Finite Group Databasebeta

← #22572857

Subgroups — #22572857

Order 8576.

350 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 115 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#16864116C211yesmaximal, characteristic, complemented
2#16864107C211yesmaximal, characteristic, complemented
3#16863749C212yesmaximal, complemented
4#16863748C211yesmaximal, characteristic, complemented
5#16863706C212yesmaximal, complemented
6#16863610C212yesmaximal
7#16863602C211yesmaximal, characteristic, complemented
8#16863520C211yesmaximal, characteristic, complemented
9#16863483C212yesmaximal, complemented
10C4×#11761613C211yesmaximal, characteristic
11#476×C67C211yesmaximal, characteristic, complemented
12C2×#11761617C2212yes—
13C2×#11761613C2211yescharacteristic
14#14524043C2211yescharacteristic
15#14524036—22no—
16#14524034C2212yes—
17#14524030C2212yes—
18#14524023—21no—
19#14524022—21no—
20#14524019—22no—
21C4×Dic134C2211yescharacteristic
22#14524008C2212yes—
23C4×D268C2211yescharacteristic
24#14524005C2212yes—
25#14524002—21no—
26C4⋊C4×D67—21no—
27#14523976C2211yescharacteristic
28Dic67⋊C8C2212yes—
29#14523968C2212yes—
30#14523966C2212yes—
31#14523966C2212yes—
32#14523964C2211yescharacteristic
33#14523963C2211yescharacteristic
34C4×C67⋊C8C2211yescharacteristic
35C8×Dic67C2212yes—
36C4⋊Q8×C67C2211yescharacteristic
37C4×D4×C67C2211yescharacteristic
38#106×C67C2212yes—
39C4⋊C8×C67C2211yescharacteristic
40#102×C67C2212yes—
41C2×Dic134D411yescharacteristic
42C2×Dic134C2312yes—
43C2×D268C2311yescharacteristic
44C2×C4×D67—21no—
45C2×C4×D67—22no—
46#11761617—24no—
47#11761613—22no—
48C2×C67⋊C8C2311yescharacteristic
49C2×C67⋊C8C2311yescharacteristic
50C4⋊Dic67—21no—
51C4⋊Dic67—21no—
52C4⋊Dic67—22no—
53C4⋊Dic67D411yescharacteristic
54C4⋊Dic67C2311yescharacteristic
55Dic67⋊C4—22no—
56Dic67⋊C4—22no—
57Dic67⋊C4D412yes—
58C22⋊Dic67—22no—
59C22⋊Dic67—22no—
60(C2×D67)⋊C4—21no—
61(C2×D67)⋊C4—22no—
62C4×Dic67—22no—
63C4×Dic67C2312yes—
64C2×Q8×C67—21no—
65C2×D4×C67C2311yescharacteristic
66C22×C4×C67—21no—
67C2×C8×C67C2312yes—
68C4⋊C4×C67—21no—
69C4⋊C4×C67C2312yes—
70C4⋊C4×C67C2311yescharacteristic
71C22⋊C4×C67—21no—
72C42×C67C2311yescharacteristic
73C23×C67—21no—
74C22×D67—21no—
75Q8×C67—81no—
76D4×C67—21no—
77D4×C67—21no—
78C2×C4×C67—21no—
79C2×C4×C67—22no—
80C2×C4×C67—42no—
81C2×C4×C67C2×D411yescharacteristic
82C2×C4×C67C2×D411yescharacteristic
83C2×C4×C67C2411yescharacteristic
84C2×Dic67—22no—
85C2×Dic67—21no—
86C2×Dic67C2×D412yes—
87C2×Dic67C2×D412yes—
88D268—21no—
89D268#4112yes—
90C4×D67—42no—
91C4×D67—82no—
92Dic134—22no—
93Dic134—22no—
94Dic134—22no—
95C8×C67—22no—
96C67⋊C8—21no—
97C67⋊C8—21no—
98C22×C67—41no—
99C22×C67—41no—
100C22×C67—41no—
101C22×C67C22×D411yescharacteristic
102C2×D67—22no—
103C2×D67—41no—
104C2×D67—41no—
105C4×C67—21no—
106C4×C67—41no—
107C4×C67—42no—
108C4×C67#13512yes—
109C4×C67C2×#4111yescharacteristic
110C4×C672−1+411yescharacteristic
