Finite Group Databasebeta

← #29583789

Subgroups — #29583789

Order 15616.

366 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 93 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#21455624C211yesmaximal, characteristic
2#21444589C211yesmaximal, characteristic
3#21444477C211yesmaximal, characteristic
4#21444382C211yesmaximal, characteristic
5#21444143C211yesmaximal, characteristic
6#21443677C211yesmaximal, characteristic
7C61×#1805C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840069—22no—
10#16840057—21no—
11#16840047—21no—
12#16839815C2211yescharacteristic
13#16839815C2211yescharacteristic
14#16839802—21no—
15#16839786C2211yescharacteristic
16#16839771—21no—
17#16839512—22no—
18#16839510—21no—
19#16839507—21no—
20#16839498—22no—
21#16839488—22no—
22C2×C4×Q8×C61C2211yescharacteristic
23C2×C4⋊C8×C61C2211yescharacteristic
24C2×#103×C61—21no—
25C61×#382C2211yescharacteristic
26C61×#340—21no—
27C2×#11758194—21no—
28#14519777D411yescharacteristic
29#14519777D411yescharacteristic
30#14519769—21no—
31#14519756—21no—
32#14519756—21no—
33C4×Dic122—21no—
34C4×Dic122—21no—
35#14519736—21no—
36#14519710—22no—
37#14519708—22no—
38C4⋊#92418—21no—
39C4⋊#92418—21no—
40C4⋊#92418—21no—
41C4⋊#92418D412yes—
42#14519701—22no—
43#14519701—22no—
44C22⋊#92418—21no—
45C22⋊#92418D412yes—
46C4×#92418—21no—
47C4×#92418—21no—
48C22×Q8×C61D411yescharacteristic
49C22×C8×C61—21no—
50C4×Q8×C61—21no—
51C4×Q8×C61—21no—
52C4×Q8×C61—21no—
53C4×Q8×C61—21no—
54C2×C4⋊C4×C61—21no—
55C2×C4⋊C4×C61—21no—
56C2×C4⋊C4×C61D411yescharacteristic
57C2×C42×C61—21no—
58C2×C42×C61C2311yescharacteristic
59C4⋊C8×C61—21no—
60C4⋊C8×C61—21no—
61#103×C61—42no—
62#95×C61—21no—
63#95×C61—21no—
64C2×Q8×C61—21no—
65C2×Q8×C61—21no—
66C2×Q8×C61—21no—
67C2×Q8×C61—21no—
68C2×Q8×C61—21no—
69C2×Q8×C61SD1612yes—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—41no—
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61#4111yescharacteristic
76C2×Dic122—21no—
77C2×Dic122—21no—
78C2×C8×C61—41no—
79C2×C8×C61—41no—
80C2×C8×C61—41no—
81C4⋊C4×C61—21no—
82C4⋊C4×C61—21no—
83C4⋊C4×C61—41no—
84C4⋊C4×C61—41no—
85C4⋊C4×C61—41no—
86C4⋊C4×C61—41no—
87C42×C61—41no—
88C42×C61—41no—
89C42×C61C2×D411yescharacteristic
90C42×C61C2×D411yescharacteristic
91C42×C61#4111yescharacteristic
92C42×C61#4111yescharacteristic
93C22⋊Dic61—41no—
94C4⋊Dic61—21no—
95C4⋊Dic61—21no—
96C4⋊Dic61—21no—
97C4⋊Dic61—21no—
98#11758196—41no—
99#11758196—41no—
100C4×Dic61—41no—
101#11758194—41no—
102#11758194—41no—
103C2×#92418—22no—
104C2×#92418—22no—
105C2×#92418—41no—
106C23×C61#12111yescharacteristic
107Q8×C61—22no—
108Q8×C61—22no—
109Q8×C61—41no—
110Q8×C61—41no—
111C2×C4×C61—22no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—81no—
118C2×C4×C61—81no—
119C2×C4×C61—81no—
120C2×C4×C61—81no—
121C2×C4×C61#12011yescharacteristic
122C2×C4×C61#12111yescharacteristic
123C2×C4×C61#12111yescharacteristic
124C2×C4×C61#12311yescharacteristic
125C2×C4×C61#12311yescharacteristic
126C2×C4×C61#12411yescharacteristic
127C2×C4×C61C2×SD1611yescharacteristic
128C2×C4×C61#13511yescharacteristic
129C2×C4×C61#13711yescharacteristic
130C2×C4×C61#13711yescharacteristic
131C2×Dic61—41no—
132C2×Dic61—41no—
133Dic122—81no—
134Dic122—81no—
135C8×C61—81no—
136#92418—42no—
137#92418—42no—
138#92418—81no—
139#10470—611nomaximal, Sylow
140C22×C61—21no—
141C22×C61—21no—
142C22×C61#39411yescharacteristic
143C22×C61#46511yescharacteristic
144C22×C61#47011yescharacteristic
145C4×C61—21no—
146C4×C61—21no—
147C4×C61—21no—
148C4×C61—21no—
149C4×C61—42no—
150C4×C61—81no—
151C4×C61—161no—
152C4×C61#44811yescharacteristic
153C4×C61#45211yescharacteristic
154C4×C61#48311yescharacteristic
