Finite Group Databasebeta

← #29576955

Subgroups — #29576955

Order 15616.

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

normal subgroups only

This group has 482 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#21455941C211yesmaximal, characteristic
2#21448470C211yesmaximal, characteristic, complemented
3#21446738C211yesmaximal, characteristic
4#21444328C211yesmaximal, characteristic
5#21443713C211yesmaximal, characteristic, complemented
6#21443674C211yesmaximal, characteristic, complemented
7C61×#1491C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9#16840075—21no—
10#16840057C412yescomplemented
11#16839962C412yescomplemented
12#16839901—21no—
13#16839860—22no—
14#16839819C2211yescharacteristic
15C2×#14519703—22no—
16#16839669—21no—
17#16839661—21no—
18#16839648—22no—
19#16839630—21no—
20#16839628—21no—
21C4×C4⋊Dic61C2211yescharacteristic
22#16839608C2211yescharacteristic
23#16839497—21no—
24#16839497—21no—
25C2×C4×Q8×C61C2211yescharacteristic
26C61×#432C2211yescharacteristic
27C4×M16×C61C2211yescharacteristic
28C61×Q8⋊C8—22no—
29C2×#11758222—21no—
30C2×#11758201—22no—
31#14519777C2×C411yescharacteristic
32#14519774C2×C411yescharacteristic
33#14519765—21no—
34#14519751—21no—
35#14519751—21no—
36C4×D244—21no—
37C4×D244—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61—22no—
40C2×C4×Dic61—21no—
41C2×C4×Dic61D412yes—
42#14519710—22no—
43C4⋊#92418C2×C412yes—
44#14519703—44no—
45#14519701—22no—
46#14519697—22no—
47C22⋊#92418—21no—
48#14519688—21no—
49#14519684—22no—
50#14519684—22no—
51#14519684—22no—
52#14519684—21no—
53#14519684—21no—
54#14519684D411yescharacteristic
55#14519684D411yescharacteristic
56C22×Q8×C61D411yescharacteristic
57C2×M16×C61—21no—
58C4×Q8×C61—21no—
59C4×Q8×C61—21no—
60C4×Q8×C61C2×C412yes—
61C4×Q8×C61D412yes—
62C2×C4⋊C4×C61—21no—
63C2×C4⋊C4×C61D411yescharacteristic
64C2×C42×C61—21no—
65C2×C42×C61C2311yescharacteristic
66C4⋊C8×C61—21no—
67C22⋊C8×C61—21no—
68#97×C61—21no—
69#97×C61—21no—
70C4×C8×C61—21no—
71C2×Q8×C61—21no—
72C2×Q8×C61—42no—
73C2×Q8×C61C22⋊C412yes—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C2×C22⋊D61—41no—
80C22×Dic61—21no—
81C22×Dic61—22no—
82C22×Dic61—21no—
83#11758222—42no—
84#11758222—42no—
85C2×D244—21no—
86C2×D244—21no—
87C2×D244—21no—
88C2×C4×D61—41no—
89C2×Dic122—21no—
90M16×C61—22no—
91M16×C61—41no—
92C2×C8×C61—41no—
93C2×C8×C61—41no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—22no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61C22⋊C412yes—
98C42×C61—22no—
99C42×C61—22no—
100C42×C61C22×C411yescharacteristic
101C42×C61C2×D411yescharacteristic
102C42×C61#4111yescharacteristic
103C42×C61#4111yescharacteristic
104#11758201—44no—
105#11758198—41no—
106#11758198—41no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109C4⋊Dic61—42no—
110C4⋊Dic61—82no—
111C4×Dic61—42no—
112C4×Dic61—42no—
113C2×#92418—22no—
114C23×C61#12011yescharacteristic
115C22×D61—41no—
116Q8×C61—22no—
117Q8×C61—22no—
118Q8×C61—42no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—22no—
124C2×C4×C61—41no—
125C2×C4×C61—42no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61C2×C22⋊C411yescharacteristic
129C2×C4×C61C4×D411yescharacteristic
130C2×C4×C61C4×D411yescharacteristic
131C2×C4×C61#12111yescharacteristic
132C2×C4×C61#12111yescharacteristic
133C2×C4×C61#12311yescharacteristic
134C22⋊D61—81no—
135C22⋊D61—81no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—42no—
139C2×Dic61—41no—
140C2×Dic61—42no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—42no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146D244—42no—
147D244—42no—
148D244—81no—
149C4×D61—82no—
150Dic122—42no—
151C8×C61—42no—
152C8×C61—81no—
153#92418—42no—
154#9330—611nomaximal, Sylow
155C22×C61—21no—
156C22×C61—21no—
157C22×C61#38611yescharacteristic
158C22×C61#45311yescharacteristic
159C22×C61#45411yescharacteristic
160C2×D61—81no—
161C2×D61—81no—
162C2×D61—81no—
163C4×C61—21no—
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—21no—
167C4×C61—21no—
168C4×C61—42no—
169C4×C61—42no—
170C4×C61#42112yes—
171Dic61—81no—
