Finite Group Databasebeta

← #360×Dic61 #29595263

Subgroups — #360×Dic61 #29595263

Order 15616.

466 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#135×Dic61C211yesmaximal, characteristic
2C2×#16839572C211yesmaximal, characteristic
3M32×Dic61C211yesmaximal, characteristic, complemented
4#106×Dic61C211yesmaximal, characteristic, complemented
5#21443926C211yesmaximal, characteristic, complemented
6#21443925C211yesmaximal, characteristic, complemented
7C2×C61×#360C211yesmaximal, characteristic
8#41×Dic61—21no—
9C2×#14519717C2211yescharacteristic
10Q16×Dic61C411yescharacteristic, complemented
11Q16×Dic61C411yescharacteristic, complemented
12SD16×Dic61—21no—
13SD16×Dic61—21no—
14D8×Dic61C411yescharacteristic, complemented
15D8×Dic61C411yescharacteristic, complemented
16#16839800C2211yescharacteristic
17C2×#14519709C2211yescharacteristic
18#16839758—21no—
19C2×C8×Dic61C2211yescharacteristic
20C4⋊C4×Dic61—21no—
21#16839572C414yescomplemented
22#16839537—21no—
23C16×Dic61—21no—
24#16839524—21no—
25C2×#135×C61C2211yescharacteristic
26C2×M32×C61C2211yescharacteristic
27C2×#106×C61C2211yescharacteristic
28C61×#360C414yescomplemented
29C22×#92418D411yescharacteristic
30Q8×Dic61—21no—
31Q8×Dic61—21no—
32D4×Dic61—21no—
33D4×Dic61—21no—
34D4×Dic61—41no—
35D4×Dic61—41no—
36#14519736—21no—
37#14519736—41no—
38C2×C4⋊Dic61—21no—
39C2×C4×Dic61—41no—
40C2×C4×Dic61—41no—
41C2×C4×Dic61D411yescharacteristic
42#14519717—21no—
43#14519717C2×C412yescomplemented
44C2×#11758185—21no—
45#14519711C2×C411yescharacteristic
46#14519711C2×C411yescharacteristic
47#14519709—21no—
48#14519709—21no—
49#14519709C2×C412yescomplemented
50#14519701—21no—
51#14519701—21no—
52#14519698—21no—
53#14519698—21no—
54C8×Dic61C2×C411yescharacteristic
55C8×Dic61C2×C411yescharacteristic
56C8×Dic61D411yescharacteristic
57C8×Dic61D411yescharacteristic
58#14519684—41no—
59C2×#41×C61—21no—
60#135×C61—21no—
61#135×C61—21no—
62#135×C61C2×C412yes—
63#135×C61C2×C412yes—
64C2×Q16×C61C2×C411yescharacteristic
65C2×SD16×C61—21no—
66C2×D8×C61C2×C411yescharacteristic
67C22×C8×C61C2311yescharacteristic
68C2×C4⋊C4×C61—21no—
69M32×C61—21no—
70M32×C61C2×C412yescomplemented
71C2×C16×C61—21no—
72#106×C61—21no—
73#106×C61C2×C412yescomplemented
74#41×C61—22no—
75#41×C61—22no—
76#41×C61—41no—
77#41×C61—41no—
78C2×Q8×C61—21no—
79C2×D4×C61—21no—
80C2×D4×C61—41no—
81C22×C4×C61—41no—
82C22×C4×C61—41no—
83C22×C4×C61C2×D411yescharacteristic
84C22×Dic61—41no—
85C22×Dic61—41no—
86C22×Dic61—41no—
87C22×Dic61D811yescharacteristic
88C22×Dic61D811yescharacteristic
89Q16×C61—21no—
90Q16×C61C4212yescomplemented
91SD16×C61—22no—
92SD16×C61—22no—
93D8×C61—21no—
94D8×C61C4212yescomplemented
95C2×C8×C61C22×C411yescharacteristic
96C2×C8×C61C22×C411yescharacteristic
97C2×C8×C61C22×C411yescharacteristic
98C2×C8×C61C2×D411yescharacteristic
99C2×C8×C61#4111yescharacteristic
100C2×C8×C61#4111yescharacteristic
101C4⋊C4×C61—22no—
102C4⋊C4×C61—41no—
103C22⋊Dic61—41no—
104C22⋊Dic61—41no—
105C22⋊Dic61—41no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61—22no—
109C4⋊Dic61—41no—
110C4⋊Dic61—41no—
111C4⋊Dic61—41no—
112C4×Dic61—41no—
113C4×Dic61—41no—
114C4×Dic61—41no—
115C4×Dic61—82no—
116C4×Dic61C22⋊C411yescharacteristic
117C4×Dic61C22⋊C411yescharacteristic
118C4×Dic61SD1611yescharacteristic
119C4×Dic61SD1611yescharacteristic
120C2×#92418—21no—
