Finite Group Databasebeta

← #29576662

Subgroups — #29576662

Order 15616.

410 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 109 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
1#21456972C211yesmaximal, characteristic
2#21454759C211yesmaximal, characteristic
3#21446792C211yesmaximal, characteristic
4#21444499C211yesmaximal, characteristic
5#21444384C211yesmaximal, characteristic
6#21443678C211yesmaximal, characteristic
7C61×#1275C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9C2×#14519777C2211yescharacteristic
10#16840167C412yescomplemented
11#16840113C412yescomplemented
12#16840098—22no—
13#16840058—22no—
14#16839890—22no—
15#16839890—22no—
16#16839788—21no—
17#16839786C2211yescharacteristic
18#16839717—21no—
19#16839628C2211yescharacteristic
20#16839525—21no—
21#16839525—21no—
22#16839510—22no—
23#16839500—22no—
24C61×#431C2211yescharacteristic
25C61×#431C2211yescharacteristic
26C2×#97×C61C2211yescharacteristic
27C22×Dic122—21no—
28C2×#11758194—21no—
29#14519777—21no—
30#14519777—21no—
31#14519777—21no—
32#14519777C2×C412yes—
33#14519777C2×C411yescharacteristic
34#14519777C2×C411yescharacteristic
35#14519769—21no—
36#14519756—21no—
37#14519756D412yes—
38C4×Dic122—21no—
39C4×Dic122D412yes—
40#14519736—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—21no—
44#14519711—24no—
45C4⋊#92418—21no—
46C4⋊#92418D412yes—
47#14519706—24no—
48#14519702—24no—
49#14519687—22no—
50C8×Dic61—22no—
51#14519684—21no—
52#14519684—21no—
53C22×C8×C61D411yescharacteristic
54C22×C8×C61D411yescharacteristic
55C2×C42×C61C2311yescharacteristic
56C4⋊C8×C61—21no—
57C4⋊C8×C61—21no—
58C22⋊C8×C61—21no—
59C22⋊C8×C61—21no—
60#97×C61—21no—
61#97×C61—21no—
62C4×C8×C61—21no—
63C4×C8×C61C2×C412yes—
64C22×C4×C61C2×D411yescharacteristic
65C22×C4×C61C2×D411yescharacteristic
66C22×C4×C61#4111yescharacteristic
67C22×Dic61—41no—
68C22×Dic61—41no—
69C2×Dic122—21no—
70C2×Dic122—22no—
71C2×Dic122—41no—
72C2×Dic122—82no—
73C2×Dic122C22⋊C412yes—
74C2×C8×C61—21no—
75C2×C8×C61—21no—
76C2×C8×C61—22no—
77C2×C8×C61—21no—
78C2×C8×C61—21no—
79C2×C8×C61—21no—
80C2×C8×C61—41no—
81C2×C8×C61Q1612yes—
82C42×C61C22×C411yescharacteristic
83C42×C61C2×D411yescharacteristic
84C42×C61C2×D411yescharacteristic
85C42×C61#4111yescharacteristic
86C22⋊Dic61—22no—
87C4⋊Dic61—21no—
88C4⋊Dic61—21no—
89C4⋊Dic61—21no—
90C4⋊Dic61—22no—
91C4⋊Dic61—22no—
92C4⋊Dic61—41no—
93C4⋊Dic61—41no—
94C4⋊Dic61—41no—
95C4⋊Dic61—41no—
96C4⋊Dic61C22⋊C412yes—
97#11758196—22no—
98#11758196—22no—
99C4×Dic61—22no—
100#11758194—41no—
101#11758194—41no—
102C2×#92418—22no—
103C23×C61#12111yescharacteristic
104C2×C4×C61—41no—
105C2×C4×C61—41no—
106C2×C4×C61C2×C22⋊C411yescharacteristic
107C2×C4×C61C4×D411yescharacteristic
108C2×C4×C61C4×D411yescharacteristic
109C2×C4×C61#12011yescharacteristic
110C2×C4×C61#12111yescharacteristic
111C2×C4×C61#12311yescharacteristic
112C2×C4×C61C2×Q1611yescharacteristic
113C2×C4×C61#13511yescharacteristic
114C2×C4×C61#13711yescharacteristic
115C2×C4×C61#13711yescharacteristic
116C2×Dic61—22no—
117C2×Dic61—22no—
118C2×Dic61—42no—
119C2×Dic61—82no—
120C2×Dic61—81no—
121C2×Dic61—81no—
122C2×Dic61—81no—
123Dic122—42no—
124Dic122—81no—
125Dic122—82no—
126Dic122—82no—
127C8×C61—22no—
128C8×C61—42no—
129C8×C61—41no—
130C8×C61—81no—
131#92418—42no—
132#9288—611nomaximal, Sylow
133C22×C61—21no—
134C22×C61—21no—
135C22×C61#38611yescharacteristic
136C22×C61#46211yescharacteristic
137C22×C61#46411yescharacteristic
138C4×C61—21no—
139C4×C61—21no—
140C4×C61—21no—
141C4×C61—21no—
142C4×C61C4×Q1611yescharacteristic
143C4×C61#44111yescharacteristic
144C4×C61#45111yescharacteristic
145C4×C61#45211yescharacteristic
146Dic61—42no—
147Dic61—42no—
148Dic61—82no—
149Dic61—161no—
150#3230—611no—
151#2961—611no—
152#1993—611no—
153#1548—611no—
154#1518—611no—
155#1286—611no—
