Finite Group Databasebeta

← #29576941

Subgroups — #29576941

Order 15616.

444 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 99 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#21455949C211yesmaximal, characteristic
2#21448470C211yesmaximal, characteristic
3#21446763C211yesmaximal, characteristic
4#21444332C211yesmaximal, characteristic
5#21443713C211yesmaximal, characteristic
6#21443674C211yesmaximal, characteristic
7C61×#1491C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840092—21no—
10#16840047C412yescomplemented
11#16839972C412yescomplemented
12#16839918—21no—
13#16839867—22no—
14#16839819C2211yescharacteristic
15#16839669—21no—
16#16839661—21no—
17#16839648—22no—
18#16839630—21no—
19#16839628—21no—
20C4×C4⋊Dic61C2211yescharacteristic
21#16839608C2211yescharacteristic
22#16839523—22no—
23#16839500—21no—
24#16839500—21no—
25C2×C4×Q8×C61C2211yescharacteristic
26C61×#432C2211yescharacteristic
27C4×M16×C61C2211yescharacteristic
28C61×Q8⋊C8—22no—
29C2×#11758202—22no—
30#14519777C2×C411yescharacteristic
31#14519777C2×C411yescharacteristic
32#14519769—21no—
33#14519756—21no—
34#14519756—21no—
35C4×Dic122—21no—
36C4×Dic122—21no—
37#14519736—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61—22no—
40C2×C4×Dic61—21no—
41C2×C4×Dic61D412yes—
42#14519708—22no—
43C4⋊#92418C2×C412yes—
44#14519701—22no—
45#14519700—22no—
46C22⋊#92418—21no—
47#14519688—21no—
48#14519684—22no—
49#14519684—22no—
50#14519684—22no—
51#14519684—21no—
52#14519684—21no—
53#14519684D411yescharacteristic
54#14519684D411yescharacteristic
55C22×Q8×C61D411yescharacteristic
56C2×M16×C61—21no—
57C4×Q8×C61—21no—
58C4×Q8×C61—21no—
59C4×Q8×C61C2×C412yes—
60C4×Q8×C61D412yes—
61C2×C4⋊C4×C61—21no—
62C2×C4⋊C4×C61D411yescharacteristic
63C2×C42×C61—21no—
64C2×C42×C61C2311yescharacteristic
65C4⋊C8×C61—21no—
66C22⋊C8×C61—21no—
67#97×C61—21no—
68#97×C61—21no—
69C4×C8×C61—21no—
70C2×Q8×C61—21no—
71C2×Q8×C61—42no—
72C2×Q8×C61C22⋊C412yes—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×Dic61—21no—
79C22×Dic61—22no—
80C22×Dic61—21no—
81C2×Dic122—21no—
82C2×Dic122—21no—
83M16×C61—22no—
84M16×C61—41no—
85C2×C8×C61—41no—
86C2×C8×C61—41no—
87C4⋊C4×C61—22no—
88C4⋊C4×C61—22no—
89C4⋊C4×C61—21no—
90C4⋊C4×C61C22⋊C412yes—
91C42×C61—22no—
92C42×C61—22no—
93C42×C61C22×C411yescharacteristic
94C42×C61C2×D411yescharacteristic
95C42×C61#4111yescharacteristic
96C42×C61#4111yescharacteristic
97C22⋊Dic61—41no—
98#11758202—44no—
99C4⋊Dic61—21no—
100C4⋊Dic61—21no—
101C4⋊Dic61—21no—
102C4⋊Dic61—21no—
103C4⋊Dic61—42no—
104C4⋊Dic61—82no—
105#11758196—41no—
106#11758196—41no—
107C4×Dic61—42no—
108C4×Dic61—41no—
109C4×Dic61—42no—
110C2×#92418—22no—
111C23×C61#12011yescharacteristic
112Q8×C61—22no—
113Q8×C61—22no—
114Q8×C61—42no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—22no—
118C2×C4×C61—22no—
119C2×C4×C61—22no—
120C2×C4×C61—41no—
121C2×C4×C61—42no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61C2×C22⋊C411yescharacteristic
125C2×C4×C61C4×D411yescharacteristic
126C2×C4×C61C4×D411yescharacteristic
127C2×C4×C61#12111yescharacteristic
128C2×C4×C61#12111yescharacteristic
129C2×C4×C61#12311yescharacteristic
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—42no—
133C2×Dic61—41no—
134C2×Dic61—42no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—42no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141Dic122—42no—
142Dic122—81no—
143C8×C61—42no—
144C8×C61—81no—
145#92418—42no—
146#9328—611nomaximal, Sylow
147C22×C61—21no—
148C22×C61—21no—
149C22×C61#38611yescharacteristic
150C22×C61#45511yescharacteristic
151C22×C61#45611yescharacteristic
152C4×C61—21no—
153C4×C61—21no—
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—42no—
158C4×C61—42no—
159C4×C61#42112yes—
160Dic61—81no—
161Dic61—82no—
162Dic61—81no—
163Dic61—81no—
164Dic61—81no—
165#3113—611no—
