Finite Group Databasebeta

← D61⋊#1317 #30603814

Subgroups — D61⋊#1317 #30603814

Order 15616.

726 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 98 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
2#21460746C212yesmaximal, complemented
3#21460745C212yesmaximal, complemented
4#381×D61C211yesmaximal, characteristic
5D61⋊(C22⋊C8)—22no—
6#16840606—22no—
7#16840605C2214yes—
8C23×C4×D61C411yescharacteristic
9C2×C22⋊C4×D61C411yescharacteristic
10C2×C22⋊C4×D61C2211yescharacteristic
11#16839625C2211yescharacteristic
12#16839624—22no—
13#16839623C411yescharacteristic
14#16839623C2211yescharacteristic
15#95×D61—22no—
16C61×#381C411yescharacteristic
17C2×D61⋊C8—42no—
18#14519834—42no—
19C22⋊#92420—24no—
20C24×D61C2×C411yescharacteristic
21C23×Dic61C2×C411yescharacteristic
22C22×C4×D61—21no—
23C22×C4×D61—21no—
24C22×C4×D61—22no—
25C22×C4×D61—21no—
26C22×C4×D61—41no—
27C22×C4×D61—41no—
28C22×C4×D61—42no—
29C22×C4×D61C814yescomplemented
30C22×C4×D61D412yes—
31C2×C22⋊Dic61C2×C411yescharacteristic
32C2×C22⋊Dic61C2311yescharacteristic
33C2×#11758198—21no—
34C2×#11758198D412yes—
35C22⋊C4×D61—41no—
36C22⋊C4×D61—41no—
37C22⋊C4×D61—42no—
38#14519684—22no—
39#14519684—22no—
40#14519684—22no—
41C23×C4×C61C2×C411yescharacteristic
42C2×C22⋊C4×C61C2×C411yescharacteristic
43C2×C22⋊C4×C61C2×C411yescharacteristic
44#95×C61—22no—
45C24×C61C22×C411yescharacteristic
46C23×D61—21no—
47C23×D61—21no—
48C23×D61—22no—
49C23×D61—44no—
50C23×D61C22⋊C411yescharacteristic
51C23×D61C22⋊C411yescharacteristic
52C23×D61C2×C812yes—
53C23×D61M1612yes—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61—41no—
58C22×C4×C61—41no—
59C22×C4×C61C22⋊C412yes—
60C22×C4×C61C2×C812yescomplemented
61C22×Dic61—21no—
62C22×Dic61—21no—
63C22×Dic61—21no—
64C22×Dic61—41no—
65C22×Dic61—41no—
66C22×Dic61C2×C812yescomplemented
67C22×Dic61C2×D412yes—
68C2×C4×D61—24no—
69C2×C4×D61—24no—
70C2×C4×D61—22no—
71C2×C4×D61—22no—
72C2×C4×D61—42no—
73C2×C4×D61—42no—
74C2×C4×D61—44no—
75C2×C4×D61—44no—
76C2×C4×D61—41no—
77C2×C4×D61—41no—
78C2×C4×D61—42no—
79C2×C4×D61—44no—
80C2×C4×D61—44no—
81C2×C4×D61—44no—
82C2×C4×D61—44no—
83C2×C4×D61—42no—
84C2×#92420—44no—
85D61⋊C8—84no—
86C22⋊C4×C61—41no—
87C22⋊C4×C61—41no—
88C22⋊Dic61—41no—
89C22⋊Dic61—41no—
90#11758198—42no—
91#11758198—42no—
92C23×C61—21no—
93C23×C61—21no—
94C23×C61—42no—
95C23×C61C2×C22⋊C411yescharacteristic
96C23×C61C22×C811yescharacteristic
97C23×C61C2×M1611yescharacteristic
98C22×D61—24no—
99C22×D61—21no—
100C22×D61—21no—
101C22×D61—22no—
102C22×D61—22no—
103C22×D61—22no—
104C22×D61—22no—
105C22×D61—44no—
106C22×D61—48no—
107C22×D61—44no—
108C22×D61—44no—
109C22×D61—44no—
110C22×D61—42no—
111C22×D61C22⋊C812yes—
112C22×D61C22⋊C812yes—
113C22×D61#9911yescharacteristic
114C22×D61#9911yescharacteristic
115C22×D61#10012yes—
116C2×C4×C61—22no—
117C2×C4×C61—22no—
118C2×C4×C61—42no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61—42no—
124C2×C4×C61—41no—
125C2×Dic61—22no—
126C2×Dic61—22no—
127C2×Dic61—42no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—42no—
131C2×Dic61—42no—
132C2×Dic61—42no—
133C2×Dic61—41no—
134C4×D61—48no—
135C4×D61—44no—
136C4×D61—44no—
137C4×D61—44no—
138C4×D61—82no—
139C4×D61—84no—
140#92420—84no—
141C2×#1317—611nomaximal, Sylow
142C22×C61—22no—
143C22×C61—21no—
144C22×C61—21no—
145C22×C61—41no—
146C22×C61—41no—
147C22×C61—42no—
148C22×C61—42no—
149C22×C61C2×C22⋊C811yescharacteristic
150C22×C61C2×#9911yescharacteristic
151C22×C61C2×#10011yescharacteristic
152C2×D61—22no—
153C2×D61—24no—
154C2×D61—24no—
155C2×D61—22no—
156C2×D61—44no—
157C2×D61—44no—
158C2×D61—42no—
159C2×D61—42no—
160C2×D61—44no—
161C2×D61—44no—
162C2×D61—42no—
163C2×D61#32312yes—
164C2×D61#32312yes—
165C2×D61#35214yes—
166C4×C61—42no—
167C4×C61—42no—
168C4×C61—81no—
169Dic61—42no—
170Dic61—42no—
171Dic61—81no—
172C2×#381—611no—
173C2×#323—612no—
