Finite Group Databasebeta

← #29568162

Subgroups — #29568162

Order 15616.

612 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839660C211yesmaximal, characteristic
2#21445513C211yesmaximal, characteristic
3#21443641C214yesmaximal, complemented
4C61×#1817C211yesmaximal, characteristic
5C22×#11758194—21no—
6C22×C4⋊Dic61C411yescharacteristic
7C22×C4⋊Dic61C411yescharacteristic
8C22×#11758196—21no—
9#16839660—21no—
10#16839660—21no—
11#16839660C412yes—
12#16839660C2212yes—
13C2×#14519684—21no—
14#16839517—21no—
15#16839517C2212yes—
16C22×C42×C61C2211yescharacteristic
17C2×C22⋊C8×C61—21no—
18C61×#336—21no—
19C61×#336C2212yes—
20C23×Dic61—21no—
21C23×Dic61—21no—
22C2×#11758194—24no—
23C2×#11758194—42no—
24C2×#11758194—42no—
25C22×#92418—22no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61—24no—
28C2×C4⋊Dic61—21no—
29C2×C4⋊Dic61—42no—
30C2×C4⋊Dic61C2×C412yes—
31C2×C4⋊Dic61C2×C412yes—
32C2×#11758196—22no—
33C2×#11758196—22no—
34C2×#11758196—42no—
35C2×#11758196—42no—
36#14519684—22no—
37#14519684—22no—
38C23×C4×C61C2×C411yescharacteristic
39C23×C4×C61D411yescharacteristic
40C23×C4×C61D411yescharacteristic
41C22×C8×C61—22no—
42C2×C42×C61—42no—
43C2×C42×C61C2×C411yescharacteristic
44C2×C42×C61D411yescharacteristic
45C2×C42×C61D411yescharacteristic
46C2×C42×C61C2311yescharacteristic
47C22⋊C8×C61—42no—
48C24×C61C22⋊C411yescharacteristic
49C22×C4×C61—22no—
50C22×C4×C61—22no—
51C22×C4×C61—22no—
52C22×C4×C61—22no—
53C22×C4×C61—22no—
54C22×C4×C61—22no—
55C22×C4×C61—22no—
56C22×C4×C61—41no—
57C22×C4×C61C22⋊C411yescharacteristic
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22⋊C411yescharacteristic
60C22×C4×C61D812yes—
61C22×C4×C61SD1612yes—
62C22×C4×C61C22×C411yescharacteristic
63C22×C4×C61C2×D411yescharacteristic
64C22×C4×C61C2×D411yescharacteristic
65C22×Dic61—24no—
66C22×Dic61—22no—
67C22×Dic61—22no—
68C22×Dic61—42no—
69C22×Dic61—42no—
70C22×Dic61—42no—
71C22×Dic61—42no—
72C22×Dic61—42no—
73C2×C8×C61—42no—
74C2×C8×C61—42no—
75C2×C8×C61—42no—
76C42×C61—41no—
77C42×C61—41no—
78C42×C61—41no—
79C42×C61—41no—
80C4⋊Dic61—24no—
81C4⋊Dic61—22no—
82C4⋊Dic61—42no—
83C4⋊Dic61—42no—
84C4⋊Dic61M1614yescomplemented
85#11758196—44no—
86#11758196—42no—
87#11758196—42no—
88#11758194—48no—
89C2×#92418—24no—
90C2×#92418—42no—
91C2×#92418—42no—
92C23×C61—21no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—21no—
96C23×C61#9912yes—
97C23×C61#10212yes—
98C23×C61#10412yes—
99C23×C61C2×C22⋊C411yescharacteristic
100C2×C4×C61—28no—
101C2×C4×C61—24no—
102C2×C4×C61—24no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—44no—
117C2×C4×C61#10112yes—
118C2×C4×C61#10212yes—
119C2×C4×C61C2×M1612yes—
120C2×C4×C61C2×D811yescharacteristic
121C2×C4×C61C2×SD1611yescharacteristic
122C2×C4×C61#13712yes—
123C2×Dic61—24no—
124C2×Dic61—48no—
125C2×Dic61—42no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130C2×Dic61—84no—
131C8×C61—82no—
132#92418—44no—
133#7610—611nomaximal, Sylow
134C22×C61—22no—
135C22×C61—22no—
136C22×C61—22no—
137C22×C61—22no—
138C22×C61—22no—
139C22×C61—22no—
140C22×C61#32714yes—
141C22×C61#40711yescharacteristic
142C22×C61C2×#9911yescharacteristic
143C22×C61C2×#10111yescharacteristic
144C22×C61C2×#10211yescharacteristic
145C22×C61#41911yescharacteristic
146C22×C61C2×#10411yescharacteristic
147C22×C61#42111yescharacteristic
148C4×C61—22no—
149C4×C61—22no—
150C4×C61—21no—
151C4×C61—22no—
152C4×C61—21no—
153C4×C61—22no—
154C4×C61—41no—
155Dic61—44no—
156Dic61—82no—
157C2×#389—611no—
158#1817D6111yescharacteristic
159#1728—611no—
160#1271—614no—
161C2×C61—24no—
162C2×C61#146311yescharacteristic
163C2×C61#148611yescharacteristic
