Finite Group Databasebeta

← #29577763

Subgroups — #29577763

Order 15616.

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

normal subgroups only

This group has 554 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#21448422C211yesmaximal, characteristic, complemented
2#21445899C211yesmaximal, characteristic, complemented
3#21444348C212yesmaximal
4#21443672C212yesmaximal, complemented
5C61×#2504C211yesmaximal, characteristic
6#16839726C2212yes—
7#16839648—24no—
8#16839625C2212yes—
9#16839622—21no—
10C4×C22⋊Dic61—21no—
11#16839531—22no—
12#16839496—24no—
13C2×#121×C61C2211yescharacteristic
14C22×C4⋊C4×C61C2211yescharacteristic
15C61×#397—22no—
16C61×#390C412yescomplemented
17C61×#386—21no—
18C61×#386—21no—
19C61×#382C412yescomplemented
20C4×C22⋊C4×C61C2211yescharacteristic
21C2×#11758194—22no—
22C2×C4⋊Dic61—22no—
23C2×C22⋊Dic61—22no—
24C2×C4×Dic61—22no—
25C22⋊#92418—22no—
26C22⋊#92418—22no—
27#14519684—22no—
28#14519684—22no—
29#14519684—22no—
30#14519684D414yes—
31C22×D4×C61—21no—
32C22×D4×C61C2×C411yescharacteristic
33C23×C4×C61D411yescharacteristic
34C23×C4×C61D411yescharacteristic
35C23×C4×C61C2311yescharacteristic
36#121×C61—22no—
37#121×C61—22no—
38C2×C4⋊C4×C61—22no—
39C2×C4⋊C4×C61—22no—
40C2×C4⋊C4×C61C2×C411yescharacteristic
41C2×C4⋊C4×C61C2×C412yes—
42C2×C4⋊C4×C61D412yes—
43C2×C22⋊C4×C61—21no—
44C2×C22⋊C4×C61—21no—
45C2×C22⋊C4×C61—22no—
46C2×C22⋊C4×C61—22no—
47C2×C42×C61C2×C412yes—
48#95×C61—22no—
49#95×C61—22no—
50#95×C61—21no—
51C24×C61—21no—
52C24×C61C2×D411yescharacteristic
53C2×D4×C61—22no—
54C2×D4×C61—42no—
55C2×D4×C61—44no—
56C2×D4×C61—41no—
57C2×D4×C61—41no—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—22no—
64C22×C4×C61—42no—
65C22×C4×C61—81no—
66C22×C4×C61—81no—
67C22×C4×C61C22⋊C412yes—
68C22×C4×C61C22⋊C412yes—
69C22×C4×C61C22×C411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61C2×D412yes—
72C22×C4×C61#4112yes—
73C22×Dic61—24no—
74C4⋊C4×C61—24no—
75C4⋊C4×C61—24no—
76C4⋊C4×C61—22no—
77C22⋊C4×C61—42no—
78C22⋊C4×C61—42no—
79C22⋊C4×C61—82no—
80C22⋊C4×C61—82no—
81C42×C61—42no—
82C22⋊Dic61—44no—
83C4⋊Dic61—82no—
84C4×Dic61—82no—
85#11758194—44no—
86C2×#92418—42no—
87C23×C61—22no—
88C23×C61—22no—
89C23×C61—41no—
90C23×C61—42no—
91C23×C61—42no—
92C23×C61—41no—
93C23×C61—42no—
94C23×C61C2×C22⋊C411yescharacteristic
95C23×C61#12011yescharacteristic
96C23×C61#12311yescharacteristic
97D4×C61—82no—
98D4×C61—82no—
99D4×C61—82no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61—22no—
103C2×C4×C61—22no—
104C2×C4×C61—42no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61—42no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61—42no—
113C2×C4×C61—41no—
114C2×C4×C61—42no—
115C2×C4×C61—81no—
116C2×C4×C61—81no—
117C2×C4×C61C4×D412yes—
118C2×C4×C61#12112yes—
119C2×Dic61—44no—
120C2×Dic61—44no—
121C2×Dic61—44no—
122#92418—82no—
123#9465—611nomaximal, Sylow
124C22×C61—21no—
125C22×C61—21no—
126C22×C61—21no—
127C22×C61—21no—
128C22×C61—21no—
129C22×C61—21no—
130C22×C61—21no—
131C22×C61—44no—
132C22×C61—84no—
133C22×C61—81no—
134C22×C61—81no—
135C22×C61—81no—
136C22×C61#38611yescharacteristic
137C22×C61#41012yes—
138C22×C61#45412yes—
139C4×C61—22no—
140C4×C61—22no—
141C4×C61—42no—
142C4×C61—42no—
143C4×C61—82no—
144C4×C61—81no—
145Dic61—84no—
