Finite Group Databasebeta

← #29577403

Subgroups — #29577403

Order 15616.

574 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 135 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#21450818C211yesmaximal, characteristic
2#21445815C211yesmaximal, characteristic, complemented
3#21444486C212yesmaximal
4#21443721C212yesmaximal, complemented
5C61×#2982C211yesmaximal, characteristic, complemented
6C2×#14519774C2211yescharacteristic
7C2×#14519766—21no—
8#16839786C2212yes—
9C2×#14519696—21no—
10#16839679C414yes—
11#16839664—22no—
12#16839663—21no—
13C4×C4⋊Dic61C2211yescharacteristic
14#16839531—21no—
15#16839505—24no—
16C2×C4×Q8×C61C2211yescharacteristic
17C8×Q8×C61—24no—
18C61×#431—22no—
19C61×#431C2212yes—
20C22×D244—21no—
21C22×D244—21no—
22C22×D244C2×C412yes—
23C2×#11758194—22no—
24#14519774—22no—
25#14519774—22no—
26#14519766—42no—
27#14519766—42no—
28C2×C4⋊Dic61—21no—
29C2×#11758198—24no—
30C2×#11758198—24no—
31C2×C4×Dic61—21no—
32C4⋊#92418—22no—
33C4⋊#92418—22no—
34#14519696—42no—
35#14519684C2×C412yes—
36C22×Q8×C61D411yescharacteristic
37C4×Q8×C61—22no—
38C4×Q8×C61—22no—
39C2×C4⋊C4×C61C2×C412yes—
40C2×C4⋊C4×C61D411yescharacteristic
41C2×C42×C61D412yes—
42C2×C42×C61C2311yescharacteristic
43C4⋊C8×C61—24no—
44C4⋊C8×C61D414yes—
45C22⋊C8×C61—22no—
46C4×C8×C61—24no—
47C4×C8×C61—22no—
48C23×D61—22no—
49C2×Q8×C61—22no—
50C2×Q8×C61—22no—
51C2×Q8×C61D812yes—
52C2×Q8×C61SD1612yes—
53C22×C4×C61C22⋊C412yes—
54C22×C4×C61C22⋊C412yes—
55C22×C4×C61C22×C411yescharacteristic
56C22×C4×C61C2×D411yescharacteristic
57C22×C4×C61C2×D411yescharacteristic
58C22×Dic61—22no—
59C2×D244—24no—
60C2×D244—44no—
61C2×D244—41no—
62C2×D244—42no—
63C2×D244—41no—
64C2×D244—41no—
65C2×D244—41no—
66C2×D244—42no—
67C2×D244—82no—
68C2×C8×C61—24no—
69C4⋊C4×C61—24no—
70C4⋊C4×C61—22no—
71C42×C61—22no—
72C42×C61—22no—
73C42×C61C2×D412yes—
74C42×C61#4112yes—
75#11758198—48no—
76#11758198—84no—
77C4⋊Dic61—42no—
78C4×Dic61—41no—
79C4×Dic61—41no—
80#11758194—42no—
81#11758194—42no—
82C2×#92418—42no—
83C23×C61C2×C22⋊C411yescharacteristic
84C22×D61—44no—
85C22×D61—44no—
86C22×D61—42no—
87C22×D61—42no—
88C22×D61—42no—
89Q8×C61—24no—
90Q8×C61—24no—
91C2×C4×C61—22no—
92C2×C4×C61—22no—
93C2×C4×C61—22no—
94C2×C4×C61—22no—
95C2×C4×C61—41no—
96C2×C4×C61—42no—
97C2×C4×C61#10214yes—
98C2×C4×C61C4×D412yes—
99C2×C4×C61#12011yescharacteristic
100C2×C4×C61#12112yes—
101C2×C4×C61#12311yescharacteristic
102C2×C4×C61C2×D811yescharacteristic
103C2×C4×C61C2×SD1611yescharacteristic
104C2×C4×C61#13512yes—
105C2×Dic61—42no—
106C2×Dic61—42no—
107C2×Dic61—42no—
108D244—84no—
109D244—82no—
110D244—82no—
111D244—82no—
112D244—82no—
113C8×C61—44no—
114#92418—82no—
115#9398—611nomaximal, Sylow
116C22×C61—21no—
117C22×C61—21no—
118C22×C61#38611yescharacteristic
119C22×C61C2×#10211yescharacteristic
120C22×C61#41611yescharacteristic
121C2×D61—84no—
122C2×D61—84no—
123C2×D61—82no—
124C2×D61—82no—
125C2×D61—82no—
126C4×C61—21no—
127C4×C61—24no—
128C4×C61—22no—
129C4×C61—42no—
130C4×C61#45412yes—
131C4×C61#46512yes—
132C4×C61#46612yes—
133Dic61—82no—
134#2982D6111yescharacteristic, complemented
135#2548—611no—
136#1801—611no—
137#1545—612no—
138#1303—612no—
139C2×C61—21no—
140C2×C61—21no—