174#1317—612no—
175#1317—612no—
176C2×C61—21no—
177C2×C61—21no—
178C2×C61—41no—
179C2×C61—41no—
180C2×C61C2×#32311yescharacteristic
181C2×C61C2×#35212yes—
182D61—22no—
183D61—42no—
184D61#131714yescomplemented
185C24×C4—611no—
186C22×C22⋊C4—611no—
187C22×C22⋊C4—611no—
188C2×C22⋊C8—1222no—
189#381—611no—
190#381—611no—
191#381—611no—
192#381—1222no—
193#381#692311yescharacteristic
194C2×#95—1222no—
195#323—614no—
196#323—1222no—
197C61C2×#131711yescharacteristic, complemented, Sylow
198C25—611no—
199C23×C4—611no—
200C23×C4—614no—
201C23×C4—612no—
202C23×C4—1222no—
203C23×C4—1221no—
204C23×C4—1221no—
205C23×C4—1221no—
206C23×C4—2441no—
207C23×C4—2441no—
208C23×C4—2442no—
209C23×C4C2×#692311yescharacteristic
210C22×C8—2442no—
211C2×C22⋊C4—611no—
212C2×C22⋊C4—612no—
213C2×C22⋊C4—611no—
214C2×C22⋊C4—1221no—
215C2×C22⋊C4—2441no—
216C2×C22⋊C4—2441no—
217C2×C22⋊C4—2442no—
218C2×C22⋊C4C2×#692311yescharacteristic
219C2×C22⋊C4C2×#692311yescharacteristic
220C22⋊C8—1224no—
221C22⋊C8—2442no—
222#95—22no—
223#95—1222no—
224#95—1222no—
225#95—1222no—
226C24—611no—
227C24—611no—
228C24—612no—
229C24—612no—
230C24—1221no—
231C24—1221no—
232C24—1222no—
233C24—2444no—
234C24C22×#692311yescharacteristic
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—612no—
241C22×C4—612no—
242C22×C4—1222no—
243C22×C4—1222no—
244C22×C4—1224no—
245C22×C4—1224no—
246C22×C4—1221no—
247C22×C4—1221no—
248C22×C4—1221no—
249C22×C4—2442no—
250C22×C4—2444no—
251C22×C4—2444no—
252C22×C4—2441no—
253C22×C4—2441no—
254C22×C4—2442no—
255C22×C4—2444no—
256C22×C4—2444no—
257C22×C4—2442no—
258C22×C4—2442no—
259C22×C4—2444no—
260C22×C4—2444no—
261C22×C4—2441no—
262C22×C4—2441no—
263C22×C4D61⋊C812yescomplemented
264C22×C4C22⋊#692312yes—
265C2×C8—4884no—
266C2×C8—2444no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—2442no—
270C22⋊C4—2442no—
271C22⋊C4—2441no—
272C22⋊C4—2441no—
273C23—21no—
274C23—21no—
275C23—42no—
276C23—611no—
277C23—612no—
278C23—611no—
279C23—612no—
280C23—612no—
281C23—1224no—
282C23—1222no—
283C23—1222no—
284C23—1222no—
285C23—1222no—
286C23—1221no—
287C23—1221no—
288C23—2442no—
289C23—2444no—
290C23—2444no—
291C23—2444no—
292C23—2448no—
293C23—2444no—
294C23C2×C22⋊#692311yescharacteristic
295C23C2×D61⋊C811yescharacteristic
296C23D61⋊M1611yescharacteristic
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—1222no—
307C2×C4—1222no—
308C2×C4—4882no—
309C2×C4—4884no—
310C2×C4—2444no—
311C2×C4—2444no—
312C2×C4—2444no—
313C2×C4—2448no—
314C2×C4—2442no—
315C2×C4—2441no—
316C2×C4—2442no—
317C2×C4—2441no—
318C2×C4—2441no—
319C2×C4—2442no—
320C2×C4—2442no—
321C8—4884no—
322C22—22no—
323C22—21no—
324C22—21no—
325C22—41no—
326C22—41no—
327C22—42no—
328C22—42no—
329C22—614no—
330C22—612no—
331C22—612no—
332C22—1224no—
333C22—1224no—
334C22—1222no—
335C22—1222no—
336C22—2442no—
337C22—2442no—
338C22—2444no—
339C22—2444no—
340C22—2444no—
341C22—2444no—
342C22—2442no—
343C22D61⋊(C22⋊C8)11yescharacteristic
344C22D61⋊#9911yescharacteristic
345C22D61⋊#10011yescharacteristic
346C4—42no—
347C4—42no—
348C4—81no—
349C4—4881no—
350C4—2442no—
351C4—2442no—
352C2—21no—
353C2—21no—
354C2—41no—
355C2—41no—
356C2—614no—
357C2—1222no—
358C2—2442no—
359C2D61⋊#32311yescharacteristic
360C2D61⋊#35212yes—
subgroupG/H
centerC22D61⋊#99
commutatorC23×C61C22×C8
FrattiniC24C22×#6923
FittingC61×#381C4
socleC22×C61C2×#99
prime radicalC24×D61C2×C4
cosocleC2×C22⋊Dic61C23
hypercenter#381#6923
hypocenterC61C2×#1317
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. #21460746C2semidirect
3. #21460745C2semidirect
29. C22×C4×D61C8semidirect
60. C22×C4×C61C2×C8semidirect
66. C22×Dic61C2×C8semidirect
184. D61#1317semidirect
197. C61C2×#1317semidirect
263. C22×C4D61⋊C8semidirect

Maximal subgroups

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

subgroupindex
D61⋊#3232
#214607462
#214607452
#381×D612
C2×#131761