146#2504D6111yescharacteristic
147#2313—611no—
148#1821—611no—
149#1510—612no—
150#1284—612no—
151C2×C61—22no—
152C2×C61—22no—
153C2×C61—41no—
154C2×C61—161no—
155C2×C61#188811yescharacteristic
156C2×C61#189911yescharacteristic
157C2×C61#191111yescharacteristic
158C2×#121C2×D6111yescharacteristic
159C22×C4⋊C4C2×D6111yescharacteristic
160#407—612no—
161#397—22no—
162#390Dic6112yescomplemented
163#387—1224no—
164#386—21no—
165#386—21no—
166#382Dic6112yescomplemented
167#381—612no—
168#380—1221no—
169C4×C22⋊C4—1221no—
170C4×C22⋊C4C2×D6111yescharacteristic
171#343—1222no—
172#327—1224no—
173C61#946511yescharacteristic, complemented, Sylow
174C22×D4—21no—
175C22×D4C2×Dic6111yescharacteristic
176C23×C4C22⋊D6111yescharacteristic
177C23×C4C22⋊D6111yescharacteristic
178C23×C4C22×D6111yescharacteristic
179C2×M16—1222no—
180#121—22no—
181#121—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—1222no—
185C2×C4⋊C4C2×Dic6111yescharacteristic
186C2×C4⋊C4C2×Dic6112yes—
187C2×C4⋊C4C22⋊D6112yes—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—1222no—
193C2×C42—1222no—
194C2×C42C2×Dic6112yes—
195C22⋊C8—1222no—
196C22⋊C8—1222no—
197#95—22no—
198#95—22no—
199#95—21no—
200#95—614no—
201#95—1222no—
202#95—1222no—
203#95—1222no—
204C24—21no—
205C24C2×C22⋊D6111yescharacteristic
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—44no—
209C2×D4—41no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4—1224no—
221C22×C4C22⋊Dic6112yes—
222C22×C4C22⋊Dic6112yes—
223C22×C4D4×D6112yes—
224C22×C4#1175822412yes—
225C22×C4C22×Dic6111yescharacteristic
226C22×C4C2×C22⋊D6111yescharacteristic
227M16—2444no—
228C2×C8—2442no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—22no—
232C4⋊C4—4882no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C22⋊C4—2444no—
238C42—42no—
239C42—4882no—
240C23—22no—
241C23—22no—
242C23—41no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—41no—
247C23C2×C22⋊Dic6111yescharacteristic
248C23#1451974511yescharacteristic
249C23#1451975811yescharacteristic
250D4—82no—
251D4—82no—
252D4—82no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—2444no—
271C2×C4—2444no—
272C2×C4—2444no—
273C2×C4D4×Dic6112yes—
274C2×C4#1451974912yes—
275C8—4882no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—84no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22#1683964211yescharacteristic
289C22#1683973612yes—
290C22#1683993412yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—82no—
296C4—81no—
297C4—4884no—
298C2—22no—
299C2—22no—
300C2—41no—
301C2—161no—
302C2#2144622311yescharacteristic
303C2#2144627811yescharacteristic
304C2#2144635211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2504C2
socleC22×C61#386
prime radicalC2×#121×C61C22
cosocleC23×C4×C61C23
hypercenter#2504D61
hypocenterC61#9465
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2504

Complements

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

NH ≅ G/Ntype
1. #21448422C2semidirect
2. #21445899C2semidirect
4. #21443672C2semidirect
16. C61×#390C4semidirect
19. C61×#382C4semidirect
162. #390Dic61semidirect
166. #382Dic61semidirect
173. C61#9465semidirect

Maximal subgroups

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

subgroupindex
#214484222
#214458992
#214443482
#214436722
C61×#25042
#946561