Finite Group Databasebeta

← #29665203

Subgroups — #29665203

Order 15616.

654 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456764C211yesmaximal, characteristic, complemented
2#21455612C211yesmaximal, characteristic
3C2×#16840043C211yesmaximal, characteristic
4#21454842C211yesmaximal, characteristic
5#21445273C214yesmaximal, complemented
6#21445183C214yesmaximal, complemented
7#21444593C211yesmaximal, characteristic, complemented
8#21444503C211yesmaximal, characteristic, complemented
9C61×#1542C211yesmaximal, characteristic, complemented
10C2×#14519777C2211yescharacteristic
11C2×C4×D244C2211yescharacteristic
12#16840141C2212yescomplemented
13#16840114C2212yescomplemented
14#16840059—22no—
15#16840049C2212yes—
16#16840043—22no—
17#16840043C2214yes—
18#16840039C2212yes—
19#16839972—22no—
20#16839917—22no—
21C8×D244—22no—
22#16839827C2214yes—
23#16839815C2211yescharacteristic
24#16839805—21no—
25C2×#14519711—21no—
26#16839786C2211yescharacteristic
27#16839766—21no—
28C2×#14519699—21no—
29C22⋊C8×D61—22no—
30#16839514C2214yescomplemented
31#16839509C2214yescomplemented
32C61×#533C2211yescharacteristic
33C61×#431C2211yescharacteristic
34C2×C4⋊C8×C61C2211yescharacteristic
35C61×#333C2212yescomplemented
36C61×#332C2212yescomplemented
37C22×Dic122—21no—
38C22×D244D411yescharacteristic
39C22×C4×D61—21no—
40C2×#11758202—24no—
41C2×#11758194—21no—
42C2×C8×D61—24no—
43#14519777—21no—
44#14519777—21no—
45#14519777—21no—
46#14519777C2312yes—
47C4×D244—21no—
48C4×D244—21no—
49C4×D244D412yes—
50C4×D244C2312yes—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61D411yescharacteristic
54C2×#11758198—21no—
55#14519711—24no—
56#14519710—22no—
57#14519708—22no—
58C4⋊#92418—21no—
59C4⋊#92418D412yes—
60C4⋊#92418C2312yes—
61#14519700—24no—
62#14519699—24no—
63#14519697—22no—
64#14519697—22no—
65C22⋊#92418D412yes—
66#14519689—24no—
67C4×#92418C2312yes—
68C22×C8×C61—21no—
69C4⋊Q8×C61C2311yescharacteristic
70C4⋊Q8×C61C2311yescharacteristic
71#125×C61—21no—
72C22⋊Q8×C61—21no—
73C22⋊Q8×C61—21no—
74C4×Q8×C61—21no—
75C4×Q8×C61—21no—
76#117×C61—21no—
77C2×C42×C61C2311yescharacteristic
78C4⋊C8×C61—21no—
79C4⋊C8×C61C2312yes—
80C4⋊C8×C61C2312yes—
81C22⋊C8×C61—21no—
82C4×C8×C61—21no—
83C23×D61—21no—
84C2×Q8×C61—21no—
85C2×Q8×C61—21no—
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61#4111yescharacteristic
89C22×Dic61—21no—
90C22×Dic61—41no—
91C2×D244—21no—
92C2×D244—22no—
93C2×D244Q1614yescomplemented
94C2×D244C2×D412yes—
95C2×C4×D61—22no—
96C2×C4×D61—22no—
97C2×C4×D61—82no—
98C2×Dic122—22no—
99C2×Dic122—41no—
100C2×Dic122—82no—
101C2×C8×C61—22no—
102C2×C8×C61—22no—
103C2×C8×C61—42no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—41no—
109C4⋊C4×C61—41no—
110C22⋊C4×C61—41no—
111C42×C61—41no—
112C42×C61C2×D411yescharacteristic
113C42×C61C2×D411yescharacteristic
114C42×C61#4111yescharacteristic
115C42×C61C2411yescharacteristic
116#11758202—84no—
117#11758198—22no—
