Finite Group Databasebeta

← #29576694

Subgroups — #29576694

Order 15616.

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

normal subgroups only

This group has 426 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#21456963C211yesmaximal, characteristic
2#21454759C211yesmaximal, characteristic
3#21444491C211yesmaximal, characteristic
4#21444382C211yesmaximal, characteristic
5#21443677C211yesmaximal, characteristic
6#21443650C211yesmaximal, characteristic
7C61×#1960C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840158C412yescomplemented
10#16840114C412yescomplemented
11#16840092—22no—
12#16840044—22no—
13#16839890—22no—
14#16839890—22no—
15#16839815C2211yescharacteristic
16#16839815C2211yescharacteristic
17#16839788—21no—
18#16839786C2211yescharacteristic
19#16839717—21no—
20#16839508—22no—
21#16839498—22no—
22C2×C4⋊Q8×C61C2211yescharacteristic
23C2×#103×C61—21no—
24C2×#103×C61—21no—
25C2×#97×C61C2211yescharacteristic
26C61×#382C2211yescharacteristic
27C2×#11758194—21no—
28#14519777C2×C411yescharacteristic
29#14519777C2×C411yescharacteristic
30#14519769—21no—
31#14519756—21no—
32#14519756D412yes—
33C4×Dic122—21no—
34C4×Dic122D412yes—
35#14519736—21no—
36#14519708—24no—
37C4⋊#92418—21no—
38C4⋊#92418—21no—
39C4⋊#92418—21no—
40C4⋊#92418D412yes—
41#14519706—24no—
42#14519700—24no—
43C22⋊#92418—21no—
44C22⋊#92418—21no—
45#14519687—22no—
46C4×#92418—21no—
47C4×#92418C2×C412yes—
48C8×Dic61—22no—
49C22×Q8×C61—21no—
50C22×C8×C61D411yescharacteristic
51C22×C8×C61D411yescharacteristic
52C4⋊Q8×C61—21no—
53C4⋊Q8×C61—21no—
54C4⋊Q8×C61—21no—
55C4⋊Q8×C61C2×C412yes—
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61—21no—
58C2×C4⋊C4×C61—21no—
59C2×C42×C61C2311yescharacteristic
60#103×C61—42no—
61#103×C61—42no—
62#97×C61—21no—
63#97×C61—21no—
64#95×C61—21no—
65#95×C61—21no—
66C2×Q8×C61—22no—
67C2×Q8×C61—41no—
68C2×Q8×C61—41no—
69C2×Q8×C61—41no—
70C2×Q8×C61—41no—
71C2×Q8×C61—41no—
72C22×C4×C61—41no—
73C22×C4×C61—41no—
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61#4111yescharacteristic
77C2×Dic122—21no—
78C2×Dic122C22⋊C412yes—
79C2×C8×C61—21no—
80C2×C8×C61—21no—
81C2×C8×C61—21no—
82C2×C8×C61—21no—
83C2×C8×C61—21no—
84C2×C8×C61SD1612yes—
85C4⋊C4×C61—22no—
86C4⋊C4×C61—22no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—41no—
91C42×C61C22×C411yescharacteristic
92C42×C61C2×D411yescharacteristic
93C42×C61C2×D411yescharacteristic
94C42×C61#4111yescharacteristic
95C22⋊Dic61—22no—
96C4⋊Dic61—21no—
97C4⋊Dic61—21no—
98C4⋊Dic61—21no—
99C4⋊Dic61C22⋊C412yes—
100#11758196—22no—
101#11758196—22no—
102C4×Dic61—22no—
103#11758194—41no—
104#11758194—41no—
105C2×#92418—22no—
106C2×#92418—22no—
107C2×#92418—41no—
108C23×C61#12111yescharacteristic
109Q8×C61—42no—
110Q8×C61—42no—
111Q8×C61—81no—
112Q8×C61—81no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—42no—
116C2×C4×C61—81no—
117C2×C4×C61—81no—
118C2×C4×C61—81no—
119C2×C4×C61—81no—
120C2×C4×C61—81no—
121C2×C4×C61C2×C22⋊C411yescharacteristic
122C2×C4×C61C4×D411yescharacteristic
123C2×C4×C61C4×D411yescharacteristic
124C2×C4×C61#12011yescharacteristic
125C2×C4×C61#12111yescharacteristic
126C2×C4×C61#12311yescharacteristic
127C2×C4×C61C2×SD1611yescharacteristic
128C2×C4×C61#13511yescharacteristic
129C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C61#13711yescharacteristic
131C2×Dic61—22no—
132C2×Dic61—22no—
133Dic122—42no—
134Dic122—81no—
135C8×C61—22no—
136C8×C61—41no—
137#92418—42no—
138#92418—42no—
139#92418—81no—
140#9292—611nomaximal, Sylow
141C22×C61—21no—
142C22×C61—21no—
143C22×C61#38611yescharacteristic
144C22×C61#46011yescharacteristic
145C22×C61#46411yescharacteristic
146C4×C61—21no—
147C4×C61—21no—
148C4×C61—21no—
149C4×C61—21no—
150C4×C61—82no—
151C4×C61—161no—
152C4×C61C4×SD1611yescharacteristic
