Finite Group Databasebeta

← #29572259

Subgroups — #29572259

Order 15616.

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

normal subgroups only

This group has 682 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#21446809C212yesmaximal, complemented
2#21444557C211yesmaximal, characteristic, complemented
3#21444414C211yesmaximal, characteristic, complemented
4#21444258C212yesmaximal
5C61×#3509C211yesmaximal, characteristic
6#16839786C2212yes—
7C4×#11758194C2212yes—
8#16839531—24no—
9#16839508—22no—
10#16839504—22no—
11Q8⋊#92418—22no—
12D4⋊#92418—22no—
13C61×#557—22no—
14C61×#556—22no—
15C61×#552—24no—
16C61×#541—22no—
17C61×#536C412yes—
18C61×#533—22no—
19C61×#533C2211yescharacteristic
20C2×C4⋊Q8×C61C2211yescharacteristic
21C61×#519C412yes—
22C4×#41×C61C2211yescharacteristic
23C2×#11758194—22no—
24C2×#11758194—22no—
25C4⋊#92418—22no—
26C4⋊#92418—22no—
27#14519688—22no—
28C4×#92418—22no—
29C2×#41×C61C2×C411yescharacteristic
30C22×Q8×C61C2×C412yes—
31C4⋊Q8×C61—22no—
32C4⋊Q8×C61—21no—
33C4⋊Q8×C61—21no—
34C4⋊Q8×C61—22no—
35C4⋊Q8×C61D412yes—
36#126×C61—28no—
37#125×C61—24no—
38#125×C61—28no—
39#125×C61D412yes—
40#124×C61—22no—
41#123×C61—24no—
42C22⋊Q8×C61—24no—
43C22⋊Q8×C61—22no—
44C22⋊Q8×C61—24no—
45C4×Q8×C61—24no—
46C4×Q8×C61—24no—
47C4×Q8×C61—24no—
48C4×Q8×C61—22no—
49C4×Q8×C61D412yes—
50C4×D4×C61—22no—
51C4×D4×C61D412yes—
52#117×C61—24no—
53#117×C61—24no—
54#117×C61—21no—
55#117×C61C2×C412yes—
56#117×C61C2×C411yescharacteristic
57C2×C4⋊C4×C61D414yes—
58C2×C42×C61—21no—
59C2×C42×C61C2311yescharacteristic
60#41×C61—42no—
61C2×Q8×C61—22no—
62C2×Q8×C61—22no—
63C2×Q8×C61—22no—
64C2×Q8×C61—44no—
65C2×Q8×C61C22⋊C411yescharacteristic
66C2×D4×C61—21no—
67C2×D4×C61C22⋊C411yescharacteristic
68C22×C4×C61—21no—
69C22×C4×C61—22no—
70C22×C4×C61C22⋊C414yes—
71C22×C4×C61C22×C411yescharacteristic
72C22×C4×C61C2×D412yes—
73C4⋊C4×C61—28no—
74C4⋊C4×C61—28no—
75C4⋊C4×C61—24no—
76C4⋊C4×C61—24no—
77C4⋊C4×C61—22no—
78C4⋊C4×C61—22no—
79C4⋊C4×C61—24no—
80C4⋊C4×C61C22⋊C412yes—
81C4⋊C4×C61C22⋊C414yes—
82C22⋊C4×C61—28no—
83C22⋊C4×C61—21no—
84C22⋊C4×C61—22no—
85C22⋊C4×C61—22no—
86C42×C61—28no—
87C42×C61—21no—
88C42×C61—22no—
89C42×C61—22no—
90C42×C61C2×D411yescharacteristic
91C42×C61C2×D411yescharacteristic
92C42×C61C2×D412yes—
93#11758194—24no—
94#11758194—42no—
95#11758194—44no—
96C2×#92418—42no—
97C2×#92418—42no—
98C23×C61—21no—
99C23×C61C2×C22⋊C411yescharacteristic
100Q8×C61—22no—
101Q8×C61—44no—
102Q8×C61—44no—
103Q8×C61—42no—
104Q8×C61—42no—
105D4×C61—22no—
106D4×C61—42no—
107C2×C4×C61—22no—
108C2×C4×C61—22no—
109C2×C4×C61—22no—
110C2×C4×C61—22no—
111C2×C4×C61—24no—
112C2×C4×C61—24no—
113C2×C4×C61—21no—
114C2×C4×C61—42no—
115C2×C4×C61—42no—
116C2×C4×C61—42no—
117C2×C4×C61—44no—
118C2×C4×C61C2×C22⋊C411yescharacteristic
119C2×C4×C61C2×C22⋊C411yescharacteristic
120C2×C4×C61#12012yes—
121C2×C4×C61#12012yes—
122#92418—44no—
123#92418—82no—
124#8474—611nomaximal, Sylow
125C22×C61—22no—
126C22×C61—21no—
127C22×C61—42no—
128C22×C61—41no—
129C22×C61#37911yescharacteristic
130C4×C61—22no—
131C4×C61—21no—
132C4×C61—42no—
133C4×C61—44no—
134C4×C61—41no—
135C4×C61—44no—
136C4×C61#41312yes—
137C4×C61#45412yes—
138C4×C61#45612yes—
