Finite Group Databasebeta

← #22572862

Subgroups — #22572862

Order 8576.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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#16864120C211yesmaximal, characteristic, complemented
2#16864103C211yesmaximal, characteristic, complemented
3#16863752C212yesmaximal, complemented
4#16863746C211yesmaximal, characteristic, complemented
5#16863701C212yesmaximal, complemented
6#16863687C211yesmaximal, characteristic, complemented
7#16863613C212yesmaximal
8#16863520C211yesmaximal, characteristic, complemented
9SD16×Dic67C212yesmaximal, complemented
10C4×#11761613C211yesmaximal, characteristic
11#461×C67C211yesmaximal, characteristic, complemented
12C2×#11761613C2211yescharacteristic
13#14524043C2211yescharacteristic
14#14524038—22no—
15#14524034C2212yes—
16#14524032C2212yes—
17#14524023—21no—
18#14524022—21no—
19#14524020—22no—
20Q8×Dic67C2212yes—
21C4×Dic134C2211yescharacteristic
22C4×D268C2211yescharacteristic
23D4×Dic67C2212yes—
24#14524001—21no—
25C2×C4⋊Dic67—21no—
26#14523976C2211yescharacteristic
27#14523975C2212yes—
28Dic67⋊C8C2212yes—
29#14523967C2212yes—
30#14523967C2212yes—
31#14523965C2212yes—
32#14523964C2211yescharacteristic
33#14523963C2211yescharacteristic
34C4×C67⋊C8C2211yescharacteristic
35C8×Dic67C2212yes—
36C2×SD16×C67C2212yes—
37C4⋊Q8×C67C2211yescharacteristic
38C4×D4×C67C2211yescharacteristic
39C4⋊C8×C67C2211yescharacteristic
40#103×C67C2212yes—
41C2×Dic134—21no—
42C2×Dic134D411yescharacteristic
43C2×D268C2311yescharacteristic
44C22×Dic67—22no—
45C2×C4×D67—21no—
46#11761613—22no—
47C2×C67⋊C8C2311yescharacteristic
48C2×C67⋊C8C2311yescharacteristic
49C4⋊Dic67—21no—
50C4⋊Dic67—22no—
51C4⋊Dic67—22no—
52C4⋊Dic67D411yescharacteristic
53C4⋊Dic67C2311yescharacteristic
54C4⋊Dic67C2312yes—
55Dic67⋊C4—22no—
56Dic67⋊C4D412yes—
57C22⋊Dic67—22no—
58(C2×D67)⋊C4—21no—
59(C2×D67)⋊C4—22no—
60(C2×D67)⋊C4—22no—
61C4×Dic67—22no—
62C4×Dic67C2312yes—
63C2×Q8×C67C2312yes—
64C2×D4×C67C2311yescharacteristic
65C22×C4×C67—21no—
66SD16×C67—24no—
67C2×C8×C67C2312yes—
68C4⋊C4×C67—21no—
69C4⋊C4×C67—21no—
70C4⋊C4×C67C2311yescharacteristic
71C22⋊C4×C67—21no—
72C42×C67C2311yescharacteristic
73C23×C67—21no—
74C22×D67—21no—
75Q8×C67—22no—
76Q8×C67—22no—
77D4×C67—21no—
78D4×C67#4112yes—
79C2×C4×C67—21no—
80C2×C4×C67—22no—
81C2×C4×C67—42no—
82C2×C4×C67C2×D411yescharacteristic
83C2×C4×C67C2×D411yescharacteristic
84C2×C4×C67C2411yescharacteristic
85C2×Dic67—21no—
86C2×Dic67—22no—
87C2×Dic67—82no—
88C2×Dic67C2×D412yes—
89C2×Dic67C2×D412yes—
90D268—21no—
91D268—21no—
92C4×D67—42no—
93Dic134—22no—
94Dic134—81no—
95C8×C67—22no—
96C67⋊C8—21no—
97C67⋊C8—21no—
98C22×C67—22no—
99C22×C67—41no—
100C22×C67—41no—
101C22×C67C22×D411yescharacteristic
102C2×D67—41no—
103C2×D67—41no—
104C2×D67—41no—
105C4×C67—21no—
