Finite Group Databasebeta

← #30606627

Subgroups — #30606627

Order 15616.

422 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 104 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⋊#362C211yesmaximal, characteristic
2#21462129C211yesmaximal, characteristic
3#21462036C211yesmaximal, characteristic
4#21461625C211yesmaximal, characteristic
5#21460892C211yesmaximal, characteristic
6#21460883C211yesmaximal, characteristic
7#441×D61C211yesmaximal, characteristic
8D61⋊M32C2211yescharacteristic
9D61⋊M32C2211yescharacteristic
10D61⋊#108C2211yescharacteristic
11#16840772C2211yescharacteristic
12#16840725—21no—
13#16840681C414yescomplemented
14#16840675C414yescomplemented
15#16840655—21no—
16#16840654—21no—
17#16840633—21no—
18C2×Q16×D61C2211yescharacteristic
19C4×Q8×D61—21no—
20#16839870C411yescharacteristic, complemented
21#16839869C411yescharacteristic, complemented
22#16839868C411yescharacteristic, complemented
23#16839867—21no—
24#16839866—21no—
25#106×D61C2211yescharacteristic
26#103×D61—21no—
27#97×D61C2211yescharacteristic
28C61×#441C411yescharacteristic, complemented
29D61⋊M16—21no—
30#14519856—21no—
31#14519852—21no—
32#14519852C2×C412yes—
33#14519851—21no—
34#14519851—21no—
35D61⋊C16—21no—
36D61⋊C16C2×C412yes—
37#14519848—21no—
38#14519847C2×C412yes—
39#14519846C2×C412yes—
40#14519845—21no—
41#14519831—41no—
42C8×#6923—41no—
43C2×Q8×D61—21no—
44C2×Dic244C2×C411yescharacteristic
45C2×#11758202—21no—
46Q16×D61—21no—
47Q16×D61—21no—
48Q16×D61C2×C412yes—
49Q16×D61C2×C412yes—
50C2×C8×D61D411yescharacteristic
51C2×C8×D61C2311yescharacteristic
52Q8×Dic61—21no—
53C4×Dic122—21no—
54C4×Dic122—41no—
55#14519742—21no—
56#14519742—41no—
57C4⋊C4×D61—21no—
58C4⋊C4×D61—41no—
59C42×D61—41no—
60C42×D61D411yescharacteristic
61#14519709C2×C411yescharacteristic
62#14519708—21no—
63#14519702—21no—
64#14519701—21no—
65#14519700—21no—
66#14519688D411yescharacteristic
67#14519687C2×C411yescharacteristic
68#14519687D411yescharacteristic
69C2×Q16×C61C2×C411yescharacteristic
70C4×Q8×C61—21no—
71#106×C61C2×C411yescharacteristic
72#103×C61—21no—
73#97×C61C2×C411yescharacteristic
74C2×Q8×C61—21no—
75Q8×D61—22no—
76Q8×D61—22no—
77Q8×D61—41no—
78Q8×D61—41no—
79C2×C4×D61—21no—
80C2×C4×D61—41no—
81C2×C4×D61—41no—
82C2×C4×D61C2×D411yescharacteristic
83C2×Dic122—21no—
84C2×Dic122—41no—
85C22⋊#6923—41no—
86#11758217—41no—
87C4⋊#6923—22no—
88C4×#6923—41no—
89#11758213—22no—
90D61⋊C8—41no—
91Q16×C61—21no—
92Q16×C61C4212yes—
93C2×C8×C61C22⋊C411yescharacteristic
94C2×C8×C61C22×C411yescharacteristic
95C4⋊C4×C61—21no—
96C4⋊C4×C61—41no—
97C42×C61—41no—
98C42×C61C22⋊C411yescharacteristic
99#11758202—22no—
100#11758202—22no—
101C4⋊Dic61—21no—
102C4⋊Dic61—41no—
103#11758196—41no—
104#11758196—41no—
