Finite Group Databasebeta

← D61⋊#1348 #30604770

Subgroups — D61⋊#1348 #30604770

Order 15616.

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

normal subgroups only

This group has 506 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⋊(C22⋊C16)C211yesmaximal, characteristic
2C2×#16840636C211yesmaximal, characteristic
3#21460808C212yesmaximal, complemented
4#21460807C212yesmaximal, complemented
5#467×D61C211yesmaximal, characteristic
6C22×C4⋊#6923—21no—
7C2×D61⋊C16—21no—
8#16840657C2212yes—
9C22⋊#11758187—21no—
10#16840636—21no—
11#16840636C2212yes—
12C2×Q16×D61C411yescharacteristic, complemented
13C22×C8×D61C2211yescharacteristic
14C22⋊Q8×D61—21no—
15#16840004C2211yescharacteristic
16#16840003C411yescharacteristic, complemented
17#16840002—21no—
18#16840001—21no—
19#16840000C2211yescharacteristic
20#107×D61C411yescharacteristic, complemented
21#103×D61—21no—
22C61×#467C411yescharacteristic, complemented
23C23×#6923—41no—
24C2×C4⋊#6923—24no—
25C2×C4⋊#6923—22no—
26C2×C4⋊#6923—42no—
27C2×C4⋊#6923—41no—
28D61⋊C16—42no—
29C2×#11758187—22no—
30C2×Q8×D61—21no—
31C22×C4×D61D411yescharacteristic
32C2×Dic244C2×C411yescharacteristic, complemented
33C2×#11758202—21no—
34Q16×D61—42no—
35C22×#92418D411yescharacteristic
36C2×C8×D61—42no—
37C2×C8×D61C2×C411yescharacteristic
38C2×C8×D61D411yescharacteristic
39C2×C8×D61D411yescharacteristic
40C2×C8×D61C2311yescharacteristic
41#14519757—21no—
42#14519756—21no—
43#14519755—41no—
44#14519754—41no—
45#14519753—21no—
46C4⋊C4×D61—21no—
47C4⋊C4×D61—41no—
48C22⋊C4×D61—41no—
49#14519711C2×C411yescharacteristic, complemented
50#14519710—21no—
51#14519702—21no—
52#14519701—21no—
53#14519700—21no—
54C2×Q16×C61C2×C411yescharacteristic, complemented
55C22×C8×C61C2×C411yescharacteristic
56C22⋊Q8×C61—21no—
57#107×C61C2×C411yescharacteristic, complemented
58#103×C61—21no—
59C23×D61D811yescharacteristic
60C22×#6923—44no—
61C22×#6923—42no—
62C22×#6923—82no—
63C22×#6923—82no—
64C2×Q8×C61—21no—
65C22×C4×C61C22⋊C411yescharacteristic
66C22×Dic61SD1611yescharacteristic
67Q8×D61—42no—
68Q8×D61—42no—
69C2×C4×D61—22no—
70C2×C4×D61—22no—
71C2×C4×D61—41no—
72C2×C4×D61—41no—
73C2×C4×D61C22⋊C411yescharacteristic
74C2×C4×D61D811yescharacteristic
75C2×C4×D61SD1611yescharacteristic
76C2×C4×D61C2×D411yescharacteristic
77C2×Dic122—21no—
78C2×Dic122—41no—
79C4⋊#6923—24no—
80C4⋊#6923—42no—
81C4⋊#6923—44no—
82Q16×C61—41no—
83C2×C8×C61—41no—
84C2×C8×C61C22⋊C411yescharacteristic
85C2×C8×C61C22×C411yescharacteristic
86C4⋊C4×C61—21no—
87C4⋊C4×C61—41no—
88C22⋊C4×C61—41no—
89C22⋊Dic61—41no—
90#11758202—42no—
91#11758198—41no—
92#11758198—41no—
93C4⋊Dic61—21no—
94C4⋊Dic61—41no—
95#11758196—41no—
96#11758196—41no—
97#11758196—41no—
98C2×#92418—41no—
99C2×#92418C22⋊C411yescharacteristic
100C2×#92418C2×D411yescharacteristic
101Dic244—41no—
102C8×D61—22no—
103C8×D61—22no—
104C8×D61—41no—
