Finite Group Databasebeta

← #29590291

Subgroups — #29590291

Order 15616.

468 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21449087C211yesmaximal, characteristic
2#21444133C211yesmaximal, characteristic
3#21443842C212yesmaximal
4#21443839C212yesmaximal
5C61×#1740C211yesmaximal, characteristic
6C2×C4×Dic122C411yescharacteristic, complemented
7#16839815—21no—
8#16839701C411yescharacteristic, complemented
9#16839698C2211yescharacteristic
10#16839695—21no—
11#16839685—21no—
12#16839682C411yescharacteristic, complemented
13#16839674C2211yescharacteristic
14C4×#11758196—21no—
15C4×C4⋊Dic61C411yescharacteristic, complemented
16C4×C22⋊Dic61—21no—
17#16839488—21no—
18#16839486C2212yes—
19C2×#117×C61—21no—
20C2×#117×C61C2211yescharacteristic
21C61×#342—21no—
22C61×#342C2212yes—
23C22×Dic122C2×C411yescharacteristic
24C4×Dic122—41no—
25C4×Dic122—41no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×#11758196—21no—
29C2×#11758196—21no—
30C2×#11758196—21no—
31C2×C22⋊Dic61—21no—
32C2×C22⋊Dic61—21no—
33C2×C4×Dic61—21no—
34C2×C4×Dic61—21no—
35C4⋊#92418—41no—
36C22⋊#92418—22no—
37C4×#92418—41no—
38#14519684—21no—
39#14519684—21no—
40#14519684—21no—
41#14519684—21no—
42#14519684C2×C411yescharacteristic
43#14519684C2×C411yescharacteristic
44C23×C4×C61D411yescharacteristic
45#117×C61—24no—
46#117×C61—41no—
47#117×C61—41no—
48#117×C61—42no—
49C2×C4⋊C4×C61—41no—
50C2×C4⋊C4×C61C2×C411yescharacteristic
51C2×C4⋊C4×C61D411yescharacteristic
52C2×C22⋊C4×C61—22no—
53C2×C22⋊C4×C61D411yescharacteristic
54C2×C22⋊C4×C61C2311yescharacteristic
55C2×C42×C61—41no—
56C2×C42×C61C2×C411yescharacteristic
57C2×C42×C61D411yescharacteristic
58C24×C61C2×D411yescharacteristic
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—22no—
62C22×C4×C61—42no—
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61SD1612yes—
68C22×C4×C61Q1612yes—
69C22×C4×C61C22×C411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C2×Dic122—41no—
76C2×Dic122—42no—
77C4⋊C4×C61—41no—
78C4⋊C4×C61—41no—
79C4⋊C4×C61—44no—
80C22⋊C4×C61—24no—
81C22⋊C4×C61—41no—
82C22⋊C4×C61—41no—
83C22⋊C4×C61—42no—
84C42×C61—21no—
85C42×C61—21no—
86C42×C61—21no—
87C42×C61—21no—
88C42×C61—44no—
89C22⋊Dic61—42no—
90C22⋊Dic61—42no—
91C4⋊Dic61—42no—
92C4⋊Dic61—41no—
93#11758196—42no—
94#11758196—42no—
95#11758196—42no—
96C4×Dic61—41no—
97C4×Dic61—41no—
98C4×Dic61—42no—
99C2×#92418—42no—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—41no—
103C23×C61—41no—
104C23×C61C2×C22⋊C411yescharacteristic
105C23×C61C2×SD1611yescharacteristic
106C23×C61C2×Q1611yescharacteristic
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—22no—
110C2×C4×C61—22no—
111C2×C4×C61—22no—
112C2×C4×C61—22no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—44no—
122C2×C4×C61—82no—
123C2×C4×C61—82no—
124C2×C4×C61#9911yescharacteristic
125C2×C4×C61#9911yescharacteristic
126C2×C4×C61#10312yes—
127C2×C4×C61#10312yes—
128C2×C4×C61#10412yes—
129C2×Dic61—22no—
130C2×Dic61—22no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141Dic122—42no—
142Dic122—42no—
143#92418—82no—
144#11715—611nomaximal, Sylow
145C22×C61—21no—
146C22×C61—21no—
147C22×C61—21no—
148C22×C61—21no—
149C22×C61—41no—
150C22×C61—41no—
151C22×C61—41no—
152C22×C61—41no—
153C22×C61—81no—
154C22×C61C2×#9911yescharacteristic
155C22×C61C2×#10311yescharacteristic
156C22×C61C2×#10411yescharacteristic
157C4×C61—22no—
158C4×C61—41no—
159C4×C61—41no—
160C4×C61—41no—
161C4×C61—41no—
162C4×C61—84no—
163C4×C61#32812yes—
164C4×C61#32812yes—
165Dic61—42no—
166Dic61—42no—
167Dic61—81no—
168Dic61—81no—
169#2396—611no—
170#1740D6111yescharacteristic
171#1458—611no—
172#1352—612no—
173#1351—612no—
