Finite Group Databasebeta

← #29593475

Subgroups — #29593475

Order 15616.

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

normal subgroups only

This group has 384 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#21457578C211yesmaximal, characteristic
2#21448075C211yesmaximal, characteristic, complemented
3#21447869C211yesmaximal, characteristic, complemented
4#21445185C211yesmaximal, characteristic, complemented
5#21443864C211yesmaximal, characteristic, complemented
6#21443775C211yesmaximal, characteristic, complemented
7C61×#1332C211yesmaximal, characteristic, complemented
8C2×#14519800—21no—
9#16840434—21no—
10C2×#14519723—21no—
11C2×#14519722—22no—
12#16840142C2211yescharacteristic
13#16840139C2211yescharacteristic
14#16840043—21no—
15#16839999—21no—
16#16839962—21no—
17#16839910—22no—
18C8×D244C2211yescharacteristic
19C4⋊#11758185C2211yescharacteristic
20#16839565—21no—
21#16839507—21no—
22C4×Q16×C61C2211yescharacteristic
23C4×C16×C61C2211yescharacteristic
24C61×#335C2211yescharacteristic
25C2×#11758226—22no—
26#14519800—42no—
27#14519800—42no—
28C2×Dic244—21no—
29C2×Dic244—21no—
30C2×#11758201—22no—
31C2×D488—21no—
32C2×D488D412yes—
33C2×C8×D61—21no—
34#14519777—21no—
35#14519774—21no—
36#14519774—41no—
37#14519752—22no—
38#14519752—42no—
39C4×D244—41no—
40C4×D244D411yescharacteristic
41#14519723—42no—
42#14519722—44no—
43C2×#11758185—21no—
44#14519711—41no—
45C4⋊#92418D411yescharacteristic
46#14519699—41no—
47#14519697—22no—
48#14519689—21no—
49C2×Q16×C61D411yescharacteristic
50C4×Q8×C61—21no—
51C2×C16×C61—21no—
52#107×C61D411yescharacteristic
53C4⋊C8×C61—21no—
54#103×C61—21no—
55C4×C8×C61C2311yescharacteristic
56C2×Q8×C61—21no—
57#11758226—44no—
58#11758226—82no—
59C2×D244—22no—
60C2×D244—42no—
61C2×D244—41no—
62C2×D244—42no—
63C2×D244—41no—
64C2×D244D811yescharacteristic
65C2×C4×D61—21no—
66C2×C4×D61—42no—
67C2×Dic122—41no—
68Q16×C61—21no—
69Q16×C61D812yes—
70C2×C8×C61—41no—
71C2×C8×C61C2×D411yescharacteristic
72C2×C8×C61C2×D411yescharacteristic
73C4⋊C4×C61—21no—
74C4⋊C4×C61—41no—
75C42×C61—41no—
76C42×C61C2×D411yescharacteristic
77#11758201—44no—
78#11758198—21no—
79#11758198—42no—
80C4⋊Dic61—41no—
81C4⋊Dic61—41no—
82C4⋊Dic61D811yescharacteristic
83C2×#92418—21no—
84Dic244—42no—
85Dic244—41no—
86Dic244—41no—
87D488—42no—
88D488—41no—
89D488—41no—
90C8×D61—41no—
91C8×D61—41no—
92C16×C61—22no—
93#11758185—41no—
94C22×D61—21no—
95C22×D61—42no—
96Q8×C61—22no—
97Q8×C61—41no—
98C2×C4×C61—41no—
99C2×C4×C61—41no—
100C2×C4×C61#12011yescharacteristic
101C2×C4×C61C2×D811yescharacteristic
102C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
103C2×Dic61—21no—
104C2×Dic61—81no—
105D244—21no—
106D244—21no—
107D244—42no—
108D244—42no—
109D244—84no—
110D244—81no—
111D244—81no—
112D244—161no—
113D244—162no—
114C4×D61—41no—
115C4×D61—41no—
116C4×D61—84no—
117Dic122—81no—
118Dic122—81no—
119C8×C61—21no—
120C8×C61—81no—
121C8×C61C2×D811yescharacteristic
122C8×C61AGL(1,ℤ/8)11yescharacteristic
123#92418—41no—
124#12321—611nomaximal, Sylow
125C22×C61#44711yescharacteristic
126C2×D61—41no—
127C2×D61—41no—
128C2×D61—41no—
129C2×D61—82no—
130C2×D61—82no—
131C2×D61—82no—
132C4×C61—21no—
133C4×C61—42no—
134C4×C61—81no—
135C4×C61#44711yescharacteristic
136C4×C61#45311yescharacteristic
137C4×C61#50811yescharacteristic
138C4×C61#51011yescharacteristic
139Dic61—41no—
140Dic61—161no—
141#3287—611no—
142#2248—611no—
143#2209—611no—
144#1669—611no—
145#1361—611no—
146#1332D6111yescharacteristic, complemented
