Finite Group Databasebeta

← #29665234

Subgroups — #29665234

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456960C211yesmaximal, characteristic, complemented
2#21455612C211yesmaximal, characteristic
3C2×#16839968C211yesmaximal, characteristic
4#21454842C211yesmaximal, characteristic
5#21445241C214yesmaximal, complemented
6#21445197C214yesmaximal, complemented
7#21444589C211yesmaximal, characteristic, complemented
8#21444490C211yesmaximal, characteristic, complemented
9C61×#1542C211yesmaximal, characteristic, complemented
10C22×#11758190—21no—
11C2×#14519777C2211yescharacteristic
12C2×C4×D244C2211yescharacteristic
13#16840158C2212yescomplemented
14#16840112C2212yescomplemented
15#16840090—22no—
16#16840059—22no—
17#16840049C2212yes—
18#16840047—22no—
19#16840039C2212yes—
20#16839968—22no—
21#16839968C2214yes—
22C8×D244—22no—
23#16839827C2214yes—
24#16839815C2211yescharacteristic
25#16839805—21no—
26#16839786C2211yescharacteristic
27#16839766—21no—
28C2×#14519702—21no—
29C22⋊C8×D61—22no—
30#16839512C2214yescomplemented
31#16839507C2214yescomplemented
32C61×#533C2211yescharacteristic
33C61×#431C2211yescharacteristic
34C2×C4⋊C8×C61C2211yescharacteristic
35C61×#333C2212yescomplemented
36C61×#332C2212yescomplemented
37C22×Dic122—21no—
38C22×D244D411yescharacteristic
39C22×C4×D61—21no—
40C2×#11758190—24no—
41C2×#11758190—44no—
42C2×#11758194—21no—
43C2×C8×D61—24no—
44#14519777—21no—
45#14519777—21no—
46#14519777—21no—
47#14519777C2312yes—
48C4×D244—21no—
49C4×D244—21no—
50C4×D244D412yes—
51C4×D244C2312yes—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61D411yescharacteristic
55C2×#11758198—21no—
56#14519710—22no—
57#14519708—22no—
58#14519708—24no—
59C4⋊#92418—21no—
60C4⋊#92418D412yes—
61C4⋊#92418C2312yes—
62#14519702—24no—
63#14519701—24no—
64#14519697—22no—
65#14519697—22no—
66C22⋊#92418D412yes—
67#14519689—24no—
68C4×#92418C2312yes—
69C22×C8×C61—21no—
70C4⋊Q8×C61C2311yescharacteristic
71C4⋊Q8×C61C2311yescharacteristic
72#125×C61—21no—
73C22⋊Q8×C61—21no—
74C22⋊Q8×C61—21no—
75C4×Q8×C61—21no—
76C4×Q8×C61—21no—
77#117×C61—21no—
78C2×C42×C61C2311yescharacteristic
79C4⋊C8×C61—21no—
80C4⋊C8×C61C2312yes—
81C4⋊C8×C61C2312yes—
82C22⋊C8×C61—21no—
83C4×C8×C61—21no—
84C23×D61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—21no—
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61#4111yescharacteristic
90C22×Dic61—21no—
91C22×Dic61—41no—
92C2×D244—21no—
93C2×D244—22no—
94C2×D244SD1614yes—
95C2×D244C2×D412yes—
96C2×C4×D61—22no—
97C2×C4×D61—22no—
98C2×C4×D61—82no—
99C2×Dic122—22no—
100C2×Dic122—42no—
101C2×Dic122—42no—
102C2×Dic122—41no—
103C2×C8×C61—22no—
104C2×C8×C61—22no—
105C2×C8×C61—42no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113C42×C61—41no—
114C42×C61C2×D411yescharacteristic
115C42×C61C2×D411yescharacteristic
116C42×C61#4111yescharacteristic
117C42×C61C2411yescharacteristic
118#11758198—22no—
119#11758198—22no—
120C4⋊Dic61—21no—
121C4⋊Dic61—22no—
