Finite Group Databasebeta

← #29576956

Subgroups — #29576956

Order 15616.

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

normal subgroups only

This group has 482 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#21455942C211yesmaximal, characteristic
2#21448465C211yesmaximal, characteristic, complemented
3#21446737C211yesmaximal, characteristic
4#21444328C211yesmaximal, characteristic
5#21444265C211yesmaximal, characteristic, complemented
6#21443711C211yesmaximal, characteristic, complemented
7C61×#1285C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9C2×C4×Dic122C2211yescharacteristic
10#16840076—21no—
11#16840058C412yescomplemented
12#16839963C412yescomplemented
13#16839903—21no—
14#16839864—22no—
15C2×#14519705—22no—
16C4×#11758194C2211yescharacteristic
17#16839667—21no—
18#16839658—21no—
19#16839646—22no—
20#16839629—21no—
21#16839626—21no—
22#16839608C2211yescharacteristic
23#16839497—21no—
24#16839497—21no—
25#16839492—22no—
26C61×#432C2211yescharacteristic
27C61×#432C2211yescharacteristic
28C4×C4⋊C4×C61C2211yescharacteristic
29C2×#11758222—21no—
30C22×Dic122D411yescharacteristic
31C2×#11758190—22no—
32C2×#11758194—21no—
33#14519777C2×C411yescharacteristic
34#14519774C2×C411yescharacteristic
35#14519765—21no—
36#14519751—21no—
37#14519751—21no—
38C4×Dic122—21no—
39C4×Dic122—21no—
40C4×Dic122C2×C412yes—
41C4×Dic122D412yes—
42C4×D244—21no—
43C4×D244—21no—
44C2×C4⋊Dic61D411yescharacteristic
45C2×#11758196—21no—
46C2×#11758196—22no—
47C2×C4×Dic61—21no—
48C2×C4×Dic61—21no—
49#14519711—22no—
50#14519705—44no—
51#14519702—22no—
52#14519699—22no—
53#14519688—21no—
54C4×#92418—21no—
55#14519684—22no—
56#14519684—22no—
57#14519684—22no—
58#14519684—21no—
59#14519684D411yescharacteristic
60#14519684D411yescharacteristic
61C2×C4⋊C4×C61—21no—
62C2×C42×C61D412yes—
63C2×C42×C61C2311yescharacteristic
64C4⋊C8×C61—21no—
65C4⋊C8×C61C2×C412yes—
66C22⋊C8×C61—21no—
67C22⋊C8×C61—21no—
68#97×C61—21no—
69#97×C61—21no—
70#95×C61—21no—
71C22×C4×C61—22no—
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C2×C22⋊D61—41no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—21no—
80#11758222—42no—
81#11758222—42no—
82C2×D244—21no—
83C2×D244—21no—
84C2×D244—21no—
85C2×C4×D61—41no—
86C2×Dic122—21no—
87C2×Dic122—21no—
88C2×Dic122—42no—
89C2×Dic122C22⋊C412yes—
90C2×C8×C61—22no—
91C2×C8×C61—41no—
92C4⋊C4×C61—42no—
93C42×C61—42no—
94C42×C61C22×C411yescharacteristic
95C42×C61C2×D411yescharacteristic
96C42×C61#4111yescharacteristic
97C42×C61#4111yescharacteristic
98#11758198—41no—
99#11758198—41no—
100C4⋊Dic61—21no—
101C4⋊Dic61—21no—
102C4⋊Dic61—21no—
103C4⋊Dic61C22⋊C412yes—
104#11758196—22no—
105#11758196—22no—
106#11758196—82no—
107C4×Dic61—22no—
108C4×Dic61—22no—
109C4×Dic61—42no—
110#11758194—22no—
111#11758194—41no—
112C2×#92418—41no—
113#11758190—44no—
114C23×C61#12011yescharacteristic
115C22×D61—41no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—22no—
119C2×C4×C61—41no—
120C2×C4×C61—42no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61C2×C22⋊C411yescharacteristic
124C2×C4×C61C4×D411yescharacteristic
125C2×C4×C61C4×D411yescharacteristic
126C2×C4×C61#12111yescharacteristic
127C2×C4×C61#12111yescharacteristic
128C2×C4×C61#12311yescharacteristic
129C22⋊D61—81no—
130C22⋊D61—81no—
131C2×Dic61—22no—
132C2×Dic61—22no—
133C2×Dic61—42no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143D244—42no—
144D244—42no—
145D244—81no—
146C4×D61—82no—
147Dic122—22no—
148Dic122—22no—
149Dic122—42no—
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—82no—
169C4×C61#42112yes—
170Dic61—42no—
171Dic61—42no—
172Dic61—81no—
173Dic61—81no—
174Dic61—81no—
175#3112—611no—
176#2320—611no—
177#1980—611no—
178#1505—611no—
179#1491—611no—
180#1300—611no—
181#1285D6111yescharacteristic, complemented
182C2×C61—21no—
183C2×C61—21no—
184C2×C61#188311yescharacteristic
185C2×C61#189711yescharacteristic
