
24C2×C22⋊Dic13C2211yescharacteristic
25C2×C22⋊Dic13C2211yescharacteristic
26C2×C4⋊Dic13C2211yescharacteristic
27C2×C4⋊Dic13C2211yescharacteristic
28C2×C4⋊Dic13C2211yescharacteristic
29C2×C4⋊Dic13C2211yescharacteristic
30C2×C4⋊Dic13C2211yescharacteristic
31#88178—24no—
32C2×#6101C2211yescharacteristic
33C2×#6101C2211yescharacteristic
34C2×#6101C2211yescharacteristic
35C2×#6101C2211yescharacteristic
36C2×#6101C2211yescharacteristic
37C2×#6101C2211yescharacteristic
38C2×#6101C2211yescharacteristic
39C2×#6101C2211yescharacteristic
40C2×C4×Dic13C2211yescharacteristic
41C2×C4×Dic13C2211yescharacteristic
42#88124—24no—
43C13×#95C2211yescharacteristic
44C13×#95C2211yescharacteristic
45#88071C2211yescharacteristic
46#88071C2211yescharacteristic
47#88071C2211yescharacteristic
48#88071C2211yescharacteristic
49#88071C2211yescharacteristic
50#88071C2211yescharacteristic
51#88071C2211yescharacteristic
52#88071C2211yescharacteristic
53C24×C13C2311yescharacteristic
54C22×C4×C13—22no—
55C22×C4×C13—22no—
56C22×C4×C13—41no—
57C22×C4×C13C2311yescharacteristic
58C22×C4×C13C2311yescharacteristic
59C22×C4×C13C2311yescharacteristic
60C22×C4×C13C2311yescharacteristic
61C22×C4×C13C2311yescharacteristic
62C22×C4×C13C2311yescharacteristic
63C22×Dic13C2311yescharacteristic
64C22×Dic13C2311yescharacteristic
65C22×Dic13C2311yescharacteristic
66C22×Dic13C2311yescharacteristic
67C22×Dic13C2311yescharacteristic
68C22×Dic13C2311yescharacteristic
69C22×Dic13C2311yescharacteristic
70C22×Dic13C2311yescharacteristic
71C2×Dic26—22no—
72C2×Dic26—22no—
73C2×Dic26—41no—
74C13×C22⋊C4—22no—
75C13×C22⋊C4Q814yescomplemented
76C42×C13—22no—
77C42×C13—41no—
78C22⋊Dic13—22no—
79C22⋊Dic13—22no—
80C22⋊Dic13—22no—
81C22⋊Dic13—41no—
82C4⋊Dic13—22no—
83C4⋊Dic13—22no—
84C4⋊Dic13—22no—
85C4⋊Dic13—22no—
86C4⋊Dic13—41no—
87#6101—22no—
88#6101—22no—
89#6101—22no—
90#6101—22no—
91#6101—22no—
92#6101—22no—
93#6101—22no—
94#6101D414yes—
95C4×Dic13—22no—
96C4×Dic13—41no—
97#2556—131nomaximal, Sylow
98C23×C13—21no—
99C23×C13—21no—
100C23×C13—21no—
101C23×C13—21no—
102C23×C13—21no—
103C23×C13—21no—
104C23×C13C2×Q812yes—
105C23×C13C2411yescharacteristic
106C2×C4×C13—21no—
107C2×C4×C13—21no—
108C2×C4×C13—21no—
109C2×C4×C13—21no—
110C2×C4×C13—21no—
111C2×C4×C13—21no—
112C2×C4×C13—21no—
113C2×C4×C13—21no—
114C2×C4×C13—21no—
115C2×C4×C13—21no—
116C2×C4×C13—21no—
117C2×C4×C13—21no—
118C2×C4×C13—21no—
119C2×C4×C13—44no—
120C2×C4×C13C2×D412yes—
121C2×C4×C13C2×Q812yes—
122C2×C4×C13C2×Q812yes—
123C2×C4×C13#4112yes—
124C2×C4×C13#4112yes—
125C2×Dic13—21no—
126C2×Dic13—21no—
127C2×Dic13—21no—
128C2×Dic13—21no—
129C2×Dic13—21no—
