Finite Group Databasebeta

← #29596019

Subgroups — #29596019

Order 15616.

480 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 93 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#21448533C211yesmaximal, characteristic
2C2×#16839558C212yesmaximal
3#21445813C212yesmaximal, complemented
4#21443946C211yesmaximal, characteristic, complemented
5C61×#1394C211yesmaximal, characteristic, complemented
6C2×#14519766C2211yescharacteristic
7C2×Q8×Dic61C2211yescharacteristic
8C2×#14519739C411yescharacteristic
9#16839785—22no—
10C2×#14519696C2212yes—
11#16839664C411yescharacteristic
12#16839651—22no—
13#16839636—22no—
14#16839631C411yescharacteristic
15#16839608C411yescharacteristic
16#16839558—24no—
17#16839530—22no—
18C2×#101×C61C2212yes—
19C61×#391C2211yescharacteristic
20C22×D244C2×C411yescharacteristic
21C22×C4×D61—21no—
22C2×#11758194—22no—
23#14519766—22no—
24#14519766D412yes—
25#14519766Q812yes—
26Q8×Dic61—42no—
27#14519739—42no—
28C2×C4⋊Dic61C2×C411yescharacteristic
29C2×C4⋊Dic61D412yes—
30C2×#11758196—22no—
31C2×#11758198—22no—
32C2×#11758198—21no—
33C2×#11758198—22no—
34C2×C4×Dic61C2×C411yescharacteristic
35C2×C4×Dic61C2×C411yescharacteristic
36C2×C4×Dic61D412yes—
37#14519706—44no—
38#14519696—22no—
39#14519696—22no—
40#14519684—21no—
41#14519684—21no—
42C22×Q8×C61C2311yescharacteristic
43C2×M16×C61—22no—
44C2×C4⋊C4×C61C2×C411yescharacteristic
45C2×C42×C61C2×C411yescharacteristic
46#101×C61—22no—
47#101×C61—22no—
48#95×C61—22no—
49C23×D61—21no—
50C2×Q8×C61—41no—
51C2×Q8×C61—41no—
52C2×Q8×C61C2×D411yescharacteristic
53C2×Q8×C61C2×Q811yescharacteristic
54C2×Q8×C61#4112yes—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61C22×C411yescharacteristic
58C22×C4×C61C2×D412yes—
59C22×Dic61—21no—
60C22×Dic61C22⋊C411yescharacteristic
61C22×Dic61C22⋊C412yes—
62C22×Dic61C22⋊C411yescharacteristic
63C2×D244—21no—
64C2×D244—41no—
65C2×D244—41no—
66C2×D244C4⋊C412yes—
67C2×C4×D61—42no—
68C2×C4×D61—82no—
69M16×C61—42no—
70M16×C61—42no—
71C2×C8×C61—42no—
72C4⋊C4×C61—41no—
73C42×C61—41no—
74#11758198—24no—
75#11758198—24no—
76#11758198—42no—
77#11758198—82no—
78C4⋊Dic61—41no—
79C4⋊Dic61—42no—
80#11758196—82no—
81C4×Dic61—21no—
82C4×Dic61—24no—
83C4×Dic61—22no—
84C4×Dic61C4⋊C412yes—
85#11758194—42no—
86#11758194—42no—
87C2×#92418—42no—
88C23×C61C2×C22⋊C411yescharacteristic
89C22×D61—22no—
90C22×D61—42no—
91C22×D61—42no—
92C22×D61—41no—
93C22×D61—41no—
94Q8×C61—41no—
95Q8×C61—41no—
96Q8×C61—42no—
97C2×C4×C61—22no—
98C2×C4×C61—21no—
99C2×C4×C61—22no—
100C2×C4×C61—21no—
101C2×C4×C61—42no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—41no—
106C2×C4×C61C2×C4⋊C411yescharacteristic
107C2×C4×C61#11711yescharacteristic
108C2×C4×C61#12112yes—
109C2×C4×C61C22⋊Q812yes—
110C2×Dic61—22no—
111C2×Dic61—21no—
112C2×Dic61—22no—
113C2×Dic61—41no—
114C2×Dic61—41no—
115C2×Dic61—41no—
116C2×Dic61—42no—
117C2×Dic61—41no—
118C2×Dic61#9912yes—
119D244—42no—
120D244—42no—
121C4×D61—84no—
122C8×C61—82no—
123#92418—82no—
124#12803—611nomaximal, Sylow
125C22×C61—21no—
126C22×C61—21no—
127C22×C61#38011yescharacteristic
128C22×C61C2×#9911yescharacteristic
129C22×C61#41011yescharacteristic
130C2×D61—44no—
131C2×D61—42no—
132C2×D61—82no—
133C2×D61—81no—
134C2×D61—81no—
135C4×C61—22no—
136C4×C61—21no—
137C4×C61—21no—
138C4×C61—22no—
139C4×C61—81no—
140C4×C61—81no—
141Dic61—22no—
142Dic61—42no—
143Dic61—41no—
144Dic61—42no—
145#2326—611no—