105#11758196—41no—
106C4×Dic61—21no—
107C4×Dic61—41no—
108C4×Dic61—41no—
109C4×Dic61—41no—
110C4×Dic61C22⋊C411yescharacteristic
111C2×#92418C22⋊C411yescharacteristic
112C2×#92418C2×D411yescharacteristic
113Dic244—21no—
114Dic244C4212yes—
115C8×D61—21no—
116C8×D61—21no—
117C8×D61C22×C411yescharacteristic
118C8×D61C22×C411yescharacteristic
119C8×D61#4111yescharacteristic
120C8×D61#4111yescharacteristic
121#11758187—22no—
122#11758187—22no—
123C22×D61AGL(1,ℤ/8)11yescharacteristic
124C2×#6923—42no—
125Q8×C61—22no—
126Q8×C61—41no—
127C2×C4×C61—21no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61C2×C22⋊C411yescharacteristic
131C2×Dic61—21no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61AGL(1,ℤ/8)11yescharacteristic
135C4×D61—42no—
136C4×D61—42no—
137C4×D61—42no—
138C4×D61—82no—
139C4×D61C4×D411yescharacteristic
140C4×D61C4×D411yescharacteristic
141C4×D61AGL(1,ℤ/8)11yescharacteristic
142C4×D61#13711yescharacteristic
143Dic122—22no—
144Dic122—41no—
145Dic122—42no—
146Dic122—42no—
147#92420—42no—
148C8×C61—21no—
149C8×C61C2×C4211yescharacteristic
150C8×C61#11711yescharacteristic
151#92418—21no—
152#92418C4×D411yescharacteristic
153#92418C4×D411yescharacteristic
154#12648—611nomaximal, Sylow
155C22×C61#41811yescharacteristic
156C2×D61—22no—
157C2×D61#44011yescharacteristic
158C2×D61#44211yescharacteristic
159#6923—82no—
160C4×C61—41no—
161C4×C61—42no—
162C4×C61—81no—
163C4×C61C4×C22⋊C411yescharacteristic
164C4×C61#41711yescharacteristic
165Dic61—41no—
166Dic61—42no—
167Dic61—81no—
168Dic61#44011yescharacteristic
169Dic61#44211yescharacteristic
170C2×#441—611no—
171C2×#362—611no—
172#2141—611no—
173C2×#347—611no—
174#1773—611no—
175#1386—611no—
176#1383—611no—
177C2×C61—21no—
178C2×C61#176211yescharacteristic
179D61—21no—
180D61#218012yes—
181C22×Q16—611no—
182C2×C4×Q8—1221no—
183C2×M32—611no—
184C2×M32—611no—
185#441—611no—
186#441—611no—
187#441—611no—
188#441—1221no—
189#441—1221no—
190#441#692311yescharacteristic, complemented
191C2×#108—611no—
192#427—611no—
193C2×#106—611no—
194C2×#103—1221no—
195#405—1221no—
196C2×#97—611no—
197#362—614no—
198#358—614no—
199#347—1221no—
200#347—1221no—
201#339—1221no—
202C61#1264811yescharacteristic, complemented, Sylow
203C22×Q8—1221no—
204C2×Q16—612no—
205C2×Q16—612no—
206C2×Q16—611no—
207C2×Q16—1221no—
208C2×Q16—1221no—
209C2×Q16—1221no—
210C2×Q16C2×#692311yescharacteristic
211C2×M16—1221no—
212C22×C8—611no—
213C22×C8—611no—
214C4×Q8—21no—
215C4×Q8—1221no—
216C4×Q8—1221no—
217C4×Q8—1221no—
218C4×Q8—2441no—
219C4×Q8—2441no—
220#117—1221no—
221C2×C4⋊C4—1221no—
222C2×C4⋊C4—2441no—
223C2×C42—611no—
224C2×C42—2441no—
225M32—612no—
226M32—1221no—
227M32—1221no—
228M32—1221no—
229C2×C16—612no—
230C2×C16—1221no—
231#108—612no—
