Finite Group Databasebeta

← #29595202

Subgroups — #29595202

Order 15616.

512 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 113 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#101×Dic61C212yesmaximal, complemented
4#21443946C211yesmaximal, characteristic, complemented
5C61×#1394C211yesmaximal, characteristic, complemented
6C2×#14519766C2211yescharacteristic
7C2×Q8×Dic61C2211yescharacteristic
8C2×#14519739C411yescharacteristic
9C2×#14519696C2212yes—
10M16×Dic61—22no—
11#16839664C411yescharacteristic
12#16839651—22no—
13#16839636—22no—
14#16839631C411yescharacteristic
15#16839608C411yescharacteristic
16#16839558C418yescomplemented
17#16839530—22no—
18C2×#101×C61C2212yes—
19C61×#391C2211yescharacteristic
20C22×D244C2×C411yescharacteristic
21C22×C4×D61—21no—
22C2×#11758194—22no—
23#14519766—21no—
24#14519766—21no—
25#14519766C2×C414yes—
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#14519696—22no—
38#14519696C2×C414yescomplemented
39#14519687—42no—
40C8×Dic61—42no—
41#14519684—21no—
42#14519684—21no—
43C22×Q8×C61C2311yescharacteristic
44C2×M16×C61—22no—
45C2×C4⋊C4×C61C2×C411yescharacteristic
46C2×C42×C61C2×C411yescharacteristic
47#101×C61—22no—
48#101×C61C2×C414yescomplemented
49#95×C61—22no—
50C23×D61—21no—
51C2×Q8×C61—41no—
52C2×Q8×C61—41no—
53C2×Q8×C61C22×C412yes—
54C2×Q8×C61#4111yescharacteristic
55C2×Q8×C61#4111yescharacteristic
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61C22×C411yescharacteristic
59C22×C4×C61C2×D412yes—
60C22×Dic61—21no—
61C22×Dic61C22⋊C411yescharacteristic
62C22×Dic61C22⋊C412yes—
63C22×Dic61C22⋊C411yescharacteristic
64C2×D244—21no—
65C2×D244—41no—
66C2×D244—41no—
67C2×D244C4212yes—
68C2×C4×D61—42no—
69C2×C4×D61—82no—
70M16×C61—24no—
71M16×C61—42no—
72C2×C8×C61—42no—
73C4⋊C4×C61—41no—
74C42×C61—41no—
75#11758198—24no—
76#11758198—24no—
77#11758198—42no—
78#11758198—82no—
79C4⋊Dic61—41no—
80C4⋊Dic61—42no—
81#11758196—82no—
82C4×Dic61—21no—
83C4×Dic61—22no—
84C4×Dic61—22no—
85C4×Dic61—22no—
86C4×Dic61C4212yes—
87#11758194—24no—
88#11758194—42no—
89C2×#92418—42no—
90C23×C61C2×C22⋊C411yescharacteristic
91C22×D61—22no—
92C22×D61—42no—
93C22×D61—42no—
94C22×D61—41no—
95C22×D61—41no—
96Q8×C61—42no—
97Q8×C61—41no—
98Q8×C61—41no—
99C2×C4×C61—22no—
100C2×C4×C61—21no—
101C2×C4×C61—22no—
102C2×C4×C61—21no—
103C2×C4×C61—42no—
104C2×C4×C61—41no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61C2×C4211yescharacteristic
109C2×C4×C61#11711yescharacteristic
110C2×C4×C61C4×D412yes—
111C2×C4×C61C4×D412yes—
112C2×Dic61—22no—
113C2×Dic61—21no—
114C2×Dic61—22no—
115C2×Dic61—41no—
116C2×Dic61—41no—
117C2×Dic61—41no—
118C2×Dic61—42no—
119C2×Dic61—41no—
120C2×Dic61#9912yes—
121D244—42no—
122D244—42no—
123C4×D61—84no—
124C8×C61—44no—
125#92418—44no—
126#12632—611nomaximal, Sylow
127C22×C61—21no—
128C22×C61—21no—
129C22×C61C4×C22⋊C411yescharacteristic
130C22×C61C2×#9911yescharacteristic
131C22×C61#41011yescharacteristic
132C2×D61—44no—
133C2×D61—42no—
134C2×D61—82no—
135C2×D61—81no—
136C2×D61—81no—
137C4×C61—22no—
138C4×C61—22no—
139C4×C61—21no—
140C4×C61—21no—
141C4×C61—81no—
142C4×C61—81no—
143Dic61—22no—
144Dic61—42no—
145Dic61—41no—
146Dic61—42no—
147#2326—611no—
148C2×#355—612no—
149C4×#101—612no—
150#1394—611no—
151#1394D6111yescharacteristic, complemented
152C2×C61—21no—
153C2×C61—21no—
154C2×C61C4×#9911yescharacteristic