186C2×C61#190511yescharacteristic
187D61—161no—
188#532—611no—
189C2×C4×Q8—611no—
190#480—1221no—
191#477—612no—
192#460—612no—
193#448—1221no—
194#440—1222no—
195#432C2×D6111yescharacteristic
196#432C2×D6111yescharacteristic
197C2×#104—1222no—
198C4×M16—611no—
199#391—1221no—
200#389—1221no—
201#387—1222no—
202#383—1221no—
203#382—1221no—
204C4×C4⋊C4C2×D6111yescharacteristic
205#376—611no—
206#327—1221no—
207#327—1221no—
208Q8⋊C8—1222no—
209C61#933011yescharacteristic, complemented, Sylow
210C2×#41—1221no—
211C22×Q8—611no—
212C2×SD16—1222no—
213C2×M16—1221no—
214C4⋊Q8—611no—
215#127—611no—
216#124—1221no—
217#121—1221no—
218#121—1221no—
219C4×Q8—612no—
220C4×Q8—612no—
221C4×Q8—1221no—
222C4×Q8—1221no—
223C4×D4—1221no—
224C4×D4—1221no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—611no—
227C2×C4⋊C4—1221no—
228C2×C4⋊C4—1222no—
229C2×C42—1221no—
230C2×C42—1221no—
231C2×C42D24412yes—
232C2×C42C22×D6111yescharacteristic
233#107—1222no—
234C4⋊C8—21no—
235C4⋊C8C4×D6112yes—
236#104—2444no—
237#103—1222no—
238#102—1222no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241#97—21no—
242#97—21no—
243#97—1221no—
244C4×C8—1221no—
245#95—21no—
246#95—611no—
247#95—611no—
248#95—1222no—
249#95—1222no—
250#95—1222no—
251#95—1221no—
252#41—2442no—
253#41—2442no—
254C2×Q8—612no—
255C2×Q8—1221no—
256C2×Q8—1221no—
257C2×Q8—2442no—
258C2×D4—1221no—
259C2×D4—1221no—
260C2×D4—1221no—
261C2×D4—2441no—
262C22×C4—22no—
263C22×C4—1221no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4—2441no—
268C22×C4C2×D24411yescharacteristic
269C22×C4D4×D6111yescharacteristic
270C22×C4D4×D6111yescharacteristic
271SD16—2444no—
272M16—1222no—
273M16—2441no—
274C2×C8—22no—
275C2×C8—41no—
276C2×C8—2441no—
277C4⋊C4—42no—
278C4⋊C4—612no—
279C4⋊C4—1222no—
280C4⋊C4—1222no—
281C4⋊C4—1221no—
282C4⋊C4—1221no—
283C4⋊C4—1221no—
284C4⋊C4—4882no—
285C22⋊C4—2441no—
286C22⋊C4—2441no—
287C42—42no—
288C42—1222no—
289C42—1222no—
290C42—2442no—
291C42C2×C4×D6111yescharacteristic
292C42#1175822211yescharacteristic
293C42D4×D6111yescharacteristic
294C42#1175822611yescharacteristic
295C23—2441no—
296C23#1451974711yescharacteristic
297Q8—1222no—
298Q8—1222no—
299Q8—2442no—
300Q8—2442no—
301D4—4881no—
302D4—4881no—
303D4—4881no—
304D4—2442no—
305D4—2442no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—1222no—
314C2×C4—1222no—
315C2×C4—4882no—
316C2×C4—2441no—
317C2×C4—2441no—
318C2×C4—2441no—
319C2×C4—2441no—
320C2×C4—2441no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4—2442no—
324C2×C4—2441no—
325C2×C4—2441no—
326C2×C4C22⋊C4×D6111yescharacteristic
327C2×C4#1451973911yescharacteristic
328C2×C4C4×D24411yescharacteristic
329C2×C4#1451975011yescharacteristic
330C2×C4#1451975211yescharacteristic
331C2×C4#1451976211yescharacteristic
332C8—42no—
333C8—81no—
334C8—2442no—
335C22—21no—
336C22—21no—
337C22—4881no—
338C22—4881no—
339C22—4881no—
340C22#1683964011yescharacteristic
341C22#1683993211yescharacteristic
342C22#1683993711yescharacteristic
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—82no—
349C4—4881no—
350C4—4881no—
351C4—4881no—
352C4—2442no—
353C4—2442no—
354C4#1683977712yes—
355C2—21no—
356C2—21no—
357C2—9761no—
358C2#2144619411yescharacteristic
359C2#2144626611yescharacteristic
360C2#2144631411yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1285C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1285D61
hypocenterC61#9330
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

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

NH ≅ G/Ntype
2. #21448465C2semidirect
5. #21444265C2semidirect
6. #21443711C2semidirect
7. C61×#1285C2semidirect
11. #16840058C4semidirect
12. #16839963C4semidirect
181. #1285D61semidirect
209. C61#9330semidirect

Maximal subgroups

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

subgroupindex
#214559422
#214484652
#214467372
#214443282
#214442652
#214437112
C61×#12852
#933061