106C4×C67—42no—
107C4×C67—41no—
108C4×C67#13512yes—
109C4×C67C2×#4111yescharacteristic
110C4×C672−1+411yescharacteristic
111Dic67—22no—
112Dic67—22no—
113Dic67—41no—
114Dic67—42no—
115C2×C67—42no—
116C2×C67#54811yescharacteristic
117C2×C67C2×#13511yescharacteristic
118C2×C67#57711yescharacteristic
119D67—81no—
120#3360—671nomaximal, Sylow
121C67#336011yescharacteristic, complemented, Sylow
122#556—671no—
123#554—671no—
124#488—672no—
125#487—671no—
126#479—672no—
127#476—671no—
128#463—672no—
129#461D6711yescharacteristic, complemented
130C8×Q8—671no—
131C4×SD16—672no—
132C4×D8—671no—
133C2×SD16C2×D6712yes—
134C2×D8—671no—
135C4⋊Q8—671no—
136C4⋊Q8C2×D6711yescharacteristic
137#126—1342no—
138#125—672no—
139#124—672no—
140C22⋊Q8—1342no—
141C22⋊Q8—1341no—
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#106—672no—
152C4⋊C8—672no—
153C4⋊C8C2×D6711yescharacteristic
154#103—672no—
155#103—672no—
156#103C2×D6712yes—
157#102—671no—
158#102—672no—
159#102—671no—
160C4×C8—672no—
161C4×C8—671no—
162C2×Q8—671no—
163C2×Q8—1341no—
164C2×Q8C22×D6712yes—
165C2×D4—671no—
166C2×D4C22×D6711yescharacteristic
167C22×C4—21no—
168C22×C4—1341no—
169C22×C4—1342no—
170SD16—24no—
171D8—1342no—
172C2×C8—671no—
173C2×C8—671no—
174C2×C8C22×D6712yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—671no—
178C4⋊C4—672no—
179C4⋊C4—672no—
180C4⋊C4—671no—
181C4⋊C4—1342no—
182C4⋊C4—1342no—
183C4⋊C4—1342no—
184C4⋊C4—1341no—
185C4⋊C4C22×D6711yescharacteristic
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—22no—
197Q8—22no—
198Q8—5361no—
199Q8—1342no—
200D4—21no—
201D4—1341no—
202D4—1341no—
203D4#1176162712yes—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—2682no—
208C2×C4—5362no—
209C2×C4—672no—
210C2×C4—672no—
211C2×C4—1341no—
212C2×C4—1342no—
213C2×C4D4×D6711yescharacteristic
214C2×C4D4×D6711yescharacteristic
215C2×C4C23×D6711yescharacteristic
216C8—22no—
217C8—1341no—
218C8—1341no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—2681no—
223C22—2681no—
224C22—2681no—
225C22C2×D4×D6711yescharacteristic
226C4—21no—
227C4—42no—
228C4—41no—
229C4—2681no—
230C4—2682no—
231C4—1342no—
232C4—1342no—
233C4#1452406412yes—
234C4C2×#1176162711yescharacteristic
235C4#1452409011yescharacteristic
236C2—42no—
237C2—5361no—
238C2#1686405211yescharacteristic
239C2C2×#1452406411yescharacteristic
240C2#1686420511yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#461×C67C2
socleC22×C67C22×D4
hypercenter#461D67
hypocenterC67#3360
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #16864120C2semidirect
2. #16864103C2semidirect
3. #16863752C2semidirect
4. #16863746C2semidirect
5. #16863701C2semidirect
6. #16863687C2semidirect
8. #16863520C2semidirect
9. SD16×Dic67C2semidirect
11. #461×C67C2semidirect
121. C67#3360semidirect
129. #461D67semidirect