Finite Group Databasebeta

← #30603821

Subgroups — #30603821

Order 15616.

734 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 90 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
1D61⋊#323C212yesmaximal, complemented
2#21460746C212yesmaximal, complemented
3#21460743C212yesmaximal, complemented
4#21448395C211yesmaximal, characteristic
5D61⋊(C22⋊C8)—22no—
6#16840606—22no—
7#16840605C2214yes—
8C23×C22⋊D61C411yescharacteristic
9C2×C22⋊C4×D61C411yescharacteristic
10C2×C22⋊C4×D61C2211yescharacteristic
11#16839638—22no—
12#16839638—22no—
13#16839623C411yescharacteristic
14#16839623C2211yescharacteristic
15#16839618C2211yescharacteristic
16C61×#379C411yescharacteristic
17C2×D61⋊C8—42no—
18#14519834—42no—
19C22⋊#92420—24no—
20C24×D61C2×C411yescharacteristic
21C22×C22⋊D61—21no—
22C22×C22⋊D61—21no—
23C22×C22⋊D61—22no—
24C22×C22⋊D61—42no—
25C22×C22⋊D61—42no—
26C22×C22⋊D61C814yescomplemented
27C23×Dic61C2×C411yescharacteristic
28C22×C4×D61—21no—
29C22×C4×D61D412yes—
30C2×C22⋊Dic61—22no—
31C2×C22⋊Dic61C2×C411yescharacteristic
32C2×C22⋊Dic61C2311yescharacteristic
33C2×#11758198—21no—
34C2×#11758198D412yes—
35C22⋊C4×D61—42no—
36C22⋊C4×D61—42no—
37#14519684—22no—
38#14519684—22no—
39C25×C61C2×C411yescharacteristic
40C2×C22⋊C4×C61—22no—
41C2×C22⋊C4×C61C2×C411yescharacteristic
42C2×C22⋊C4×C61C2×C411yescharacteristic
43C24×C61—21no—
44C24×C61—21no—
45C24×C61—41no—
46C24×C61—41no—
47C24×C61—82no—
48C24×C61C2×C812yescomplemented
49C24×C61C22×C411yescharacteristic
50C23×D61—22no—
51C23×D61—22no—
52C23×D61—82no—
53C23×D61C22⋊C411yescharacteristic
54C23×D61C22⋊C411yescharacteristic
55C23×D61C2×C812yes—
56C23×D61M1612yes—
57C22×C4×C61—21no—
58C22×C4×C61C22⋊C412yes—
59C2×C22⋊D61—24no—
60C2×C22⋊D61—24no—
61C2×C22⋊D61—48no—
62C2×C22⋊D61—42no—
63C2×C22⋊D61—42no—
64C2×C22⋊D61—42no—
65C2×C22⋊D61—42no—
66C2×C22⋊D61—48no—
67C22×Dic61—21no—
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—41no—
71C22×Dic61—41no—
72C22×Dic61C2×C812yescomplemented
73C22×Dic61C2×D412yes—
74C2×C4×D61—22no—
75C2×C4×D61—22no—
76C2×C4×D61—42no—
77C2×C4×D61—82no—
78C2×C4×D61—82no—
79C2×#92420—44no—
80D61⋊C8—84no—
81C22⋊C4×C61—41no—
82C22⋊C4×C61—41no—
83C22⋊C4×C61—82no—
84C22⋊Dic61—41no—
85C22⋊Dic61—41no—
86C22⋊Dic61—82no—
87#11758198—42no—
88#11758198—42no—
89C23×C61—21no—
90C23×C61—21no—
91C23×C61—22no—
92C23×C61—42no—
93C23×C61—42no—
94C23×C61—42no—
95C23×C61—42no—
96C23×C61—42no—
97C23×C61—41no—
98C23×C61—41no—
99C23×C61—82no—
100C23×C61—82no—
101C23×C61—84no—
102C23×C61—84no—
103C23×C61C2×C22⋊C411yescharacteristic
104C23×C61C22×C811yescharacteristic
105C23×C61C2×M1611yescharacteristic