122C4⋊Dic61—22no—
123C4⋊Dic61—41no—
124C4⋊Dic61—41no—
125C4⋊Dic61C2×D412yes—
126#11758194—42no—
127C2×#92418—22no—
128C2×#92418C2×D414yes—
129#11758190—48no—
130C8×D61—84no—
131C23×C61#12111yescharacteristic
132C22×D61—22no—
133C22×D61—24no—
134C22×D61—81no—
135C22×D61—81no—
136Q8×C61—42no—
137Q8×C61—81no—
138C2×C4×C61—22no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61#12111yescharacteristic
143C2×C4×C61#12111yescharacteristic
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61C2×SD1612yes—
146C2×C4×C61#13512yes—
147C2×C4×C61C22×D411yescharacteristic
148C2×C4×C61C22×D411yescharacteristic
149C2×C4×C61C2×#4111yescharacteristic
150C2×Dic61—22no—
151C2×Dic61—42no—
152C2×Dic61—81no—
153C2×Dic61—82no—
154D244—22no—
155D244—24no—
156D244C2×SD1614yes—
157C4×D61—44no—
158C4×D61—82no—
159Dic122—42no—
160Dic122—42no—
161Dic122—82no—
162C8×C61—42no—
163C8×C61—42no—
164#92418—24no—
165#92418—42no—
166#19390—611nomaximal, Sylow
167C22×C61—21no—
168C22×C61—21no—
169C22×C61#46512yes—
170C22×C61C2×#12111yescharacteristic
171C2×D61—24no—
172C2×D61—42no—
173C2×D61—42no—
174C2×D61—84no—
175C4×C61—21no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—81no—
179C4×C61—81no—
180C4×C61#46512yes—
181C4×C61C22×SD1611yescharacteristic
182C4×C61C2×#13511yescharacteristic
183C4×C61#57712yes—
184Dic61—42no—
185Dic61—82no—
186#3229—611no—
187#3077—611no—
188C2×#461—611no—
189#2972—611no—
190#1679—614no—
191#1671—614no—
192#1570—611no—
193#1546—611no—
194#1542D6111yescharacteristic, complemented
195C2×C61—22no—
196C2×C61#304211yescharacteristic
197C2×C61C2×#46511yescharacteristic
198C2×C61#306911yescharacteristic
199D61—44no—
200C22×SD16—1221no—
201#533C2×D6111yescharacteristic
202C2×C4⋊Q8—611no—
203C2×C4×D4—611no—
204#498—612no—
205#487—612no—
206#483—1222no—
207#478—1222no—
208#476—612no—
209#475—1222no—
210#474—612no—
211#461—614no—
212#461—1222no—
213C8×D4—614no—
214C8×D4—1222no—
215#431—611no—
216#431C2×D6111yescharacteristic
217#428—1221no—
218#423—611no—
219C2×C4⋊C8C2×D6111yescharacteristic
220#418—1221no—
221C2×#103—1221no—
222C2×C22⋊C8—1222no—
223#334—614no—
224#333C2×D6112yescomplemented
225#332—614no—
226#332C2×D6112yescomplemented
227C61#1939011yescharacteristic, complemented, Sylow
228C22×Q8—1221no—
229C22×D4—611no—
230C23×C4—1221no—
231C2×SD16—1224no—
232C2×SD16—2444no—
233C2×M16—1221no—
234C22×C8—21no—
235C22×C8—1224no—
236C4⋊Q8—612no—
237C4⋊Q8—1221no—
238C4⋊Q8—1221no—
239C4⋊Q8—1221no—
240C4⋊Q8C22×D6111yescharacteristic
241C4⋊Q8C22×D6111yescharacteristic
242#125—21no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×D4—612no—
248C4×D4—612no—
249C4×D4—1221no—
250C4×D4—1221no—
251#117—21no—
252C2×C4⋊C4—611no—
253C2×C4⋊C4—1221no—
254C2×C4⋊C4—1221no—
255C2×C22⋊C4—1221no—
256C2×C42C22×D6111yescharacteristic
257#107—1222no—
258#106—1222no—
259#106—1224no—
260C4⋊C8—21no—
261C4⋊C8—612no—
262C4⋊C8—612no—
263C4⋊C8—1221no—
264C4⋊C8C22×D6112yes—
265C4⋊C8C22×D6112yes—
266#103—1224no—
