Finite Group Databasebeta

← #22571354

Subgroups — #22571354

Order 8576.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 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#16863951C211yesmaximal, characteristic
2C4⋊Q8×D67C211yesmaximal, characteristic
3#16863792C211yesmaximal, characteristic, complemented
4#16863748C211yesmaximal, characteristic, complemented
5#16863747C211yesmaximal, characteristic, complemented
6#16863728C214yesmaximal
7#16863680C214yesmaximal, complemented
8#16863447C211yesmaximal, characteristic
9#498×C67C211yesmaximal, characteristic, complemented
10C2×Q8×D67—21no—
11#14524043C2211yescharacteristic
12#14524043C2211yescharacteristic
13#14524042—21no—
14#14524042C2212yes—
15#14524041—21no—
16#14524034—21no—
17#14524023C2212yes—
18#14524019—21no—
19Q8×Dic67—21no—
20#14524008C2212yes—
21#14524001—21no—
22C4⋊C4×D67—21no—
23C4⋊C4×D67C2212yes—
24C42×D67C2211yescharacteristic
25#14523974C2214yes—
26Dic67⋊C8C2212yes—
27#14523968C2214yes—
28#14523966C2214yes—
29(C2×D67)⋊C8C2212yes—
30C4×C67⋊C8C2211yescharacteristic
31C4⋊Q8×C67C2211yescharacteristic
32C4⋊Q8×C67C2211yescharacteristic
33#106×C67C2214yes—
34C4×C8×C67C2211yescharacteristic
35C2×Dic134—21no—
36C2×Dic134—22no—
37C2×Dic134C2312yes—
38Q8×D67—44no—
39C2×C4×D67—22no—
40C2×C4×D67D411yescharacteristic
41C2×C4×D67D411yescharacteristic
42C2×C4×D67C2311yescharacteristic
43C2×C67⋊C8C2312yes—
44C4⋊Dic67—21no—
45C4⋊Dic67—21no—
46C4⋊Dic67—21no—
47C4⋊Dic67—22no—
48C4⋊Dic67C2312yes—
49Dic67⋊C4—22no—
50Dic67⋊C4—22no—
51Dic67⋊C4—22no—
52(C2×D67)⋊C4—22no—
53C4×Dic67—22no—
54C4×Dic67D411yescharacteristic
55C4×Dic67D411yescharacteristic
56C4×Dic67C2311yescharacteristic
57C2×Q8×C67—21no—
58C2×Q8×C67—21no—
59C2×C8×C67C2312yes—
60C4⋊C4×C67—21no—
61C4⋊C4×C67—21no—
62C4⋊C4×C67C2312yes—
63C4⋊C4×C67C2312yes—
64C42×C67C2311yescharacteristic
65C22×D67C2×D411yescharacteristic
66Q8×C67—42no—
67Q8×C67—42no—
68C2×C4×C67—22no—
69C2×C4×C67—22no—
70C2×C4×C67C2×D411yescharacteristic
71C2×C4×C67C2×D411yescharacteristic
72C2×C4×C67C2411yescharacteristic
73C2×Dic67—22no—
74C2×Dic67—22no—
75C2×Dic67C2×D411yescharacteristic
76C2×Dic67C2×D411yescharacteristic
77C2×Dic67C2×D411yescharacteristic
78C4×D67—22no—
79C4×D67—22no—
80C4×D67—44no—
81C4×D67SD1614yes—
82Dic134—24no—
83Dic134—42no—
84Dic134—44no—
85C8×C67—22no—
86C67⋊C8—22no—
87C22×C67C22×D411yescharacteristic
88C2×D67—21no—
89C2×D67—21no—
90C2×D67C2×SD1612yes—
91C4×C67—42no—
92C4×C67—42no—
93C4×C67C2×SD1612yes—
94C4×C67#13712yes—
95C4×C672−1+412yes—
96Dic67—21no—
97Dic67—21no—
98Dic67—42no—
99Dic67—42no—
100Dic67C2×SD1612yes—
101C2×C67#53611yescharacteristic
102C2×C67C22×SD1611yescharacteristic
103C2×C67C2×#13711yescharacteristic
104D67—22no—
105#3229—671nomaximal, Sylow
106C67#322911yescharacteristic, complemented, Sylow
107#533—671no—
108C2×C4⋊Q8—671no—
109#498—671no—
110#498D6711yescharacteristic, complemented
111#487—671no—
112#487—671no—
113#483—674no—
114#475—674no—
115#431—671no—
116C22×Q8—1341no—
117C4⋊Q8—672no—
118C4⋊Q8—671no—
119C4⋊Q8—671no—
120C4⋊Q8—1341no—
121C4⋊Q8—1341no—
122C4⋊Q8C2×D6711yescharacteristic
123C4⋊Q8C2×D6711yescharacteristic
124#125—1341no—
125C22⋊Q8—672no—
126C22⋊Q8—1341no—
127C4×Q8—672no—
128C4×Q8—1341no—
129#117—1341no—
130C2×C4⋊C4—672no—
131C2×C4⋊C4—1341no—
132C2×C42—671no—
133#106—674no—
134#106C2×D6714yes—
135C4⋊C8—672no—
136#103—674no—
137#103—674no—
138C22⋊C8—672no—
139C4×C8—671no—
140C4×C8C2×D6711yescharacteristic
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—2684no—
144C2×Q8—672no—
145C2×Q8—1342no—
146C2×Q8—1341no—
147C22×C4—671no—
148C22×C4—671no—
149C22×C4—671no—
150C22×C4—1342no—
151C2×C8—672no—
152C2×C8C22×D6712yes—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—672no—
156C4⋊C4—1342no—
157C4⋊C4—1342no—
158C4⋊C4—1342no—
159C4⋊C4—1342no—
160C4⋊C4—1341no—
161C4⋊C4—1341no—
162C4⋊C4—1341no—
163C4⋊C4C22×D6712yes—
164C4⋊C4C22×D6712yes—
165C22⋊C4—1342no—
166C42—671no—
167C42—671no—
168C42—671no—
169C42—1342no—
170C42C22×D6711yescharacteristic
171C23—671no—
172Q8—42no—
173Q8—42no—
174Q8—2684no—
175Q8—2682no—
176Q8—1344no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—2684no—
180C2×C4—671no—
181C2×C4—674no—
182C2×C4—671no—
183C2×C4—671no—
184C2×C4—1342no—
185C2×C4—1342no—
186C2×C4—1342no—
187C2×C4—1342no—
188C2×C4D4×D6711yescharacteristic
189C2×C4D4×D6711yescharacteristic
190C2×C4C23×D6711yescharacteristic
191C8—22no—
192C8—1342no—
193C22—672no—
194C22—1341no—
195C22—1341no—
196C22C2×D4×D6711yescharacteristic
197C4—42no—
198C4—42no—
199C4—2682no—
200C4—2682no—
201C4—672no—
202C4—1341no—
203C4—1341no—
204C4SD16×D6712yes—
205C4#1452407212yes—
206C4#1452409112yes—
207C2—1342no—
208C2#1686397111yescharacteristic
209C2C2×SD16×D6711yescharacteristic
210C2C2×#1452407211yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#498×C67C2
socleC22×C67C22×D4
hypercenter#498D67
hypocenterC67#3229
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
3. #16863792C2semidirect
4. #16863748C2semidirect
5. #16863747C2semidirect
7. #16863680C2semidirect
9. #498×C67C2semidirect
106. C67#3229semidirect
110. #498D67semidirect