Finite Group Databasebeta

← #22571377

Subgroups — #22571377

Order 8576.

334 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 121 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#16863930C211yesmaximal, characteristic
2#16863795C212yesmaximal, complemented
3#16863753C212yesmaximal, complemented
4#16863731C211yesmaximal, characteristic, complemented
5#16863726C211yesmaximal, characteristic, complemented
6#16863714C211yesmaximal, characteristic
7#16863709C211yesmaximal, characteristic
8#16863700C212yesmaximal, complemented
9#16863693C212yesmaximal, complemented
10#16863449C211yesmaximal, characteristic, complemented
11C2×#125×C67C211yesmaximal, characteristic, complemented
12#14524037—21no—
13#14524035C2211yescharacteristic
14#14524031C2211yescharacteristic
15#14524026—21no—
16#14524022C2211yescharacteristic
17#14524017C2211yescharacteristic
18C4×Dic134C2211yescharacteristic
19C4×D268C2211yescharacteristic
20C4×C22⋊D67—21no—
21#14524000—21no—
22#14523976C2212yes—
23#14523976C2212yes—
24#14523974C2212yes—
25#14523974C2212yes—
26C4⋊(C67⋊C8)C2212yes—
27#14523966C2212yes—
28#14523966C2212yes—
29#14523963C2212yes—
30#14523963C2212yes—
31C22⋊(C67⋊C8)C2211yescharacteristic
32C22⋊(C67⋊C8)C2211yescharacteristic
33C4×C67⋊C8C2212yes—
34#125×C67—21no—
35#125×C67—21no—
36#125×C67C2212yes—
37#125×C67C2212yes—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67C2211yescharacteristic
41C2×C4⋊C4×C67C2211yescharacteristic
42C2×C42×C67C2211yescharacteristic
43C2×Dic134C2311yescharacteristic
44C2×D268C2311yescharacteristic
45C2×C22⋊D67—21no—
46C2×C4×D67—21no—
47C2×C67⋊C8C2312yes—
48C2×C67⋊C8C2312yes—
49C4⋊Dic67C2311yescharacteristic
50C4⋊Dic67C2311yescharacteristic
51Dic67⋊C4—21no—
52Dic67⋊C4—21no—
53Dic67⋊C4—21no—
54Dic67⋊C4—21no—
55C22⋊Dic67—21no—
56(C2×D67)⋊C4—21no—
57(C2×D67)⋊C4—21no—
58(C2×D67)⋊C4—21no—
59(C2×D67)⋊C4—21no—
60C4×Dic67—21no—
61C22×C4×C67—21no—
62C22×C4×C67—21no—
63C22×C4×C67D411yescharacteristic
64C22×C4×C67D411yescharacteristic
65C22×C4×C67C2311yescharacteristic
66C4⋊C4×C67—22no—
67C4⋊C4×C67—22no—
68C4⋊C4×C67—22no—
69C4⋊C4×C67—22no—
70C4⋊C4×C67—21no—
71C4⋊C4×C67—21no—
72C4⋊C4×C67C2312yes—
73C4⋊C4×C67C2312yes—
74C42×C67D411yescharacteristic
75C42×C67D411yescharacteristic
76C42×C67C2311yescharacteristic
77C42×C67C2311yescharacteristic
78C23×C67C2×D411yescharacteristic
79C22×D67—21no—
80C2×C4×C67—22no—
81C2×C4×C67—22no—
82C2×C4×C67—22no—
83C2×C4×C67—22no—
84C2×C4×C67—41no—
85C2×C4×C67—42no—
86C2×C4×C67—42no—
87C2×C4×C67C2×D411yescharacteristic
88C2×C4×C67C2×D411yescharacteristic
89C2×C4×C67C2×D411yescharacteristic
90C2×C4×C67C2×D411yescharacteristic
91C2×C4×C67C2×D411yescharacteristic
92C2×C4×C67C2411yescharacteristic
93C22⋊D67—42no—
94C2×Dic67—21no—
95C2×Dic67—21no—
96C2×Dic67—21no—
97D268—22no—
98C4×D67—42no—
99Dic134—22no—
100C67⋊C8—22no—
101C67⋊C8—22no—
102C22×C67—21no—
103C22×C67—21no—
104C22×C67#13512yes—
105C22×C67C22×D411yescharacteristic
106C2×D67—41no—
107C2×D67—41no—
108C2×D67—41no—
109C4×C67—21no—
110C4×C67—21no—
111C4×C67—21no—
112C4×C67—21no—