155C2×C61#213311yescharacteristic
156C2×C61#213411yescharacteristic
157D61—82no—
158C2×#124—611no—
159C2×C4×Q8—611no—
160C2×C4×D4—611no—
161C2×#101—612no—
162C2×#101C2×D6112yes—
163C4×M16—1222no—
164#391C2×D6111yescharacteristic
165#390—611no—
166#388—1222no—
167#385—1222no—
168#384—611no—
169#376—611no—
170#355—618no—
171#343—1222no—
172C61#1263211yescharacteristic, complemented, Sylow
173C22×Q8C22×D6111yescharacteristic
174C22×D4—611no—
175C23×C4—1221no—
176C2×M16—22no—
177C2×M16—1222no—
178#124—614no—
179#124—1221no—
180#124—1221no—
181C4×Q8—2442no—
182C4×D4—2442no—
183C2×C4⋊C4—611no—
184C2×C4⋊C4—612no—
185C2×C4⋊C4—1222no—
186C2×C4⋊C4C4×D6111yescharacteristic
187C2×C22⋊C4—1222no—
188C2×C22⋊C4—1222no—
189C2×C22⋊C4—1221no—
190C2×C42—611no—
191C2×C42—611no—
192C2×C42—612no—
193C2×C42C4×D6111yescharacteristic
194#101—22no—
195#101—614no—
196#101—1222no—
197#101C4×D6114yescomplemented
198#97—2442no—
199C4×C8—2442no—
200#95—22no—
201#95—1221no—
202#95—1221no—
203C24—1221no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8C2×C4×D6112yes—
207C2×Q8#1175822411yescharacteristic
208C2×Q8#1175822611yescharacteristic
209C2×D4—612no—
210C2×D4—1221no—
211C2×D4—2441no—
212C2×D4—2441no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—611no—
216C22×C4—611no—
217C22×C4—612no—
218C22×C4—1221no—
219C22×C4—4882no—
220C22×C4—2442no—
221C22×C4C2×C4×D6111yescharacteristic
222C22×C4D4×D6112yes—
223M16—24no—
224M16—42no—
225M16—1224no—
226M16—2442no—
227C2×C8—42no—
228C2×C8—2442no—
229C4⋊C4—41no—
230C4⋊C4—4882no—
231C4⋊C4—2441no—
232C4⋊C4—2442no—
233C22⋊C4—1224no—
234C22⋊C4—1224no—
235C22⋊C4—4882no—
236C22⋊C4—2442no—
237C42—41no—
238C42—612no—
239C42—1222no—
240C42—1222no—
241C42—1222no—
242C42—1221no—
243C23—1222no—
244C23—2442no—
245C23—2442no—
246C23—2441no—
247C23—2441no—
248C23C22⋊C4×D6111yescharacteristic
249Q8—42no—
250Q8—41no—
251Q8—41no—
252D4—2442no—
253D4—2442no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—612no—
264C2×C4—1222no—
265C2×C4—1221no—
266C2×C4—1222no—
267C2×C4—4884no—
268C2×C4—2441no—
269C2×C4—2441no—
270C2×C4—2441no—
271C2×C4—2442no—
272C2×C4—2441no—
273C2×C4C42×D6111yescharacteristic
274C2×C4#1451973511yescharacteristic
275C2×C4#1451973712yes—
276C2×C4#1451973912yes—
277C8—44no—
278C8—2444no—
279C22—21no—
280C22—21no—
281C22—4882no—
282C22—4881no—
283C22—4881no—
284C22—2444no—
285C22—2442no—
286C22#1683960911yescharacteristic
287C22#99×D6111yescharacteristic
288C22#1683973411yescharacteristic
289C4—22no—
290C4—22no—
291C4—21no—
292C4—21no—
293C4—81no—
294C4—81no—
295C4—1222no—
296C4—2442no—
297C4—2441no—
298C4—2442no—
299C2—21no—
300C2—21no—
301C2—4882no—
302C2#2144560611yescharacteristic
303C2#2144750711yescharacteristic
304C2#2144750911yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC2×C4×C61C2×C42
FrattiniC22×Q8C22×D61
FittingC61×#1394C2
socleC22×C61C2×#99
prime radicalC22×D244C2×C4
cosocleC22×Q8×C61C23
hypercenter#1394D61
hypocenterC61#12632
LCSG ▷ C2×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. #101×Dic61C2semidirect
4. #21443946C2semidirect
5. C61×#1394C2semidirect
16. #16839558C4semidirect
38. #14519696C2×C4semidirect
48. #101×C61C2×C4semidirect
151. #1394D61semidirect
172. C61#12632semidirect
197. #101C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214485332
C2×#168395582
#101×Dic612
#214439462
C61×#13942
#1263261