111Dic67—22no—
112Dic67—22no—
113Dic67—42no—
114Dic67—41no—
115C2×C67—81no—
116C2×C67#54811yescharacteristic
117C2×C67C2×#13511yescharacteristic
118C2×C67#57711yescharacteristic
119D67—42no—
120#3360—671nomaximal, Sylow
121C67#336011yescharacteristic, complemented, Sylow
122#556—671no—
123#554—671no—
124#488—672no—
125#487—671no—
126#479—672no—
127#476D6711yescharacteristic, complemented
128#463—672no—
129#461—671no—
130C8×Q8—671no—
131C4×SD16—672no—
132C4×D8—671no—
133C2×SD16—672no—
134C2×D8—671no—
135C4⋊Q8—671no—
136C4⋊Q8C2×D6711yescharacteristic
137#126—1342no—
138#125—672no—
139#124—672no—
140C22⋊Q8—1341no—
141C22⋊Q8—1342no—
142C22⋊Q8—1341no—
143C4×Q8—671no—
144C4×Q8—672no—
145C4×D4—672no—
146C4×D4—671no—
147C4×D4C2×D6711yescharacteristic
148#117—1341no—
149C2×C4⋊C4—1341no—
150#107—671no—
151#106C2×D6712yes—
152C4⋊C8—672no—
153C4⋊C8C2×D6711yescharacteristic
154#103—672no—
155#103—672no—
156#103—672no—
157#102—671no—
158#102—671no—
159#102C2×D6712yes—
160C4×C8—672no—
161C4×C8—671no—
162C2×Q8—21no—
163C2×Q8—671no—
164C2×Q8—672no—
165C2×D4—671no—
166C2×D4C22×D6711yescharacteristic
167C22×C4—21no—
168C22×C4—1342no—
169C22×C4—1341no—
170SD16—1344no—
171D8—1342no—
172C2×C8—671no—
173C2×C8—671no—
174C2×C8C22×D6712yes—
175C4⋊C4—21no—
176C4⋊C4—671no—
177C4⋊C4—672no—
178C4⋊C4—671no—
179C4⋊C4—1342no—
180C4⋊C4—1342no—
181C4⋊C4—1342no—
182C4⋊C4—1341no—
183C4⋊C4—1341no—
184C4⋊C4C22×D6711yescharacteristic
185C4⋊C4C22×D6712yes—
186C22⋊C4—21no—
187C22⋊C4—1342no—
188C22⋊C4—1342no—
189C22⋊C4—1342no—
190C22⋊C4—1341no—
191C42—672no—
192C42—1342no—
193C42C22×D6711yescharacteristic
194C23—21no—
195C23—1341no—
196Q8—81no—
197Q8—1342no—
198Q8—1342no—
199Q8—1342no—
200D4—21no—
201D4—21no—
202D4—672no—
203D4—1341no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—2682no—
208C2×C4—5362no—
209C2×C4—672no—
210C2×C4—672no—
211C2×C4—1342no—
212C2×C4—1341no—
213C2×C4D4×D6711yescharacteristic
214C2×C4D4×D6711yescharacteristic
215C2×C4C23×D6711yescharacteristic
216C8—22no—
217C8—1341no—
218C8—1341no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—2681no—
223C22—2681no—
224C22—1342no—
225C22C2×D4×D6711yescharacteristic
226C4—21no—
227C4—41no—
228C4—42no—
229C4—2682no—
230C4—2681no—
231C4—1342no—
232C4—1342no—
233C4#1452406412yes—
234C4#41×D6711yescharacteristic
235C4#1452409111yescharacteristic
236C2—81no—
237C2—2682no—
238C2#1686405611yescharacteristic
239C2C2×#1452406411yescharacteristic
240C2#1686420611yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#476×C67C2
socleC22×C67C22×D4
hypercenter#476D67
hypocenterC67#3360
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #16864116C2semidirect
2. #16864107C2semidirect
3. #16863749C2semidirect
4. #16863748C2semidirect
5. #16863706C2semidirect
7. #16863602C2semidirect
8. #16863520C2semidirect
9. #16863483C2semidirect
11. #476×C67C2semidirect
121. C67#3360semidirect
127. #476D67semidirect