Finite Group Databasebeta

← #30607235

Subgroups — #30607235

Order 15616.

656 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 80 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#21461655C211yesmaximal, characteristic
2#21461648C211yesmaximal, characteristic
3#21460903C214yesmaximal, complemented
4#21448309C211yesmaximal, characteristic
5C2×#14519838—21no—
6C2×#14519837C2212yes—
7C2×#14519836—21no—
8C2×#14519835C2212yes—
9#16840734—21no—
10#16840705—21no—
11C22×#11758222C2211yescharacteristic
12C2×#14519736C411yescharacteristic, complemented
13#16839668—21no—
14#16839664—21no—
15#16839641C412yescomplemented
16#16839641C2212yes—
17C4×#11758198—21no—
18C4×#11758198—21no—
19C22×C42×C61C411yescharacteristic, complemented
20C2×#11758217—22no—
21C2×C22⋊#6923—22no—
22#14519838—42no—
23#14519837—24no—
24#14519836—42no—
25#14519835—24no—
26#14519834—42no—
27(C2×#6923)⋊C4—42no—
28C2×#11758222—24no—
29C2×#11758222—42no—
30C2×#11758222—42no—
31C22×Dic122D411yescharacteristic
32C22×D244D411yescharacteristic
33C22×C22⋊D61C2×C411yescharacteristic
34C22×C22⋊D61C2311yescharacteristic
35C22×C4×D61D411yescharacteristic
36C22×C4×D61D411yescharacteristic
37#14519736—42no—
38C2×C4⋊Dic61—21no—
39C2×C22⋊Dic61C2×C412yes—
40C2×#11758198—22no—
41C2×#11758198—22no—
42C2×C4×Dic61—21no—
43#14519684—22no—
44#14519684—22no—
45C23×C4×C61C2×C412yes—
46C23×C4×C61C2×C411yescharacteristic
47C2×C42×C61—21no—
48C2×C42×C61—21no—
49C2×C42×C61—42no—
50C24×C61C22×C411yescharacteristic
51C23×D61C22⋊C411yescharacteristic
52C23×D61C2×D411yescharacteristic
53C22×#6923—42no—
54C22×C4×C61—22no—
55C22×C4×C61—22no—
56C22×C4×C61—22no—
57C22×C4×C61—22no—
58C22×C4×C61—44no—
59C22×C4×C61—42no—
60C22×C4×C61C22⋊C411yescharacteristic
61C22×C4×C61C22⋊C411yescharacteristic
62C2×C22⋊D61—22no—
63C2×C22⋊D61—41no—
64C2×C22⋊D61—41no—
65C2×C22⋊D61—41no—
66C2×C22⋊D61—41no—
67C2×C22⋊D61#4114yes—
68C22×Dic61—22no—
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61C2×D411yescharacteristic
71#11758222—416no—
72C2×D244—22no—
73C2×D244—22no—
74C2×D244—41no—
75C2×C4×D61—22no—
76C2×C4×D61—22no—
77C2×C4×D61—22no—
78C2×C4×D61—22no—
79C2×C4×D61—41no—
80C2×C4×D61—41no—
81C2×Dic122—22no—
82C2×Dic122—22no—
83C2×Dic122—41no—
84C22⋊#6923—42no—
85C22⋊#6923—42no—
86#11758217—42no—
87#11758217—42no—
88C2×#92420—42no—
89C42×C61—42no—
90C42×C61—42no—
91C22⋊Dic61—42no—
92#11758198—44no—
93#11758198—44no—
94C4⋊Dic61—42no—
95C4×Dic61—42no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—42no—
99C23×C61C2×C22⋊C411yescharacteristic
100C23×C61#11712yes—
101C22×D61—21no—
102C22×D61—21no—
103C22×D61—21no—
104C22×D61—21no—
105C22×D61—21no—
106C22×D61—42no—
107C22×D61—42no—
108C22×D61#12312yes—
109C2×#6923—82no—
110C2×#6923—84no—
111C2×C4×C61—28no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—24no—
117C2×C4×C61—42no—
118C2×C4×C61—44no—
119C2×C4×C61—42no—
120C2×C4×C61—44no—
121C2×C4×C61#9912yes—
122C2×C4×C61#9912yes—
123C22⋊D61—44no—
124C22⋊D61—44no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—42no—
131C2×Dic61—42no—
132C2×Dic61—42no—
133C2×Dic61#12312yes—
134D244—44no—
135C4×D61—44no—
136C4×D61—44no—
137Dic122—44no—
138#92420—82no—
139#12853—611nomaximal, Sylow
140C22×C61—22no—
141C22×C61—22no—
142C22×C61—42no—
143C22×C61—42no—
144C22×C61—42no—
145C22×C61#38111yescharacteristic
146C22×C61C2×#9911yescharacteristic
147C22×C61C2×#9911yescharacteristic
148C2×D61—22no—
149C2×D61—42no—
150C2×D61—42no—
151C2×D61—42no—
152C2×D61—41no—
153C2×D61—42no—
154C2×D61—81no—
155C2×D61—81no—
156#6923—162no—
157C4×C61—22no—
158C4×C61—22no—
159C4×C61—44no—
160Dic61—22no—
