Finite Group Databasebeta

← #22564686

Subgroups — #22564686

Order 8576.

544 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C2×#14524019C211yesmaximal, characteristic
2C2×C4⋊C4×D67C211yesmaximal, characteristic
3#16863330C214yesmaximal, complemented
4#16863303C211yesmaximal, characteristic, complemented
5#16863270C214yesmaximal
6#16863266C211yesmaximal, characteristic, complemented
7#16863265C211yesmaximal, characteristic, complemented
8C4×(C2×D67)⋊C4C211yesmaximal, characteristic
9#391×C67C211yesmaximal, characteristic, complemented
10C22×C4×D67C2211yescharacteristic
11C22×C4×D67C2212yes—
12#14524019—24no—
13C2×C4⋊Dic67C2211yescharacteristic
14C2×C4⋊Dic67C2211yescharacteristic
15C2×C4⋊Dic67C2214yes—
16C2×Dic67⋊C4C2212yes—
17C2×Dic67⋊C4C2212yes—
18C2×Dic67⋊C4C2214yes—
19C4⋊C4×D67—24no—
20C2×(C2×D67)⋊C4C2212yes—
21C2×(C2×D67)⋊C4C2212yes—
22C2×C4×Dic67C2211yescharacteristic
23C67⋊#95C2212yes—
24C67⋊#95C2214yes—
25C22×Q8×C67C2211yescharacteristic
26C2×C4⋊C4×C67C2211yescharacteristic
27C2×C42×C67C2211yescharacteristic
28#95×C67C2214yes—
29C23×D67C2311yescharacteristic
30C22×Dic67C2312yes—
31C22×Dic67C2312yes—
32C22×Dic67C2311yescharacteristic
33C22×Dic67C2312yes—
34C2×C4×D67—22no—
35C2×C4×D67—22no—
36C2×C4×D67—24no—
37C2×C4×D67—44no—
38C2×C4×D67D414yes—
39C4⋊Dic67—22no—
40C4⋊Dic67—22no—
41C4⋊Dic67—44no—
42Dic67⋊C4—24no—
43Dic67⋊C4—24no—
44Dic67⋊C4—44no—
45(C2×D67)⋊C4—24no—
46(C2×D67)⋊C4Q818yescomplemented
47C4×Dic67—22no—
48C2×Q8×C67—22no—
49C2×Q8×C67—22no—
50C2×Q8×C67—22no—
51C22×C4×C67C2312yes—
52C22×C4×C67C2312yes—
53C22×C4×C67C2312yes—
54C22×C4×C67C2311yescharacteristic
55C4⋊C4×C67—22no—
56C42×C67—22no—
57C23×C67C2411yescharacteristic
58C22×D67—21no—
59C22×D67—21no—
60C22×D67—22no—
61C22×D67C2×D412yes—
62C22×D67C2×Q814yes—
63Q8×C67—44no—
64C2×C4×C67—22no—
65C2×C4×C67—22no—
66C2×C4×C67—22no—
67C2×C4×C67—22no—
68C2×C4×C67—22no—
69C2×C4×C67—22no—
70C2×C4×C67—24no—
71C2×C4×C67C2×D412yes—
72C2×C4×C67C2×Q814yes—
73C2×C4×C67#4112yes—
74C2×C4×C67#4112yes—
75C2×Dic67—22no—
76C2×Dic67—21no—
77C2×Dic67—21no—
78C2×Dic67—22no—
79C2×Dic67—22no—
80C2×Dic67—22no—
81C2×Dic67—24no—
82C2×Dic67—24no—
83C2×Dic67C2×D412yes—
84C2×Dic67C2×Q814yes—
85C4×D67—28no—
86C4×D67—48no—
87C22×C67C22×D411yescharacteristic
88C22×C67C22×Q812yes—
89C22×C67C2×#4111yescharacteristic
90C22×C67C2×#4111yescharacteristic
91C22×C67H(5,𝔽2)12yes—
92C2×D67—22no—
93C2×D67—22no—
94C2×D67—24no—
95C2×D67—24no—
96C2×D67C4⋊Q814yes—
97C4×C67—24no—
98C4×C67—42no—
99C4×C67—42no—
100C4×C67#12314yes—
101Dic67—22no—
102Dic67—24no—
103Dic67—42no—
104Dic67—42no—
105C2×C67C2×#12311yescharacteristic