166#2320—611no—
167#1986—611no—
168#1506—611no—
169#1491D6111yescharacteristic
170#1300—611no—
171#1285—611no—
172C2×C61—21no—
173C2×C61—21no—
174C2×C61#188211yescharacteristic
175C2×C61#189711yescharacteristic
176C2×C61#190511yescharacteristic
177#533—611no—
178C2×C4×Q8C2×D6111yescharacteristic
179#483—1221no—
180#475—612no—
181#462—612no—
182#451—1221no—
183#441—1222no—
184#432—611no—
185#432C2×D6111yescharacteristic
186C4×M16C2×D6111yescharacteristic
187#391—1221no—
188#389—1221no—
189#387—1222no—
190#383—1221no—
191#382—1221no—
192C4×C4⋊C4—611no—
193#376—611no—
194#339—1222no—
195#328—1221no—
196#328—1221no—
197Q8⋊C8—22no—
198C61#932811yescharacteristic, complemented, Sylow
199C22×Q8D24411yescharacteristic
200C2×Q16—1222no—
201C2×M16—21no—
202C4⋊Q8—611no—
203C4⋊Q8—611no—
204#125—1221no—
205C22⋊Q8—1221no—
206C22⋊Q8—1221no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—1221no—
210C4×Q8—1221no—
211C4×Q8C4×D6112yes—
212C4×Q8C22⋊D6112yes—
213#117—1221no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—1222no—
216C2×C4⋊C4—1221no—
217C2×C4⋊C4D24411yescharacteristic
218C2×C42—21no—
219C2×C42—612no—
220C2×C42—1221no—
221C2×C42C22×D6111yescharacteristic
222#106—1222no—
223C4⋊C8—21no—
224C4⋊C8—612no—
225#103—1222no—
226#103—1222no—
227C22⋊C8—21no—
228C22⋊C8—1221no—
229#97—21no—
230#97—21no—
231#97—1221no—
232C4×C8—21no—
233#95—611no—
234#95—611no—
235#95—1222no—
236#95—1222no—
237#95—1222no—
238#95—1221no—
239#95—1221no—
240C2×Q8—21no—
241C2×Q8—42no—
242C2×Q8—1221no—
243C2×Q8—1221no—
244C2×Q8#1175819812yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—1221no—
248C22×C4—1221no—
249C22×C4—1222no—
250C22×C4C2×D24411yescharacteristic
251C22×C4D4×D6111yescharacteristic
252C22×C4D4×D6111yescharacteristic
253Q16—2444no—
254M16—22no—
255M16—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—1222no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—21no—
262C4⋊C4—1221no—
263C4⋊C4—1221no—
264C4⋊C4—1221no—
265C4⋊C4—1221no—
266C4⋊C4—4882no—
267C4⋊C4—2442no—
268C4⋊C4—2441no—
269C4⋊C4—2441no—
270C4⋊C4#1175819812yes—
271C22⋊C4—2441no—
272C42—22no—
273C42—22no—
274C42—2442no—
275C42—2442no—
276C42—2441no—
277C42C2×C4×D6111yescharacteristic
278C42#1175822411yescharacteristic
279C42#1175822411yescharacteristic
280C42C2×C22⋊D6111yescharacteristic
281C23#1451974711yescharacteristic
282Q8—22no—
283Q8—22no—
284Q8—42no—
285Q8—4881no—
286Q8—2442no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—2441no—
297C2×C4—2441no—
298C2×C4—2442no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2442no—
302C2×C4—2442no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4C2×#1175819811yescharacteristic
308C2×C4#1451973711yescharacteristic
309C2×C4#1451973711yescharacteristic
310C2×C4#1451974911yescharacteristic
311C2×C4#1451974911yescharacteristic
312C2×C4#1451976111yescharacteristic
313C8—42no—
314C8—81no—
315C8—2442no—
316C22—21no—
317C22—21no—
318C22#1683963911yescharacteristic
319C22#1683994111yescharacteristic
320C22#1683994611yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—42no—
328C4—4881no—
329C4—4882no—
330C4—4881no—
331C4—4881no—
332C4—4881no—
333C4#1683977812yes—
334C2—21no—
335C2—21no—
336C2#2144618611yescharacteristic
337C2#2144626511yescharacteristic
338C2#2144631111yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1491C2
socleC22×C61#386
prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenter#1491D61
hypocenterC61#9328
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1491

Complements

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

NH ≅ G/Ntype
10. #16840047C4semidirect
11. #16839972C4semidirect
198. C61#9328semidirect

Maximal subgroups

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

subgroupindex
#214559492
#214484702
#214467632
#214443322
#214437132
#214436742
C61×#14912
#932861