Finite Group Databasebeta

← #29576059

Subgroups — #29576059

Order 15616.

560 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 127 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#21451632C211yesmaximal, characteristic
2#21448437C211yesmaximal, characteristic
3#21444392C212yesmaximal
4#21443687C212yesmaximal
5C2×C61×#324C211yesmaximal, characteristic
6C2×#14519756C2211yescharacteristic
7C2×#14519736—21no—
8#16839701C412yescomplemented
9#16839695—22no—
10#16839682C412yescomplemented
11#16839661—24no—
12#16839635—21no—
13#16839625—21no—
14#16839622—21no—
15#16839622C2211yescharacteristic
16#16839622C2212yes—
17#16839500—24no—
18C22×C4⋊C4×C61C2211yescharacteristic
19C2×C22⋊C8×C61C2212yes—
20C61×#324—22no—
21C22×Dic122C2×C411yescharacteristic
22#14519756—22no—
23#14519756—22no—
24#14519736—42no—
25#14519736—42no—
26C2×C4⋊Dic61—22no—
27C2×C4⋊Dic61—22no—
28C2×C4⋊Dic61—21no—
29C2×C4⋊Dic61—22no—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×#11758196—21no—
32C2×#11758196—22no—
33C2×C22⋊Dic61—22no—
34C2×C22⋊Dic61—21no—
35C2×C22⋊Dic61—21no—
36C2×C4×Dic61—22no—
37#14519684—22no—
38#14519684—22no—
39#14519684—22no—
40#14519684—22no—
41#14519684C2×C412yes—
42#14519684D414yes—
43C23×C4×C61D411yescharacteristic
44C23×C4×C61D411yescharacteristic
45C23×C4×C61C2311yescharacteristic
46C22×C8×C61—22no—
47C2×C4⋊C4×C61—22no—
48C2×C4⋊C4×C61—22no—
49C2×C4⋊C4×C61C2×C412yes—
50C2×C4⋊C4×C61D412yes—
51C22⋊C8×C61—22no—
52C22⋊C8×C61—22no—
53C24×C61C2×D411yescharacteristic
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61—81no—
61C22×C4×C61—81no—
62C22×C4×C61C22⋊C412yes—
63C22×C4×C61C22⋊C412yes—
64C22×C4×C61SD1612yes—
65C22×C4×C61Q1612yes—
66C22×C4×C61C22×C411yescharacteristic
67C22×C4×C61C2×D412yes—
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61#4112yes—
70C22×Dic61—21no—
71C22×Dic61—21no—
72C22×Dic61—22no—
73C22×Dic61—24no—
74C2×Dic122—22no—
75C2×Dic122—81no—
76C2×C8×C61—42no—
77C2×C8×C61—82no—
78C4⋊C4×C61—24no—
79C4⋊C4×C61—24no—
80C22⋊Dic61—41no—
81C22⋊Dic61—41no—
82C22⋊Dic61—42no—
83C22⋊Dic61—41no—
84C22⋊Dic61—42no—
85C22⋊Dic61—41no—
86C4⋊Dic61—22no—
87C4⋊Dic61—44no—
88C4⋊Dic61—81no—
89C4⋊Dic61—82no—
90C4⋊Dic61—82no—
91#11758196—42no—
92#11758196—82no—
93C4×Dic61—44no—
94C23×C61—41no—
95C23×C61—41no—
96C23×C61C2×C22⋊C411yescharacteristic
97C23×C61#12011yescharacteristic
98C23×C61#12311yescharacteristic
99C23×C61C2×SD1611yescharacteristic
100C23×C61C2×Q1611yescharacteristic
101C23×C61AGL(1,ℤ/8)11yescharacteristic
102C23×C61#13711yescharacteristic
103C2×C4×C61—22no—
104C2×C4×C61—22no—
105C2×C4×C61—22no—
106C2×C4×C61—22no—
107C2×C4×C61—22no—
108C2×C4×C61—22no—
109C2×C4×C61—22no—
110C2×C4×C61—41no—
111C2×C4×C61—42no—
112C2×C4×C61—42no—
113C2×C4×C61—41no—
114C2×C4×C61—42no—
115C2×C4×C61—42no—
116C2×C4×C61—81no—
117C2×C4×C61#10314yes—
118C2×C4×C61C4×D412yes—
119C2×C4×C61#12112yes—
120C2×Dic61—42no—
121C2×Dic61—42no—
122C2×Dic61—44no—
123C2×Dic61—42no—
124C2×Dic61—44no—
125C2×Dic61—42no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—42no—
129C2×Dic61—44no—
130Dic122—82no—
131C8×C61—82no—
132#9200—611nomaximal, Sylow
133C22×C61—21no—
134C22×C61—21no—
135C22×C61—21no—
136C22×C61—21no—
137C22×C61—21no—
138C22×C61—21no—
139C22×C61—41no—
140C22×C61—41no—
141C22×C61—41no—
142C22×C61—41no—
143C22×C61#38611yescharacteristic
144C22×C61C2×#10311yescharacteristic
145C22×C61#41711yescharacteristic
146C22×C61#45011yescharacteristic
147C22×C61#45111yescharacteristic
148C22×C61#48311yescharacteristic
149C22×C61#48411yescharacteristic
150C4×C61—24no—
151C4×C61—22no—
152C4×C61—22no—
153C4×C61—42no—
154Dic61—84no—
155Dic61—82no—
156Dic61—81no—
