Finite Group Databasebeta

← #29593462

Subgroups — #29593462

Order 15616.

442 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 69 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#21457494C211yesmaximal, characteristic
2#21447861C211yesmaximal, characteristic, complemented
3#21445181C211yesmaximal, characteristic, complemented
4#21443865C211yesmaximal, characteristic, complemented
5#21443781C211yesmaximal, characteristic, complemented
6#21443775C211yesmaximal, characteristic, complemented
7C61×#2248C211yesmaximal, characteristic, complemented
8C2×D8×D61—21no—
9#16840397—21no—
10C2×#14519720—21no—
11#16840039—21no—
12#16839995—21no—
13#16839954—21no—
14#16839894—22no—
15C4×D488C2211yescharacteristic
16C8×D244C2211yescharacteristic
17C4⋊#11758185C2211yescharacteristic, complemented
18#16839566—21no—
19#16839515C2211yescharacteristic, complemented
20#16839508—21no—
21C2×SD32×C61—22no—
22C61×#494C2211yescharacteristic, complemented
23C61×#493C2211yescharacteristic
24C4×C16×C61C2211yescharacteristic, complemented
25C2×D4×D61—22no—
26C2×D4×D61—41no—
27C2×D488D411yescharacteristic
28C2×#11758199—22no—
29D8×D61—42no—
30D8×D61—42no—
31C2×C8×D61—21no—
32#14519748—22no—
33#14519748—41no—
34#14519745—42no—
35C4×D244—21no—
36C4×D244D411yescharacteristic
37#14519720—42no—
38C2×#11758185—21no—
39#14519711D411yescharacteristic
40#14519710—41no—
41C4⋊#92418—21no—
42C4⋊#92418D411yescharacteristic
43#14519699—21no—
44#14519698—22no—
45#14519697—41no—
46#14519689—21no—
47C2×Q16×C61—21no—
48C2×Q16×C61—21no—
49C2×D8×C61—21no—
50C2×D8×C61D412yes—
51C4⋊Q8×C61—21no—
52#127×C61—21no—
53SD32×C61—44no—
54C2×C16×C61—21no—
55C4×C8×C61C2311yescharacteristic
56C23×D61—42no—
57C2×Q8×C61—41no—
58C2×D4×C61—22no—
59C2×D4×C61—41no—
60C2×D4×C61—41no—
61C2×C22⋊D61—42no—
62C2×C22⋊D61—42no—
63D4×D61—44no—
64D4×D61—82no—
65D4×D61—82no—
66D4×D61—82no—
67C2×D244—21no—
68C2×D244D811yescharacteristic
69C2×C4×D61—21no—
70C2×C4×D61—41no—
71Q16×C61—42no—
72Q16×C61—41no—
73Q16×C61—41no—
74D8×C61—42no—
75D8×C61—41no—
76D8×C61—41no—
77C2×C8×C61C2×D411yescharacteristic
78C2×C8×C61C2×D411yescharacteristic
79C4⋊C4×C61—41no—
80C4⋊C4×C61—41no—
81C42×C61C2×D411yescharacteristic
82C22⋊Dic61—42no—
83#11758199—44no—
84#11758198—21no—
85#11758198—41no—
86C4⋊Dic61—21no—
87C4⋊Dic61D811yescharacteristic
88C2×#92418—21no—
89C2×#92418—41no—
90D488—21no—
91D488D812yes—
92C8×D61—41no—
93C8×D61—41no—
94C16×C61—22no—
95#11758185—41no—
96C23×C61—42no—
97C22×D61—21no—
98C22×D61—41no—
99C22×D61—84no—
100C22×D61—84no—
101C22×D61—162no—
102Q8×C61—81no—
103Q8×C61—81no—
104D4×C61—42no—
105D4×C61—42no—
106D4×C61—81no—
107D4×C61—81no—
108D4×C61—161no—
109C2×C4×C61—81no—
110C2×C4×C61#12011yescharacteristic
111C2×C4×C61C2×D811yescharacteristic
112C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
113C22⋊D61—84no—
114C22⋊D61—84no—
115C2×Dic61—21no—
116C2×Dic61—41no—
117D244—22no—
118D244—21no—
119D244—41no—
120D244SD3212yescomplemented
121C4×D61—41no—
122C4×D61—41no—
123C4×D61—81no—
124C4×D61—81no—
125C8×C61—21no—
126C8×C61C2×D811yescharacteristic
127C8×C61AGL(1,ℤ/8)11yescharacteristic
128#92418—41no—
129#92418—81no—
130#12320—611nomaximal, Sylow
131C22×C61—82no—
132C22×C61—82no—
133C22×C61—82no—
134C22×C61#44711yescharacteristic
135C2×D61—22no—
136C2×D61—41no—
137C2×D61—41no—
138C2×D61—42no—
139C2×D61—81no—
140C2×D61—81no—
141C2×D61—168no—
142C4×C61—21no—
143C4×C61—161no—
144C4×C61#44711yescharacteristic
145C4×C61#45311yescharacteristic
146C4×C61C2×SD3211yescharacteristic
147C4×C61#50911yescharacteristic
148Dic61—41no—
149Dic61—81no—
150D4×D8—611no—
151#2248D6111yescharacteristic, complemented
152#2208—611no—
153#1669—611no—
154#1361—611no—
155#1331—611no—
156D4⋊C16—611no—
157C2×C61—162no—
158C2×C61#161911yescharacteristic
