Finite Group Databasebeta

← #22569740

Subgroups — #22569740

Order 8576.

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

normal subgroups only

This group has 458 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
1C2×#14524076C211yesmaximal, characteristic
2C22×Dic268C211yesmaximal, characteristic, complemented
3#16863972C211yesmaximal, characteristic
4#16863873C211yesmaximal, characteristic, complemented
5#16863619C214yesmaximal, complemented
6#16863607C214yesmaximal, complemented
7#16863406C211yesmaximal, characteristic, complemented
8#16863393C211yesmaximal, characteristic
9#424×C67C211yesmaximal, characteristic, complemented
10C2×#11761628C2211yescharacteristic
11C22×Dic134C2211yescharacteristic
12C22×Dic134C2211yescharacteristic
13#14524076—24no—
14C2×Dic268—24no—
15C2×Dic268C2212yes—
16C2×Dic268C2214yes—
17C2×#11761617C2212yes—
18#14524043C2212yes—
19#14524031C2212yes—
20#14524027—22no—
21#14524023—22no—
22C4×Dic134C2214yes—
23#14524008—22no—
24#14524002C2211yescharacteristic
25#14523999—22no—
26#14523968C2212yes—
27#14523968C2214yes—
28#14523965C2212yes—
29C2×M16×C67C2211yescharacteristic
30C22×C8×C67C2211yescharacteristic
31#117×C67C2211yescharacteristic
32C4⋊C8×C67C2214yes—
33#11761628—22no—
34#11761628—22no—
35C2×Dic134—21no—
36C2×Dic134—22no—
37C2×Dic134—22no—
38C2×Dic134—22no—
39C2×Dic134—21no—
40C2×Dic134D414yes—
41C2×Dic134C2312yes—
42C2×Dic134C2311yescharacteristic
43C2×Dic134C2312yes—
44C2×D268C2311yescharacteristic
45C2×C22⋊D67—21no—
46C22×Dic67—21no—
47C22×Dic67—21no—
48C2×C4×D67—21no—
49Dic268—28no—
50Dic268—24no—
51#11761617—24no—
52C4⋊Dic67—24no—
53C4⋊Dic67C2312yes—
54Dic67⋊C4—24no—
55C22⋊Dic67—21no—
56(C2×D67)⋊C4—24no—
57C4×Dic67—24no—
58C4×Dic67—21no—
59C22×C4×C67C2311yescharacteristic
60M16×C67—22no—
61C2×C8×C67—22no—
62C2×C8×C67C2312yes—
63C2×C8×C67C2312yes—
64C4⋊C4×C67D412yes—
65C22⋊C4×C67D412yes—
66C42×C67C2312yes—
67C23×C67C2×D411yescharacteristic
68C22×D67—21no—
69C2×C4×C67C2×D412yes—
70C2×C4×C67C2×D414yes—
71C2×C4×C67C2×D411yescharacteristic
72C2×C4×C67#4112yes—
73C2×C4×C67C2411yescharacteristic
74C22⋊D67—42no—
75C2×Dic67—24no—
76C2×Dic67—22no—
77C2×Dic67—22no—
78C2×Dic67—22no—
79C2×Dic67—21no—
80C2×Dic67—42no—
81D268—21no—
82D268—21no—
83C4×D67—42no—
84Dic134—24no—
85Dic134—22no—
86Dic134—22no—
87Dic134—21no—
88Dic134—22no—
89Dic134—21no—
90Dic134—42no—
91Dic134C2×D414yes—
92C8×C67—22no—
93C8×C67—22no—
94C22×C67—22no—
95C22×C67#12112yes—
96C22×C67C22×D411yescharacteristic
97C2×D67—41no—
98C2×D67—42no—
99C4×C67—24no—
100C4×C67#12112yes—
101C4×C67C22×D411yescharacteristic
102C4×C67C2×#4111yescharacteristic
103Dic67—24no—