153C4×C61#44011yescharacteristic
154C4×C61#44811yescharacteristic
155C4×C61#44911yescharacteristic
156Dic61—42no—
157Dic61—42no—
158#3229—611no—
159#2961—611no—
160#1960D6111yescharacteristic
161#1546—611no—
162#1518—611no—
163#1286—611no—
164#1275—611no—
165C2×C61—21no—
166C2×C61—21no—
167C2×C61#188011yescharacteristic
168C2×C61#190211yescharacteristic
169C2×C61#190911yescharacteristic
170#533—611no—
171C2×C4⋊Q8C2×D6111yescharacteristic
172#498—612no—
173#487—612no—
174#483—1222no—
175#475—1222no—
176#446—1222no—
177#446—1222no—
178#431—611no—
179#431—611no—
180#424—1221no—
181#423—611no—
182C2×#103—21no—
183C2×#103—21no—
184#405—1221no—
185C2×#97C2×D6111yescharacteristic
186#382C2×D6111yescharacteristic
187#332—1222no—
188#328—1222no—
189C61#929211yescharacteristic, complemented, Sylow
190C22×Q8—21no—
191C2×M16—1221no—
192C22×C8C22⋊D6111yescharacteristic
193C22×C8C22⋊D6111yescharacteristic
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—611no—
198C4⋊Q8—611no—
199C4⋊Q8C4×D6112yes—
200#125—1221no—
201C22⋊Q8—612no—
202C22⋊Q8—1221no—
203C4×Q8—612no—
204C4×Q8—1221no—
205#117—1221no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C42C22×D6111yescharacteristic
210#106—1224no—
211C4⋊C8—612no—
212C4⋊C8—1224no—
213C4⋊C8—1221no—
214C4⋊C8—1221no—
215C4⋊C8—1221no—
216#103—42no—
217#103—42no—
218#103—1224no—
219C22⋊C8—1221no—
220C22⋊C8—1221no—
221#97—21no—
222#97—21no—
223#97—1222no—
224C4×C8—612no—
225C4×C8—1222no—
226C4×C8—1221no—
227#95—21no—
228#95—21no—
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—612no—
236C2×Q8—1221no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4D4×D6111yescharacteristic
240C22×C4#1175822411yescharacteristic
241C22×C4C2×C22⋊D6111yescharacteristic
242M16—2441no—
243M16—2441no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—1222no—
250C2×C8—1222no—
251C2×C8—2441no—
252C2×C8#1175820012yes—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—612no—
260C4⋊C4—1221no—
261C4⋊C4—1221no—
262C4⋊C4—1221no—
263C4⋊C4—1222no—
264C4⋊C4—1222no—
265C22⋊C4—1222no—
266C42—1222no—
267C42C2×C4×D6111yescharacteristic
268C42#1175822211yescharacteristic
269C42D4×D6111yescharacteristic
270C42D4×D6111yescharacteristic
271C23#1451974811yescharacteristic
272Q8—42no—
273Q8—42no—
274Q8—81no—
275Q8—81no—
276Q8—4881no—
277Q8—2442no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—1222no—
287C2×C4—1222no—
288C2×C4C22⋊C4×D6111yescharacteristic
289C2×C4#1451973711yescharacteristic
290C2×C4C4×C22⋊D6111yescharacteristic
291C2×C4#1451974511yescharacteristic
292C2×C4#1451975011yescharacteristic
293C2×C4#1451975911yescharacteristic
294C2×C4C2×#1175820011yescharacteristic
295C2×C4#1451979711yescharacteristic
296C2×C4#1451980311yescharacteristic
297C2×C4#1451980711yescharacteristic
298C8—22no—
299C8—41no—
300C8—4881no—
301C8—2442no—
302C8—2442no—
303C22—21no—
304C22—21no—
305C22#1683963811yescharacteristic
306C22#1683995911yescharacteristic
307C22#1683998111yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—82no—
313C4—161no—
314C4—2442no—
315C4—2442no—
316C4C4×#1175820011yescharacteristic
317C4#1683986211yescharacteristic
318C4#1683990011yescharacteristic
319C4#1683990511yescharacteristic
320C2—21no—
321C2—21no—
322C2#2144617711yescharacteristic
323C2#2144628911yescharacteristic
324C2#2144633811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1960C2
socleC22×C61#386
prime radicalC23×C61#121
cosocleC2×C42×C61C23
hypercenter#1960D61
hypocenterC61#9292
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
9. #16840158C4semidirect
10. #16840114C4semidirect
189. C61#9292semidirect

Maximal subgroups

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

subgroupindex
#214569632
#214547592
#214444912
#214443822
#214436772
#214436502
C61×#19602
#929261