121C2×#92418—21no—
122C2×#92418C22⋊C411yescharacteristic
123C2×#92418C22⋊C411yescharacteristic
124C16×C61—22no—
125#11758185—22no—
126C23×C61—41no—
127C23×C61C2×D811yescharacteristic
128Q8×C61—22no—
129Q8×C61—41no—
130D4×C61—22no—
131D4×C61—42no—
132D4×C61—42no—
133D4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61—81no—
139C2×C4×C61—81no—
140C2×C4×C61C2×C22⋊C411yescharacteristic
141C2×C4×C61C4×D411yescharacteristic
142C2×C4×C61C4×D411yescharacteristic
143C2×C4×C61C2×SD1611yescharacteristic
144C2×C4×C61#13511yescharacteristic
145C2×C4×C61#13511yescharacteristic
146C2×Dic61—22no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—42no—
151C2×Dic61—82no—
152C2×Dic61—81no—
153C2×Dic61—81no—
154C2×Dic61—82no—
155C2×Dic61#10211yescharacteristic
156C2×Dic61#10211yescharacteristic
157C2×Dic61#10211yescharacteristic
158C2×Dic61#10211yescharacteristic
159C8×C61C2×C4211yescharacteristic
160C8×C61#11711yescharacteristic
161C8×C61C4×D411yescharacteristic
162C8×C61C4×D411yescharacteristic
163#92418—21no—
164#92418—21no—
165C4×#360—611nomaximal, Sylow
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—42no—
169C22×C61—81no—
170C22×C61—81no—
171C22×C61C2×#10211yescharacteristic
172C22×C61C4×D811yescharacteristic
173C22×C61C4×D811yescharacteristic
174C4×C61—42no—
175C4×C61—82no—
176C4×C61C4×C22⋊C411yescharacteristic
177C4×C61#41611yescharacteristic
178C4×C61C4×SD1611yescharacteristic
179C4×C61C4×SD1611yescharacteristic
180Dic61—21no—
181Dic61—21no—
182Dic61—21no—
183Dic61—82no—
184Dic61—81no—
185Dic61—81no—
186Dic61#36012yesdirect product factor
187C4×#135—611no—
188C2×#360—611no—
189C2×#360D6111yescharacteristic
190C4×M32—611no—
191C4×#106—611no—
192#1386—611no—
193#1386—611no—
194C2×C61—21no—
195C2×C61—21no—
196C2×C61—82no—
197C2×C61C4×#10211yescharacteristic
198C2×C61C2×#36011yescharacteristic
199C2×C61#214511yescharacteristic
200C2×#135C2×D6111yescharacteristic
201C4×#41—1221no—
202C2×M32—611no—
203C2×M32C2×D6111yescharacteristic
204C4×Q16—611no—
205C4×Q16—611no—
206C4×SD16—1221no—
207C4×SD16—1221no—
208C4×D8—611no—
209C4×D8—611no—
210#427—611no—
211C2×#106—611no—
212C2×#106C2×D6111yescharacteristic
213#416—1221no—
214C2×C4×C8—611no—
215C4×C4⋊C4—1221no—
216#360—614no—
217#360Dic6114yesdirect product factor
218#346—1221no—
219C4×C16—1221no—
220#340—1221no—
221C61C4×#36011yescharacteristic, complemented, Sylow
222C2×#41—21no—
223#135—21no—
224#135—21no—
225#135C4×D6112yes—
226#135C2×Dic6112yes—
227C2×Q16C4×D6111yescharacteristic
228C2×SD16—21no—
229C2×D8C4×D6111yescharacteristic
230C22×C8—611no—
231C22×C8C22×D6111yescharacteristic
232C4×Q8—1221no—
233C4×Q8—1221no—
234C4×D4—1221no—
235C4×D4—1221no—
236C4×D4—2441no—
237C4×D4—2441no—
238#117—1221no—
239#117—2441no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—1221no—
242C2×C42—611no—
243C2×C42—2441no—
244C2×C42—2441no—
245M32—21no—
246M32—612no—
247M32—1221no—
248M32C2×Dic6112yescomplemented
249C2×C16—21no—
250C2×C16—1221no—
251#107—611no—
252#107—611no—
253#106—21no—
254#106—612no—
255#106—1221no—
256#106—1221no—
257#106C2×Dic6112yescomplemented
258#103—1221no—
259#103—1221no—
260#102—1221no—
261#102—1221no—
262C4×C8—611no—
263C4×C8—611no—
264C4×C8—611no—
265C4×C8—611no—
266#95—2441no—
267#41—22no—
268#41—22no—
269#41—41no—
270#41—41no—
271C2×Q8—21no—
272C2×D4—21no—
273C2×D4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—611no—
