Finite Group Databasebeta

← #22570824

Subgroups — #22570824

Order 8576.

362 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 115 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#16864011C212yesmaximal, complemented
2#16863793C211yesmaximal, characteristic, complemented
3#16863779C211yesmaximal, characteristic, complemented
4#16863754C212yesmaximal, complemented
5#16863655C211yesmaximal, characteristic, complemented
6#16863646C211yesmaximal, characteristic
7#16863624C212yesmaximal, complemented
8#16863504C211yesmaximal, characteristic, complemented
9C4×#11761616C212yesmaximal, complemented
10#16863353C211yesmaximal, characteristic, complemented
11#442×C67C211yesmaximal, characteristic, complemented
12C2×Dic268C2211yescharacteristic
13C2×#11761616C2212yes—
14C2×#11761611C2211yescharacteristic
15C2×C8×D67C2211yescharacteristic
16#14524042C2212yes—
17#14524036—22no—
18#14524030C2212yes—
19#14524029—22no—
20#14524026—22no—
21#14524023C2212yes—
22#14524022—22no—
23#14524014C2212yes—
24C4×Dic134C2212yes—
25#14524000C2211yescharacteristic
26#14523975C2211yescharacteristic
27#14523974C2212yes—
28#14523968C2212yes—
29#14523966C2212yes—
30#14523964C2212yes—
31#14523953C2211yescharacteristic
32C4×C67⋊C8C2211yescharacteristic
33C8×Dic67C2211yescharacteristic
34C2×D8×C67C2211yescharacteristic
35C4×D4×C67C2212yes—
36#106×C67C2211yescharacteristic
37#102×C67C2212yes—
38#97×C67C2211yescharacteristic
39C2×Dic134—22no—
40C2×Dic134C2312yes—
41C2×C22⋊D67—22no—
42C2×C4×D67C2311yescharacteristic
43Dic268—22no—
44#11761616—24no—
45#11761611—22no—
46C2×C67⋊C8C2311yescharacteristic
47C2×C67⋊C8C2311yescharacteristic
48C8×D67—22no—
49C4⋊Dic67—22no—
50C4⋊Dic67C2312yes—
51Dic67⋊C4—22no—
52Dic67⋊C4—22no—
53Dic67⋊C4D412yes—
54C22⋊Dic67—22no—
55C22⋊Dic67—22no—
56C22⋊Dic67—22no—
57(C2×D67)⋊C4—22no—
58(C2×D67)⋊C4D412yes—
59C4×Dic67—22no—
60C4×Dic67C2311yescharacteristic
61C2×D4×C67C2312yes—
62C22×C4×C67—22no—
63D8×C67—22no—
64C2×C8×C67C2311yescharacteristic
65C2×C8×C67C2311yescharacteristic
66C4⋊C4×C67C2312yes—
67C22⋊C4×C67—22no—
68C42×C67C2311yescharacteristic
69C23×C67—22no—
70C22×D67C2×D411yescharacteristic
71D4×C67—22no—
72D4×C67—22no—
73C2×C4×C67—22no—
74C2×C4×C67—82no—
75C2×C4×C67C2×D412yes—
76C2×C4×C67C2411yescharacteristic
77C22⋊D67—82no—
78C2×Dic67—22no—
79C2×Dic67—22no—
80C2×Dic67C2×D412yes—
81C2×Dic67C2×D411yescharacteristic
82C4×D67—21no—
83C4×D67—21no—
84Dic134—22no—
85Dic134—22no—
86Dic134—82no—
87C8×C67—21no—
88C8×C67#4112yes—
89C67⋊C8—21no—
90C67⋊C8#4112yes—
91C22×C67—42no—
92C22×C67—42no—
93C22×C67—42no—
94C22×C67C22×D411yescharacteristic
95C2×D67—21no—
96C2×D67—22no—
97C4×C67—22no—
98C4×C67—42no—
99C4×C67C2×#4111yescharacteristic
