Finite Group Databasebeta

← #22569861

Subgroups — #22569861

Order 8576.

460 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 141 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, complemented
2C2×#14524050C211yesmaximal, characteristic
3#16863967C211yesmaximal, characteristic
4#16863652C214yesmaximal, complemented
5#16863635C214yesmaximal, complemented
6#16863432C211yesmaximal, characteristic, complemented
7#16863405C211yesmaximal, characteristic, complemented
8#16863404C211yesmaximal, characteristic, complemented
9#536×C67C211yesmaximal, characteristic
10C22×Dic134C2211yescharacteristic
11C2×#11761618—24no—
12C2×#11761618C2212yes—
13C2×#11761618C2212yes—
14#14524050—24no—
15C2×#11761612C2212yes—
16C2×#11761612C2211yescharacteristic
17C22×C67⋊C8C2212yes—
18C22×C67⋊C8C2211yescharacteristic
19#14524041—21no—
20#14524030—21no—
21#14524024—22no—
22#14524022C2214yes—
23#14524002C2211yescharacteristic
24#14523976C2212yes—
25#14523974C2212yes—
26#14523967C2214yes—
27#14523966C2214yes—
28C2×#41×C67C2211yescharacteristic
29C22×Q8×C67C2211yescharacteristic
30C4⋊Q8×C67—21no—
31#124×C67—21no—
32#123×C67—22no—
33C22⋊Q8×C67C2214yes—
34#117×C67C2211yescharacteristic
35C2×Dic134—22no—
36C2×Dic134—22no—
37C2×Dic134—21no—
38C2×Dic134C2312yes—
39C22×Dic67—21no—
40#11761618—24no—
41#11761618—24no—
42#11761612—22no—
43#11761612—24no—
44C2×C67⋊C8—24no—
45C2×C67⋊C8D414yes—
46C2×C67⋊C8C2312yes—
47C2×C67⋊C8C2311yescharacteristic
48C2×C67⋊C8C2311yescharacteristic
49C4⋊Dic67C2312yes—
50Dic67⋊C4—24no—
51C22⋊Dic67—21no—
52C22⋊Dic67—24no—
53C4×Dic67—21no—
54#41×C67—22no—
55#41×C67—22no—
56C2×Q8×C67—21no—
57C2×Q8×C67—22no—
58C2×Q8×C67—22no—
59C2×Q8×C67D411yescharacteristic
60C2×Q8×C67C2312yes—
61C2×D4×C67D412yes—
62C2×D4×C67D411yescharacteristic
63C22×C4×C67—21no—
64C22×C4×C67C2312yes—
65C22×C4×C67C2311yescharacteristic
66C4⋊C4×C67—24no—
67C4⋊C4×C67C2312yes—
68C22⋊C4×C67—21no—
69C22⋊C4×C67—24no—
70C42×C67—21no—
71C23×C67C2×D412yes—
72C23×C67C2×D411yescharacteristic
73Q8×C67—22no—
74Q8×C67—22no—
75Q8×C67—22no—
76Q8×C67—42no—
77D4×C67—24no—
78D4×C67—22no—
79C2×C4×C67—22no—
80C2×C4×C67—22no—
81C2×C4×C67—22no—
82C2×C4×C67—22no—
83C2×C4×C67—42no—
84C2×C4×C67C2×D412yes—
85C2×C4×C67C2×D412yes—
86C2×C4×C67C2×D411yescharacteristic
87C2×C4×C67#4112yes—
88C2×C4×C67C2411yescharacteristic
89C2×Dic67—22no—
90C2×Dic67—22no—
91C2×Dic67—42no—
92Dic134—22no—
93Dic134—22no—
94Dic134—42no—
95C67⋊C8—22no—
96C67⋊C8C2×D412yes—
97C67⋊C8C2×D412yes—
98C22×C67—22no—
99C22×C67—22no—
100C22×C67—24no—
101C22×C67#12112yes—
102C22×C67C22×D411yescharacteristic
103C4×C67—22no—
104C4×C67—42no—
105C4×C67—42no—
106C4×C67#12112yes—