118#11758198—22no—
119C4⋊Dic61—21no—
120C4⋊Dic61—22no—
121C4⋊Dic61—22no—
122C4⋊Dic61—41no—
123C4⋊Dic61—41no—
124C4⋊Dic61C2×D412yes—
125#11758194—42no—
126C2×#92418—22no—
127C2×#92418C2×D414yes—
128C8×D61—84no—
129C23×C61#12111yescharacteristic
130C22×D61—22no—
131C22×D61—24no—
132C22×D61—81no—
133C22×D61—81no—
134Q8×C61—42no—
135Q8×C61—81no—
136C2×C4×C61—22no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61#12111yescharacteristic
141C2×C4×C61#12111yescharacteristic
142C2×C4×C61#12111yescharacteristic
143C2×C4×C61C2×Q1612yes—
144C2×C4×C61#13512yes—
145C2×C4×C61C22×D411yescharacteristic
146C2×C4×C61C22×D411yescharacteristic
147C2×C4×C61C2×#4111yescharacteristic
148C2×Dic61—22no—
149C2×Dic61—42no—
150C2×Dic61—81no—
151C2×Dic61—82no—
152D244—22no—
153D244—24no—
154D244C2×Q1614yes—
155C4×D61—44no—
156C4×D61—82no—
157Dic122—44no—
158Dic122—82no—
159C8×C61—42no—
160C8×C61—42no—
161#92418—24no—
162#92418—42no—
163#19387—611nomaximal, Sylow
164C22×C61—21no—
165C22×C61—21no—
166C22×C61#46712yes—
167C22×C61C2×#12111yescharacteristic
168C2×D61—24no—
169C2×D61—42no—
170C2×D61—42no—
171C2×D61—84no—
172C4×C61—21no—
173C4×C61—21no—
174C4×C61—21no—
175C4×C61—81no—
176C4×C61—81no—
177C4×C61#46712yes—
178C4×C61C22×Q1611yescharacteristic
179C4×C61C2×#13511yescharacteristic
180C4×C61#57612yes—
181Dic61—42no—
182Dic61—82no—
183#3209—611no—
184#3077—611no—
185C2×#474—611no—
186#2972—611no—
187#1683—614no—
188#1669—614no—
189#1571—611no—
190#1549—611no—
191#1542D6111yescharacteristic, complemented
192C2×C61—22no—
193C2×C61#304111yescharacteristic
194C2×C61C2×#46711yescharacteristic
195C2×C61#306911yescharacteristic
196D61—44no—
197#533C2×D6111yescharacteristic
198C2×C4⋊Q8—611no—
199C2×C4×D4—611no—
200#494—612no—
201#487—612no—
202#478—1222no—
203#476—612no—
204#474—612no—
205#474—614no—
206#474—1222no—
207#462—1222no—
208#451—1222no—
209C8×D4—614no—
210C8×D4—1222no—
211#431—611no—
212#431C2×D6111yescharacteristic
213#428—1221no—
214C2×#107—1221no—
215#423—611no—
216C2×C4⋊C8C2×D6111yescharacteristic
217#418—1221no—
218C2×#102—1221no—
219C2×C22⋊C8—1222no—
220#335—614no—
221#333C2×D6112yescomplemented
222#332—614no—
223#332C2×D6112yescomplemented
224C61#1938711yescharacteristic, complemented, Sylow
225C22×Q8—1221no—
226C22×D4—611no—
227C23×C4—1221no—
228C2×Q16—1224no—
229C2×M16—1221no—
230C22×C8—21no—
231C22×C8—1224no—
232C4⋊Q8—612no—
233C4⋊Q8—1221no—
234C4⋊Q8—1221no—
235C4⋊Q8—1221no—
236C4⋊Q8C22×D6111yescharacteristic
237C4⋊Q8C22×D6111yescharacteristic
238#125—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×D4—612no—
244C4×D4—612no—
245C4×D4—1221no—
246C4×D4—1221no—
247#117—21no—
248C2×C4⋊C4—611no—
249C2×C4⋊C4—1221no—
250C2×C4⋊C4—1221no—
251C2×C22⋊C4—1221no—
252C2×C42C22×D6111yescharacteristic
253#107—1222no—
254#107—1224no—
255#106—1222no—
256C4⋊C8—21no—
257C4⋊C8—612no—
258C4⋊C8—612no—
259C4⋊C8—1221no—
260C4⋊C8C22×D6112yes—
261C4⋊C8C22×D6112yes—
262#103—1224no—
263#102—1222no—