277C22×C4—611no—
278C22×C4—2441no—
279C22×C4—2441no—
280C22×C4—2441no—
281C22×C4D4×D6111yescharacteristic
282Q16—21no—
283Q16C4×Dic6112yescomplemented
284SD16—22no—
285SD16—22no—
286D8—21no—
287D8C4×Dic6112yescomplemented
288C2×C8—611no—
289C2×C8—611no—
290C2×C8—1221no—
291C2×C8—1221no—
292C2×C8C2×C4×D6111yescharacteristic
293C2×C8C2×C4×D6111yescharacteristic
294C2×C8D4×D6111yescharacteristic
295C2×C8#1175822411yescharacteristic
296C2×C8#1175822411yescharacteristic
297C2×C8C22×Dic6111yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—41no—
300C4⋊C4—1221no—
301C4⋊C4—1221no—
302C4⋊C4—1222no—
303C4⋊C4—2441no—
304C4⋊C4—2441no—
305C4⋊C4—2441no—
306C22⋊C4—2441no—
307C22⋊C4—2441no—
308C22⋊C4—2441no—
309C42—611no—
310C42—611no—
311C42—611no—
312C42—611no—
313C42—4882no—
314C42—2441no—
315C42—2441no—
316C42—2441no—
317C16—22no—
318C16—1222no—
319C23—41no—
320C23D8×D6111yescharacteristic
321Q8—22no—
322Q8—41no—
323D4—22no—
324D4—42no—
325D4—42no—
326D4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—611no—
334C2×C4—611no—
335C2×C4—611no—
336C2×C4—611no—
337C2×C4—1222no—
338C2×C4—4882no—
339C2×C4—4882no—
340C2×C4—4881no—
341C2×C4—4881no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4—2442no—
346C2×C4C22⋊C4×D6111yescharacteristic
347C2×C4#1451973711yescharacteristic
348C2×C4D4×Dic6111yescharacteristic
349C2×C4SD16×D6111yescharacteristic
350C2×C4#1451979611yescharacteristic
351C2×C4#1451979811yescharacteristic
352C8—1221no—
353C8—1221no—
354C8C2×C4×Dic6111yescharacteristic
355C8#1451973311yescharacteristic
356C8#1451973711yescharacteristic
357C8D4×Dic6111yescharacteristic
358C22—21no—
359C22—21no—
360C22—42no—
361C22—81no—
362C22—81no—
363C22#102×D6111yescharacteristic
364C22#1683984211yescharacteristic
365C22D8×Dic6111yescharacteristic
366C4—42no—
367C4—82no—
368C4—612no—
369C4—1221no—
370C4—1221no—
371C4—1221no—
372C4—4882no—
373C4—4881no—
374C4—4881no—
375C4C22⋊C4×Dic6111yescharacteristic
376C4#1683975511yescharacteristic
377C4SD16×Dic6111yescharacteristic
378C4#1683985311yescharacteristic
379C2—21no—
380C2—21no—
381C2—82no—
382C2#102×Dic6111yescharacteristic
383C2#360×D6111yescharacteristic
384C2#2144755311yescharacteristic
subgroupG/H
centerC22#102×D61
commutatorC8×C61C2×C42
FrattiniC22×C8C22×D61
FittingC2×C61×#360C2
socleC22×C61C2×#102
prime radicalC2×#135×C61C22
cosocleC22×C8×C61C23
hypercenterC2×#360D61
hypocenterC61C4×#360
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#360

Complements

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

NH ≅ G/Ntype
3. M32×Dic61C2semidirect
4. #106×Dic61C2semidirect
5. #21443926C2semidirect
6. #21443925C2semidirect
10. Q16×Dic61C4semidirect
11. Q16×Dic61C4semidirect
14. D8×Dic61C4semidirect
15. D8×Dic61C4semidirect
21. #16839572C4semidirect
28. C61×#360C4semidirect
43. #14519717C2×C4semidirect
49. #14519709C2×C4semidirect
70. M32×C61C2×C4semidirect
73. #106×C61C2×C4semidirect
90. Q16×C61C42semidirect
94. D8×C61C42semidirect
186. Dic61#360direct (N × H)
217. #360Dic61direct (N × H)
221. C61C4×#360semidirect
248. M32C2×Dic61semidirect
257. #106C2×Dic61semidirect
283. Q16C4×Dic61semidirect
287. D8C4×Dic61semidirect

Maximal subgroups

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

subgroupindex
#135×Dic612
C2×#168395722
M32×Dic612
#106×Dic612
#214439262
#214439252
C2×C61×#3602
C4×#36061