Finite Group Databasebeta

← #29577319

Subgroups — #29577319

Order 15616.

462 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 95 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#21457009C211yesmaximal, characteristic
2#21448625C211yesmaximal, characteristic, complemented
3#21446738C211yesmaximal, characteristic
4#21444367C211yesmaximal, characteristic
5#21443684C211yesmaximal, characteristic, complemented
6#21443676C211yesmaximal, characteristic, complemented
7C61×#1528C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9#16840172C412yescomplemented
10#16840128C412yescomplemented
11#16840080—21no—
12#16840075—21no—
13#16840059—22no—
14#16839819C2211yescharacteristic
15C2×#14519703—21no—
16C2×#14519703—21no—
17#16839661—21no—
18#16839648—21no—
19#16839648—21no—
20#16839648—21no—
21#16839648—21no—
22#16839630C2211yescharacteristic
23C4⋊C4×Dic61—21no—
24C4×C4⋊Dic61C2211yescharacteristic
25#16839497—21no—
26#16839497—21no—
27C2×#125×C61C2211yescharacteristic
28C61×#432C2211yescharacteristic
29C4×M16×C61C2211yescharacteristic
30C61×#330—22no—
31C2×#11758222—21no—
32#14519777C2×C411yescharacteristic
33#14519774C2×C411yescharacteristic
34#14519765—21no—
35#14519751—21no—
36#14519751—21no—
37C4×D244—21no—
38C4×D244—21no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C2×C4×Dic61—21no—
43C2×C4×Dic61—21no—
44C2×C4×Dic61D411yescharacteristic
45C2×C4×Dic61D411yescharacteristic
46#14519710—22no—
47#14519708—22no—
48C4⋊#92418—21no—
49#14519703—42no—
50#14519703—42no—
51#14519697—22no—
52#14519697—22no—
53C22⋊#92418—21no—
54#14519688C2×C412yes—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59#14519684—21no—
60#14519684—21no—
61#14519684—21no—
62#14519684D411yescharacteristic
63#14519684D411yescharacteristic
64C2×M16×C61—21no—
65#125×C61—21no—
66#125×C61—21no—
67#125×C61C2×C412yes—
68#125×C61D412yes—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61D411yescharacteristic
72C2×C4⋊C4×C61D411yescharacteristic
73C2×C42×C61C2311yescharacteristic
74C4⋊C8×C61—21no—
75C22⋊C8×C61—21no—
76#97×C61—21no—
77#97×C61—21no—
78C4×C8×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C2×C22⋊D61—41no—
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89#11758222—41no—
90#11758222—41no—
91#11758222—41no—
92#11758222—41no—
93C2×D244—21no—
94C2×D244—21no—
95C2×D244—21no—
96C2×C4×D61—41no—
97C2×Dic122—21no—
98M16×C61—22no—
99M16×C61—41no—
100C2×C8×C61—41no—
101C2×C8×C61—41no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61C22⋊C412yes—
109C4⋊C4×C61C22⋊C412yes—
110C42×C61C22×C411yescharacteristic
111C42×C61C2×D411yescharacteristic
112C42×C61#4111yescharacteristic
113C42×C61#4111yescharacteristic
114#11758198—41no—
115#11758198—41no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118C4⋊Dic61—42no—
119C4⋊Dic61—42no—
120C4⋊Dic61—81no—
121C4×Dic61—42no—
122C4×Dic61—41no—
123C4×Dic61—41no—
124C4×Dic61—81no—
125C2×#92418—22no—
126C23×C61#12011yescharacteristic
127C22×D61—41no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61C2×C22⋊C411yescharacteristic
139C2×C4×C61C4×D411yescharacteristic
140C2×C4×C61C4×D411yescharacteristic
141C2×C4×C61#12111yescharacteristic
142C2×C4×C61#12111yescharacteristic
143C2×C4×C61#12311yescharacteristic
144C22⋊D61—81no—
145C22⋊D61—81no—
146C2×Dic61—22no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159D244—42no—
160D244—41no—
161D244—41no—
162D244—81no—
163C4×D61—81no—
164C4×D61—81no—
165Dic122—41no—
166Dic122—41no—
167C8×C61—42no—
168C8×C61—81no—
169#92418—42no—
170#9382—611nomaximal, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61#38611yescharacteristic
174C22×C61#45311yescharacteristic
175C22×C61#45411yescharacteristic
176C2×D61—81no—
177C2×D61—81no—
178C2×D61—81no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—21no—
185C4×C61—42no—
186C4×C61—42no—
187Dic61—42no—
188Dic61—81no—
189Dic61—81no—
190Dic61—81no—
191Dic61—81no—
192#3234—611no—
193#2336—611no—
194#1980—611no—
195#1528D6111yescharacteristic, complemented
196#1514—611no—
197#1288—611no—
198#1285—611no—
199C2×C61—21no—
200C2×C61—21no—
201C2×C61#188311yescharacteristic
202C2×C61#190611yescharacteristic
