Finite Group Databasebeta

← #29656277

Subgroups — #29656277

Order 15616.

1004 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C2×#16840492C211yesmaximal, characteristic, complemented
2#21456276C211yesmaximal, characteristic
3#21455942C212yesmaximal
4#21444960C214yesmaximal, complemented
5#21444898C214yesmaximal, complemented
6#21444265C212yesmaximal, complemented
7C61×#1558C211yesmaximal, characteristic, complemented
8C2×#14519808—22no—
9#16840492—24no—
10#16840492—22no—
11#16840492C2214yes—
12#16840312C2211yescharacteristic
13C2×#14519776—22no—
14C2×#14519775—22no—
15C2×C4×Dic122C2212yes—
16C2×#14519742—22no—
17#16840154—22no—
18#16840141C2212yes—
19#16840136C2212yes—
20#16840076—22no—
21#16840058C2214yes—
22#16839963C2214yes—
23#16839903—22no—
24#16839864—24no—
25C4×#11758194C2211yescharacteristic
26#16839492C2218yes—
27C2×C4⋊Q8×C61C2211yescharacteristic
28C61×#432C2212yes—
29C61×#333C2214yes—
30C2×#11758222—21no—
31C22×Dic122—22no—
32C22×Dic122D412yes—
33#14519808—48no—
34C2×#11758202—24no—
35C2×#11758190—24no—
36C2×#11758201—24no—
37C2×#11758194—21no—
38#14519777C2311yescharacteristic
39#14519776—28no—
40#14519776—28no—
41#14519775—28no—
42#14519775—28no—
43#14519774C2311yescharacteristic
44#14519765—21no—
45#14519751—22no—
46C4×Dic122—22no—
47C4×Dic122—22no—
48C4×Dic122D414yes—
49C4×Dic122C2314yes—
50#14519742—28no—
51#14519742—28no—
52C4×D244—22no—
53C2×C4⋊Dic61D412yes—
54C2×#11758196—24no—
55C2×#11758196—22no—
56C2×C4×Dic61—22no—
57C2×C4×Dic61—22no—
58#14519711—24no—
59#14519702—24no—
60#14519699—24no—
61#14519688—21no—
62C4×#92418C2312yes—
63C22×Q8×C61D412yes—
64C4⋊Q8×C61—22no—
65C4⋊Q8×C61D412yes—
66C4⋊Q8×C61C2312yes—
67C2×C4⋊C4×C61—22no—
68C2×C42×C61C2311yescharacteristic
69C4⋊C8×C61C2314yes—
70C22⋊C8×C61—22no—
71#97×C61—22no—
72C2×Q8×C61—22no—
73C2×Q8×C61—44no—
74C2×Q8×C61C2×D414yes—
75C22×C4×C61—22no—
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D412yes—
78C2×C22⋊D61—41no—
79C22×Dic61—22no—
80C22×Dic61—22no—
81#11758222—44no—
82C2×D244—21no—
83C2×D244—22no—
84C2×C4×D61—41no—
85C2×Dic122—22no—
86C2×Dic122—28no—
87C2×Dic122—21no—
88C2×Dic122—28no—
89C2×Dic122—84no—
90C2×Dic122C2×D414yes—
91C2×C8×C61—24no—
92C4⋊C4×C61—24no—
93C4⋊C4×C61—24no—
94C42×C61C2×D411yescharacteristic
95C42×C61C2×D412yes—
96C42×C61C2411yescharacteristic
97#11758202—48no—
98#11758201—48no—
99#11758198—42no—
100C4⋊Dic61—22no—
101C4⋊Dic61—22no—
102C4⋊Dic61C2×D414yes—
103#11758196—216no—
104#11758196—24no—
105#11758196—24no—
106C4×Dic61—28no—
107C4×Dic61—24no—
108C4×Dic61—24no—
109#11758194—24no—
110C2×#92418—22no—
111#11758190—48no—
112C23×C61#12011yescharacteristic
113C22×D61—41no—
114Q8×C61—44no—
115Q8×C61—82no—
116C2×C4×C61—22no—
117C2×C4×C61—24no—
118C2×C4×C61—24no—
119C2×C4×C61—44no—
120C2×C4×C61#12012yes—
121C2×C4×C61#12011yescharacteristic
122C2×C4×C61C22×D411yescharacteristic
123C2×C4×C61C22×D412yes—
124C22⋊D61—82no—
125C2×Dic61—28no—
126C2×Dic61—24no—
127C2×Dic61—24no—
128C2×Dic61—44no—
129C2×Dic61—41no—
130D244—42no—
131D244—44no—
132C4×D61—82no—
133Dic122—28no—
134Dic122—24no—
135Dic122—42no—
136Dic122—88no—
137Dic122#13718yescomplemented
138C8×C61—44no—
139#92418—24no—
140#18546—611nomaximal, Sylow
141C22×C61—22no—
142C22×C61#45312yes—
143C22×C61C2×#12011yescharacteristic
144C2×D61—82no—
145C2×D61—81no—
146C4×C61—22no—
147C4×C61—21no—
148C4×C61—21no—
149C4×C61—44no—
150C4×C61C2×#13714yes—
151Dic61—28no—
152Dic61—44no—
153Dic61—81no—
154C2×Q82—611no—
155#3152—611no—
156#3112—612no—
157#1633—614no—