157Dic61—81no—
158#2627—611no—
159#2316—611no—
160#1520—612no—
161C2×#324D6111yescharacteristic
162#1289—612no—
163C2×C61—21no—
164C2×C61—21no—
165C2×C61—21no—
166C2×C61—21no—
167C2×C61—21no—
168C2×C61—21no—
169C2×C61#187211yescharacteristic
170C2×C61#191211yescharacteristic
171C2×C61#191311yescharacteristic
172C2×C22⋊Q8—611no—
173C2×#117—1221no—
174C22×C4⋊C4C2×D6111yescharacteristic
175C2×C22⋊C8C2×D6112yes—
176#400—612no—
177#398—1222no—
178#395—612no—
179#389—1224no—
180#385—1221no—
181#381—1221no—
182#380—611no—
183#380—612no—
184#380—1221no—
185#328—1224no—
186#324—22no—
187C61#920011yescharacteristic, complemented, Sylow
188C22×Q8—611no—
189C23×C4D24411yescharacteristic
190C23×C4D24411yescharacteristic
191C23×C4C22×D6111yescharacteristic
192C22×C8—22no—
193C22⋊Q8—1222no—
194C22⋊Q8—1222no—
195#117—2442no—
196#117—2442no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—611no—
200C2×C4⋊C4—1222no—
201C2×C4⋊C4—1222no—
202C2×C4⋊C4—1222no—
203C2×C4⋊C4—1221no—
204C2×C4⋊C4—1222no—
205C2×C4⋊C4—1221no—
206C2×C4⋊C4C4×D6112yes—
207C2×C4⋊C4C22⋊D6112yes—
208C2×C22⋊C4—1222no—
209C2×C22⋊C4—1221no—
210C2×C22⋊C4—1221no—
211C2×C42—1222no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214#95—612no—
215#95—614no—
216#95—1222no—
217#95—1222no—
218#95—1222no—
219#95—1222no—
220C24C2×D24411yescharacteristic
221C2×Q8—1222no—
222C2×Q8—4881no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4—1222no—
232C22×C4—1221no—
233C22×C4—1221no—
234C22×C4—1224no—
235C22×C4#1175819012yes—
236C22×C4Dic24412yes—
237C22×C4#1175819812yes—
238C22×C4#1175819812yes—
239C22×C4C2×C4×D6111yescharacteristic
240C22×C4D4×D6112yes—
241C22×C4#1175822412yes—
242C22×C4C2×C22⋊D6111yescharacteristic
243C2×C8—42no—
244C2×C8—82no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—1222no—
248C4⋊C4—4882no—
249C4⋊C4—4882no—
250C4⋊C4—4881no—
251C4⋊C4—4882no—
252C4⋊C4—2444no—
253C4⋊C4—2442no—
254C22⋊C4—2441no—
255C22⋊C4—2441no—
256C22⋊C4—2441no—
257C22⋊C4—2442no—
258C22⋊C4—2442no—
259C22⋊C4—2441no—
260C42—2444no—
261C23—41no—
262C23—41no—
263C23C2×#1175819811yescharacteristic
264C23#1451974711yescharacteristic
265C23#1451976111yescharacteristic
266C23C2×#1175819011yescharacteristic
267C23C2×Dic24411yescharacteristic
268C23#1451980511yescharacteristic
269C23#1451981011yescharacteristic
270Q8—4882no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—81no—
285C2×C4—2442no—
286C2×C4—2444no—
287C2×C4—2442no—
288C2×C4—2442no—
289C2×C4—2444no—
290C2×C4—2442no—
291C2×C4—2442no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4—2444no—
295C2×C4#1451970214yes—
296C2×C4#1451973712yes—
297C2×C4#1451974912yes—
298C8—82no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22#1683963911yescharacteristic
310C22C2×#1451970211yescharacteristic
311C22#1683976411yescharacteristic
312C22#1683991511yescharacteristic
313C22#1683992011yescharacteristic
314C22#1684009311yescharacteristic
315C22#1684009811yescharacteristic
316C4—22no—
317C4—22no—
318C4—24no—
319C4—42no—
320C4—4884no—
321C4—4882no—
322C4—4881no—
323C4—4881no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2#2144612411yescharacteristic
331C2#2144635411yescharacteristic
332C2#2144635911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#324C2
socleC22×C61#386
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenterC2×#324D61
hypocenterC61#9200
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
8. #16839701C4semidirect
10. #16839682C4semidirect
187. C61#9200semidirect

Maximal subgroups

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

subgroupindex
#214516322
#214484372
#214443922
#214436872
C2×C61×#3242
#920061