Finite Group Databasebeta

← #29574950

Subgroups — #29574950

Order 15616.

578 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#16840157C211yesmaximal, characteristic
2#21448993C211yesmaximal, characteristic
3#21446792C212yesmaximal, complemented
4#21443701C212yesmaximal, complemented
5C2×C61×#329C211yesmaximal, characteristic
6C22×#11758202—21no—
7C22×#11758200—21no—
8C2×#14519777C2211yescharacteristic
9#16840157—22no—
10#16840157C414yescomplemented
11C2×#14519688C2211yescharacteristic
12#16839669—21no—
13#16839655—22no—
14#16839642—21no—
15#16839637—22no—
16#16839628C2212yes—
17#16839525—22no—
18#16839525—22no—
19C2×#124×C61C2211yescharacteristic
20C2×#97×C61C2212yes—
21C61×#329—22no—
22C22×Dic122—21no—
23C23×Dic61—21no—
24C2×#11758202—24no—
25C2×#11758202—42no—
26C2×#11758202—42no—
27C2×#11758200—24no—
28C2×#11758200—42no—
29C2×#11758200—42no—
30C22×#92418—21no—
31#14519777—22no—
32#14519777—21no—
33#14519777C2×C412yescomplemented
34C2×C4⋊Dic61—22no—
35C2×C4⋊Dic61—21no—
36C2×C4⋊Dic61D412yes—
37C2×C4⋊Dic61D412yes—
38C2×C22⋊Dic61—22no—
39C2×C22⋊Dic61—21no—
40C2×C4×Dic61—21no—
41#14519688—21no—
42#14519688C2×C412yescomplemented
43#14519684—22no—
44#14519684—22no—
45#14519684—22no—
46C22×Q8×C61D411yescharacteristic
47C22×D4×C61D411yescharacteristic
48C22×C8×C61—22no—
49#124×C61—22no—
50#124×C61C2×C412yes—
51#124×C61D412yes—
52C2×C22⋊C4×C61—22no—
53C2×C42×C61C2311yescharacteristic
54#97×C61—22no—
55#97×C61—22no—
56C24×C61—21no—
57C2×Q8×C61—21no—
58C2×Q8×C61—41no—
59C2×Q8×C61—41no—
60C2×Q8×C61C22⋊C412yes—
61C2×D4×C61—21no—
62C2×D4×C61—41no—
63C2×D4×C61—41no—
64C2×D4×C61C22⋊C412yes—
65C22×C4×C61—21no—
66C22×C4×C61C2×D411yescharacteristic
67C22×C4×C61C2×D412yes—
68C22×Dic61—22no—
69C22×Dic61—22no—
70C22×Dic61—41no—
71C22×Dic61—81no—
72C22×Dic61—82no—
73C2×Dic122—22no—
74C2×Dic122—41no—
75C2×Dic122—41no—
76C2×Dic122—41no—
77C2×Dic122—41no—
78C2×Dic122—41no—
79C2×C8×C61—24no—
80C2×C8×C61—42no—
81C2×C8×C61—42no—
82C22⋊C4×C61—24no—
83C22⋊C4×C61—24no—
84C42×C61C22×C411yescharacteristic
85C42×C61C2×D411yescharacteristic
86C42×C61#4112yes—
87C22⋊Dic61—81no—
88C22⋊Dic61—82no—
89#11758202—48no—
90#11758200—48no—
91C4⋊Dic61—24no—
92C4⋊Dic61—42no—
93C4⋊Dic61—42no—
94C4⋊Dic61—42no—
95C4⋊Dic61—81no—
96C4×Dic61—81no—
97C2×#92418—22no—
98C2×#92418—22no—
99C2×#92418—22no—
100C23×C61—22no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61—42no—
104C23×C61—42no—
105C23×C61#12011yescharacteristic
106Q8×C61—42no—
107Q8×C61—81no—
108D4×C61—42no—
109D4×C61—81no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—22no—
113C2×C4×C61—22no—
114C2×C4×C61—22no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61C2×C22⋊C411yescharacteristic
118C2×C4×C61C4×D412yes—
119C2×C4×C61#12112yes—
120C2×C4×C61#12311yescharacteristic
121C2×Dic61—42no—
122C2×Dic61—42no—
123C2×Dic61—42no—
124C2×Dic61—42no—
125C2×Dic61—42no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—84no—
129C2×Dic61—81no—
130C2×Dic61—81no—
131Dic122—42no—
132Dic122—42no—
133Dic122—81no—
134Dic122—81no—
135C8×C61—44no—
136#92418—24no—
137#9003—611nomaximal, Sylow
138C22×C61—44no—
139C22×C61—42no—
140C22×C61—81no—
141C22×C61—81no—
142C22×C61—82no—
143C22×C61#38611yescharacteristic
144C22×C61#45311yescharacteristic
145C22×C61#45311yescharacteristic
146C22×C61#45512yes—
147C22×C61#45612yes—
148C4×C61—22no—
149C4×C61—22no—
150C4×C61—21no—
151C4×C61—21no—
152C4×C61—42no—
153Dic61—82no—
154Dic61—82no—
155Dic61—82no—
156C2×#497—611no—
157#2379—611no—
158#1993—612no—
159C2×#329D6111yescharacteristic
160#1295—612no—
161C2×C61—82no—
162C2×C61#165911yescharacteristic