130C2×Dic13—21no—
131C2×Dic13—21no—
132C2×Dic13—21no—
133C2×Dic13—21no—
134C2×Dic13—21no—
135C2×Dic13—21no—
136C2×Dic13—21no—
137C2×Dic13—21no—
138C2×Dic13—21no—
139C2×Dic13—21no—
140C2×Dic13—21no—
141C2×Dic13—21no—
142C2×Dic13—21no—
143C2×Dic13—21no—
144C2×Dic13—21no—
145C2×Dic13—21no—
146C2×Dic13C2×D412yes—
147C2×Dic13C2×D412yes—
148C2×Dic13#4112yes—
149Dic26—44no—
150C2×#125—131no—
151C2×C22⋊Q8—131no—
152#399—131no—
153#398—131no—
154#396—131no—
155#395—131no—
156#391—131no—
157#389—131no—
158#389—131no—
159#388—131no—
160#388—131no—
161#387—131no—
162#385—131no—
163C4×C4⋊C4—131no—
164C4×C22⋊C4D1311yescharacteristic
165C22×C13—22no—
166C22×C13—22no—
167C22×C13—22no—
168C22×C13—41no—
169C22×C13—41no—
170C22×C13—41no—
171C22×C13—41no—
172C22×C13C22×D411yescharacteristic
173C22×C13C22×Q811yescharacteristic
174C22×C13C2×#4111yescharacteristic
175C22×C13C2×#4111yescharacteristic
176C22×C13C2×#4111yescharacteristic
177C22×C13H(5,𝔽2)11yescharacteristic
178C22×C132−1+411yescharacteristic
179C4×C13—22no—
180C4×C13—22no—
181C4×C13—22no—
182C4×C13—22no—
183C4×C13—22no—
184C4×C13—41no—
185Dic13—22no—
186Dic13—22no—
187Dic13—22no—
188Dic13—41no—
189Dic13—41no—
190Dic13—41no—
191Dic13—41no—
192Dic13—41no—
193C22×Q8—131no—
194C23×C4C2×D1311yescharacteristic
195#125—264no—
196C22⋊Q8—264no—
197C2×C4⋊C4—131no—
198C2×C4⋊C4—131no—
199C2×C4⋊C4—131no—
200C2×C4⋊C4—131no—
201C2×C4⋊C4—131no—
202C2×C4⋊C4—131no—
203C2×C4⋊C4—131no—
204C2×C4⋊C4—131no—
205C2×C4⋊C4—131no—
206C2×C4⋊C4—131no—
207C2×C4⋊C4—131no—
208C2×C4⋊C4—131no—
209C2×C4⋊C4—131no—
210C2×C22⋊C4—131no—
211C2×C22⋊C4—131no—
212C2×C22⋊C4—131no—
213C2×C22⋊C4—131no—
214C2×C22⋊C4C2×D1311yescharacteristic
215C2×C22⋊C4C2×D1311yescharacteristic
216C2×C42—131no—
217C2×C42—131no—
218C2×C42C2×D1311yescharacteristic
219C2×C42C2×D1311yescharacteristic
220#95—131no—
221#95—131no—
222#95—131no—
223#95—131no—
224#95—131no—
225#95—131no—
226#95—131no—
227#95—131no—
228#95C2×D1311yescharacteristic
229#95C2×D1311yescharacteristic
230C2×C13—42no—
231C2×C13#53211yescharacteristic
232C2×C13#53311yescharacteristic
233C2×C13#53911yescharacteristic
234C2×C13#54611yescharacteristic
235C2×C13D4×Q811yescharacteristic
236C2×C13#55211yescharacteristic
237C2×C13#55411yescharacteristic
238C24C22×D1311yescharacteristic
239C2×Q8—262no—
240C2×Q8—262no—
241C2×Q8—521no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—41no—
245C22×C4—131no—
246C22×C4—131no—
247C22×C4—131no—
248C22×C4—131no—
249C22×C4—131no—
250C22×C4—131no—
251C22×C4—131no—
252C22×C4—131no—
253C22×C4C22×D1311yescharacteristic
254C22×C4C22×D1311yescharacteristic
255C22×C4C22×D1311yescharacteristic