106C2×C67C2×C4⋊Q811yescharacteristic
107C2×C67#53511yescharacteristic
108C2×C67#55412yes—
109C2×C67#55412yes—
110D67—24no—
111#2502—671nomaximal, Sylow
112C67#250211yescharacteristic, complemented, Sylow
113C2×C22⋊Q8—671no—
114C22×C4⋊C4—671no—
115#398—674no—
116#391—671no—
117#391D6711yescharacteristic, complemented
118#385—674no—
119#384—671no—
120#384—671no—
121C4×C22⋊C4—671no—
122C22×Q8C2×D6711yescharacteristic
123C23×C4—672no—
124C23×C4—671no—
125C22⋊Q8—1344no—
126C2×C4⋊C4—674no—
127C2×C4⋊C4—674no—
128C2×C4⋊C4—672no—
129C2×C4⋊C4—671no—
130C2×C4⋊C4—672no—
131C2×C4⋊C4—671no—
132C2×C4⋊C4—1344no—
133C2×C4⋊C4C2×D6711yescharacteristic
134C2×C22⋊C4—672no—
135C2×C22⋊C4—672no—
136C2×C42—671no—
137C2×C42C2×D6711yescharacteristic
138#95—672no—
139#95—674no—
140#95C2×D6714yes—
141C24—671no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—22no—
145C22×C4—2684no—
146C22×C4—672no—
147C22×C4—671no—
148C22×C4—674no—
149C22×C4—672no—
150C22×C4—672no—
151C22×C4—1344no—
152C22×C4—1342no—
153C22×C4—1342no—
154C22×C4C22×D6712yes—
155C22×C4C22×D6712yes—
156C22×C4C22×D6712yes—
157C22×C4C22×D6711yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—2684no—
160C4⋊C4—2684no—
161C4⋊C4—1344no—
162C4⋊C4—1344no—
163C4⋊C4—1342no—
164C4⋊C4—1342no—
165C22⋊C4—678no—
166C22⋊C4—1344no—
167C42—22no—
168C42—1342no—
169C23—672no—
170C23—674no—
171C23—1341no—
172C23—1341no—
173C23—1342no—
174C23C23×D6711yescharacteristic
175Q8—44no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—2688no—
184C2×C4—672no—
185C2×C4—674no—
186C2×C4—1342no—
187C2×C4—1342no—
188C2×C4—1344no—
189C2×C4—1344no—
190C2×C4—1341no—
191C2×C4—1342no—
192C2×C4—1341no—
193C2×C4—1342no—
194C2×C4—1348no—
195C2×C4D4×D6712yes—
196C2×C4Q8×D6714yes—
197C2×C4#1176162712yes—
198C2×C4#1176162812yes—
199C22—674no—
200C22—1344no—
201C22—1344no—
202C22—1342no—
203C22—1342no—
204C22C2×D4×D6711yescharacteristic
205C22C2×Q8×D6712yes—
206C22C2×#1176162711yescharacteristic
207C22C2×#1176162811yescharacteristic
208C22#1452408912yes—
209C4—24no—
210C4—42no—
211C4—42no—
212C4—2682no—
213C4—2682no—
214C4—1344no—
215C4—1342no—
216C4#1452402514yes—
217C2—1344no—
218C2C2×#1452402511yescharacteristic
219C2C4⋊Q8×D6711yescharacteristic
220C2#1686396411yescharacteristic
221C2#1686410412yes—
222C2#1686410612yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#391×C67C2
hypercenter#391D67
hypocenterC67#2502
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
3. #16863330C2semidirect
4. #16863303C2semidirect
6. #16863266C2semidirect
7. #16863265C2semidirect
9. #391×C67C2semidirect
46. (C2×D67)⋊C4Q8semidirect
112. C67#2502semidirect
117. #391D67semidirect