161Dic61—41no—
162Dic61—82no—
163#2297—611no—
164#1785—611no—
165#1782—611no—
166#1390—614no—
167C2×C61—21no—
168C2×C61—21no—
169C2×C61—42no—
170C2×C61#181911yescharacteristic
171C2×C61#212012yes—
172D61—42no—
173D61—81no—
174C22×#41—611no—
175C2×#117—611no—
176C22×C42#692311yescharacteristic, complemented
177C2×#101—1221no—
178C2×#100—612no—
179C2×#99—612no—
180C2×#99—1221no—
181#407—1221no—
182#391—1221no—
183#390—1221no—
184#386—612no—
185#386—612no—
186#381—1221no—
187C4×C22⋊C4—1221no—
188C4×C22⋊C4—1221no—
189C61#1285311yescharacteristic, complemented, Sylow
190C2×#41—1224no—
191C2×#41—2442no—
192C2×#41—2442no—
193C22×Q8—611no—
194C22×D4—611no—
195C22×D4—611no—
196C22×D4—611no—
197C23×C4—611no—
198C23×C4—611no—
199C23×C4C2×#692312yes—
200C23×C4C2×#692311yescharacteristic
201C2×M16—1222no—
202#117—2442no—
203C2×C4⋊C4—1221no—
204C2×C22⋊C4—612no—
205C2×C22⋊C4—1222no—
206C2×C22⋊C4—1222no—
207C2×C22⋊C4—1222no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—42no—
211C2×C42—1221no—
212#101—2442no—
213#100—1224no—
214#99—1224no—
215#99—2442no—
216C22⋊C8—2442no—
217#95—1222no—
218#95—1222no—
219#95—2442no—
220C24—611no—
221C24—611no—
222C24C22×#692311yescharacteristic
223#41—24416no—
224C2×Q8—1222no—
225C2×Q8—1222no—
226C2×Q8—2441no—
227C2×D4—614no—
228C2×D4—1222no—
229C2×D4—1222no—
230C2×D4—1222no—
231C2×D4—2441no—
232C2×D4—2441no—
233C2×D4—2441no—
234C2×D4—2441no—
235C2×D4—2441no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—611no—
243C22×C4—611no—
244C22×C4—1222no—
245C22×C4—1222no—
246C22×C4—1222no—
247C22×C4—1222no—
248C22×C4—1222no—
249C22×C4—2441no—
250C22×C4—2441no—
251C22×C4—2442no—
252C22×C4C22⋊#692311yescharacteristic
253C22×C4C22⋊#692311yescharacteristic
254M16—2442no—
255M16—2442no—
256C2×C8—2442no—
257C4⋊C4—2442no—
258C22⋊C4—2444no—
259C22⋊C4—2444no—
260C22⋊C4—2442no—
261C22⋊C4—2442no—
262C22⋊C4—2442no—
263C42—42no—
264C42—42no—
265C42—2442no—
266C23—21no—
267C23—21no—
268C23—42no—
269C23—612no—
270C23—1221no—
271C23—1221no—
272C23—1221no—
273C23—1221no—
274C23—1221no—
275C23—2442no—
276C23—2442no—
277C23C2×C22⋊#692311yescharacteristic
278C23#1451985612yes—
279Q8—2444no—
280D4—2444no—
281D4—2444no—
282D4—2444no—
283C2×C4—28no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—612no—
294C2×C4—1221no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—4882no—
300C2×C4—4884no—
301C2×C4—2444no—
302C2×C4—2444no—
303C2×C4—2442no—
304C2×C4—2442no—
305C2×C4—2442no—
306C2×C4#1451983512yes—
307C2×C4#1451983612yes—
308C8—4882no—
309C22—22no—
310C22—22no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—1222no—
315C22—4881no—
316C22—4881no—
317C22—2442no—
318C22—2442no—
319C22—2442no—
320C22—2441no—
321C22—2442no—
322C22#1684070511yescharacteristic
323C22C2×#1451983511yescharacteristic
324C22C2×#1451983611yescharacteristic
325C4—22no—
326C4—22no—
327C4—44no—
328C4—1222no—
329C4—9762no—
330C4—4882no—
331C4—2441no—
332C2—21no—
333C2—21no—
334C2—42no—
335C2—4881no—
336C2—2442no—
337C2#2146171411yescharacteristic
338C2#2146207012yes—
subgroupG/H
centerC22C2×#14519835
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC22×C42×C61C4
socleC22×C61C2×#99
prime radicalC22×#11758222C22
cosocleC22×C22⋊D61C23
hypercenterC22×C42#6923
hypocenterC61#12853
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
3. #21460903C2semidirect
12. C2×#14519736C4semidirect
15. #16839641C4semidirect
19. C22×C42×C61C4semidirect
176. C22×C42#6923semidirect
189. C61#12853semidirect

Maximal subgroups

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

subgroupindex
#214616552
#214616482
#214609032
#214483092
#1285361