256C22×C4C22×D1311yescharacteristic
257C22×C4C22×D1311yescharacteristic
258C22×C4C22×D1311yescharacteristic
259C4⋊C4—134no—
260C4⋊C4—262no—
261C4⋊C4—262no—
262C4⋊C4—262no—
263C4⋊C4—262no—
264C4⋊C4—262no—
265C4⋊C4—262no—
266C4⋊C4—262no—
267C4⋊C4—262no—
268C4⋊C4—262no—
269C4⋊C4—262no—
270C4⋊C4—262no—
271C4⋊C4—521no—
272C22⋊C4—22no—
273C22⋊C4—262no—
274C22⋊C4—262no—
275C22⋊C4—262no—
276C22⋊C4—521no—
277C22⋊C4Dic2614yescomplemented
278C42—22no—
279C42—41no—
280C42—262no—
281C42—521no—
282C13#255611yescharacteristic, complemented, Sylow
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23C2×Dic2612yes—
290C23C23×D1311yescharacteristic
291Q8—524no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—44no—
306C2×C4—132no—
307C2×C4—132no—
308C2×C4—132no—
309C2×C4—261no—
310C2×C4—261no—
311C2×C4—261no—
312C2×C4—261no—
313C2×C4—261no—
314C2×C4—261no—
315C2×C4—261no—
316C2×C4—261no—
317C2×C4—261no—
318C2×C4—261no—
319C2×C4—261no—
320C2×C4—261no—
321C2×C4—261no—
322C2×C4—261no—
323C2×C4—261no—
324C2×C4—261no—
325C2×C4—261no—
326C2×C4—261no—
327C2×C4—261no—
328C2×C4—261no—
329C2×C4—261no—
330C2×C4C2×Dic2612yes—
331C2×C4C2×Dic2612yes—
332C2×C4#612712yes—
333C2×C4#612712yes—
334C2×C4D4×D1312yes—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22C22×Dic2611yescharacteristic
343C22C2×#612711yescharacteristic
344C22C2×#612711yescharacteristic
345C22C2×D4×D1311yescharacteristic
346C22#8825111yescharacteristic
347C22#41×D1311yescharacteristic
348C22#8825711yescharacteristic
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—41no—
355C4—262no—
356C4—262no—
357C4—262no—
358C4—521no—
359C4—521no—
360C4—521no—
361C4—521no—
362C4—521no—
363C2—42no—
364C2#1171917511yescharacteristic
365C2#1171919511yescharacteristic
366C2#1171923411yescharacteristic
367C2D4×Dic2611yescharacteristic
368C2#1171924411yescharacteristic
369C2#1171925311yescharacteristic
370C2#1171926911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC4×C13×C22⋊C4C2
prime radicalC24×C13C23
hypercenterC4×C22⋊C4D13
hypocenterC13#2556
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#88124C2semidirect
9. #11718594C2semidirect
10. C4×#6101C2semidirect
11. #11718589C2semidirect
12. #11718322C2semidirect
13. #11718317C2semidirect
14. #11718308C2semidirect
15. #11718307C2semidirect
75. C13×C22⋊C4Q8semidirect
277. C22⋊C4Dic26semidirect
282. C13#2556semidirect

Maximal subgroups

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

subgroupindex
C2×#881782
C2×#881242
C4×C13×C22⋊C42
#117188712
#117186122
#117186102
#117186092
#117185942
C4×#61012
#117185892
#117183222
#117183172
#117183082
#117183072
#255613