164C2×C61#149111yescharacteristic
165C2×C61C2×#32711yescharacteristic
166C2×C61#151111yescharacteristic
167C2×C61#152811yescharacteristic
168C2×C61#154611yescharacteristic
169C22×M16—1221no—
170C22×C4⋊C4—611no—
171C22×C4⋊C4—611no—
172C22×C4⋊C4—1221no—
173C22×C42C2×D6111yescharacteristic
174C2×C22⋊C8—21no—
175#389—612no—
176#389—612no—
177#389—1221no—
178#389—1221no—
179C2×#95—1221no—
180#336—21no—
181#336—612no—
182#336—1221no—
183#336C2×D6112yes—
184C61#761011yescharacteristic, complemented, Sylow
185C23×C4—1221no—
186C23×C4—1221no—
187C23×C4C4×D6111yescharacteristic
188C23×C4D24411yescharacteristic
189C23×C4C22⋊D6111yescharacteristic
190C2×M16—1224no—
191C2×M16—2442no—
192C2×M16—2442no—
193C22×C8—22no—
194C22×C8—1222no—
195C2×C4⋊C4—612no—
196C2×C4⋊C4—612no—
197C2×C4⋊C4—1224no—
198C2×C4⋊C4—1221no—
199C2×C4⋊C4—1221no—
200C2×C4⋊C4—1222no—
201C2×C4⋊C4—1222no—
202C2×C4⋊C4—2442no—
203C2×C4⋊C4—2442no—
204C2×C4⋊C4—2442no—
205C2×C42—42no—
206C2×C42C4×D6111yescharacteristic
207C2×C42D24411yescharacteristic
208C2×C42C22⋊D6111yescharacteristic
209C2×C42C22×D6111yescharacteristic
210C22⋊C8—42no—
211#95—1222no—
212#95—1222no—
213C24#1175819811yescharacteristic
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—41no—
222C22×C4—1224no—
223C22×C4—1222no—
224C22×C4—1222no—
225C22×C4—2442no—
226C22×C4—2442no—
227C22×C4—2442no—
228C22×C4—2442no—
229C22×C4—2442no—
230C22×C4#1175819012yes—
231C22×C4D48812yes—
232C22×C4#1175819811yescharacteristic
233C22×C4#1175819811yescharacteristic
234C22×C4#1175819811yescharacteristic
235C22×C4C2×C4×D6111yescharacteristic
236C22×C4C2×D24411yescharacteristic
237C22×C4C2×C22⋊D6111yescharacteristic
238M16—2448no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—1224no—
243C2×C8—2442no—
244C2×C8—2442no—
245C4⋊C4—614no—
246C4⋊C4—1224no—
247C4⋊C4—1222no—
248C4⋊C4—2442no—
249C4⋊C4—2442no—
250C4⋊C4—2444no—
251C4⋊C4—2442no—
252C4⋊C4—2442no—
253C42—41no—
254C42—41no—
255C42—41no—
256C42—41no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23#1451969112yes—
262C23#1451969912yes—
263C23#1451970312yes—
264C23C2×#1175819811yescharacteristic
265C2×C4—28no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—44no—
282C2×C4—1224no—
283C2×C4—4884no—
284C2×C4—2442no—
285C2×C4—2442no—
286C2×C4—2442no—
287C2×C4—2448no—
288C2×C4—2442no—
289C2×C4—2442no—
290C2×C4#1451969512yes—
291C2×C4#1451969912yes—
292C2×C4M16×D6112yes—
293C2×C4C2×D48811yescharacteristic
294C2×C4C2×#1175819011yescharacteristic
295C2×C4#1451980712yes—
296C8—82no—
297C8—2444no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22#1683949714yes—
305C22#1683972511yescharacteristic
306C22C2×#1451969111yescharacteristic
307C22C2×#1451969511yescharacteristic
308C22C2×#1451969911yescharacteristic
309C22#1683977011yescharacteristic
310C22C2×#1451970311yescharacteristic
311C22#1683978011yescharacteristic
312C4—22no—
313C4—22no—
314C4—21no—
315C4—22no—
316C4—21no—
317C4—22no—
318C4—41no—
319C4—4882no—
320C4—2444no—
321C2—24no—
322C2#2144414811yescharacteristic
323C2#2144424511yescharacteristic
324C2#2144426511yescharacteristic
325C2C2×#1683949711yescharacteristic
326C2#2144435111yescharacteristic
327C2#2144442211yescharacteristic
328C2#2144449011yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1817C2
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1817D61
hypocenterC61#7610
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
3. #21443641C2semidirect
84. C4⋊Dic61M16semidirect
184. C61#7610semidirect

Maximal subgroups

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

subgroupindex
C2×#168396602
#214455132
#214436412
C61×#18172
#761061