100C4×C67C2×#4111yescharacteristic
101Dic67—21no—
102Dic67—22no—
103Dic67—42no—
104Dic67—42no—
105C2×C67—82no—
106C2×C67#53211yescharacteristic
107C2×C67#57712yes—
108D67—41no—
109#3173—671nomaximal, Sylow
110C67#317311yescharacteristic, complemented, Sylow
111#542—672no—
112#498—671no—
113#495—671no—
114#489—672no—
115#470—671no—
116#469—671no—
117#465—672no—
118#442D6711yescharacteristic, complemented
119#441—671no—
120C4×SD16—672no—
121#405—671no—
122C2×Q16—671no—
123C2×SD16—672no—
124C2×D8C2×D6711yescharacteristic
125C2×M16—671no—
126C22×C8—671no—
127C4⋊Q8—672no—
128#126—1342no—
129#124—672no—
130#124—1342no—
131#123—1342no—
132C22⋊Q8—672no—
133C22⋊Q8—1342no—
134#121—672no—
135C4×Q8—672no—
136C4×D4C2×D6712yes—
137#117—671no—
138#106—671no—
139#106—672no—
140#106C2×D6711yescharacteristic
141#103—672no—
142#103—672no—
143#102—672no—
144#102C2×D6712yes—
145#97—671no—
146#97C2×D6711yescharacteristic
147C4×C8—671no—
148C4×C8—671no—
149C2×Q8—672no—
150C2×Q8—1342no—
151C2×D4—1342no—
152C2×D4C22×D6712yes—
153C22×C4—22no—
154C22×C4—671no—
155Q16—1342no—
156SD16—1344no—
157D8—22no—
158M16—1342no—
159C2×C8—671no—
160C2×C8—671no—
161C2×C8—1342no—
162C2×C8C22×D6711yescharacteristic
163C2×C8C22×D6711yescharacteristic
164C4⋊C4—672no—
165C4⋊C4—672no—
166C4⋊C4—1342no—
167C4⋊C4—1342no—
168C4⋊C4—1342no—
169C4⋊C4C22×D6712yes—
170C22⋊C4—22no—
171C22⋊C4—672no—
172C22⋊C4—1342no—
173C22⋊C4—1342no—
174C22⋊C4—1342no—
175C22⋊C4—1342no—
176C42—671no—
177C42—1342no—
178C42C22×D6711yescharacteristic
179C23—22no—
180C23—671no—
181Q8—5362no—
182Q8—1342no—
183Q8—1342no—
184D4—22no—
185D4—22no—
186D4—5362no—
187C2×C4—22no—
188C2×C4—82no—
189C2×C4—672no—
190C2×C4—671no—
191C2×C4—1342no—
192C2×C4—1342no—
193C2×C4—1341no—
194C2×C4—1341no—
195C2×C4D4×D6712yes—
196C2×C4C23×D6711yescharacteristic
197C8—21no—
198C8—672no—
199C8—1341no—
200C8#1176162812yes—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—1341no—
205C22—1342no—
206C22C2×D4×D6711yescharacteristic
207C4—22no—
208C4—42no—
209C4—2682no—
210C4—2682no—
211C4—1341no—
212C4—1342no—
213C4#41×D6711yescharacteristic
214C4C2×#1176162811yescharacteristic
215C2—82no—
216C2—2681no—
217C2#1686394311yescharacteristic
218C2#1686420612yes—
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#442×C67C2
socleC22×C67C22×D4
hypercenter#442D67
hypocenterC67#3173
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #16864011C2semidirect
2. #16863793C2semidirect
3. #16863779C2semidirect
4. #16863754C2semidirect
5. #16863655C2semidirect
7. #16863624C2semidirect
8. #16863504C2semidirect
9. C4×#11761616C2semidirect
10. #16863353C2semidirect
11. #442×C67C2semidirect
110. C67#3173semidirect
118. #442D67semidirect