105C8×D61—41no—
106#11758187—42no—
107C23×C61#10211yescharacteristic
108C22×D61—22no—
109C22×D61—22no—
110C22×D61#10211yescharacteristic
111C22×D61SD3212yes—
112C22×D61Q3212yes—
113C22×D61C2×D811yescharacteristic
114C2×#6923—44no—
115C2×#6923—82no—
116C2×#6923—88no—
117C2×#6923—82no—
118Q8×C61—41no—
119Q8×C61—41no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61#10211yescharacteristic
125C2×C4×C61C2×C22⋊C411yescharacteristic
126C2×Dic61—22no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61#10211yescharacteristic
130C2×Dic61C2×SD1611yescharacteristic
131C4×D61—22no—
132C4×D61—21no—
133C4×D61—21no—
134C4×D61—22no—
135C4×D61—82no—
136C4×D61—82no—
137C4×D61#9912yes—
138C4×D61#10412yes—
139Dic122—41no—
140Dic122—41no—
141Dic122—82no—
142C8×C61—21no—
143C8×C61—21no—
144#92418—21no—
145#92418—21no—
146C2×#1348—611nomaximal, Sylow
147C22×C61—21no—
148C22×C61—21no—
149C22×C61#35812yes—
150C22×C61C2×#10211yescharacteristic
151C2×D61—21no—
152C2×D61—21no—
153C2×D61—24no—
154C2×D61—24no—
155C2×D61#32712yes—
156C2×D61#35812yes—
157C2×D61#36012yes—
158C2×D61C2×SD3211yescharacteristic
159C2×D61C2×Q3211yescharacteristic
160#6923—84no—
161C4×C61—21no—
162C4×C61—81no—
163C4×C61—81no—
164C4×C61C2×#9911yescharacteristic
165C4×C61C2×#10411yescharacteristic
166Dic61—21no—
167Dic61—81no—
168Dic61—81no—
169Dic61#32712yes—
170C2×#467—611no—
171C2×C22⋊C16—611no—
172C2×#340—611no—
173#1348—612no—
174#1348—612no—
175C2×C61—22no—
176C2×C61C2×#32711yescharacteristic
177C2×C61C2×#35811yescharacteristic
178C2×C61C2×#36011yescharacteristic
179D61—21no—
180D61—21no—
181D61#134814yescomplemented
182C22×Q16—611no—
183C23×C8—611no—
184C2×C22⋊Q8—1221no—
185C22×C4⋊C4—1221no—
186C22×C16—1221no—
187#467—611no—
188#467—611no—
189#467—611no—
190#467—1221no—
191#467—1221no—
192#467#692311yescharacteristic, complemented
193C2×#107—611no—
194C2×#103—1221no—
195C22⋊C16—612no—
196C22⋊C16—1221no—
197#340—612no—
198#340—1221no—
199C61C2×#134811yescharacteristic, complemented, Sylow
200C22×Q8—1221no—
201C23×C4—611no—
202C23×C4—2441no—
203C2×Q16—611no—
204C2×Q16—1221no—
205C2×Q16—2442no—
206C2×Q16C2×#692311yescharacteristic, complemented
207C22×C8—611no—
208C22×C8—611no—
209C22×C8—611no—
210C22×C8—611no—
211C22×C8—611no—
212C22×C8—2442no—
213C22×C8C2×#692311yescharacteristic
214C22⋊Q8—21no—
215C22⋊Q8—1221no—
216C22⋊Q8—1221no—
217C22⋊Q8—1221no—
218C22⋊Q8—2441no—
219C22⋊Q8—2441no—
220C2×C4⋊C4—1221no—
221C2×C4⋊C4—1222no—
222C2×C4⋊C4—1224no—
223C2×C4⋊C4—2441no—
224C2×C4⋊C4—2441no—
225C2×C4⋊C4—2442no—
226C2×C22⋊C4—2441no—
227C2×C16—1222no—
228C2×C16—2442no—
229#107—611no—
230#107—1221no—
231#107C2×#692311yescharacteristic, complemented
232#103—21no—
233#103—1221no—
234#103—1221no—
235#103—1221no—
236C24—611no—
237C2×Q8—21no—
238C2×Q8—1221no—
239C2×Q8—2441no—
240C2×Q8—2442no—
241C2×Q8—2442no—
242C22×C4—611no—