158#1623—614no—
159#1558D6111yescharacteristic, complemented
160#1491—612no—
161C2×C61—22no—
162C2×C61#300412yes—
163C2×C61C2×#45311yescharacteristic
164D61—161no—
165C2×#137—1222no—
166Q82—614no—
167Q82—1224no—
168Q82—1222no—
169#532—611no—
170C2×C4⋊Q8—1222no—
171C2×C4⋊Q8—1222no—
172C2×C4⋊Q8C2×D6111yescharacteristic
173C2×C4×Q8—612no—
174C2×C4×Q8—1222no—
175#497—1222no—
176#494—612no—
177#492—612no—
178#480—1222no—
179#477—614no—
180#460—614no—
181#448—1222no—
182#440—1224no—
183#432C2×D6112yes—
184C4×M16—611no—
185#333C2×D6114yes—
186Q8⋊C8—618no—
187C61#1854611yescharacteristic, complemented, Sylow
188C2×#41—1221no—
189C22×Q8—612no—
190C22×Q8—1222no—
191C22×Q8D24412yes—
192#137—2448no—
193C2×Q16—1224no—
194C2×SD16—1224no—
195C2×SD16—1224no—
196C2×M16—1221no—
197C4⋊Q8—22no—
198C4⋊Q8—611no—
199C4⋊Q8—1228no—
200C4⋊Q8—1228no—
201C4⋊Q8—1228no—
202C4⋊Q8—1228no—
203C4⋊Q8D24412yes—
204C4⋊Q8C22×D6112yes—
205#127—611no—
206#124—1221no—
207#121—1222no—
208C4×Q8—614no—
209C4×Q8—614no—
210C4×Q8—1228no—
211C4×Q8—1228no—
212C4×Q8—1222no—
213C4×Q8—1222no—
214C4×D4—1222no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—612no—
217C2×C4⋊C4—1224no—
218C2×C4⋊C4—1222no—
219C2×C42—1222no—
220C2×C42—1222no—
221C2×C42C22×D6111yescharacteristic
222#107—1224no—
223C4⋊C8C22×D6114yes—
224#103—1224no—
225#102—1224no—
226C22⋊C8—22no—
227#97—22no—
228#97—1221no—
229C4×C8—612no—
230#41—2444no—
231C2×Q8—22no—
232C2×Q8—44no—
233C2×Q8—614no—
234C2×Q8—1228no—
235C2×Q8—1228no—
236C2×Q8—1222no—
237C2×Q8—1221no—
238C2×Q8—4884no—
239C2×Q8C2×D24414yes—
240C2×D4—1222no—
241C2×D4—1221no—
242C2×D4—2441no—
243C22×C4—22no—
244C22×C4—1222no—
245C22×C4—1222no—
246C22×C4—2441no—
247C22×C4C2×D24411yescharacteristic
248C22×C4D4×D6112yes—
249Q16—2448no—
250SD16—2448no—
251SD16—2448no—
252M16—1224no—
253C2×C8—24no—
254C2×C8—1222no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—614no—
258C4⋊C4—12216no—
259C4⋊C4—1224no—
260C4⋊C4—1222no—
261C4⋊C4—1224no—
262C4⋊C4—1222no—
263C22⋊C4—2442no—
264C42—1228no—
265C42—1224no—
266C42—1224no—
267C42C2×D24411yescharacteristic
268C42D4×D6112yes—
269C42C23×D6111yescharacteristic
270C23—2441no—
271C23#1451974711yescharacteristic
272Q8—44no—
273Q8—82no—
274Q8—618no—
275Q8—1228no—
276Q8—1224no—
277Q8—4888no—
278Q8—2442no—
279D4—4882no—
280D4—2442no—
281D4—2444no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—44no—
286C2×C4—1228no—
287C2×C4—1224no—
288C2×C4—1224no—
289C2×C4—4882no—
290C2×C4—2441no—
291C2×C4—2444no—
292C2×C4#1451974712yes—
293C2×C4#1451974711yescharacteristic
294C2×C4C2×D4×D6112yes—
295C2×C4C22×D24411yescharacteristic
296C8—44no—
297C8—1224no—
298C22—22no—
299C22—4881no—
300C22—4882no—
301C22#1683993212yes—
302C22C2×#1451974711yescharacteristic
303C4—21no—
304C4—21no—
305C4—22no—
306C4—44no—
307C4—1228no—
308C4—4881no—
309C4—2444no—
310C4#137×D6114yes—
311C2—22no—
312C2—9761no—
313C2#2145505512yes—
314C2C2×#1683993211yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1558C2
socleC22×C61C2×#120
prime radical#16840312C22
hypercenter#1558D61
hypocenterC61#18546
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
1. C2×#16840492C2semidirect
4. #21444960C2semidirect
5. #21444898C2semidirect
6. #21444265C2semidirect
7. C61×#1558C2semidirect
137. Dic122#137semidirect
159. #1558D61semidirect
187. C61#18546semidirect

Maximal subgroups

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

subgroupindex
C2×#168404922
#214562762
#214559422
#214449602
#214448982
#214442652
C61×#15582
#1854661