Finite Group Databasebeta

← #29574693

Subgroups — #29574693

Order 15616.

542 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C4×#14519766C211yesmaximal, characteristic
2#21444209C212yesmaximal
3#21444143C211yesmaximal, characteristic, complemented
4#21443651C212yesmaximal, complemented
5C61×#2983C211yesmaximal, characteristic, complemented
6C2×#14519766C411yescharacteristic
7C2×C4×D244C2211yescharacteristic
8#16839815C2212yes—
9#16839669C411yescharacteristic
10#16839663C411yescharacteristic
11#16839652—22no—
12#16839627C411yescharacteristic
13C4×#11758198—22no—
14C42×Dic61C2211yescharacteristic
15#16839489—24no—
16#16839488—22no—
17C2×C4×Q8×C61C2211yescharacteristic
18C61×#446—24no—
19C61×#432—22no—
20C61×#431C2212yes—
21C22×D244C2×C411yescharacteristic
22C22×C4×D61—21no—
23#14519766—22no—
24#14519766—22no—
25C4×D244—22no—
26C4×D244—22no—
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×#11758198—22no—
29C2×#11758198—22no—
30C2×#11758198—21no—
31C2×C4×Dic61—21no—
32C2×C4×Dic61—21no—
33C2×C4×Dic61C2×C411yescharacteristic
34C2×C4×Dic61C2×C411yescharacteristic
35C4⋊#92418—22no—
36C22⋊#92418—22no—
37C4×#92418—22no—
38#14519684—22no—
39#14519684—22no—
40C22×Q8×C61C2×C411yescharacteristic
41C4×Q8×C61—22no—
42C4×Q8×C61—22no—
43C2×C4⋊C4×C61C2×C411yescharacteristic
44C2×C4⋊C4×C61D412yes—
45C2×C42×C61D412yes—
46C2×C42×C61C2311yescharacteristic
47C4⋊C8×C61—24no—
48C4⋊C8×C61D414yes—
49C22⋊C8×C61—22no—
50#97×C61—24no—
51C4×C8×C61—22no—
52C23×D61—21no—
53C2×Q8×C61—22no—
54C2×Q8×C61—21no—
55C2×Q8×C61—21no—
56C2×Q8×C61—22no—
57C22×C4×C61C22⋊C412yes—
58C22×C4×C61C22⋊C412yes—
59C22×C4×C61C22×C411yescharacteristic
60C22×C4×C61C2×D411yescharacteristic
61C22×C4×C61C2×D411yescharacteristic
62C22×Dic61—21no—
63C22×Dic61—21no—
64C22×Dic61—21no—
65C2×D244—22no—
66C2×D244—22no—
67C2×D244—21no—
68C2×D244M1612yes—
69C2×C4×D61—44no—
70C2×C4×D61—41no—
71C2×C4×D61—41no—
72C2×C8×C61—24no—
73C4⋊C4×C61—22no—
74C4⋊C4×C61—24no—
75C42×C61—22no—
76C42×C61—22no—
77C42×C61C2×D412yes—
78C42×C61#4112yes—
79#11758198—28no—
80#11758198—44no—
81#11758198—42no—
82C4⋊Dic61—22no—
83C4×Dic61—21no—
84C4×Dic61—22no—
85C4×Dic61—42no—
86C4×Dic61—42no—
87C4×Dic61M1612yes—
88C2×#92418—42no—
89C23×C61C2×C22⋊C411yescharacteristic
90C22×D61—22no—
91C22×D61—42no—
92C22×D61—42no—
93C22×D61—41no—
94C22×D61—41no—
95Q8×C61—42no—
96Q8×C61—42no—
97C2×C4×C61—22no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61—41no—
103C2×C4×C61—42no—
104C2×C4×C61C4×D412yes—
105C2×C4×C61#12011yescharacteristic
106C2×C4×C61#12112yes—
107C2×C4×C61#12311yescharacteristic
108C2×C4×C61C2×M1611yescharacteristic
109C2×C4×C61C2×M1611yescharacteristic
110C2×C4×C61#13112yes—
111C2×Dic61—22no—
112C2×Dic61—22no—
113C2×Dic61—41no—
114C2×Dic61—41no—
115C2×Dic61—41no—
116C2×Dic61—41no—
117C2×Dic61—42no—
118C2×Dic61—41no—
119D244—24no—
120D244—42no—
121C4×D61—82no—
122C4×D61—82no—
123C8×C61—44no—
124#92418—82no—
125#8964—611nomaximal, Sylow
126C22×C61—21no—
127C22×C61—21no—
128C22×C61#38611yescharacteristic
129C22×C61#40711yescharacteristic
130C22×C61#40811yescharacteristic
131C2×D61—44no—
132C2×D61—42no—
133C2×D61—82no—
134C2×D61—81no—
135C2×D61—81no—
136C4×C61—22no—
137C4×C61—21no—
138C4×C61—42no—
139C4×C61—42no—
140C4×C61#43512yes—
141C4×C61#43612yes—
142C4×C61#45412yes—
143Dic61—42no—
144Dic61—42no—
145Dic61—81no—
146#2983D6111yescharacteristic, complemented
147C4×#124—611no—
148#1478—612no—
149#1461—611no—