107C4×C67C22×D411yescharacteristic
108C4×C67C2×#4111yescharacteristic
109Dic67—42no—
110Dic67—42no—
111C2×C67—22no—
112C2×C67—42no—
113C2×C67C2×#12111yescharacteristic
114C2×C67#57812yes—
115#3059—671nomaximal, Sylow
116C67#305911yescharacteristic, complemented, Sylow
117C22×Q16—671no—
118C2×#131—671no—
119#536—671no—
120#536D6711yescharacteristic
121#470—674no—
122#467—674no—
123#427—671no—
124#419—671no—
125#419—671no—
126C2×#41C2×D6711yescharacteristic
127C22×Q8—671no—
128C22×Q8C2×D6711yescharacteristic
129C2×Q16—672no—
130C2×Q16—672no—
131C2×Q16—1344no—
132#131—1344no—
133C2×M16—671no—
134C2×M16—672no—
135C22×C8—671no—
136C22×C8—672no—
137C4⋊Q8—21no—
138C4⋊Q8—1341no—
139#124—21no—
140#124—1341no—
141#123—22no—
142#123—1342no—
143C22⋊Q8—674no—
144C22⋊Q8C2×D6714yes—
145#117—671no—
146#117C2×D6711yescharacteristic
147#107—672no—
148#106—672no—
149#103—674no—
150#103—674no—
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—672no—
157C2×Q8—1342no—
158C2×Q8—1342no—
159C2×Q8—1341no—
160C2×Q8C22⋊D6711yescharacteristic
161C2×Q8C22×D6712yes—
162C2×D4C22⋊D6712yes—
163C2×D4C22⋊D6711yescharacteristic
164C22×C4—21no—
165C22×C4—1341no—
166C22×C4C22×D6712yes—
167C22×C4C22×D6711yescharacteristic
168Q16—1344no—
169Q16—1344no—
170M16—1344no—
171M16—1342no—
172C2×C8—674no—
173C2×C8—671no—
174C2×C8—671no—
175C2×C8—672no—
176C2×C8—1344no—
177C4⋊C4—24no—
178C4⋊C4—672no—
179C4⋊C4—1344no—
180C4⋊C4C22×D6712yes—
181C22⋊C4—21no—
182C22⋊C4—24no—
183C22⋊C4—1341no—
184C22⋊C4—1344no—
185C42—21no—
186C42—1341no—
187C23C2×C22⋊D6712yes—
188C23C2×C22⋊D6711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—2682no—
194Q8—1342no—
195Q8—1342no—
196D4—24no—
197D4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—2682no—
204C2×C4—1342no—
205C2×C4—1342no—
206C2×C4D4×D6712yes—
207C2×C4C2×C22⋊D6712yes—
208C2×C4C2×C22⋊D6711yescharacteristic
209C2×C4#1176162712yes—
210C2×C4C23×D6711yescharacteristic
211C8—672no—
212C8—672no—
213C8—1342no—
214C22—22no—
215C22—22no—
216C22—24no—
217C22#1452401512yes—
218C22C22×C22⋊D6711yescharacteristic
219C4—22no—
220C4—42no—
221C4—42no—
222C4—2682no—
223C4—2682no—
224C4#1452401512yes—
225C4C2×D4×D6711yescharacteristic
226C4C2×#1176162711yescharacteristic
227C2—22no—
228C2—42no—
229C2C2×#1452401511yescharacteristic
230C2#1686421112yes—
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#536×C67C2
socleC22×C67C22×D4
hypercenter#536D67
hypocenterC67#3059
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C22×#11761618C2semidirect
4. #16863652C2semidirect
5. #16863635C2semidirect
6. #16863432C2semidirect
7. #16863405C2semidirect
8. #16863404C2semidirect
116. C67#3059semidirect