106C22×D61—21no—
107C22×D61—21no—
108C22×D61—24no—
109C22×D61—24no—
110C22×D61—42no—
111C22×D61—42no—
112C22×D61—82no—
113C22×D61—82no—
114C22×D61—84no—
115C22×D61—82no—
116C22×D61—82no—
117C22×D61C22⋊C812yes—
118C22×D61C22⋊C812yes—
119C22×D61#9912yes—
120C22×D61#10012yes—
121C2×C4×C61—22no—
122C2×C4×C61—42no—
123C2×C4×C61—42no—
124C2×C4×C61—41no—
125C22⋊D61—48no—
126C22⋊D61—48no—
127C2×Dic61—22no—
128C2×Dic61—22no—
129C2×Dic61—42no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134C2×Dic61—42no—
135C2×Dic61—41no—
136C4×D61—44no—
137C4×D61—82no—
138C4×D61—84no—
139#92420—84no—
140#11426—611nomaximal, Sylow
141C22×C61—22no—
142C22×C61—21no—
143C22×C61—21no—
144C22×C61—41no—
145C22×C61—41no—
146C22×C61—42no—
147C22×C61—42no—
148C22×C61—42no—
149C22×C61—42no—
150C22×C61—42no—
151C22×C61—82no—
152C22×C61—84no—
153C22×C61—84no—
154C22×C61—82no—
155C22×C61C2×C22⋊C811yescharacteristic
156C22×C61C2×#9911yescharacteristic
157C22×C61C2×#10011yescharacteristic
158C2×D61—22no—
159C2×D61—22no—
160C2×D61—22no—
161C2×D61—42no—
162C2×D61—42no—
163C2×D61—42no—
164C2×D61—42no—
165C2×D61—42no—
166C2×D61—84no—
167C2×D61—84no—
168C2×D61#32314yes—
169C4×C61—42no—
170C4×C61—81no—
171Dic61—42no—
172Dic61—42no—
173Dic61—81no—
174#2308—611no—
175C2×#323—612no—
176#1317—612no—
177#1316—612no—
178C2×C61—21no—
179C2×C61—21no—
180C2×C61—41no—
181C2×C61—41no—
182C2×C61—82no—
183C2×C61C2×#32311yescharacteristic
184C2×C61#212112yes—
185D61—24no—
186D61—42no—
187C23×D4—611no—
188C22×C22⋊C4—611no—
189C22×C22⋊C4—611no—
190C2×C22⋊C8—1222no—
191#386—1222no—
192#386—1222no—
193#381—611no—
194#381—611no—
195#379—611no—
196#379#692311yescharacteristic
197#323—614no—
198#323—1222no—
199C61#1142611yescharacteristic, complemented, Sylow
200C25—611no—
201C25C2×#692311yescharacteristic
202C22×D4—614no—
203C22×D4—1222no—
204C22×D4—1221no—
205C22×D4—1221no—
206C22×D4—2442no—
207C22×D4—2442no—
208C23×C4—611no—
209C23×C4—612no—
210C23×C4—1221no—
211C22×C8—2442no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—611no—
214C2×C22⋊C4—612no—
215C2×C22⋊C4—611no—
216C2×C22⋊C4—1222no—
217C2×C22⋊C4—1221no—
218C2×C22⋊C4—2442no—
219C2×C22⋊C4—2442no—
220C2×C22⋊C4C2×#692311yescharacteristic
221C2×C22⋊C4C2×#692311yescharacteristic
222C22⋊C8—1224no—
223C22⋊C8—2442no—
224#95—1222no—
225#95—1222no—
226C24—21no—
227C24—21no—
228C24—41no—
229C24—41no—
230C24—82no—
231C24—611no—
232C24—612no—
233C24—612no—
234C24—611no—
235C24—1222no—
236C24—1222no—
237C24—4882no—
238C24D61⋊C812yescomplemented
239C24C22×#692311yescharacteristic
240C2×D4—1224no—
241C2×D4—1224no—
242C2×D4—2442no—