104Dic67—42no—
105Dic67—41no—
106Dic67—42no—
107C2×C67—22no—
108C2×C67C2×#12111yescharacteristic
109C2×C67#57812yes—
110D67—81no—
111#3044—671nomaximal, Sylow
112C67#304411yescharacteristic, complemented, Sylow
113C2×#137—671no—
114C22×Q16—671no—
115#536—671no—
116#519—671no—
117#464—674no—
118#462—674no—
119#424D6711yescharacteristic, complemented
120#419—671no—
121#416—671no—
122C2×#41—671no—
123C22×Q8—671no—
124C22×Q8—671no—
125#137—1344no—
126C2×Q16—674no—
127C2×Q16—672no—
128C2×Q16—1344no—
129C2×SD16—672no—
130C2×M16C2×D6711yescharacteristic
131C22×C8C2×D6711yescharacteristic
132C4⋊Q8—672no—
133#124—672no—
134#123—1342no—
135C22⋊Q8—1342no—
136C4×Q8—674no—
137C4×Q8—1342no—
138#117—671no—
139#117—1342no—
140#117C2×D6711yescharacteristic
141C4⋊C8C2×D6714yes—
142#103—672no—
143#103—674no—
144#102—672no—
145#41—1342no—
146#41—1342no—
147C2×Q8—674no—
148C2×Q8—672no—
149C2×Q8—672no—
150C2×Q8—671no—
151C2×Q8—1342no—
152C2×Q8—1342no—
153C2×Q8—1341no—
154C2×Q8—1341no—
155C2×Q8—1342no—
156C2×D4—671no—
157C2×D4—1341no—
158C22×C4—1341no—
159C22×C4—1341no—
160C22×C4—1341no—
161C22×C4C22×D6711yescharacteristic
162Q16—1348no—
163Q16—1344no—
164SD16—1344no—
165M16—22no—
166C2×C8—22no—
167C2×C8C22×D6712yes—
168C2×C8C22×D6712yes—
169C4⋊C4—672no—
170C4⋊C4—1344no—
171C4⋊C4—1344no—
172C4⋊C4D26812yes—
173C22⋊C4—1341no—
174C22⋊C4—1344no—
175C22⋊C4D26812yes—
176C42—1341no—
177C42—1344no—
178C42C22×D6712yes—
179C23—1341no—
180C23C2×D26811yescharacteristic
181Q8—2682no—
182Q8—674no—
183Q8—1342no—
184Q8—1341no—
185Q8—1342no—
186Q8—1341no—
187Q8—1344no—
188Q8—1342no—
189D4—2682no—
190D4—1341no—
191D4—1341no—
192C2×C4—2682no—
193C2×C4—2682no—
194C2×C4—1342no—
195C2×C4—1342no—
196C2×C4—1341no—
197C2×C4—1344no—
198C2×C4—1342no—
199C2×C4D4×D6712yes—
200C2×C4C2×D26814yes—
201C2×C4C2×D26811yescharacteristic
202C2×C4#1176162912yes—
203C2×C4C23×D6711yescharacteristic
204C8—22no—
205C8—22no—
206C22—22no—
207C22—2681no—
208C22—2682no—
209C22#1452401812yes—
210C22C22×D26811yescharacteristic
211C4—24no—
212C4—2681no—
213C4—2682no—
214C4—2682no—
215C4—1344no—
216C4#1452401812yes—
217C4C2×D4×D6711yescharacteristic
218C4C2×#1176162911yescharacteristic
219C2—22no—
220C2—5361no—
221C2C2×#1452401811yescharacteristic
222C2#1686421312yes—
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#424×C67C2
socleC22×C67C22×D4
hypercenter#424D67
hypocenterC67#3044
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. C22×Dic268C2semidirect
4. #16863873C2semidirect
5. #16863619C2semidirect
6. #16863607C2semidirect
7. #16863406C2semidirect
9. #424×C67C2semidirect
112. C67#3044semidirect
119. #424D67semidirect