159C2×C61#218711yescharacteristic
160C2×C61#219411yescharacteristic
161D61—42no—
162D61—82no—
163C22×D8—1221no—
164D42—1221no—
165C2×SD32—22no—
166C2×SD32—1221no—
167#494C2×D6111yescharacteristic, complemented
168#493C2×D6111yescharacteristic
169#474—1221no—
170#466—1221no—
171#459—1221no—
172#447—1222no—
173C4×D8—611no—
174C8×D4—611no—
175C4⋊C16—611no—
176#358—1221no—
177C4×C16C2×D6111yescharacteristic, complemented
178#335—611no—
179#332—1221no—
180C61#1232011yescharacteristic, complemented, Sylow
181C22×D4—1222no—
182C22×D4—2441no—
183C2×Q16—21no—
184C2×Q16—21no—
185C2×D8—21no—
186C2×D8—611no—
187C2×D8—1222no—
188C2×D8—2442no—
189C2×D8—2442no—
190C2×D8C22⋊D6112yes—
191C22×C8—1221no—
192C4⋊Q8—21no—
193#127—21no—
194#121—1222no—
195#121—2441no—
196#120—2442no—
197C4×D4—611no—
198C4×D4—1221no—
199SD32—44no—
200SD32—2442no—
201C2×C16—21no—
202C2×C16—1221no—
203#107—611no—
204#107—2441no—
205C4⋊C8—611no—
206C4⋊C8—1221no—
207#102—1222no—
208#102—1221no—
209#102—2441no—
210C22⋊C8—1221no—
211C4×C8C22×D6111yescharacteristic
212C24—2442no—
213C2×Q8—41no—
214C2×D4—22no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—611no—
218C2×D4—1221no—
219C2×D4—4882no—
220C2×D4—4882no—
221C2×D4—4882no—
222C2×D4—2442no—
223C2×D4—2444no—
224C2×D4—2442no—
225C22×C4—1221no—
226C22×C4—2441no—
227Q16—42no—
228Q16—41no—
229Q16—41no—
230D8—42no—
231D8—41no—
232D8—41no—
233D8—612no—
234D8—1221no—
235D8—2444no—
236C2×C8—1221no—
237C2×C8—2441no—
238C2×C8—2441no—
239C2×C8—2441no—
240C2×C8D4×D6111yescharacteristic
241C2×C8C2×C22⋊D6111yescharacteristic
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—611no—
245C4⋊C4—1221no—
246C22⋊C4—1221no—
247C22⋊C4—2442no—
248C22⋊C4—2441no—
249C42D4×D6111yescharacteristic
250C16—22no—
251C16—2441no—
252C23—42no—
253C23—1221no—
254C23—9762no—
255C23—4884no—
256C23—4884no—
257C23—2441no—
258Q8—81no—
259Q8—81no—
260D4—42no—
261D4—42no—
262D4—81no—
263D4—81no—
264D4—161no—
265D4—612no—
266D4—1221no—
267D4—1222no—
268D4—4884no—
269D4—4884no—
270D4—2441no—
271C2×C4—81no—
272C2×C4—1221no—
273C2×C4—4881no—
274C2×C4—4881no—
275C2×C4—2441no—
276C2×C4—2441no—
277C2×C4—2441no—
278C2×C4#1451974511yescharacteristic
279C2×C4D8×D6111yescharacteristic
280C2×C4#1451980111yescharacteristic
281C8—21no—
282C8—4881no—
283C8—2441no—
284C8D8×D6111yescharacteristic
285C8#1451980111yescharacteristic
286C22—82no—
287C22—82no—
288C22—82no—
289C22—1222no—
290C22—9768no—
291C22—4881no—
292C22—4881no—
293C22—2442no—
294C22—2441no—
295C22—2441no—
296C22#1683989411yescharacteristic
297C4—21no—
298C4—161no—
299C4—4881no—
300C4—2441no—
301C4#1683989411yescharacteristic
302C4#1683992911yescharacteristic
303C4SD32×D6111yescharacteristic
304C4#1684019911yescharacteristic
305C2—162no—
306C2—4882no—
307C2—2442no—
308C2#2144487111yescharacteristic
309C2#2144774611yescharacteristic
310C2#2144778911yescharacteristic
subgroupG/H
centerC22#16839894
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#2248C2
socleC22×C61#447
prime radical#21457494C2
hypercenter#2248D61
hypocenterC61#12320
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2248

Complements

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

NH ≅ G/Ntype
2. #21447861C2semidirect
3. #21445181C2semidirect
4. #21443865C2semidirect
5. #21443781C2semidirect
6. #21443775C2semidirect
7. C61×#2248C2semidirect
17. C4⋊#11758185C22semidirect
19. #16839515C22semidirect
22. C61×#494C22semidirect
24. C4×C16×C61C22semidirect
120. D244SD32semidirect
151. #2248D61semidirect
167. #494C2×D61semidirect
177. C4×C16C2×D61semidirect
180. C61#12320semidirect

Maximal subgroups

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

subgroupindex
#214574942
#214478612
#214451812
#214438652
#214437812
#214437752
C61×#22482
#1232061