Finite Group Databasebeta

← #22571656

Subgroups — #22571656

Order 8576.

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

normal subgroups only

This group has 346 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#16864008C211yesmaximal, characteristic, complemented
2#16864001C211yesmaximal, characteristic
3#16863704C212yesmaximal, complemented
4#16863697C212yesmaximal, complemented
5#16863642C212yesmaximal, complemented
6#16863625C212yesmaximal, complemented
7#16863602C212yesmaximal, complemented
8#16863597C212yesmaximal, complemented
9#424×C67C211yesmaximal, characteristic, complemented
10C2×#11761617C2212yes—
11C2×#11761617C2212yes—
12#14524043C2211yescharacteristic
13#14524040C2211yescharacteristic
14#14524036—22no—
15#14524035C2211yescharacteristic
16#14524035C2211yescharacteristic
17#14524033—22no—
18#14524028—22no—
19#14524022C2212yes—
20#14524017C2212yes—
21#14524016—22no—
22C4×Dic134C2212yes—
23C4×D268C2212yes—
24#14523977C2212yes—
25#14523975C2212yes—
26#14523968C2212yes—
27#14523968C2212yes—
28#14523965C2212yes—
29#14523965C2212yes—
30C2×M16×C67C2211yescharacteristic
31C22×C8×C67C2211yescharacteristic
32#117×C67C2211yescharacteristic
33C4⋊C8×C67C2212yes—
34C4⋊C8×C67C2212yes—
35C2×Dic134C2312yes—
36C2×D268—22no—
37C2×D268C2312yes—
38C2×C22⋊D67—22no—
39C2×C4×D67—22no—
40#11761617—42no—
41#11761617—42no—
42C4⋊Dic67—22no—
43C4⋊Dic67C2312yes—
44C4⋊Dic67C2312yes—
45Dic67⋊C4—22no—
46Dic67⋊C4—22no—
47Dic67⋊C4—22no—
48Dic67⋊C4—22no—
49C22⋊Dic67—22no—
50(C2×D67)⋊C4—22no—
51(C2×D67)⋊C4—22no—
52C4×Dic67—22no—
53C22×C4×C67C2311yescharacteristic
54M16×C67—41no—
55C2×C8×C67—41no—
56C2×C8×C67C2312yes—
57C2×C8×C67C2312yes—
58C4⋊C4×C67D412yes—
59C22⋊C4×C67D412yes—
60C42×C67C2311yescharacteristic
61C42×C67C2311yescharacteristic
62C23×C67C2×D411yescharacteristic
63C22×D67—22no—
64C2×C4×C67—21no—
65C2×C4×C67—21no—
66C2×C4×C67C2×D412yes—
67C2×C4×C67C2×D412yes—
68C2×C4×C67C2×D411yescharacteristic
69C2×C4×C67C2411yescharacteristic
70C22⋊D67—82no—
71C2×Dic67—22no—
72C2×Dic67—22no—
73C2×Dic67—22no—
74D268—22no—
75D268—22no—
76D268—82no—
77C4×D67—82no—
78Dic134—22no—
79Dic134—22no—
80C8×C67—22no—
81C8×C67—22no—
82C22×C67—21no—
83C22×C67—22no—
84C22×C67C22×D411yescharacteristic
85C2×D67—42no—
86C2×D67—42no—
87C2×D67—42no—
88C4×C67—21no—
89C4×C67—22no—
90C4×C67—22no—
91C4×C67H(5,𝔽2)11yescharacteristic
92C4×C672−1+411yescharacteristic
93Dic67—42no—
94Dic67—42no—
95Dic67—42no—
96C2×C67—41no—
97C2×C67#53711yescharacteristic
98C2×C67#57712yes—
99D67—82no—
100#3260—671nomaximal, Sylow
101C67#326011yescharacteristic, complemented, Sylow
102#541—671no—
103#540—671no—
104#479—672no—
105#478—672no—
106#468—672no—
107#465—672no—
108#461—672no—
109#460—672no—
110#424D6711yescharacteristic, complemented
111C2×SD16—672no—
112C2×SD16—672no—
113C2×M16C2×D6711yescharacteristic
114C22×C8C2×D6711yescharacteristic
115C4⋊Q8—671no—
116#127—671no—
117#126—1342no—
118#125—671no—
119#125—671no—
120#125—1342no—
121#123—1342no—
122C22⋊Q8—672no—
123#121—672no—
124#121—1342no—
125C4×Q8—672no—
126C4×D4—672no—
127#117C2×D6711yescharacteristic
128#107—672no—
129#106—672no—
130C4⋊C8C2×D6712yes—
131C4⋊C8C2×D6712yes—
132#103—672no—
133#103—672no—
134#102—672no—
135#102—672no—
136C2×Q8—672no—
137C2×D4—672no—
138C2×D4—1342no—
139C2×D4—1342no—
140C22×C4—1342no—
141C22×C4C22×D6711yescharacteristic
142SD16—2682no—
143SD16—2682no—
144M16—41no—
145C2×C8—41no—
146C2×C8C22×D6712yes—
147C2×C8C22×D6712yes—
148C4⋊C4—672no—
149C4⋊C4—672no—
150C4⋊C4—1342no—
151C4⋊C4—1342no—
152C4⋊C4—1342no—
153C4⋊C4—1342no—
154C4⋊C4—1342no—
155C4⋊C4D26812yes—
156C22⋊C4—1342no—
157C22⋊C4—1342no—
158C22⋊C4—1342no—
159C22⋊C4D26812yes—
160C42—1342no—
161C42C22×D6711yescharacteristic
162C42C22×D6711yescharacteristic
163C23—1342no—
164C23C2×D26811yescharacteristic
165Q8—1342no—
166Q8—1342no—
167D4—5362no—
168D4—5362no—
169D4—1342no—
170D4—1342no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—5362no—
174C2×C4—1342no—
175C2×C4—1342no—
176C2×C4—1342no—
177C2×C4C2×D26812yes—
178C2×C4C2×D26812yes—
179C2×C4C2×D26811yescharacteristic
180C2×C4C23×D6711yescharacteristic
181C8—22no—
182C8—22no—
183C22—22no—
184C22—21no—
185C22—2682no—
186C22—2682no—
187C22—2682no—
188C22C22×D26811yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—2682no—
193C4—2682no—
194C4—2682no—
195C4#1452408811yescharacteristic
196C4#1452409111yescharacteristic
197C2—41no—
198C2—5362no—
199C2#1686397711yescharacteristic
200C2#1686420812yes—
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#424×C67C2
socleC22×C67C22×D4
hypercenter#424D67
hypocenterC67#3260
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #16864008C2semidirect
3. #16863704C2semidirect
4. #16863697C2semidirect
5. #16863642C2semidirect
6. #16863625C2semidirect
7. #16863602C2semidirect
8. #16863597C2semidirect
9. #424×C67C2semidirect
101. C67#3260semidirect
110. #424D67semidirect