146C2×#355—612no—
147#1801—612no—
148#1394—611no—
149#1394D6111yescharacteristic, complemented
150C2×C61—21no—
151C2×C61—21no—
152C2×C61#179911yescharacteristic
153C2×C61#213311yescharacteristic
154C2×C61#213411yescharacteristic
155D61—82no—
156C2×#124—611no—
157C2×C4×Q8—611no—
158C2×C4×D4—611no—
159#423—1222no—
160C2×#101—612no—
161C2×#101C2×D6112yes—
162#391C2×D6111yescharacteristic
163#390—611no—
164#388—1222no—
165#385—1222no—
166#384—611no—
167#376—611no—
168#355—1224no—
169#343—1222no—
170C61#1280311yescharacteristic, complemented, Sylow
171C22×Q8C22×D6111yescharacteristic
172C22×D4—611no—
173C23×C4—1221no—
174C2×M16—22no—
175C2×M16—1222no—
176#124—612no—
177#124—612no—
178#124—1222no—
179C4×Q8—2442no—
180C4×D4—2442no—
181C2×C4⋊C4—611no—
182C2×C4⋊C4—612no—
183C2×C4⋊C4—1222no—
184C2×C4⋊C4C4×D6111yescharacteristic
185C2×C22⋊C4—1222no—
186C2×C22⋊C4—1222no—
187C2×C22⋊C4—1221no—
188C2×C42—611no—
189C2×C42—611no—
190C2×C42—612no—
191C2×C42C4×D6111yescharacteristic
192C4⋊C8—2444no—
193#101—22no—
194#101—22no—
195#101—1222no—
196#101—1222no—
197#95—22no—
198#95—1221no—
199#95—1221no—
200C24—1221no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8#1175822212yes—
204C2×Q8D4×D6111yescharacteristic
205C2×Q8Q8×D6111yescharacteristic
206C2×D4—612no—
207C2×D4—1221no—
208C2×D4—2441no—
209C2×D4—2441no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—611no—
213C22×C4—611no—
214C22×C4—612no—
215C22×C4—1221no—
216C22×C4—4882no—
217C22×C4—2442no—
218C22×C4C2×C4×D6111yescharacteristic
219C22×C4D4×D6112yes—
220M16—42no—
221M16—42no—
222M16—2442no—
223M16—2442no—
224C2×C8—42no—
225C2×C8—2442no—
226C4⋊C4—41no—
227C4⋊C4—4882no—
228C4⋊C4—2441no—
229C4⋊C4—2442no—
230C22⋊C4—1224no—
231C22⋊C4—1224no—
232C22⋊C4—4882no—
233C22⋊C4—2442no—
234C42—41no—
235C42—612no—
236C42—1224no—
237C42—1222no—
238C42—1221no—
239C23—1222no—
240C23—2442no—
241C23—2442no—
242C23—2441no—
243C23—2441no—
244C23C22⋊C4×D6111yescharacteristic
245Q8—42no—
246Q8—41no—
247Q8—41no—
248D4—2442no—
249D4—2442no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—612no—
260C2×C4—1222no—
261C2×C4—1221no—
262C2×C4—1222no—
263C2×C4—4884no—
264C2×C4—2441no—
265C2×C4—2441no—
266C2×C4—2441no—
267C2×C4—2442no—
268C2×C4—2441no—
269C2×C4C4⋊C4×D6111yescharacteristic
270C2×C4#1451973411yescharacteristic
271C2×C4#1451975012yes—
272C2×C4#1451975512yes—
273C8—82no—
274C8—4882no—
275C22—21no—
276C22—21no—
277C22—4882no—
278C22—4881no—
279C22—4881no—
280C22—2444no—
281C22—2442no—
282C22#1683962111yescharacteristic
283C22#99×D6111yescharacteristic
284C22#1683973411yescharacteristic
285C4—22no—
286C4—21no—
287C4—21no—
288C4—22no—
289C4—81no—
290C4—81no—
291C4—1222no—
292C4—2442no—
293C4—2441no—
294C4—2442no—
295C2—21no—
296C2—21no—
297C2—4882no—
298C2#2144580611yescharacteristic
299C2#2144750711yescharacteristic
300C2#2144750911yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC22×C4×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC61×#1394C2
socleC22×C61C2×#99
prime radicalC22×D244C2×C4
cosocleC22×Q8×C61C23
hypercenter#1394D61
hypocenterC61#12803
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
3. #21445813C2semidirect
4. #21443946C2semidirect
5. C61×#1394C2semidirect
149. #1394D61semidirect
170. C61#12803semidirect

Maximal subgroups

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

subgroupindex
#214485332
C2×#168395582
#214458132
#214439462
C61×#13942
#1280361