174C2×C61—21no—
175C2×C61—21no—
176C2×C61—41no—
177C2×C61—41no—
178C2×C61C2×#32811yescharacteristic
179C2×C61#212011yescharacteristic
180C2×C61#212611yescharacteristic
181C2×C4×Q8—611no—
182C2×#117—21no—
183C2×#117C2×D6111yescharacteristic
184#431—1221no—
185#400—611no—
186#399—611no—
187#398—1221no—
188#396—1221no—
189#395—611no—
190#393—611no—
191C4×C4⋊C4—611no—
192C4×C4⋊C4—1221no—
193C4×C22⋊C4—1221no—
194#342—21no—
195#342C2×D6112yes—
196#324—1221no—
197#323—612no—
198C61#1171511yescharacteristic, complemented, Sylow
199C22×Q8—611no—
200C23×C4C22⋊D6111yescharacteristic
201C4×Q8—2441no—
202C4×Q8—2441no—
203#117—24no—
204#117—41no—
205#117—41no—
206#117—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—611no—
209C2×C4⋊C4—1221no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1221no—
212C2×C4⋊C4—1221no—
213C2×C4⋊C4C4×D6111yescharacteristic
214C2×C4⋊C4D24411yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—1221no—
217C2×C22⋊C4—1221no—
218C2×C22⋊C4C22⋊D6111yescharacteristic
219C2×C22⋊C4C22×D6111yescharacteristic
220C2×C42—41no—
221C2×C42—1221no—
222C2×C42—1221no—
223C2×C42C4×D6111yescharacteristic
224C2×C42D24411yescharacteristic
225C4⋊C8—2441no—
226C22⋊C8—1222no—
227C4×C8—2441no—
228#95—611no—
229#95—611no—
230#95—1221no—
231#95—1221no—
232#95—1221no—
233#95—1221no—
234C24C2×C22⋊D6111yescharacteristic
235C2×Q8—2441no—
236C2×Q8—2442no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—42no—
241C22×C4—1221no—
242C22×C4—1221no—
243C22×C4—1221no—
244C22×C4—1221no—
245C22×C4#1175819811yescharacteristic
246C22×C4#1175819811yescharacteristic
247C22×C4#1175819811yescharacteristic
248C22×C4#1175819811yescharacteristic
249C22×C4#1175820012yes—
250C22×C4#1175820212yes—
251C22×C4C2×C4×D6111yescharacteristic
252C22×C4C2×D24411yescharacteristic
253C2×C8—2442no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—44no—
257C4⋊C4—2441no—
258C4⋊C4—2442no—
259C4⋊C4—2442no—
260C4⋊C4—2442no—
261C4⋊C4—2442no—
262C22⋊C4—24no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—42no—
266C22⋊C4—2442no—
267C22⋊C4—2442no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—44no—
273C42—2441no—
274C42—2441no—
275C42—2442no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23C2×#1175819811yescharacteristic
281C23C2×#1175820011yescharacteristic
282C23C2×#1175820211yescharacteristic
283Q8—2442no—
284Q8—2442no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—1222no—
303C2×C4—1222no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4#1451969111yescharacteristic
315C2×C4#1451969111yescharacteristic
316C2×C4#1451970012yes—
317C2×C4#1451970012yes—
318C2×C4#1451970512yes—
319C8—4882no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22C2×#1451969111yescharacteristic
330C22C2×#1451970011yescharacteristic
331C22C2×#1451970511yescharacteristic
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—84no—
338C4—4881no—
339C4—4881no—
340C4—2442no—
341C4—2442no—
342C4#1683949812yes—
343C4#1683949812yes—
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2C2×#1683949811yescharacteristic
349C2#2144745811yescharacteristic
350C2#2144748211yescharacteristic
subgroupG/H
centerC2×C4#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1740C2
socleC22×C61C2×#99
prime radicalC24×C61C2×D4
cosocleC2×C22⋊C4×C61C23
hypercenter#1740D61
hypocenterC61#11715
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
6. C2×C4×Dic122C4semidirect
8. #16839701C4semidirect
12. #16839682C4semidirect
15. C4×C4⋊Dic61C4semidirect
198. C61#11715semidirect

Maximal subgroups

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

subgroupindex
#214490872
#214441332
#214438422
#214438392
C61×#17402
#1171561