156#1275D6111yescharacteristic
157C2×C61—21no—
158C2×C61—21no—
159C2×C61#187811yescharacteristic
160C2×C61#190311yescharacteristic
161C2×C61#190911yescharacteristic
162#533—611no—
163C2×C4⋊Q8—611no—
164#500—612no—
165#487—612no—
166#484—1222no—
167#477—1222no—
168#446—1222no—
169#446—1222no—
170#431C2×D6111yescharacteristic
171#431C2×D6111yescharacteristic
172#424—1221no—
173#423—611no—
174#405—1221no—
175C2×#97C2×D6111yescharacteristic
176#382—611no—
177#340—1221no—
178#340—1221no—
179#333—1222no—
180#328—1222no—
181C61#928811yescharacteristic, complemented, Sylow
182C22×Q8—1221no—
183C2×M16—1221no—
184C22×C8D24411yescharacteristic
185C22×C8D24411yescharacteristic
186C4⋊Q8—611no—
187C4⋊Q8—611no—
188C4⋊Q8—612no—
189C4⋊Q8—1221no—
190C4⋊Q8—1221no—
191C4⋊Q8—1221no—
192#125—1221no—
193C22⋊Q8—612no—
194C22⋊Q8—1221no—
195C4×Q8—612no—
196C4×Q8—1221no—
197#117—1221no—
198C2×C4⋊C4—1221no—
199C2×C4⋊C4—1221no—
200C2×C4⋊C4—1221no—
201C2×C42C22×D6111yescharacteristic
202#107—1224no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—612no—
206C4⋊C8—1224no—
207C4⋊C8—1221no—
208#103—1224no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211#97—21no—
212#97—21no—
213#97—1222no—
214C4×C8—21no—
215C4×C8—1222no—
216C4×C8C4×D6112yes—
217#95—1221no—
218#95—1221no—
219C2×Q8—612no—
220C2×Q8—1221no—
221C2×Q8—1222no—
222C2×Q8—4882no—
223C2×Q8—2441no—
224C22×C4—2441no—
225C22×C4—2441no—
226C22×C4C2×D24411yescharacteristic
227C22×C4D4×D6111yescharacteristic
228C22×C4#1175822611yescharacteristic
229M16—2441no—
230M16—2441no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—22no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—41no—
238C2×C8—1222no—
239C2×C8Dic24412yes—
240C4⋊C4—612no—
241C4⋊C4—1221no—
242C4⋊C4—1221no—
243C4⋊C4—1221no—
244C4⋊C4—1222no—
245C4⋊C4—1222no—
246C4⋊C4—1222no—
247C4⋊C4—1222no—
248C4⋊C4—2441no—
249C4⋊C4—2441no—
250C4⋊C4—2441no—
251C4⋊C4—2441no—
252C22⋊C4—1222no—
253C42—1222no—
254C42C2×C4×D6111yescharacteristic
255C42#1175822211yescharacteristic
256C42D4×D6111yescharacteristic
257C42D4×D6111yescharacteristic
258C23#1451975211yescharacteristic
259Q8—4882no—
260Q8—4882no—
261Q8—4881no—
262Q8—2442no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—1222no—
266C2×C4—1222no—
267C2×C4—4881no—
268C2×C4—4881no—
269C2×C4—4881no—
270C2×C4—4882no—
271C2×C4—2442no—
272C2×C4C22⋊C4×D6111yescharacteristic
273C2×C4#1451973911yescharacteristic
274C2×C4C4×D24411yescharacteristic
275C2×C4#1451974711yescharacteristic
276C2×C4#1451975011yescharacteristic
277C2×C4#1451976211yescharacteristic
278C2×C4C2×Dic24411yescharacteristic
279C2×C4#1451979911yescharacteristic
280C2×C4#1451981011yescharacteristic
281C2×C4#1451981011yescharacteristic
282C8—22no—
283C8—42no—
284C8—41no—
285C8—81no—
286C8—2442no—
287C22—21no—
288C22—21no—
289C22#1683964011yescharacteristic
290C22#1683997311yescharacteristic
291C22#1683998511yescharacteristic
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—9761no—
297C4—4882no—
298C4—2442no—
299C4—2442no—
300C4C4×Dic24411yescharacteristic
301C4#1683987011yescharacteristic
302C4#1683992011yescharacteristic
303C4#1683992611yescharacteristic
304C2—21no—
305C2—21no—
306C2#2144615911yescharacteristic
307C2#2144630011yescharacteristic
308C2#2144634011yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1275C2
socleC22×C61#386
prime radicalC23×C61#121
cosocleC2×C42×C61C23
hypercenter#1275D61
hypocenterC61#9288
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1275

Complements

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

NH ≅ G/Ntype
10. #16840167C4semidirect
11. #16840113C4semidirect
181. C61#9288semidirect

Maximal subgroups

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

subgroupindex
#214569722
#214547592
#214467922
#214444992
#214443842
#214436782
C61×#12752
#928861