155C4×C61#48511yescharacteristic
156Dic61—81no—
157Dic61—81no—
158#3078—611no—
159#1805D6111yescharacteristic
160#1570—611no—
161#1542—611no—
162#1518—611no—
163#1461—611no—
164#1286—611no—
165C2×C61—21no—
166C2×C61—21no—
167C2×C61#203511yescharacteristic
168C2×C61#204911yescharacteristic
169C2×C61#205311yescharacteristic
170#533—611no—
171C2×C4×Q8C2×D6111yescharacteristic
172#479—1222no—
173#477—1221no—
174#475—1221no—
175#431—611no—
176#431—611no—
177#428—1221no—
178#423—611no—
179C2×C4⋊C8C2×D6111yescharacteristic
180#419—1221no—
181C2×#103—21no—
182#382C2×D6111yescharacteristic
183#340—21no—
184#334—1222no—
185#333—1221no—
186#332—1221no—
187#328—1222no—
188#324—1222no—
189C61#1047011yescharacteristic, complemented, Sylow
190C22×Q8C22⋊D6111yescharacteristic
191C2×M16—1221no—
192C22×C8—21no—
193C4⋊Q8—611no—
194C4⋊Q8—611no—
195#125—1221no—
196C22⋊Q8—1221no—
197C22⋊Q8—1221no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—1221no—
203C4×Q8—1221no—
204#117—1221no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C22⋊D6111yescharacteristic
208C2×C42—21no—
209C2×C42C22×D6111yescharacteristic
210#107—1222no—
211#106—1222no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—612no—
215C4⋊C8—1221no—
216C4⋊C8—1221no—
217C4⋊C8—1221no—
218#103—42no—
219#103—1222no—
220#103—1222no—
221C22⋊C8—612no—
222C22⋊C8—1221no—
223C4×C8—1221no—
224C4×C8—1221no—
225#95—21no—
226#95—21no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—1221no—
233C2×Q8—1221no—
234C2×Q8#1175820012yes—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—41no—
238C22×C4D4×D6111yescharacteristic
239C22×C4#1175822411yescharacteristic
240C22×C4C2×C22⋊D6111yescharacteristic
241M16—2441no—
242M16—2441no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—1222no—
247C2×C8—1222no—
248C2×C8—2441no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—1221no—
256C4⋊C4—1221no—
257C4⋊C4—1221no—
258C4⋊C4—1221no—
259C4⋊C4—2441no—
260C4⋊C4—2441no—
261C22⋊C4—2441no—
262C42—41no—
263C42—41no—
264C42—2441no—
265C42#1175822211yescharacteristic
266C42#1175822211yescharacteristic
267C42D4×D6111yescharacteristic
268C42D4×D6111yescharacteristic
269C23#1451974911yescharacteristic
270Q8—22no—
271Q8—22no—
272Q8—41no—
273Q8—41no—
274Q8—4881no—
275Q8—4881no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—2441no—
287C2×C4—2441no—
288C2×C4#1451974511yescharacteristic
289C2×C4#1451975011yescharacteristic
290C2×C4#1451975011yescharacteristic
291C2×C4#1451975911yescharacteristic
292C2×C4#1451976011yescharacteristic
293C2×C4#1451976311yescharacteristic
294C2×C4C2×#1175820011yescharacteristic
295C2×C4#1451979711yescharacteristic
296C2×C4#1451980711yescharacteristic
297C2×C4#1451980811yescharacteristic
298C8—81no—
299C8—4881no—
300C8—2442no—
301C8—2442no—
302C22—21no—
303C22—21no—
304C22#1683967711yescharacteristic
305C22#1683998911yescharacteristic
306C22#1684001811yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—42no—
312C4—81no—
313C4—161no—
314C4—4881no—
315C4—4881no—
316C4#1683990011yescharacteristic
317C4#1683992211yescharacteristic
318C4#1684009211yescharacteristic
319C4#1684010311yescharacteristic
320C2—21no—
321C2—21no—
322C2#2144701511yescharacteristic
323C2#2144709711yescharacteristic
324C2#2144712311yescharacteristic
subgroupG/H
centerC22#16839677
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1805C2
socleC22×C61#394
prime radicalC23×C61#121
hypercenter#1805D61
hypocenterC61#10470
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
189. C61#10470semidirect

Maximal subgroups

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

subgroupindex
#214556242
#214445892
#214444772
#214443822
#214441432
#214436772
C61×#18052
#1047061