Finite Group Databasebeta

← #29589288

Subgroups — #29589288

Order 15616.

480 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 89 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#21448362C211yesmaximal, characteristic
2#21444136C212yesmaximal
3#21443762C212yesmaximal, complemented
4#21443753C211yesmaximal, characteristic, complemented
5C61×#1319C211yesmaximal, characteristic, complemented
6C2×#14519766C411yescharacteristic
7C2×Q8×Dic61C411yescharacteristic
8C2×#14519739C2211yescharacteristic
9C22⋊C8×D61—22no—
10#16839664C411yescharacteristic
11#16839654—22no—
12#16839627C2211yescharacteristic
13#16839609—22no—
14C42×Dic61C411yescharacteristic
15#16839488C2212yes—
16#16839485—22no—
17C61×#391C2211yescharacteristic
18C61×#324C2212yes—
19C22×D244C2×C411yescharacteristic
20C22×C4×D61D412yes—
21C2×C8×D61—42no—
22#14519766—22no—
23#14519766C814yescomplemented
24Q8×Dic61—41no—
25Q8×Dic61—41no—
26#14519739—41no—
27#14519739—41no—
28C2×C4⋊Dic61—21no—
29C2×C4⋊Dic61C2×C411yescharacteristic
30C2×#11758198—22no—
31C2×#11758198—22no—
32C2×#11758198D412yes—
33C2×C4×Dic61—21no—
34C2×C4×Dic61—21no—
35C2×C4×Dic61—21no—
36C2×C4×Dic61C2×C411yescharacteristic
37C2×C4×Dic61C2×C411yescharacteristic
38C22⋊#92418—22no—
39#14519689—42no—
40#14519684—22no—
41C22×Q8×C61C2×C411yescharacteristic
42C2×C4⋊C4×C61C2311yescharacteristic
43C2×C42×C61C2×C411yescharacteristic
44C22⋊C8×C61—22no—
45#95×C61—22no—
46C23×D61C22⋊C412yes—
47C2×Q8×C61—21no—
48C2×Q8×C61—41no—
49C2×Q8×C61—41no—
50C2×Q8×C61C2×C812yescomplemented
51C22×C4×C61—21no—
52C22×C4×C61—21no—
53C22×C4×C61C22×C411yescharacteristic
54C22×C4×C61C2×D412yes—
55C22×Dic61—21no—
56C22×Dic61—21no—
57C22×Dic61C22⋊C411yescharacteristic
58C22×Dic61C22⋊C411yescharacteristic
59C2×D244—41no—
60C2×D244—41no—
61C2×D244C2×C812yes—
62C2×D244M1612yes—
63C2×C4×D61—22no—
64C2×C4×D61—22no—
65C2×C4×D61—44no—
66C2×C8×C61—42no—
67C4⋊C4×C61—41no—
68C42×C61—41no—
69#11758198—28no—
70#11758198—42no—
71#11758198—44no—
72C4⋊Dic61—42no—
73C4⋊Dic61—41no—
74C4×Dic61—21no—
75C4×Dic61—21no—
76C4×Dic61—21no—
77C4×Dic61—42no—
78C4×Dic61—42no—
79C4×Dic61—41no—
80C4×Dic61—41no—
81C4×Dic61C2×C812yescomplemented
82C2×#92418—42no—
83C8×D61—84no—
84C23×C61C2×C22⋊C411yescharacteristic
85C22×D61—22no—
86C22×D61—22no—
87C22×D61—42no—
88C22×D61—42no—
89C22×D61C22⋊C814yes—
90Q8×C61—42no—
91Q8×C61—42no—
92C2×C4×C61—21no—
93C2×C4×C61—21no—
94C2×C4×C61—22no—
95C2×C4×C61—22no—
96C2×C4×C61—22no—
97C2×C4×C61—41no—
98C2×C4×C61—41no—
99C2×C4×C61—41no—
100C2×C4×C61—41no—
101C2×C4×C61—42no—
102C2×C4×C61C22×C811yescharacteristic
103C2×C4×C61C2×M1611yescharacteristic
104C2×Dic61—22no—
105C2×Dic61—22no—
106C2×Dic61—21no—
107C2×Dic61—41no—
108C2×Dic61—41no—
109C2×Dic61—41no—
110C2×Dic61—42no—
111C2×Dic61—41no—
112C2×Dic61#9911yescharacteristic
113C2×Dic61#9911yescharacteristic
114D244—42no—
115D244—42no—
116C4×D61—44no—
117C4×D61—82no—
118C8×C61—82no—
119#92418—82no—
120#11477—611nomaximal, Sylow
121C22×C61—21no—
122C22×C61—21no—
123C22×C61C2×C22⋊C811yescharacteristic
124C22×C61C2×#9911yescharacteristic
125C22×C61#41311yescharacteristic
126C2×D61—24no—
127C2×D61—42no—
128C2×D61—42no—
129C2×D61—44no—
130C2×D61—82no—
131C4×C61—21no—
132C4×C61—21no—
133C4×C61—42no—
134C4×C61—42no—
135C4×C61—42no—
136Dic61—21no—
137Dic61—42no—
138Dic61—41no—
139Dic61—81no—
140Dic61#35412yes—
141C4×#124—611no—
142#1459—612no—
143#1324—612no—