147D4⋊C16—611no—
148C2×C61#161911yescharacteristic
149C2×C61#218911yescharacteristic
150C2×C61#219411yescharacteristic
151D61—81no—
152D61—162no—
153C2×#135—1221no—
154#550—1221no—
155C2×Q32—1221no—
156C2×SD32—1222no—
157#494—611no—
158#493—611no—
159#474—1221no—
160#466—1221no—
161#460—1221no—
162#449—1222no—
163C4×Q16C2×D6111yescharacteristic
164C8×D4—611no—
165C4⋊C16—611no—
166#358—1221no—
167C4×C16C2×D6111yescharacteristic
168#335C2×D6111yescharacteristic
169#332—1221no—
170C61#1232111yescharacteristic, complemented, Sylow
171C2×#41—1222no—
172#135—2442no—
173#135—2442no—
174C2×Q16—1221no—
175C2×Q16—1221no—
176C2×Q16D24411yescharacteristic
177C2×SD16—1222no—
178C2×D8—612no—
179C2×D8—1221no—
180C22×C8—1221no—
181C4⋊Q8—1221no—
182#127—1221no—
183#127—2441no—
184#121—1222no—
185#121—2442no—
186C4×Q8—21no—
187C4×D4—611no—
188C4×D4—2441no—
189Q32—2442no—
190SD32—2444no—
191C2×C16—21no—
192C2×C16—1221no—
193#107—2441no—
194#107D24411yescharacteristic
195C4⋊C8—21no—
196C4⋊C8—611no—
197#103—21no—
198#102—1222no—
199#102—2441no—
200C22⋊C8—1221no—
201C4×C8C22×D6111yescharacteristic
202#41—4882no—
203#41—2444no—
204C2×Q8—21no—
205C2×Q8—2441no—
206C2×D4—611no—
207C2×D4—1222no—
208C2×D4—2442no—
209C2×D4—2442no—
210C2×D4—2441no—
211C2×D4—2441no—
212C22×C4—1221no—
213C22×C4—2442no—
214Q16—21no—
215Q16—2442no—
216Q16—2441no—
217Q16—2441no—
218Q16D48812yes—
219SD16—2444no—
220D8—2442no—
221D8—2441no—
222D8—2441no—
223C2×C8—41no—
224C2×C8—1221no—
225C2×C8—2441no—
226C2×C8—2441no—
227C2×C8C2×D24411yescharacteristic
228C2×C8D4×D6111yescharacteristic
229C4⋊C4—21no—
230C4⋊C4—41no—
231C4⋊C4—611no—
232C4⋊C4—2441no—
233C4⋊C4—2441no—
234C22⋊C4—1221no—
235C22⋊C4—2442no—
236C42—41no—
237C42D4×D6111yescharacteristic
238C16—22no—
239C16—2441no—
240C23—1221no—
241C23—2442no—
242Q8—22no—
243Q8—41no—
244Q8—4881no—
245Q8—4881no—
246D4—1221no—
247D4—1221no—
248D4—9761no—
249D4—9762no—
250D4—4884no—
251D4—4881no—
252D4—4881no—
253D4—2442no—
254D4—2442no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—1221no—
258C2×C4—4884no—
259C2×C4—4881no—
260C2×C4—2441no—
261C2×C4—2441no—
262C2×C4#1451974711yescharacteristic
263C2×C4D8×D6111yescharacteristic
264C2×C4#1451980411yescharacteristic
265C8—21no—
266C8—81no—
267C8—2441no—
268C8C2×D48811yescharacteristic
269C8#1451980511yescharacteristic
270C22—4882no—
271C22—4882no—
272C22—4882no—
273C22—2441no—
274C22—2441no—
275C22—2441no—
276C22#1683989711yescharacteristic
277C4—21no—
278C4—42no—
279C4—81no—
280C4—9761no—
281C4—2441no—
282C4#1683989811yescharacteristic
283C4#1683993111yescharacteristic
284C4#1684019311yescharacteristic
285C4#1684020411yescharacteristic
286C2—9762no—
287C2—4881no—
288C2#2144487511yescharacteristic
289C2#2144776211yescharacteristic
290C2#2144779111yescharacteristic
subgroupG/H
centerC22#16839897
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#1332C2
socleC22×C61#447
prime radical#21457578C2
hypercenter#1332D61
hypocenterC61#12321
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1332

Complements

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

NH ≅ G/Ntype
2. #21448075C2semidirect
3. #21447869C2semidirect
4. #21445185C2semidirect
5. #21443864C2semidirect
6. #21443775C2semidirect
7. C61×#1332C2semidirect
146. #1332D61semidirect
170. C61#12321semidirect

Maximal subgroups

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

subgroupindex
#214575782
#214480752
#214478692
#214451852
#214438642
#214437752
C61×#13322
#1232161