243C2×D4—2442no—
244C2×D4—2448no—
245C2×D4—2442no—
246C2×D4—2442no—
247C2×D4—2448no—
248C22×C4—21no—
249C22×C4—612no—
250C22×C4—612no—
251C22×C4—1222no—
252C22×C4—1222no—
253C22×C4—1221no—
254C22×C4—1221no—
255C22×C4—1221no—
256C22×C4—4882no—
257C22×C4—4882no—
258C22×C4—2442no—
259C22×C4—2441no—
260C22×C4—2441no—
261C22×C4C22⋊#692312yes—
262C2×C8—4884no—
263C2×C8—2444no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—82no—
267C22⋊C4—4882no—
268C22⋊C4—2442no—
269C22⋊C4—2442no—
270C22⋊C4—2441no—
271C22⋊C4—2441no—
272C23—21no—
273C23—21no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—41no—
281C23—41no—
282C23—82no—
283C23—82no—
284C23—84no—
285C23—84no—
286C23—612no—
287C23—612no—
288C23—612no—
289C23—612no—
290C23—1224no—
291C23—1224no—
292C23—1221no—
293C23—1221no—
294C23—4882no—
295C23—4882no—
296C23—4884no—
297C23—4882no—
298C23—4882no—
299C23—2442no—
300C23—2442no—
301C23C2×C22⋊#692311yescharacteristic
302C23C2×D61⋊C811yescharacteristic
303C23D61⋊M1611yescharacteristic
304D4—2448no—
305D4—2448no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—1222no—
311C2×C4—1222no—
312C2×C4—4882no—
313C2×C4—4884no—
314C2×C4—2444no—
315C2×C4—2442no—
316C2×C4—2441no—
317C2×C4—2442no—
318C2×C4—2441no—
319C2×C4—2441no—
320C2×C4—2442no—
321C2×C4—2442no—
322C8—4884no—
323C22—22no—
324C22—21no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—82no—
334C22—84no—
335C22—84no—
336C22—82no—
337C22—614no—
338C22—1222no—
339C22—1222no—
340C22—1222no—
341C22—4884no—
342C22—4884no—
343C22—2442no—
344C22—2442no—
345C22—2442no—
346C22—2442no—
347C22—2442no—
348C22D61⋊(C22⋊C8)11yescharacteristic
349C22D61⋊#9911yescharacteristic
350C22D61⋊#10011yescharacteristic
351C4—42no—
352C4—81no—
353C4—4881no—
354C4—2442no—
355C4—2442no—
356C2—21no—
357C2—21no—
358C2—41no—
359C2—41no—
360C2—82no—
361C2—1224no—
362C2—2442no—
363C2D61⋊#32311yescharacteristic
364C2#2146207512yes—
subgroupG/H
centerC22D61⋊#99
commutatorC23×C61C22×C8
FrattiniC24C22×#6923
FittingC61×#379C4
socleC22×C61C2×#99
prime radicalC23×C22⋊D61C4
cosocleC2×C22⋊Dic61C23
hypercenter#379#6923
hypocenterC61#11426
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #379

Complements

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

NH ≅ G/Ntype
1. D61⋊#323C2semidirect
2. #21460746C2semidirect
3. #21460743C2semidirect
26. C22×C22⋊D61C8semidirect
48. C24×C61C2×C8semidirect
72. C22×Dic61C2×C8semidirect
199. C61#11426semidirect
238. C24D61⋊C8semidirect

Maximal subgroups

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

subgroupindex
D61⋊#3232
#214607462
#214607432
#214483952
#1142661