163C2×C61#166011yescharacteristic
164C2×C61#166112yes—
165C2×C61#188212yes—
166C2×C61#188711yescharacteristic
167C22×Q16—1221no—
168C22×SD16—1221no—
169C2×C4⋊Q8—611no—
170C2×#124C2×D6111yescharacteristic
171#497—614no—
172#497—1222no—
173C2×#97—611no—
174C2×#97C2×D6112yes—
175#391—1221no—
176#388—1222no—
177#386—1221no—
178#385—1222no—
179#382—612no—
180#340—1222no—
181#340—1222no—
182#329—22no—
183C61#900311yescharacteristic, complemented, Sylow
184C22×Q8—1221no—
185C22×Q8D24411yescharacteristic
186C22×D4D24411yescharacteristic
187C23×C4—1221no—
188C2×Q16—1224no—
189C2×Q16—2442no—
190C2×Q16—2442no—
191C2×SD16—1224no—
192C2×SD16—2442no—
193C2×SD16—2442no—
194C22×C8—22no—
195C22×C8—1221no—
196C4⋊Q8—612no—
197C4⋊Q8—1222no—
198C4⋊Q8—1221no—
199#124—22no—
200#124C4×D6112yes—
201#124C22⋊D6112yes—
202C2×C4⋊C4—612no—
203C2×C4⋊C4—612no—
204C2×C4⋊C4—1222no—
205C2×C4⋊C4—1221no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—1222no—
208C2×C22⋊C4—1221no—
209C2×C42—1221no—
210C2×C42C22×D6111yescharacteristic
211#97—22no—
212#97—22no—
213#97—612no—
214#97—1221no—
215#95—1222no—
216#95—1222no—
217#95—1222no—
218C24—21no—
219C2×Q8—21no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—1222no—
223C2×Q8—2441no—
224C2×Q8—2441no—
225C2×Q8—2441no—
226C2×Q8—2441no—
227C2×Q8—2441no—
228C2×Q8#1175819812yes—
229C2×D4—21no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4#1175819812yes—
233C22×C4—21no—
234C22×C4—1222no—
235C22×C4—1222no—
236C22×C4—4882no—
237C22×C4—4881no—
238C22×C4—2441no—
239C22×C4C2×D24411yescharacteristic
240C22×C4D4×D6112yes—
241Q16—2448no—
242SD16—2448no—
243C2×C8—24no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—1222no—
247C2×C8—1222no—
248C2×C8—1222no—
249C4⋊C4—1224no—
250C4⋊C4—4881no—
251C4⋊C4—2442no—
252C4⋊C4—2442no—
253C4⋊C4—2442no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—4882no—
257C22⋊C4—4881no—
258C42—4881no—
259C42C2×C4×D6111yescharacteristic
260C42#1175822412yes—
261C42C2×C22⋊D6111yescharacteristic
262C23—22no—
263C23—42no—
264C23—42no—
265C23—41no—
266C23—41no—
267C23#1451974711yescharacteristic
268Q8—42no—
269Q8—81no—
270Q8—4881no—
271Q8—4881no—
272Q8—2442no—
273Q8—2442no—
274D4—42no—
275D4—81no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—4884no—
284C2×C4—4881no—
285C2×C4—4881no—
286C2×C4—2442no—
287C2×C4—2442no—
288C2×C4—2442no—
289C2×C4—2442no—
290C2×C4—2442no—
291C2×C4—2442no—
292C2×C4—2442no—
293C2×C4C2×#1175819811yescharacteristic
294C2×C4#1451973712yes—
295C2×C4#1451974912yes—
296C2×C4#1451976111yescharacteristic
297C8—44no—
298C8—1224no—
299C22—44no—
300C22—42no—
301C22—81no—
302C22—81no—
303C22—82no—
304C22#1683963911yescharacteristic
305C22#1683993211yescharacteristic
306C22#1683993211yescharacteristic
307C22#1683994112yes—
308C22#1683994612yes—
309C4—22no—
310C4—22no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—4882no—
315C4—4882no—
316C4—4882no—
317C2—82no—
318C2#2144512611yescharacteristic
319C2#2144513111yescharacteristic
320C2#2144513712yes—
321C2#2144618612yes—
322C2#2144621511yescharacteristic
subgroupG/H
centerC23#14519747
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC2×C61×#329C2
socleC23×C61#120
prime radicalC22×D4×C61D4
cosocleC2×C42×C61C23
hypercenterC2×#329D61
hypocenterC61#9003
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
3. #21446792C2semidirect
4. #21443701C2semidirect
10. #16840157C4semidirect
33. #14519777C2×C4semidirect
42. #14519688C2×C4semidirect
183. C61#9003semidirect

Maximal subgroups

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

subgroupindex
C2×#168401572
#214489932
#214467922
#214437012
C2×C61×#3292
#900361