264#102—1222no—
265#102—1224no—
266C22⋊C8—21no—
267C22⋊C8—612no—
268C22⋊C8—1224no—
269C4×C8—21no—
270C4×C8—612no—
271C24—1221no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—1222no—
275C2×Q8—4882no—
276C2×Q8—2441no—
277C2×D4—612no—
278C2×D4—614no—
279C2×D4—1221no—
280C2×D4—1222no—
281C22×C4—1221no—
282C22×C4—1222no—
283C22×C4—1222no—
284C22×C4—4882no—
285C22×C4—2441no—
286C22×C4#1175822211yescharacteristic
287C22×C4D4×D6111yescharacteristic
288C22×C4D4×D6111yescharacteristic
289Q16—4884no—
290M16—2442no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—42no—
294C2×C8—614no—
295C2×C8—1222no—
296C2×C8—4884no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—612no—
304C4⋊C4—1221no—
305C4⋊C4—1222no—
306C4⋊C4—1222no—
307C4⋊C4—2441no—
308C4⋊C4—2441no—
309C22⋊C4—41no—
310C22⋊C4—1222no—
311C22⋊C4—1222no—
312C42—41no—
313C42#1175822211yescharacteristic
314C42D4×D6111yescharacteristic
315C42D4×D6111yescharacteristic
316C42C23×D6111yescharacteristic
317C23—1224no—
318C23—1222no—
319C23—4881no—
320C23—4881no—
321C23#1451975011yescharacteristic
322Q8—42no—
323Q8—81no—
324Q8—4882no—
325Q8—2444no—
326D4—614no—
327D4—1224no—
328D4—1222no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—1222no—
334C2×C4—4882no—
335C2×C4—4882no—
336C2×C4—4881no—
337C2×C4—2442no—
338C2×C4—2444no—
339C2×C4#1451975011yescharacteristic
340C2×C4#1451975011yescharacteristic
341C2×C4#1451975011yescharacteristic
342C2×C4Q16×D6112yes—
343C2×C4#1451979812yes—
344C2×C4C2×D4×D6111yescharacteristic
345C2×C4C2×D4×D6111yescharacteristic
346C2×C4C2×#1175822211yescharacteristic
347C8—42no—
348C8—42no—
349C8—1224no—
350C8—2442no—
351C22—21no—
352C22—21no—
353C22—1224no—
354C22—4884no—
355C22—2442no—
356C22—2442no—
357C22#1684000212yes—
358C22C2×#1451975011yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—81no—
363C4—81no—
364C4—4882no—
365C4—2442no—
366C4#1684000212yes—
367C4C2×Q16×D6111yescharacteristic
368C4C2×#1451979811yescharacteristic
369C4#1684056412yes—
370C2—22no—
371C2—2444no—
372C2#2145532511yescharacteristic
373C2C2×#1684000211yescharacteristic
374C2#2145555911yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1542C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#1542D61
hypocenterC61#19387
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
1. #21456764C2semidirect
5. #21445273C2semidirect
6. #21445183C2semidirect
7. #21444593C2semidirect
8. #21444503C2semidirect
9. C61×#1542C2semidirect
12. #16840141C22semidirect
13. #16840114C22semidirect
30. #16839514C22semidirect
31. #16839509C22semidirect
35. C61×#333C22semidirect
36. C61×#332C22semidirect
93. C2×D244Q16semidirect
191. #1542D61semidirect
221. #333C2×D61semidirect
223. #332C2×D61semidirect
224. C61#19387semidirect

Maximal subgroups

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

subgroupindex
#214567642
#214556122
C2×#168400432
#214548422
#214452732
#214451832
#214445932
#214445032
C61×#15422
#1938761