150#1276—612no—
151C2×C61—21no—
152C2×C61—21no—
153C2×C61#187011yescharacteristic
154C2×C61#188411yescharacteristic
155C2×C61#188811yescharacteristic
156D61—82no—
157C2×#124—611no—
158C2×C4×Q8C2×D6111yescharacteristic
159C2×C4×D4—611no—
160#446—24no—
161#432—22no—
162#431—612no—
163#431C2×D6112yes—
164#391—611no—
165#390—611no—
166#388—1222no—
167#382—611no—
168C4×C22⋊C4—1222no—
169C43—611no—
170D4⋊C8—1224no—
171#324—1222no—
172C61#896411yescharacteristic, complemented, Sylow
173C22×Q8C4×D6111yescharacteristic
174C22×D4—611no—
175C23×C4—1221no—
176#124—1222no—
177#124—1222no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×D4—1222no—
181C4×D4—1222no—
182C2×C4⋊C4—611no—
183C2×C4⋊C4C4×D6111yescharacteristic
184C2×C4⋊C4D24412yes—
185C2×C22⋊C4—1222no—
186C2×C22⋊C4—1222no—
187C2×C22⋊C4—1221no—
188C2×C42—611no—
189C2×C42—611no—
190C2×C42—1221no—
191C2×C42—1221no—
192C2×C42C22⋊D6112yes—
193C2×C42C22×D6111yescharacteristic
194C4⋊C8—24no—
195C4⋊C8—1222no—
196C4⋊C8C22⋊D6114yes—
197C22⋊C8—22no—
198C22⋊C8—1222no—
199#97—24no—
200C4×C8—22no—
201C4×C8—1222no—
202#95—1222no—
203#95—1222no—
204C24—1221no—
205C2×Q8—22no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—22no—
209C2×D4—612no—
210C2×D4—1222no—
211C2×D4—1221no—
212C2×D4—1222no—
213C22×C4—1221no—
214C22×C4—1221no—
215C22×C4—1221no—
216C22×C4—2444no—
217C22×C4—2441no—
218C22×C4—2441no—
219C22×C4#1175819812yes—
220C22×C4#1175819812yes—
221C22×C4C2×C4×D6111yescharacteristic
222C22×C4C2×D24411yescharacteristic
223C22×C4C2×C22⋊D6111yescharacteristic
224C2×C8—24no—
225C2×C8—2442no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4—1222no—
229C22⋊C4—1228no—
230C22⋊C4—2444no—
231C22⋊C4—2442no—
232C42—22no—
233C42—22no—
234C42—612no—
235C42—1221no—
236C42—1222no—
237C42—2442no—
238C42—2442no—
239C42#1175822212yes—
240C42C2×C22⋊D6112yes—
241C23—1222no—
242C23—2442no—
243C23—2442no—
244C23—2441no—
245C23—2441no—
246C23C2×#1175819811yescharacteristic
247Q8—42no—
248Q8—42no—
249D4—1224no—
250D4—2442no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—1222no—
259C2×C4—1222no—
260C2×C4—4882no—
261C2×C4—4882no—
262C2×C4—2442no—
263C2×C4—2441no—
264C2×C4—2441no—
265C2×C4—2441no—
266C2×C4—2441no—
267C2×C4—2441no—
268C2×C4C4×C22⋊D6112yes—
269C2×C4#1451974611yescharacteristic
270C2×C4#1451975112yes—
271C2×C4#1451976011yescharacteristic
272C2×C4M16×D6111yescharacteristic
273C2×C4M16×D6111yescharacteristic
274C2×C4#1451978512yes—
275C8—44no—
276C8—4882no—
277C22—21no—
278C22—21no—
279C22—4882no—
280C22—4881no—
281C22—4881no—
282C22—2444no—
283C22—2442no—
284C22#1683964111yescharacteristic
285C22#1683972511yescharacteristic
286C22#1683973011yescharacteristic
287C4—22no—
288C4—21no—
289C4—42no—
290C4—42no—
291C4—4881no—
292C4—2442no—
293C4—2442no—
294C4#1683983112yes—
295C4#1683983812yes—
296C4#1683993512yes—
297C2—21no—
298C2—21no—
299C2—4882no—
300C2#2144611611yescharacteristic
301C2#2144620211yescharacteristic
302C2#2144622211yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2983C2
socleC22×C61#386
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#2983D61
hypocenterC61#8964
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
3. #21444143C2semidirect
4. #21443651C2semidirect
5. C61×#2983C2semidirect
146. #2983D61semidirect
172. C61#8964semidirect

Maximal subgroups

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

subgroupindex
C4×#145197662
#214442092
#214441432
#214436512
C61×#29832
#896461