243C22×C4—611no—
244C22×C4—611no—
245C22×C4—611no—
246C22×C4—611no—
247C22×C4—1222no—
248C22×C4—1222no—
249C22×C4—4882no—
250C22×C4—4882no—
251C22×C4—2441no—
252C22×C4—2441no—
253C22×C4—2442no—
254C22×C4—2444no—
255C22×C4C22⋊#692311yescharacteristic
256Q16—41no—
257Q16—2441no—
258Q16—2442no—
259C2×C8—41no—
260C2×C8—611no—
261C2×C8—611no—
262C2×C8—1222no—
263C2×C8—1222no—
264C2×C8—2441no—
265C2×C8—2441no—
266C2×C8—2441no—
267C2×C8C22⋊#692311yescharacteristic
268C2×C8C22×#692311yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—1224no—
272C4⋊C4—1221no—
273C4⋊C4—2441no—
274C4⋊C4—2441no—
275C4⋊C4—2441no—
276C4⋊C4—2444no—
277C4⋊C4—2442no—
278C4⋊C4—2441no—
279C22⋊C4—41no—
280C22⋊C4—2441no—
281C22⋊C4—2441no—
282C22⋊C4—2441no—
283C16—2442no—
284C23—611no—
285C23—611no—
286C23—612no—
287C23—612no—
288C23—1222no—
289C23—1222no—
290C23#1451983911yescharacteristic
291Q8—41no—
292Q8—41no—
293Q8—4882no—
294Q8—2441no—
295Q8—2441no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—611no—
301C2×C4—612no—
302C2×C4—612no—
303C2×C4—611no—
304C2×C4—1222no—
305C2×C4—1222no—
306C2×C4—1222no—
307C2×C4—1221no—
308C2×C4—1221no—
309C2×C4—4882no—
310C2×C4—4882no—
311C2×C4—4882no—
312C2×C4—4888no—
313C2×C4—4882no—
314C2×C4—2444no—
315C2×C4—2441no—
316C2×C4—2441no—
317C2×C4#1451983911yescharacteristic
318C2×C4C2×C22⋊#692311yescharacteristic
319C8—21no—
320C8—21no—
321C8—1221no—
322C8—1221no—
323C22—21no—
324C22—21no—
325C22—612no—
326C22—612no—
327C22—612no—
328C22—611no—
329C22—611no—
330C22—1224no—
331C22—1224no—
332C22—1221no—
333C22—1221no—
334C22#1684067512yes—
335C22C2×#1451983911yescharacteristic
336C4—21no—
337C4—81no—
338C4—81no—
339C4—612no—
340C4—1221no—
341C4—4884no—
342C4—4881no—
343C4—4881no—
344C4D61⋊#9911yescharacteristic
345C4D61⋊#10411yescharacteristic
346C2—22no—
347C2—614no—
348C2—1221no—
349C2—1221no—
350C2D61⋊#32711yescharacteristic
351C2C2×#1684067511yescharacteristic
352C2D61⋊#36011yescharacteristic
subgroupG/H
centerC22C2×#14519839
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#467C4
socleC22×C61C2×#102
prime radicalC23×D61D8
cosocleC2×C8×D61C23
hypercenter#467#6923
hypocenterC61C2×#1348
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #467

Complements

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

NH ≅ G/Ntype
3. #21460808C2semidirect
4. #21460807C2semidirect
12. C2×Q16×D61C4semidirect
16. #16840003C4semidirect
20. #107×D61C4semidirect
22. C61×#467C4semidirect
32. C2×Dic244C2×C4semidirect
49. #14519711C2×C4semidirect
54. C2×Q16×C61C2×C4semidirect
57. #107×C61C2×C4semidirect
181. D61#1348semidirect
192. #467#6923semidirect
199. C61C2×#1348semidirect
206. C2×Q16C2×#6923semidirect
231. #107C2×#6923semidirect

Maximal subgroups

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

subgroupindex
D61⋊(C22⋊C16)2
C2×#168406362
#214608082
#214608072
#467×D612
C2×#134861