Finite Group Databasebeta

← #29577890

Subgroups — #29577890

Order 15616.

544 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 103 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#21454238C211yesmaximal, characteristic
2#21451396C211yesmaximal, characteristic
3#21444265C212yesmaximal, complemented
4#21443685C212yesmaximal, complemented
5C61×#1464C211yesmaximal, characteristic
6C22×#11758194—21no—
7C2×C4×Dic122C2211yescharacteristic
8#16839718—24no—
9C4×#11758194—21no—
10C4×#11758194C2212yes—
11#16839692C411yescharacteristic
12#16839687C411yescharacteristic
13#16839682C411yescharacteristic
14#16839674C411yescharacteristic
15#16839650—21no—
16#16839646—21no—
17#16839643—21no—
18#16839635—21no—
19#16839630C2211yescharacteristic
20#16839492—24no—
21C2×#117×C61C2211yescharacteristic
22C61×#432C2212yes—
23C61×#324—21no—
24C61×#323—21no—
25C22×Dic122C2×C411yescharacteristic
26C2×#11758194—22no—
27C2×#11758194—24no—
28C2×#11758194—24no—
29C2×#11758194—22no—
30C22×#92418—22no—
31C4×Dic122—22no—
32C4×Dic122—22no—
33C2×C4⋊Dic61C2×C411yescharacteristic
34C2×#11758196—21no—
35C2×#11758196—21no—
36C2×#11758196—21no—
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—21no—
39C2×C4×Dic61—21no—
40#14519688—22no—
41#14519688—22no—
42C4×#92418—22no—
43C4×#92418D414yes—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684C2×C411yescharacteristic
51#14519684C2×C411yescharacteristic
52C23×C4×C61D411yescharacteristic
53#117×C61—22no—
54#117×C61—22no—
55C2×C4⋊C4×C61C2×C411yescharacteristic
56C2×C4⋊C4×C61D411yescharacteristic
57C2×C22⋊C4×C61C2×C411yescharacteristic
58C2×C22⋊C4×C61D411yescharacteristic
59C2×C42×C61D411yescharacteristic
60C2×C42×C61C2311yescharacteristic
61C4⋊C8×C61—22no—
62C22⋊C8×C61—22no—
63#97×C61—22no—
64C24×C61C22⋊C411yescharacteristic
65C22×C4×C61—22no—
66C22×C4×C61—22no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22×C411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C2×Dic122—22no—
80C2×Dic122—42no—
81C2×C8×C61—42no—
82C4⋊C4×C61—22no—
83C4⋊C4×C61—22no—
84C22⋊C4×C61—22no—
85C22⋊C4×C61—22no—
86C42×C61—21no—
87C42×C61—21no—
88C42×C61C2×D412yes—
89C42×C61#4112yes—
90C22⋊Dic61—42no—
91C22⋊Dic61—42no—
92C4⋊Dic61—22no—
93#11758196—42no—
94#11758196—42no—
95#11758196—41no—
96#11758196—41no—
97C4×Dic61—42no—
98#11758194—28no—
99#11758194—24no—
100#11758194—24no—
101C2×#92418—28no—
102C2×#92418—24no—
103C23×C61—21no—
104C23×C61—21no—
105C23×C61—41no—
106C23×C61—41no—
107C23×C61#10412yes—
108C23×C61C2×C22⋊C411yescharacteristic
109C2×C4×C61—22no—
110C2×C4×C61—22no—
111C2×C4×C61—24no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61#10412yes—
123C2×C4×C61C4×D412yes—
124C2×C4×C61#12011yescharacteristic
125C2×C4×C61#12112yes—
126C2×C4×C61#12311yescharacteristic
127C2×Dic61—22no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139Dic122—24no—
140Dic122—42no—
141C8×C61—82no—
142#92418—28no—
143#9491—611nomaximal, Sylow
144C22×C61—21no—
145C22×C61—21no—
146C22×C61—22no—
147C22×C61—42no—
148C22×C61—41no—
149C22×C61—41no—
150C22×C61—81no—
151C22×C61#38611yescharacteristic
152C22×C61C2×#10411yescharacteristic
153C22×C61C2×#10411yescharacteristic
154C4×C61—22no—
155C4×C61—21no—
156C4×C61—22no—
157C4×C61—21no—
158C4×C61—22no—
159C4×C61—41no—
160C4×C61—41no—
161Dic61—42no—
162Dic61—81no—
163Dic61—81no—
164Dic61—81no—
165#2873—611no—
166#2599—611no—
167#1491—612no—
168#1464D6111yescharacteristic
169#1288—612no—
170C2×C61—21no—
171C2×C61—21no—
