Finite Group Databasebeta

← #29593879

Subgroups — #29593879

Order 15616.

344 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 71 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#21458794C211yesmaximal, characteristic
2#21448043C211yesmaximal, characteristic, complemented
3#21447957C211yesmaximal, characteristic, complemented
4#21445213C211yesmaximal, characteristic, complemented
5#21443866C211yesmaximal, characteristic, complemented
6#21443781C211yesmaximal, characteristic, complemented
7C61×Q8⋊C16C211yesmaximal, characteristic, complemented
8#16840434—21no—
9#16840167C2211yescharacteristic
10#16840139—21no—
11#16840139C2211yescharacteristic
12#16839972—21no—
13#16839958—22no—
14#16839958—21no—
15C4×D488—21no—
16C4×D488C2211yescharacteristic
17#16839588—21no—
18#16839564—21no—
19#16839564—22no—
20#16839515C2211yescharacteristic
21#16839507—21no—
22C8×Q8×C61C2211yescharacteristic
23C61×C4⋊C16C2211yescharacteristic
24C4×C16×C61C2211yescharacteristic
25C2×#11758226—41no—
26C2×#11758202—41no—
27C2×D488—21no—
28C2×D488—22no—
29C2×D488D411yescharacteristic
30C2×D488D412yes—
31#14519777—21no—
32#14519774—22no—
33#14519774—21no—
34#14519752—42no—
35#14519752—41no—
36C4×D244—22no—
37C4×D244—21no—
38#14519711—21no—
39#14519711—21no—
40#14519711D411yescharacteristic
41C4⋊#92418—21no—
42#14519700—41no—
43#14519699—22no—
44#14519699—21no—
45C4×Q8×C61D411yescharacteristic
46C2×C16×C61—21no—
47C2×C16×C61—21no—
48C4⋊C8×C61—21no—
49C4⋊C8×C61D411yescharacteristic
50C4×C8×C61—21no—
51C4×C8×C61C2311yescharacteristic
52C2×Q8×C61D811yescharacteristic
53#11758226—82no—
54#11758226—82no—
55C2×D244—21no—
56C2×D244—22no—
57C2×D244—42no—
58C2×D244—41no—
59C2×D244—42no—
60C2×D244—41no—
61C2×C4×D61—42no—
62C2×C4×D61—41no—
63C2×Dic122—41no—
64C2×C8×C61—21no—
65C2×C8×C61C2×D411yescharacteristic
66C2×C8×C61C2×D411yescharacteristic
67C4⋊C4×C61—21no—
68C4⋊C4×C61D811yescharacteristic
69C42×C61—21no—
70C42×C61C2×D411yescharacteristic
71#11758202—82no—
72#11758198—42no—
73#11758198—41no—
74C4⋊Dic61—21no—
75C4⋊Dic61—41no—
76C4⋊Dic61—41no—
77C2×#92418—41no—
78D488—21no—
79D488—21no—
80D488—42no—
81D488—41no—
82D488—41no—
83D488—82no—
84C16×C61—41no—
85C16×C61—41no—
86C22×D61—42no—
87C22×D61—41no—
88Q8×C61—21no—
89Q8×C61D1612yes—
90C2×C4×C61—21no—
91C2×C4×C61—21no—
92C2×C4×C61#12011yescharacteristic
93C2×C4×C61C2×D811yescharacteristic
94C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
95C2×Dic61—41no—
96C2×Dic61—81no—
97D244—41no—
98D244—42no—
99D244—42no—
100D244—41no—
101D244—84no—
102D244—81no—
103D244—81no—
104D244—161no—
105D244—162no—
106C4×D61—84no—
107C4×D61—81no—
108C4×D61—81no—
109Dic122—81no—
110Dic122—81no—
111C8×C61—21no—
112C8×C61—41no—
113C8×C61#13511yescharacteristic
114C8×C61#13711yescharacteristic
115#92418—81no—
116#12380—611nomaximal, Sylow
117C22×C61#44711yescharacteristic
118C2×D61—81no—
119C2×D61—82no—
120C2×D61—82no—
121C2×D61—81no—
122C2×D61—82no—
123C2×D61—81no—
124C4×C61—22no—
125C4×C61—21no—
126C4×C61—41no—
127C4×C61#45211yescharacteristic
128C4×C61#45511yescharacteristic
129C4×C61C2×D1611yescharacteristic
130C4×C61#50911yescharacteristic
131Dic61—81no—
132Dic61—161no—
133#3391—611no—
134#2240—611no—
135#2224—611no—
136#1674—611no—
137#1361—611no—
138Q8⋊C16D6111yescharacteristic, complemented
139#1331—611no—
140C2×C61#164111yescharacteristic