139#3509D6111yescharacteristic
140#1997—612no—
141#1561—611no—
142#1526—611no—
143#1489—612no—
144C2×C61—41no—
145C2×C61—81no—
146C2×C61#178511yescharacteristic
147C2×C61#179312yes—
148#557—22no—
149#556—22no—
150#552—24no—
151#541—22no—
152#536Dic6112yes—
153#533—22no—
154#533C2×D6111yescharacteristic
155C2×C4⋊Q8C2×D6111yescharacteristic
156#519Dic6112yes—
157C4×#41C2×D6111yescharacteristic
158#423—612no—
159C4×M16—612no—
160#343—1224no—
161#332—1222no—
162#330—1222no—
163Q8⋊C8—1222no—
164D4⋊C8—1222no—
165C61#847411yescharacteristic, complemented, Sylow
166C2×#41C2×Dic6111yescharacteristic
167C22×Q8C2×Dic6112yes—
168C2×M16—1222no—
169C2×M16—1222no—
170C4⋊Q8—22no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—22no—
174C4⋊Q8C22⋊D6112yes—
175#126—28no—
176#125—24no—
177#125—28no—
178#125C22⋊D6112yes—
179#124—22no—
180#123—24no—
181C22⋊Q8—24no—
182C22⋊Q8—22no—
183C22⋊Q8—24no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—22no—
188C4×Q8C22⋊D6112yes—
189C4×D4—22no—
190C4×D4C22⋊D6112yes—
191#117—24no—
192#117—24no—
193#117—21no—
194#117C2×Dic6111yescharacteristic
195#117C2×Dic6112yes—
196C2×C4⋊C4C22⋊D6114yes—
197C2×C42—21no—
198C2×C42C22×D6111yescharacteristic
199C4⋊C8—1222no—
200C4⋊C8—1222no—
201#97—1222no—
202C4×C8—1222no—
203#41—42no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—44no—
208C2×Q8C22⋊Dic6111yescharacteristic
209C2×D4—21no—
210C2×D4C22⋊Dic6111yescharacteristic
211C22×C4—21no—
212C22×C4—22no—
213C22×C4C22⋊Dic6114yes—
214C22×C4C22×Dic6111yescharacteristic
215C22×C4C2×C22⋊D6112yes—
216M16—1224no—
217M16—2444no—
218M16—2442no—
219C2×C8—2442no—
220C2×C8—2442no—
221C4⋊C4—28no—
222C4⋊C4—28no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4C22⋊Dic6112yes—
229C4⋊C4C22⋊Dic6114yes—
230C22⋊C4—28no—
231C22⋊C4—21no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—28no—
235C42—21no—
236C42—22no—
237C42—22no—
238C42C2×C22⋊D6112yes—
239C42C2×C22⋊D6111yescharacteristic
240C42C2×C22⋊D6111yescharacteristic
241C23—21no—
242C23C2×C22⋊Dic6111yescharacteristic
243Q8—22no—
244Q8—44no—
245Q8—42no—
246Q8—44no—
247Q8—42no—
248D4—22no—
249D4—42no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4C2×C22⋊Dic6111yescharacteristic
262C2×C4C2×C22⋊Dic6111yescharacteristic
263C2×C4#1451974612yes—
264C2×C4#1451974612yes—
265C8—4882no—
266C8—2444no—
267C22—21no—
268C22—22no—
269C22—41no—
270C22—42no—
271C22#1683961811yescharacteristic
272C4—21no—
273C4—22no—
274C4—41no—
275C4—44no—
276C4—42no—
277C4—44no—
278C4#1683974612yes—
279C4#1683993512yes—
280C4#1683994512yes—
281C2—41no—
282C2—81no—
283C2#2144574711yescharacteristic
284C2#2144577812yes—
subgroupG/H
centerC22#16839618
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#3509C2
socleC22×C61#379
prime radicalC2×#41×C61C2×C4
cosocleC2×C42×C61C23
hypercenter#3509D61
hypocenterC61#8474
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3509

Complements

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

NH ≅ G/Ntype
1. #21446809C2semidirect
2. #21444557C2semidirect
3. #21444414C2semidirect
165. C61#8474semidirect

Maximal subgroups

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

subgroupindex
#214468092
#214445572
#214444142
#214442582
C61×#35092
#847461