203C2×C61#191211yescharacteristic
204D61—161no—
205#532—611no—
206C2×#125C2×D6111yescharacteristic
207#501—612no—
208#490—612no—
209#481—1221no—
210#480—1221no—
211#478—1222no—
212#432—611no—
213#432C2×D6111yescharacteristic
214C2×#104—1221no—
215C2×#104—1221no—
216C4×M16C2×D6111yescharacteristic
217#389—1221no—
218#387—1221no—
219#387—1221no—
220#387—1221no—
221#387—1221no—
222#383—611no—
223C4×C4⋊C4—611no—
224C4×C4⋊C4—1221no—
225#330—22no—
226#327—1221no—
227#327—1221no—
228C61#938211yescharacteristic, complemented, Sylow
229C2×#41—1221no—
230C2×M16—21no—
231C4⋊Q8—611no—
232#127—611no—
233#125—21no—
234#125—21no—
235#125C4×D6112yes—
236#125C22⋊D6112yes—
237#124—1221no—
238#121—1221no—
239#121—1221no—
240C4×D4—1221no—
241C4×D4—1221no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—1221no—
245C2×C4⋊C4—1221no—
246C2×C4⋊C4—1221no—
247C2×C4⋊C4D24411yescharacteristic
248C2×C4⋊C4D24411yescharacteristic
249C2×C42—611no—
250C2×C42—611no—
251C2×C42—1221no—
252C2×C42—1221no—
253C2×C42C22×D6111yescharacteristic
254#107—1222no—
255#106—1222no—
256C4⋊C8—21no—
257C4⋊C8—1221no—
258#104—2442no—
259#104—2442no—
260#102—1222no—
261#102—1222no—
262C22⋊C8—21no—
263C22⋊C8—1221no—
264#97—21no—
265#97—21no—
266#97—612no—
267C4×C8—21no—
268#95—611no—
269#95—611no—
270#95—1221no—
271#95—1221no—
272#95—1221no—
273#95—1221no—
274#95—1221no—
275#95—1221no—
276#95—1221no—
277#41—2441no—
278#41—2441no—
279#41—2441no—
280#41—2441no—
281C2×Q8—1221no—
282C2×D4—1221no—
283C2×D4—1221no—
284C2×D4—1221no—
285C2×D4—2441no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—1221no—
289C22×C4—1221no—
290C22×C4—1221no—
291C22×C4—1221no—
292C22×C4—2441no—
293C22×C4C2×D24411yescharacteristic
294C22×C4D4×D6111yescharacteristic
295C22×C4D4×D6111yescharacteristic
296M16—22no—
297M16—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—1222no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—1221no—
308C4⋊C4—1221no—
309C4⋊C4—4881no—
310C4⋊C4—2442no—
311C4⋊C4—2442no—
312C4⋊C4#1175819812yes—
313C4⋊C4#1175819812yes—
314C22⋊C4—2441no—
315C22⋊C4—2441no—
316C42—4881no—
317C42—2441no—
318C42—2441no—
319C42—2442no—
320C42C2×C4×D6111yescharacteristic
321C42#1175822411yescharacteristic
322C42#1175822411yescharacteristic
323C42C2×C22⋊D6111yescharacteristic
324C23—2441no—
325C23#1451974711yescharacteristic
326Q8—2441no—
327Q8—2441no—
328D4—4881no—
329D4—4881no—
330D4—4881no—
331D4—2442no—
332D4—2441no—
333D4—2441no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—1222no—
345C2×C4—4881no—
346C2×C4—4881no—
347C2×C4—2441no—
348C2×C4—2441no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4—2441no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2441no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4—2441no—
359C2×C4C2×#1175819811yescharacteristic
360C2×C4#1451973711yescharacteristic
361C2×C4#1451973711yescharacteristic
362C2×C4#1451974911yescharacteristic
363C2×C4#1451974911yescharacteristic
364C2×C4#1451976111yescharacteristic
365C8—42no—
366C8—81no—
367C8—2442no—
368C22—21no—
369C22—21no—
370C22—4881no—
371C22—4881no—
372C22—4881no—
373C22#1683963911yescharacteristic
374C22#1683993111yescharacteristic
375C22#1683993611yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—42no—
383C4—42no—
384C4—4881no—
385C4—4881no—
386C4—4881no—
387C4—4881no—
388C4—2442no—
389C2—21no—
390C2—21no—
391C2—9761no—
392C2#2144619311yescharacteristic
393C2#2144631811yescharacteristic
394C2#2144635411yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1528C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1528D61
hypocenterC61#9382
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1528

Complements

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

NH ≅ G/Ntype
2. #21448625C2semidirect
5. #21443684C2semidirect
6. #21443676C2semidirect
7. C61×#1528C2semidirect
9. #16840172C4semidirect
10. #16840128C4semidirect
195. #1528D61semidirect
228. C61#9382semidirect

Maximal subgroups

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

subgroupindex
#214570092
#214486252
#214467382
#214443672
#214436842
#214436762
C61×#15282
#938261