267#103—1224no—
268#102—1222no—
269#102—1222no—
270C22⋊C8—21no—
271C22⋊C8—612no—
272C22⋊C8—1224no—
273C4×C8—21no—
274C4×C8—612no—
275C24—1221no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—1222no—
279C2×Q8—2442no—
280C2×Q8—2442no—
281C2×Q8—2441no—
282C2×D4—612no—
283C2×D4—614no—
284C2×D4—1221no—
285C2×D4—1222no—
286C22×C4—1221no—
287C22×C4—1222no—
288C22×C4—1222no—
289C22×C4—4882no—
290C22×C4—2441no—
291C22×C4#1175822211yescharacteristic
292C22×C4D4×D6111yescharacteristic
293C22×C4D4×D6111yescharacteristic
294SD16—2448no—
295M16—2442no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—42no—
299C2×C8—614no—
300C2×C8—1222no—
301C2×C8—4884no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—612no—
309C4⋊C4—1221no—
310C4⋊C4—1222no—
311C4⋊C4—1222no—
312C4⋊C4—2441no—
313C4⋊C4—2441no—
314C22⋊C4—41no—
315C22⋊C4—1222no—
316C22⋊C4—1222no—
317C42—41no—
318C42#1175822211yescharacteristic
319C42D4×D6111yescharacteristic
320C42D4×D6111yescharacteristic
321C42C23×D6111yescharacteristic
322C23—1224no—
323C23—1222no—
324C23—4881no—
325C23—4881no—
326C23#1451975011yescharacteristic
327Q8—42no—
328Q8—81no—
329Q8—4882no—
330Q8—2442no—
331Q8—2442no—
332D4—614no—
333D4—1224no—
334D4—1222no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—1222no—
340C2×C4—4882no—
341C2×C4—4882no—
342C2×C4—4881no—
343C2×C4—2442no—
344C2×C4—2444no—
345C2×C4#1451975011yescharacteristic
346C2×C4#1451975011yescharacteristic
347C2×C4#1451975011yescharacteristic
348C2×C4SD16×D6112yes—
349C2×C4#1451980012yes—
350C2×C4C2×D4×D6111yescharacteristic
351C2×C4C2×D4×D6111yescharacteristic
352C2×C4C2×#1175822211yescharacteristic
353C8—42no—
354C8—42no—
355C8—1224no—
356C8—2442no—
357C22—21no—
358C22—21no—
359C22—1224no—
360C22—4884no—
361C22—2442no—
362C22—2442no—
363C22#1683999412yes—
364C22C2×#1451975011yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—81no—
369C4—81no—
370C4—4882no—
371C4—2442no—
372C4#1683999412yes—
373C4C2×SD16×D6111yescharacteristic
374C4C2×#1451980011yescharacteristic
375C4#1684056912yes—
376C2—22no—
377C2—2444no—
378C2#2145533411yescharacteristic
379C2C2×#1683999411yescharacteristic
380C2#2145555911yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1542C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#1542D61
hypocenterC61#19390
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. #21456960C2semidirect
5. #21445241C2semidirect
6. #21445197C2semidirect
7. #21444589C2semidirect
8. #21444490C2semidirect
9. C61×#1542C2semidirect
13. #16840158C22semidirect
14. #16840112C22semidirect
30. #16839512C22semidirect
31. #16839507C22semidirect
35. C61×#333C22semidirect
36. C61×#332C22semidirect
194. #1542D61semidirect
224. #333C2×D61semidirect
226. #332C2×D61semidirect
227. C61#19390semidirect

Maximal subgroups

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

subgroupindex
#214569602
#214556122
C2×#168399682
#214548422
#214452412
#214451972
#214445892
#214444902
C61×#15422
#1939061