141C2×C61#188811yescharacteristic
142C2×C61#189311yescharacteristic
143C2×C61#191411yescharacteristic
144D61—162no—
145C2×#127—611no—
146C2×#124—1221no—
147C2×C4×Q8C2×D6111yescharacteristic
148C8×Q8—24no—
149#431—22no—
150#431C2×D6112yes—
151#423—612no—
152C2×#101—1221no—
153#394—614no—
154#390—1221no—
155#390—1222no—
156C4×C4⋊C4—611no—
157#343—1221no—
158#331—1224no—
159C61#939811yescharacteristic, complemented, Sylow
160C22×Q8D24411yescharacteristic
161C22×D4—612no—
162C22×D4—1221no—
163C22×D4—1221no—
164C2×M16—1222no—
165#127—1222no—
166#127—1222no—
167#124—2442no—
168#124—2442no—
169C4×Q8—22no—
170C4×Q8—22no—
171C2×C4⋊C4—1221no—
172C2×C4⋊C4C4×D6112yes—
173C2×C4⋊C4D24411yescharacteristic
174C2×C22⋊C4—1224no—
175C2×C22⋊C4—1224no—
176C2×C42—1221no—
177C2×C42C22⋊D6112yes—
178C2×C42C22×D6111yescharacteristic
179C4⋊C8—24no—
180C4⋊C8—1222no—
181C4⋊C8—1222no—
182C4⋊C8C22⋊D6114yes—
183#101—2442no—
184C22⋊C8—22no—
185C4×C8—24no—
186C4×C8—22no—
187#95—612no—
188C24—1222no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8#1175819012yes—
192C2×Q8D48812yes—
193C2×D4—1224no—
194C2×D4—4882no—
195C2×D4—2444no—
196C2×D4—2441no—
197C2×D4—2442no—
198C2×D4—2441no—
199C2×D4—2441no—
200C2×D4—2441no—
201C2×D4—2442no—
202C22×C4—1222no—
203C22×C4#1175819812yes—
204C22×C4#1175819812yes—
205C22×C4C2×C4×D6111yescharacteristic
206C22×C4C2×D24411yescharacteristic
207C22×C4C2×C22⋊D6111yescharacteristic
208M16—2442no—
209M16—2442no—
210C2×C8—24no—
211C2×C8—2442no—
212C4⋊C4—24no—
213C4⋊C4—22no—
214C4⋊C4—2442no—
215C22⋊C4—4884no—
216C22⋊C4—2448no—
217C42—22no—
218C42—22no—
219C42—2441no—
220C42—2441no—
221C42#1175822212yes—
222C42C2×C22⋊D6112yes—
223C23—2444no—
224C23—2444no—
225C23—2442no—
226C23—2442no—
227C23—2442no—
228C23C2×#1175819811yescharacteristic
229Q8—24no—
230Q8—24no—
231D4—4884no—
232D4—4882no—
233D4—4882no—
234D4—4882no—
235D4—4882no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—2442no—
243C2×C4—2442no—
244C2×C4—2442no—
245C2×C4#1451969914yes—
246C2×C4C4×C22⋊D6112yes—
247C2×C4#1451974611yescharacteristic
248C2×C4#1451975112yes—
249C2×C4#1451976011yescharacteristic
250C2×C4C2×D48811yescharacteristic
251C2×C4C2×#1175819011yescharacteristic
252C2×C4#1451979912yes—
253C8—44no—
254C8—4882no—
255C22—21no—
256C22—21no—
257C22—4884no—
258C22—4884no—
259C22—4882no—
260C22—4882no—
261C22—4882no—
262C22#1683964111yescharacteristic
263C22C2×#1451969911yescharacteristic
264C22#1683975911yescharacteristic
265C4—21no—
266C4—24no—
267C4—22no—
268C4—42no—
269C4—4882no—
270C4#1683993512yes—
271C4#1683999112yes—
272C4#1683999812yes—
273C2—21no—
274C2—21no—
275C2—9762no—
276C2#2144622211yescharacteristic
277C2#2144624711yescharacteristic
278C2#2144636611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2982C2
socleC22×C61#386
prime radicalC2×#14519774C22
cosocleC2×C42×C61C23
hypercenter#2982D61
hypocenterC61#9398
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #21445815C2semidirect
4. #21443721C2semidirect
5. C61×#2982C2semidirect
134. #2982D61semidirect
159. C61#9398semidirect

Maximal subgroups

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

subgroupindex
#214508182
#214458152
#214444862
#214437212
C61×#29822
#939861