141C2×C61#218411yescharacteristic
142C2×C61#219411yescharacteristic
143D61—161no—
144D61—162no—
145#550—1221no—
146#500—611no—
147#493—611no—
148#493—1221no—
149#462—1221no—
150#459—1222no—
151#459—1221no—
152C8×Q8C2×D6111yescharacteristic
153C4×D8—611no—
154C4×D8—1221no—
155#366—1221no—
156C4⋊C16C2×D6111yescharacteristic
157#357—1221no—
158#357—1222no—
159C4×C16C2×D6111yescharacteristic
160#335—611no—
161#332—1221no—
162C61#1238011yescharacteristic, complemented, Sylow
163C2×#41—2441no—
164C2×Q16—2441no—
165C2×D8—611no—
166C2×D8—612no—
167C2×D8—1222no—
168C2×D8—1221no—
169C4⋊Q8—1221no—
170#127—1222no—
171#127—1221no—
172#121—2442no—
173#121—2441no—
174C4×Q8D24411yescharacteristic
175C4×D4—1222no—
176C4×D4—1221no—
177C2×C16—21no—
178C2×C16—21no—
179#107—611no—
180#107—1221no—
181#107—1221no—
182C4⋊C8—21no—
183C4⋊C8—1221no—
184C4⋊C8D24411yescharacteristic
185#103—2441no—
186#102—1222no—
187#102—1221no—
188C4×C8—21no—
189C4×C8C22×D6111yescharacteristic
190#41—4882no—
191#41—4882no—
192C2×Q8—2441no—
193C2×Q8D48811yescharacteristic
194C2×D4—1221no—
195C2×D4—1222no—
196C2×D4—2442no—
197C2×D4—2442no—
198C2×D4—2441no—
199C2×D4—2441no—
200C22×C4—2442no—
201C22×C4—2441no—
202Q16—4882no—
203D8—1221no—
204D8—1221no—
205D8—4882no—
206D8—2442no—
207D8—2441no—
208D8—2441no—
209C2×C8—21no—
210C2×C8—2441no—
211C2×C8D4×D6111yescharacteristic
212C2×C8D4×D6111yescharacteristic
213C4⋊C4—21no—
214C4⋊C4—1221no—
215C4⋊C4—2441no—
216C4⋊C4—2441no—
217C4⋊C4D48811yescharacteristic
218C22⋊C4—2442no—
219C22⋊C4—2441no—
220C42—21no—
221C42C2×D24411yescharacteristic
222C16—41no—
223C16—41no—
224C23—2442no—
225C23—2441no—
226Q8—21no—
227Q8—4881no—
228Q8—4881no—
229Q8D97612yes—
230D4—9761no—
231D4—9762no—
232D4—4884no—
233D4—4881no—
234D4—4881no—
235D4—2441no—
236D4—2442no—
237D4—2441no—
238D4—2442no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—4884no—
242C2×C4—4881no—
243C2×C4—4881no—
244C2×C4—4881no—
245C2×C4—2441no—
246C2×C4#1451974711yescharacteristic
247C2×C4C2×D48811yescharacteristic
248C2×C4#1451980511yescharacteristic
249C8—21no—
250C8—41no—
251C8—4881no—
252C8#1451979811yescharacteristic
253C8#1451980911yescharacteristic
254C22—4882no—
255C22—4881no—
256C22—4882no—
257C22—4881no—
258C22—4881no—
259C22—4882no—
260C22#1683989811yescharacteristic
261C4—22no—
262C4—21no—
263C4—41no—
264C4—9761no—
265C4—4881no—
266C4#1683992711yescharacteristic
267C4#1683994111yescharacteristic
268C4C2×D97611yescharacteristic
269C4#1684020011yescharacteristic
270C2—9762no—
271C2—9761no—
272C2#2144501411yescharacteristic
273C2#2144773311yescharacteristic
274C2#2144779211yescharacteristic
subgroupG/H
centerC22#16839898
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×Q8⋊C16C2
socleC22×C61#447
prime radical#21458794C2
hypercenterQ8⋊C16D61
hypocenterC61#12380
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ Q8⋊C16

Complements

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

NH ≅ G/Ntype
2. #21448043C2semidirect
3. #21447957C2semidirect
4. #21445213C2semidirect
5. #21443866C2semidirect
6. #21443781C2semidirect
7. C61×Q8⋊C16C2semidirect
138. Q8⋊C16D61semidirect
162. C61#12380semidirect

Maximal subgroups

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

subgroupindex
#214587942
#214480432
#214479572
#214452132
#214438662
#214437812
C61×Q8⋊C162
#1238061