172Dic61—82no—
173Dic61—81no—
174Dic61—81no—
175#3112—611no—
176#2320—611no—
177#1980—611no—
178#1505—611no—
179#1491D6111yescharacteristic, complemented
180#1300—611no—
181#1285—611no—
182C2×C61—21no—
183C2×C61—21no—
184C2×C61#188311yescharacteristic
185C2×C61#189711yescharacteristic
186C2×C61#190511yescharacteristic
187D61—161no—
188#532—611no—
189C2×C4×Q8C2×D6111yescharacteristic
190#480—1221no—
191#477—612no—
192#460—612no—
193#448—1221no—
194#440—1222no—
195#432—611no—
196#432C2×D6111yescharacteristic
197C2×#104—1222no—
198C4×M16C2×D6111yescharacteristic
199#391—1221no—
200#389—1221no—
201#387—1222no—
202#383—1221no—
203#382—1221no—
204C4×C4⋊C4—611no—
205#376—611no—
206#327—1221no—
207#327—1221no—
208Q8⋊C8—22no—
209C61#933011yescharacteristic, complemented, Sylow
210C2×#41—1221no—
211C22×Q8D24411yescharacteristic
212C2×SD16—1222no—
213C2×M16—21no—
214C4⋊Q8—611no—
215#127—611no—
216#124—1221no—
217#121—1221no—
218#121—1221no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8C4×D6112yes—
222C4×Q8C22⋊D6112yes—
223C4×D4—1221no—
224C4×D4—1221no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—1222no—
227C2×C4⋊C4—1221no—
228C2×C4⋊C4D24411yescharacteristic
229C2×C42—21no—
230C2×C42—612no—
231C2×C42—1221no—
232C2×C42C22×D6111yescharacteristic
233#107—1222no—
234C4⋊C8—21no—
235C4⋊C8—612no—
236#104—2444no—
237#103—1222no—
238#102—1222no—
239C22⋊C8—21no—
240C22⋊C8—1221no—
241#97—21no—
242#97—21no—
243#97—1221no—
244C4×C8—21no—
245#95—611no—
246#95—611no—
247#95—1222no—
248#95—1222no—
249#95—1222no—
250#95—1221no—
251#95—1221no—
252#41—2442no—
253#41—2442no—
254C2×Q8—21no—
255C2×Q8—42no—
256C2×Q8—1221no—
257C2×Q8#1175819812yes—
258C2×D4—1221no—
259C2×D4—1221no—
260C2×D4—1221no—
261C2×D4—2441no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—1222no—
267C22×C4—2441no—
268C22×C4C2×D24411yescharacteristic
269C22×C4D4×D6111yescharacteristic
270C22×C4D4×D6111yescharacteristic
271SD16—2444no—
272M16—22no—
273M16—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—1222no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—1221no—
281C4⋊C4—1221no—
282C4⋊C4—4882no—
283C4⋊C4—2442no—
284C4⋊C4#1175819812yes—
285C22⋊C4—2441no—
286C22⋊C4—2441no—
287C42—22no—
288C42—22no—
289C42—2442no—
290C42—2442no—
291C42C2×C4×D6111yescharacteristic
292C42#1175822411yescharacteristic
293C42#1175822411yescharacteristic
294C42C2×C22⋊D6111yescharacteristic
295C23—2441no—
296C23#1451974711yescharacteristic
297Q8—22no—
298Q8—22no—
299Q8—42no—
300Q8—2442no—
301D4—4881no—
302D4—4881no—
303D4—4881no—
304D4—2442no—
305D4—2442no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—4882no—
316C2×C4—2441no—
317C2×C4—2441no—
318C2×C4—2442no—
319C2×C4—2441no—
320C2×C4—2441no—
321C2×C4—2442no—
322C2×C4—2441no—
323C2×C4—2441no—
324C2×C4—2442no—
325C2×C4—2441no—
326C2×C4C2×#1175819811yescharacteristic
327C2×C4#1451973711yescharacteristic
328C2×C4#1451973711yescharacteristic
329C2×C4#1451974911yescharacteristic
330C2×C4#1451974911yescharacteristic
331C2×C4#1451976111yescharacteristic
332C8—42no—
333C8—81no—
334C8—2442no—
335C22—21no—
336C22—21no—
337C22—4881no—
338C22—4881no—
339C22—4881no—
340C22#1683963911yescharacteristic
341C22#1683993111yescharacteristic
342C22#1683993611yescharacteristic
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—42no—
350C4—4881no—
351C4—4882no—
352C4—4881no—
353C4—4881no—
354C4#1683977812yes—
355C2—21no—
356C2—21no—
357C2—9761no—
358C2#2144619311yescharacteristic
359C2#2144626511yescharacteristic
360C2#2144631111yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1491C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1491D61
hypocenterC61#9330
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
2. #21448470C2semidirect
5. #21443713C2semidirect
6. #21443674C2semidirect
7. C61×#1491C2semidirect
10. #16840057C4semidirect
11. #16839962C4semidirect
179. #1491D61semidirect
209. C61#9330semidirect

Maximal subgroups

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

subgroupindex
#214559412
#214484702
#214467382
#214443282
#214437132
#214436742
C61×#14912
#933061