Finite Group Databasebeta

← #29572572

Subgroups — #29572572

Order 15616.

728 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 139 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#21446797C212yesmaximal, complemented
2#21446648C211yesmaximal, characteristic
3#21444557C212yesmaximal, complemented
4#21444554C211yesmaximal, characteristic
5C61×#3550C211yesmaximal, characteristic
6#16839786C2212yes—
7#16839786C2211yescharacteristic
8C2×#14519704—22no—
9C42×Dic61C2211yescharacteristic
10#16839521—24no—
11#16839511—21no—
12#16839508—22no—
13#16839508—22no—
14#16839506—21no—
15C61×#563C414yescomplemented
16C61×#559—21no—
17Q82×C61—21no—
18C61×#556—22no—
19D4×Q8×C61—22no—
20C61×#542—24no—
21C61×#536—22no—
22C61×#533C2212yes—
23C61×#532C2211yescharacteristic
24C2×#11758194—21no—
25C2×#11758194—22no—
26C2×C4×Dic61—21no—
27C2×C4×Dic61—22no—
28C4⋊#92418—22no—
29C4⋊#92418—22no—
30C4⋊#92418—22no—
31#14519704—44no—
32C2×#41×C61D412yes—
33C22×Q8×C61—22no—
34C4⋊Q8×C61—24no—
35C4⋊Q8×C61—24no—
36C4⋊Q8×C61D411yescharacteristic
37C4⋊Q8×C61D412yes—
38C4⋊Q8×C61D412yes—
39#127×C61D411yescharacteristic
40#126×C61—28no—
41#125×C61C2×C414yes—
42#124×C61—24no—
43#124×C61—24no—
44#124×C61C2×C412yes—
45#123×C61—24no—
46#123×C61—24no—
47C22⋊Q8×C61—24no—
48C22⋊Q8×C61—22no—
49C22⋊Q8×C61—22no—
50C22⋊Q8×C61—28no—
51#121×C61—22no—
52C4×Q8×C61—24no—
53C4×Q8×C61—22no—
54C4×Q8×C61—22no—
55C4×D4×C61—24no—
56C4×D4×C61—22no—
57#117×C61—22no—
58#117×C61D414yes—
59C2×C42×C61C2311yescharacteristic
60#41×C61—42no—
61#41×C61—42no—
62C2×Q8×C61—24no—
63C2×Q8×C61—22no—
64C2×Q8×C61—22no—
65C2×Q8×C61—82no—
66C2×Q8×C61—82no—
67C2×Q8×C61C22⋊C412yes—
68C2×D4×C61—22no—
69C2×D4×C61—22no—
70C2×D4×C61C22⋊C412yes—
71C22×C4×C61—24no—
72C22×C4×C61—22no—
73C22×C4×C61C2×D412yes—
74C22×C4×C61C2×D411yescharacteristic
75C22×Dic61—41no—
76C4⋊C4×C61—28no—
77C4⋊C4×C61—24no—
78C4⋊C4×C61—24no—
79C4⋊C4×C61—22no—
80C4⋊C4×C61—22no—
81C4⋊C4×C61—22no—
82C4⋊C4×C61C22⋊C414yes—
83C4⋊C4×C61C22⋊C414yes—
84C22⋊C4×C61—24no—
85C22⋊C4×C61—28no—
86C22⋊C4×C61—24no—
87C22⋊C4×C61—24no—
88C42×C61—24no—
89C42×C61—24no—
90C42×C61C22×C411yescharacteristic
91C42×C61C2×D412yes—
92C42×C61C2×D411yescharacteristic
93C4×Dic61—22no—
94C4×Dic61—24no—
95C4×Dic61—42no—
96C4×Dic61—41no—
97#11758194—42no—
98#11758194—44no—
99C2×#92418—41no—
100C2×#92418—42no—
101C23×C61—22no—
102C23×C61#12011yescharacteristic
103Q8×C61—42no—
104Q8×C61—42no—
105Q8×C61—42no—
106Q8×C61—42no—
107Q8×C61—42no—
108Q8×C61—84no—
109D4×C61—42no—
110D4×C61—42no—
111D4×C61—42no—
112D4×C61—82no—
113C2×C4×C61—21no—
114C2×C4×C61—22no—
115C2×C4×C61—21no—
116C2×C4×C61—22no—
117C2×C4×C61—24no—
118C2×C4×C61—22no—
119C2×C4×C61—24no—
120C2×C4×C61—42no—
121C2×C4×C61—42no—
122C2×C4×C61—84no—
123C2×C4×C61C2×C22⋊C412yes—
124C2×C4×C61C2×C22⋊C411yescharacteristic
125C2×C4×C61#12011yescharacteristic
126C2×C4×C61#12012yes—
127C2×Dic61—42no—
128C2×Dic61—44no—
129#92418—81no—
130#92418—82no—
131#8531—611nomaximal, Sylow
132C22×C61—21no—
133C22×C61—22no—
134C22×C61—42no—
135C22×C61—44no—
136C22×C61#37911yescharacteristic
137C4×C61—21no—
138C4×C61—22no—
139C4×C61—42no—
140C4×C61—44no—
141C4×C61—44no—
142C4×C61#45311yescharacteristic
143C4×C61#45412yes—
144C4×C61#45511yescharacteristic
145C4×C61#45612yes—
146Dic61—44no—
147#3550D6111yescharacteristic
148#1994—612no—
149#1962—611no—