232#108—1221no—
233#107—612no—
234#106—611no—
235#106—1221no—
236#106—1221no—
237#106C2×#692311yescharacteristic
238#103—21no—
239#103—1221no—
240#103—1221no—
241#103—1221no—
242#97—611no—
243#97—611no—
244#97—611no—
245#97—2441no—
246#97C2×#692311yescharacteristic
247C4×C8—2441no—
248C2×Q8—21no—
249C2×Q8—1222no—
250C2×Q8—1222no—
251C2×Q8—1221no—
252C2×Q8—2441no—
253C2×Q8—2441no—
254C2×Q8—2441no—
255C22×C4—611no—
256C22×C4—1221no—
257C22×C4—2441no—
258C22×C4—2441no—
259Q16—21no—
260Q16—612no—
261Q16—1222no—
262Q16—1222no—
263Q16—1221no—
264Q16C4×#692312yes—
265M16—1222no—
266M16—2441no—
267C2×C8—611no—
268C2×C8—611no—
269C2×C8—611no—
270C2×C8—611no—
271C2×C8—611no—
272C2×C8—611no—
273C2×C8—1221no—
274C2×C8—1221no—
275C2×C8—2441no—
276C2×C8C22⋊#692311yescharacteristic
277C2×C8C22×#692311yescharacteristic
278C4⋊C4—21no—
279C4⋊C4—41no—
280C4⋊C4—1222no—
281C4⋊C4—1221no—
282C4⋊C4—2441no—
283C4⋊C4—2441no—
284C4⋊C4—2441no—
285C4⋊C4—2441no—
286C22⋊C4—2441no—
287C42—41no—
288C42—611no—
289C42—1221no—
290C42—2441no—
291C42—2441no—
292C42—2441no—
293C42—2441no—
294C42C22⋊#692311yescharacteristic
295C16—1222no—
296C16—1222no—
297C23—611no—
298Q8—22no—
299Q8—41no—
300Q8—1222no—
301Q8—2442no—
302Q8—2442no—
303Q8—2441no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—611no—
308C2×C4—611no—
309C2×C4—611no—
310C2×C4—611no—
311C2×C4—611no—
312C2×C4—1221no—
313C2×C4—4882no—
314C2×C4—2442no—
315C2×C4—2442no—
316C2×C4—2442no—
317C2×C4—2442no—
318C2×C4—2441no—
319C2×C4—2441no—
320C2×C4C2×C22⋊#692311yescharacteristic
321C8—21no—
322C8—611no—
323C8—611no—
324C8—1221no—
325C8—2442no—
326C8C2×C4×#692311yescharacteristic
327C8#1451985611yescharacteristic
328C22—611no—
329C22—611no—
330C22—1222no—
331C22#1684075411yescharacteristic
332C4—41no—
333C4—42no—
334C4—81no—
335C4—611no—
336C4—611no—
337C4—4881no—
338C4—4882no—
339C4—2441no—
340C4—2442no—
341C4C4×C22⋊#692311yescharacteristic
342C4#1684075311yescharacteristic
343C2—21no—
344C2—612no—
345C2—1221no—
346C2#2146158911yescharacteristic
subgroupG/H
centerC2#21461589
commutatorC8×C61C2×C42
FrattiniC2×C8C22×#6923
FittingC61×#441C4
socleC2×C61#1762
prime radicalC22×D61AGL(1,ℤ/8)
cosocleC2×C8×D61C23
hypercenter#441#6923
hypocenterC61#12648
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #441

Complements

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

NH ≅ G/Ntype
13. #16840681C4semidirect
14. #16840675C4semidirect
20. #16839870C4semidirect
21. #16839869C4semidirect
22. #16839868C4semidirect
28. C61×#441C4semidirect
190. #441#6923semidirect
202. C61#12648semidirect

Maximal subgroups

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

subgroupindex
D61⋊#3622
#214621292
#214620362
#214616252
#214608922
#214608832
#441×D612
#1264861