113C4×C67—21no—
114C4×C67—42no—
115C4×C67—42no—
116C4×C672−1+412yes—
117Dic67—41no—
118Dic67—41no—
119Dic67—41no—
120C2×C67—22no—
121C2×C67#53611yescharacteristic
122C2×C67C2×#13511yescharacteristic
123C2×C67#57511yescharacteristic
124D67—81no—
125#3232—671nomaximal, Sylow
126C67#323211yescharacteristic, complemented, Sylow
127#529—671no—
128C2×#125D6711yescharacteristic, complemented
129#499—672no—
130#488—672no—
131#484—671no—
132#483—671no—
133#481—671no—
134#480—671no—
135#479—672no—
136#478—672no—
137#431—671no—
138#126—1341no—
139#125—21no—
140#125—21no—
141#125—671no—
142#125C2×D6712yes—
143#125C2×D6712yes—
144#124—671no—
145#123—1341no—
146C22⋊Q8—671no—
147#121—671no—
148C4×Q8—671no—
149C4×D4—671no—
150C4×D4—1341no—
151#117—1341no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4C2×D6711yescharacteristic
155C2×C4⋊C4C2×D6711yescharacteristic
156C2×C42C2×D6711yescharacteristic
157#107—672no—
158#107—672no—
159#106—672no—
160#106—672no—
161C4⋊C8—672no—
162#103—672no—
163#103—672no—
164#102—672no—
165#102—672no—
166C22⋊C8—671no—
167C22⋊C8—671no—
168C4×C8—672no—
169C2×Q8—671no—
170C2×D4—671no—
171C2×D4—1341no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—1341no—
175C22×C4C22⋊D6711yescharacteristic
176C22×C4C22⋊D6711yescharacteristic
177C22×C4C22×D6711yescharacteristic
178C2×C8—672no—
179C2×C8—672no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—671no—
187C4⋊C4—671no—
188C4⋊C4—1341no—
189C4⋊C4—1341no—
190C4⋊C4—1341no—
191C4⋊C4—1341no—
192C4⋊C4C22×D6712yes—
193C4⋊C4C22×D6712yes—
194C22⋊C4—1341no—
195C22⋊C4—1341no—
196C22⋊C4—1341no—
197C22⋊C4—1341no—
198C22⋊C4—1341no—
199C42—1341no—
200C42C22⋊D6711yescharacteristic
201C42C22⋊D6711yescharacteristic
202C42C22×D6711yescharacteristic
203C42C22×D6711yescharacteristic
204C23—1341no—
205C23C2×C22⋊D6711yescharacteristic
206Q8—1342no—
207D4—2682no—
208D4—1342no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—2682no—
217C2×C4—1341no—
218C2×C4—1341no—
219C2×C4—1341no—
220C2×C4C2×C22⋊D6711yescharacteristic
221C2×C4C2×C22⋊D6711yescharacteristic
222C2×C4C2×C22⋊D6711yescharacteristic
223C2×C4C2×C22⋊D6711yescharacteristic
224C2×C4C2×C22⋊D6711yescharacteristic
225C2×C4C23×D6711yescharacteristic
226C8—1342no—
227C8—1342no—
228C22—21no—
229C22—21no—
230C22—2681no—
231C22—2681no—
232C22—2681no—
233C22#1452406312yes—
234C22C22×C22⋊D6711yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—42no—
241C4—42no—
242C4—2681no—
243C4—2681no—
244C4—2681no—
245C4#1452409112yes—
246C2—22no—
247C2—5361no—
248C2#1686397011yescharacteristic
249C2C2×#1452406311yescharacteristic
250C2#1686419411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
FittingC2×#125×C67C2
socleC22×C67C22×D4
hypercenterC2×#125D67
hypocenterC67#3232
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #16863795C2semidirect
3. #16863753C2semidirect
4. #16863731C2semidirect
5. #16863726C2semidirect
8. #16863700C2semidirect
9. #16863693C2semidirect
10. #16863449C2semidirect
11. C2×#125×C67C2semidirect
126. C67#3232semidirect
128. C2×#125D67semidirect