Finite Group Databasebeta

← #22571401

Subgroups — #22571401

Order 8576.

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

normal subgroups only

This group has 368 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#16863946C211yesmaximal, characteristic
2C2×#14524035C211yesmaximal, characteristic, complemented
3#16863805C212yesmaximal, complemented
4#16863761C212yesmaximal, complemented
5#16863715C212yesmaximal, complemented
6#16863710C212yesmaximal, complemented
7#16863697C214yesmaximal, complemented
8#432×C67C211yesmaximal, characteristic, complemented
9C2×#11761628—21no—
10#14524043C2211yescharacteristic
11#14524040C2211yescharacteristic
12#14524035—21no—
13#14524035—21no—
14#14524035C2212yes—
15#14524035C2212yes—
16#14524031—21no—
17#14524017—21no—
18#14524017C2212yes—
19C4×D268—21no—
20C4×D268C2212yes—
21C2×C4⋊Dic67C2211yescharacteristic
22C2×C4⋊Dic67C2211yescharacteristic
23C2×Dic67⋊C4—21no—
24C2×Dic67⋊C4—21no—
25#14523977C2214yes—
26#14523975C2214yes—
27#14523965C2214yes—
28#14523965C2214yes—
29C2×C42×C67C2211yescharacteristic
30C4⋊C8×C67C2212yes—
31C22⋊C8×C67C2212yes—
32#97×C67C2212yes—
33#11761628—44no—
34C2×Dic134—21no—
35C2×D268—21no—
36C2×D268—21no—
37C2×D268C2312yes—
38C2×C22⋊D67—22no—
39C22×Dic67—21no—
40C22×Dic67—21no—
41C2×C4×D67—22no—
42C4⋊Dic67—21no—
43C4⋊Dic67—21no—
44C4⋊Dic67—21no—
45C4⋊Dic67C2312yes—
46C4⋊Dic67C2312yes—
47C4⋊Dic67C2312yes—
48Dic67⋊C4—22no—
49Dic67⋊C4—22no—
50Dic67⋊C4—22no—
51Dic67⋊C4—22no—
52(C2×D67)⋊C4—22no—
53(C2×D67)⋊C4—22no—
54C22×C4×C67D411yescharacteristic
55C22×C4×C67D411yescharacteristic
56C22×C4×C67C2311yescharacteristic
57C2×C8×C67C2314yes—
58C42×C67D411yescharacteristic
59C42×C67D411yescharacteristic
60C42×C67C2311yescharacteristic
61C42×C67C2311yescharacteristic
62C23×C67C2×D411yescharacteristic
63C22×D67—22no—
64C2×C4×C67—22no—
65C2×C4×C67—22no—
66C2×C4×C67—41no—
67C2×C4×C67C2×D411yescharacteristic
68C2×C4×C67C2×D411yescharacteristic
69C2×C4×C67C2×D411yescharacteristic
70C2×C4×C67C2×D411yescharacteristic
71C2×C4×C67C2×D411yescharacteristic
72C2×C4×C67C2411yescharacteristic
73C22⋊D67—44no—
74C2×Dic67—22no—
75C2×Dic67—22no—
76C2×Dic67—22no—
77C2×Dic67—42no—
78C2×Dic67—42no—
79D268—24no—
80D268—42no—
81D268—42no—
82C4×D67—44no—
83Dic134—42no—
84C8×C67—24no—
85C22×C67—21no—
86C22×C67—21no—
87C22×C67AGL(1,ℤ/8)12yes—
88C22×C67C22×D411yescharacteristic
89C2×D67—42no—
90C2×D67—42no—
91C2×D67—42no—
92C4×C67—21no—
93C4×C67—21no—
94C4×C67—21no—
95C4×C67—21no—
96C4×C67—21no—
97C4×C672−1+412yes—
98Dic67—42no—
99Dic67—42no—
100Dic67—42no—
101C2×C67—22no—
102C2×C67#53611yescharacteristic
103C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
104C2×C67#57511yescharacteristic
105D67—82no—
106#3234—671nomaximal, Sylow
107C67#323411yescharacteristic, complemented, Sylow
108#532—671no—
109C2×#125—671no—
110#501—672no—
111#490—672no—
112#481—672no—
113#480—672no—
114#478—674no—
115#432D6711yescharacteristic, complemented
116C2×#41—1341no—
117C4⋊Q8—671no—
118#127—671no—
119#125—672no—
120#125—672no—
121#125—1341no—
122#125—1341no—
123#124—1341no—
124#121—672no—
125#121—1341no—
126C4×D4—672no—
127C4×D4—1341no—
128C2×C4⋊C4—671no—
129C2×C4⋊C4—671no—
130C2×C4⋊C4—1341no—
131C2×C4⋊C4—1341no—
132C2×C42C2×D6711yescharacteristic
133#107—674no—
134#106—674no—
135C4⋊C8C2×D6712yes—
136#102—674no—
137#102—674no—
138C22⋊C8C2×D6712yes—
139#97C2×D6712yes—
140#41—2684no—
141C2×Q8—1341no—
142C2×D4—672no—
143C2×D4—1342no—
144C2×D4—1341no—
145C2×D4—1341no—
146C22×C4—1341no—
147C22×C4—1341no—
148C22×C4—1342no—
149C22×C4D26811yescharacteristic
150C22×C4D26811yescharacteristic
151C22×C4C22×D6711yescharacteristic
152C2×C8C22×D6714yes—
153C4⋊C4—672no—
154C4⋊C4—672no—
155C4⋊C4—672no—
156C4⋊C4—1341no—
157C4⋊C4—1341no—
158C4⋊C4—1342no—
159C4⋊C4—1342no—
160C4⋊C4—1342no—
161C4⋊C4—1342no—
162C4⋊C4—1341no—
163C22⋊C4—1342no—
164C22⋊C4—1342no—
165C42D26811yescharacteristic
166C42D26811yescharacteristic
167C42C22×D6711yescharacteristic
168C42C22×D6711yescharacteristic
169C23—1342no—
170C23C2×D26811yescharacteristic
171Q8—2682no—
172D4—2682no—
173D4—2682no—
174D4—2684no—
175D4—1344no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—2684no—
180C2×C4—2682no—
181C2×C4—2682no—
182C2×C4—1342no—
183C2×C4—1342no—
184C2×C4—1342no—
185C2×C4C2×D26811yescharacteristic
186C2×C4C2×D26811yescharacteristic
187C2×C4C2×D26811yescharacteristic
188C2×C4C2×D26811yescharacteristic
189C2×C4C2×D26811yescharacteristic
190C2×C4C23×D6711yescharacteristic
191C8—24no—
192C22—21no—
193C22—21no—
194C22—2682no—
195C22—2682no—
196C22—2682no—
197C22#1452407112yes—
198C22C22×D26811yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—21no—
203C4—21no—
204C4—2682no—
205C4—2682no—
206C4—2682no—
207C4#1452409012yes—
208C2—22no—
209C2—5362no—
210C2#1686397211yescharacteristic
211C2C2×#1452407111yescharacteristic
212C2#1686419611yescharacteristic
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#432×C67C2
socleC22×C67C22×D4
hypercenter#432D67
hypocenterC67#3234
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. C2×#14524035C2semidirect
3. #16863805C2semidirect
4. #16863761C2semidirect
5. #16863715C2semidirect
6. #16863710C2semidirect
7. #16863697C2semidirect
8. #432×C67C2semidirect
107. C67#3234semidirect
115. #432D67semidirect