172C2×C61—42no—
173C2×C61#189111yescharacteristic
174C2×C61#189611yescharacteristic
175C2×C61#190911yescharacteristic
176C22×M16—1221no—
177C2×C4×Q8—611no—
178C2×#117C2×D6111yescharacteristic
179#432C2×D6112yes—
180#405—1224no—
181C4×M16—612no—
182C4×M16—1221no—
183#398—611no—
184#397—611no—
185#395—611no—
186#393—611no—
187#388—1221no—
188#387—1221no—
189#387—1221no—
190#385—1221no—
191#383—611no—
192Q8⋊C8—1224no—
193#324—21no—
194#323—21no—
195C61#949111yescharacteristic, complemented, Sylow
196C22×Q8—611no—
197C23×C4D24411yescharacteristic
198C2×M16—1222no—
199C2×M16—1224no—
200C2×M16—1224no—
201C2×M16—1222no—
202C22×C8—1222no—
203C4×Q8—1222no—
204C4×Q8—1222no—
205#117—22no—
206#117—22no—
207C2×C4⋊C4—611no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4—1221no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4C4×D6111yescharacteristic
212C2×C4⋊C4C22⋊D6111yescharacteristic
213C2×C22⋊C4—1221no—
214C2×C22⋊C4—1221no—
215C2×C22⋊C4C4×D6111yescharacteristic
216C2×C22⋊C4C22⋊D6111yescharacteristic
217C2×C42—1221no—
218C2×C42D24411yescharacteristic
219C2×C42C22×D6111yescharacteristic
220C4⋊C8—22no—
221C22⋊C8—22no—
222#97—22no—
223#97—1222no—
224#97—1222no—
225C4×C8—614no—
226C4×C8—1222no—
227#95—611no—
228#95—611no—
229#95—1221no—
230#95—1221no—
231#95—1221no—
232#95—1221no—
233#95—1221no—
234#95—1221no—
235C24#1175819811yescharacteristic
236C2×Q8—1222no—
237C2×Q8—2442no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—1221no—
243C22×C4—1221no—
244C22×C4—1221no—
245C22×C4—1221no—
246C22×C4#1175819811yescharacteristic
247C22×C4#1175819811yescharacteristic
248C22×C4#1175819811yescharacteristic
249C22×C4C2×C4×D6111yescharacteristic
250C22×C4C2×D24411yescharacteristic
251C22×C4C2×C22⋊D6111yescharacteristic
252M16—1228no—
253M16—1224no—
254M16—1224no—
255C2×C8—42no—
256C2×C8—1228no—
257C2×C8—1224no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—1222no—
261C4⋊C4—2442no—
262C4⋊C4—2442no—
263C4⋊C4—2441no—
264C4⋊C4—2441no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—2442no—
268C22⋊C4—2442no—
269C42—21no—
270C42—21no—
271C42—2442no—
272C42D4×D6112yes—
273C42#1175822412yes—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23#1451970312yes—
279C23C2×#1175819811yescharacteristic
280Q8—1224no—
281Q8—2442no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—1222no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4#1451970312yes—
308C2×C4#1451973712yes—
309C2×C4#1451974711yescharacteristic
310C2×C4#1451974912yes—
311C2×C4#1451976111yescharacteristic
312C8—82no—
313C8—1228no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1683963911yescharacteristic
322C22C2×#1451970311yescharacteristic
323C22C2×#1451970311yescharacteristic
324C4—22no—
325C4—21no—
326C4—22no—
327C4—21no—
328C4—22no—
329C4—41no—
330C4—41no—
331C4—4881no—
332C4—4881no—
333C4—4881no—
334C4—2442no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2#2144623511yescharacteristic
339C2#2144626011yescharacteristic
340C2#2144633911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1464C2
socleC22×C61#386
prime radicalC24×C61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1464D61
hypocenterC61#9491
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #21444265C2semidirect
4. #21443685C2semidirect
195. C61#9491semidirect

Maximal subgroups

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

subgroupindex
#214542382
#214513962
#214442652
#214436852
C61×#14642
#949161