Finite Group Databasebeta

← #22561476

Subgroups — #22561476

Order 8576.

256 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C22×#11761618C211yesmaximal, characteristic
2#16863423C211yesmaximal, characteristic
3#16863404C211yesmaximal, characteristic
4C2×#14523962C211yesmaximal, characteristic
5C2×#14523961C211yesmaximal, characteristic
6#16863160C211yesmaximal, characteristic
7#419×C67C211yesmaximal, characteristic
8C22×Dic134C2211yescharacteristic
9C2×#11761618—22no—
10C2×#11761618—22no—
11C2×#11761618—22no—
12C2×#11761612C2211yescharacteristic
13C2×#11761612C2211yescharacteristic
14C22×C67⋊C8C2211yescharacteristic
15C4⋊(C67⋊C8)—22no—
16#14523966—22no—
17#14523962—22no—
18#14523961—22no—
19C22×Q8×C67C2211yescharacteristic
20C2×M16×C67C2211yescharacteristic
21#117×C67C2211yescharacteristic
22#103×C67—22no—
23C2×Dic134—21no—
24C2×Dic134—21no—
25C2×Dic134—21no—
26C2×Dic134—21no—
27C2×Dic134—21no—
28C2×Dic134D412yes—
29C22×Dic67—21no—
30#11761618—28no—
31#11761612—21no—
32#11761612—21no—
33#11761612—21no—
34#11761612—21no—
35C2×C67⋊C8—21no—
36C2×C67⋊C8—21no—
37C2×C67⋊C8—21no—
38C2×C67⋊C8—21no—
39C2×C67⋊C8D412yes—
40C2×Q8×C67—21no—
41C2×Q8×C67—21no—
42C2×Q8×C67—21no—
43C2×Q8×C67—21no—
44C2×Q8×C67—21no—
45C2×Q8×C67D412yes—
46C22×C4×C67—21no—
47C22×C4×C67C2311yescharacteristic
48M16×C67—21no—
49M16×C67—21no—
50C2×C8×C67—21no—
51C4⋊C4×C67D412yes—
52C22⋊C4×C67—21no—
53C42×C67—21no—
54C23×C67#4111yescharacteristic
55Q8×C67—22no—
56Q8×C67—22no—
57Q8×C67—41no—
58Q8×C67—41no—
59C2×C4×C67—22no—
60C2×C4×C67—22no—
61C2×C4×C67—41no—
62C2×C4×C67—41no—
63C2×C4×C67C2×D411yescharacteristic
64C2×C4×C67C2×D411yescharacteristic
65C2×C4×C67C2×D411yescharacteristic
66C2×C4×C67C2×D411yescharacteristic
67C2×C4×C67#4111yescharacteristic
68C2×C4×C67#4111yescharacteristic
69C2×Dic67—22no—
70C2×Dic67—41no—
71C2×Dic67—41no—
72Dic134—22no—
73Dic134—22no—
74Dic134—41no—
75Dic134—41no—
76C8×C67—41no—
77C67⋊C8—22no—
78C67⋊C8—41no—
79C67⋊C8—41no—
80C22×C67—22no—
81C22×C67#12111yescharacteristic
82C22×C67#12311yescharacteristic
83C22×C67#12311yescharacteristic
84C4×C67—42no—
85C4×C67—42no—
86C4×C67#12011yescharacteristic
87C4×C67#12111yescharacteristic
88C4×C67#12111yescharacteristic
89C4×C67#12411yescharacteristic
90Dic67—42no—
91C2×C67—21no—
92C2×C67—21no—
93C2×C67#39411yescharacteristic
94C2×C67#47312yes—
95#2047—671nomaximal, Sylow
96C67#204711yescharacteristic, complemented, Sylow
97C22×Q16—671no—
98#424—671no—
99#419—671no—
100#419D6711yescharacteristic
101C2×#101—671no—
102C2×#101—671no—
103#341—671no—
104C22×Q8—671no—
105C22×Q8C2×D6711yescharacteristic
106C2×Q16—1342no—
107C2×Q16—1342no—
108C2×Q16—1342no—
109C2×M16—671no—
110C2×M16—671no—
111C2×M16C2×D6711yescharacteristic
112C22×C8—671no—
113#117C2×D6711yescharacteristic
114C4⋊C8—1342no—
115#103—22no—
116#103—1342no—
117#101—1342no—
118#101—1342no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—672no—
125C2×Q8—1341no—
126C2×Q8—1341no—
127C2×Q8—1341no—
128C2×Q8—1341no—
129C2×Q8—1341no—
130C2×Q8D26812yes—
131C22×C4—21no—
132C22×C4—1341no—
133C22×C4C22×D6711yescharacteristic
134Q16—1348no—
135M16—21no—
136M16—21no—
137M16—1341no—
138M16—1341no—
139M16—1341no—
140M16—1341no—
141C2×C8—21no—
142C2×C8—672no—
143C2×C8—1341no—
144C2×C8—1341no—
145C2×C8—1341no—
146C2×C8—1341no—
147C4⋊C4C22⋊D6712yes—
148C22⋊C4—21no—
149C42—21no—
150C23#1176162711yescharacteristic
151Q8—22no—
152Q8—22no—
153Q8—41no—
154Q8—41no—
155Q8—2681no—
156Q8—2681no—
157Q8—1342no—
158Q8—1342no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—2681no—
164C2×C4—2681no—
165C2×C4—1342no—
166C2×C4D4×D6711yescharacteristic
167C2×C4D4×D6711yescharacteristic
168C2×C4C2×C22⋊D6711yescharacteristic
169C2×C4C2×D26811yescharacteristic
170C2×C4#1176162711yescharacteristic
171C2×C4#1176162811yescharacteristic
172C8—41no—
173C8—2681no—
174C8—2681no—
175C8—1342no—
176C22—22no—
177C22#1452401511yescharacteristic
178C22#1452402511yescharacteristic
179C22#1452402711yescharacteristic
180C4—42no—
181C4—42no—
182C4—2682no—
183C4#1452401311yescharacteristic
184C4#1452401411yescharacteristic
185C4#1452401711yescharacteristic
186C4#1452402911yescharacteristic
187C2—21no—
188C2—21no—
189C2#1686331211yescharacteristic
190C2#1686366912yes—
subgroupG/H
centerC22#14524015
commutatorC22×C4×C67C23
FrattiniC22×C4C22×D67
Fitting#419×C67C2
socleC22×C67#121
hypercenter#419D67
hypocenterC67#2047
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
96. C67#2047semidirect