150#1561—612no—
151#1560—611no—
152C2×C61—41no—
153C2×C61—82no—
154C2×C61#179211yescharacteristic
155C2×C61#179312yes—
156#563Dic6114yescomplemented
157#559—21no—
158Q82—21no—
159#556—22no—
160D4×Q8—22no—
161#542—24no—
162#536—22no—
163#533C2×D6112yes—
164#532C2×D6111yescharacteristic
165#423—612no—
166#423—611no—
167C2×#104—1222no—
168C43—611no—
169#339—1224no—
170#333—1221no—
171#332—1222no—
172#332—1222no—
173#331—1221no—
174C61#853111yescharacteristic, complemented, Sylow
175C2×#41C22⋊D6112yes—
176C22×Q8—22no—
177C2×M16—1222no—
178C2×M16—1221no—
179C4⋊Q8—24no—
180C4⋊Q8—24no—
181C4⋊Q8C22⋊D6111yescharacteristic
182C4⋊Q8C22⋊D6112yes—
183C4⋊Q8C22⋊D6112yes—
184#127C22⋊D6111yescharacteristic
185#126—28no—
186#125C2×Dic6114yes—
187#124—24no—
188#124—24no—
189#124C2×Dic6112yes—
190#123—24no—
191#123—24no—
192C22⋊Q8—24no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—28no—
196#121—22no—
197C4×Q8—24no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×D4—24no—
201C4×D4—22no—
202#117—22no—
203#117C22⋊D6114yes—
204C2×C42—1222no—
205C2×C42—1221no—
206C2×C42C22×D6111yescharacteristic
207C4⋊C8—1222no—
208C4⋊C8—1222no—
209C4⋊C8—1222no—
210#104—2444no—
211#41—42no—
212#41—42no—
213C2×Q8—24no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—82no—
217C2×Q8—82no—
218C2×Q8C22⋊Dic6112yes—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4C22⋊Dic6112yes—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—2441no—
225C22×C4C2×C22⋊D6112yes—
226C22×C4C2×C22⋊D6111yescharacteristic
227M16—2442no—
228M16—2444no—
229C2×C8—2441no—
230C2×C8—2442no—
231C4⋊C4—28no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4C22⋊Dic6114yes—
238C4⋊C4C22⋊Dic6114yes—
239C22⋊C4—24no—
240C22⋊C4—28no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C42—24no—
244C42—24no—
245C42—1222no—
246C42—1224no—
247C42—2441no—
248C42—2442no—
249C42C22×Dic6111yescharacteristic
250C42C2×C22⋊D6112yes—
251C42C2×C22⋊D6111yescharacteristic
252C23—22no—
253C23#1451974611yescharacteristic
254Q8—42no—
255Q8—42no—
256Q8—42no—
257Q8—42no—
258Q8—42no—
259Q8—84no—
260D4—42no—
261D4—42no—
262D4—42no—
263D4—82no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—84no—
274C2×C4—2442no—
275C2×C4—2444no—
276C2×C4C2×C22⋊Dic6112yes—
277C2×C4C2×C22⋊Dic6111yescharacteristic
278C2×C4#1451974611yescharacteristic
279C2×C4#1451974612yes—
280C8—4881no—
281C8—4882no—
282C22—21no—
283C22—22no—
284C22—42no—
285C22—44no—
286C22#1683961811yescharacteristic
287C4—21no—
288C4—22no—
289C4—42no—
290C4—44no—
291C4—44no—
292C4—2444no—
293C4#1683993011yescharacteristic
294C4#1683993512yes—
295C4#1683994011yescharacteristic
296C4#1683994512yes—
297C2—41no—
298C2—82no—
299C2#2144577311yescharacteristic
300C2#2144577812yes—
subgroupG/H
centerC22#16839618
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#3550C2
socleC22×C61#379
prime radicalC61×#532C22
cosocleC2×C42×C61C23
hypercenter#3550D61
hypocenterC61#8531
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3550

Complements

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

NH ≅ G/Ntype
1. #21446797C2semidirect
3. #21444557C2semidirect
15. C61×#563C4semidirect
156. #563Dic61semidirect
174. C61#8531semidirect

Maximal subgroups

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

subgroupindex
#214467972
#214466482
#214445572
#214445542
C61×#35502
#853161