144#1319—611no—
145#1319D6111yescharacteristic, complemented
146C2×C61—21no—
147C2×C61—21no—
148C2×C61#145911yescharacteristic
149C2×C61C2×#35411yescharacteristic
150C2×C61#213211yescharacteristic
151D61—44no—
152C2×#124—611no—
153C2×C4×Q8—611no—
154C2×C4×D4—611no—
155C2×C22⋊C8—1222no—
156#391C2×D6111yescharacteristic
157#390—611no—
158#388—1222no—
159#382—611no—
160C4×C22⋊C4—1222no—
161C43—611no—
162#324—612no—
163#324C2×D6112yes—
164#323—1222no—
165C61#1147711yescharacteristic, complemented, Sylow
166C22×Q8C4×D6111yescharacteristic
167C22×D4—611no—
168C23×C4—612no—
169C22×C8—2442no—
170#124—614no—
171#124—1222no—
172C4×Q8—2441no—
173C4×Q8—2441no—
174C4×D4—2441no—
175C4×D4—2441no—
176C2×C4⋊C4—611no—
177C2×C4⋊C4—1221no—
178C2×C4⋊C4C22×D6111yescharacteristic
179C2×C22⋊C4—612no—
180C2×C22⋊C4—1222no—
181C2×C22⋊C4—1222no—
182C2×C42—611no—
183C2×C42—611no—
184C2×C42—1221no—
185C2×C42—1221no—
186C2×C42—1221no—
187C2×C42C4×D6111yescharacteristic
188C22⋊C8—22no—
189C22⋊C8—1222no—
190C22⋊C8—2442no—
191#95—22no—
192#95—1222no—
193C24—612no—
194C2×Q8—21no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8C8×D6112yescomplemented
198C2×D4—612no—
199C2×D4—612no—
200C2×D4—2441no—
201C2×D4—2441no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—611no—
205C22×C4—611no—
206C22×C4—1222no—
207C22×C4—1222no—
208C22×C4—1221no—
209C22×C4—1221no—
210C22×C4—2444no—
211C22×C4C2×C4×D6111yescharacteristic
212C22×C4D4×D6112yes—
213C2×C8—42no—
214C2×C8—4884no—
215C2×C8—2442no—
216C4⋊C4—41no—
217C4⋊C4—2441no—
218C4⋊C4—2442no—
219C22⋊C4—1228no—
220C22⋊C4—2442no—
221C22⋊C4—2444no—
222C42—41no—
223C42—612no—
224C42—1221no—
225C42—1221no—
226C42—1221no—
227C42—2442no—
228C42—2441no—
229C42—2441no—
230C42—2442no—
231C23—614no—
232C23—1222no—
233C23—1222no—
234C23—2442no—
235C23—2442no—
236C23C22⋊C4×D6111yescharacteristic
237Q8—42no—
238Q8—42no—
239D4—2442no—
240D4—2442no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—611no—
252C2×C4—611no—
253C2×C4—1222no—
254C2×C4—1222no—
255C2×C4—1221no—
256C2×C4—4882no—
257C2×C4—2444no—
258C2×C4—2442no—
259C2×C4—2441no—
260C2×C4—2441no—
261C2×C4—2441no—
262C2×C4—2441no—
263C2×C4C2×C8×D6111yescharacteristic
264C2×C4M16×D6111yescharacteristic
265C8—82no—
266C8—4882no—
267C22—21no—
268C22—21no—
269C22—1224no—
270C22—4882no—
271C22—2444no—
272C22—2442no—
273C22—2442no—
274C22C22⋊C8×D6111yescharacteristic
275C22#99×D6111yescharacteristic
276C22#1683974311yescharacteristic
277C4—21no—
278C4—21no—
279C4—42no—
280C4—42no—
281C4—42no—
282C4—612no—
283C4—1221no—
284C4—4881no—
285C4—2442no—
286C4—2441no—
287C2—21no—
288C2—21no—
289C2—2444no—
290C2#2144413411yescharacteristic
291C2#354×D6111yescharacteristic
292C2#2144750311yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC2×C4×C61C22×C8
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1319C2
socleC22×C61C2×#99
prime radicalC22×D244C2×C4
cosocleC2×C4⋊C4×C61C23
hypercenter#1319D61
hypocenterC61#11477
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
3. #21443762C2semidirect
4. #21443753C2semidirect
5. C61×#1319C2semidirect
23. #14519766C8semidirect
50. C2×Q8×C61C2×C8semidirect
81. C4×Dic61C2×C8semidirect
145. #1319D61semidirect
165. C61#11477semidirect
197. C2×Q8C8×D61semidirect

Maximal subgroups

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

subgroupindex
#214483622
#214441362
#214437622
#214437532
C61×#13192
#1147761