Finite Group Databasebeta

← #22571670

Subgroups — #22571670

Order 8576.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#16864017C212yesmaximal, complemented
2#16863692C212yesmaximal, complemented
3#16863677C212yesmaximal, complemented
4#16863642C212yesmaximal, complemented
5#16863637C211yesmaximal, characteristic, complemented
6#16863632C211yesmaximal, characteristic
7#16863619C212yesmaximal, complemented
8#16863612C212yesmaximal, complemented
9#424×C67C211yesmaximal, characteristic, complemented
10C2×Dic268C2211yescharacteristic
11C2×#11761617C2212yes—
12C2×D536C2211yescharacteristic
13#14524043C2212yes—
14#14524038—22no—
15#14524031C2212yes—
16#14524029—22no—
17#14524025—22no—
18#14524022C2212yes—
19#14524021—22no—
20#14524017C2212yes—
21C4×Dic134C2212yes—
22C4×D268C2212yes—
23#14523977C2211yescharacteristic
24#14523977C2211yescharacteristic
25#14523977C2212yes—
26#14523968C2212yes—
27#14523968C2212yes—
28#14523965C2212yes—
29#14523965C2212yes—
30C2×M16×C67C2211yescharacteristic
31C22×C8×C67C2211yescharacteristic
32#117×C67C2211yescharacteristic
33C4⋊C8×C67C2212yes—
34C4⋊C8×C67C2212yes—
35C2×Dic134—22no—
36C2×Dic134C2312yes—
37C2×D268C2312yes—
38C2×C22⋊D67—22no—
39C2×C4×D67—22no—
40Dic268—41no—
41#11761617—42no—
42D536—41no—
43C4⋊Dic67—22no—
44C4⋊Dic67C2312yes—
45C4⋊Dic67C2312yes—
46Dic67⋊C4—22no—
47Dic67⋊C4—22no—
48C22⋊Dic67—22no—
49(C2×D67)⋊C4—22no—
50(C2×D67)⋊C4—22no—
51(C2×D67)⋊C4—22no—
52(C2×D67)⋊C4—22no—
53C4×Dic67—22no—
54C22×C4×C67C2311yescharacteristic
55M16×C67—41no—
56C2×C8×C67—41no—
57C2×C8×C67C2311yescharacteristic
58C2×C8×C67C2311yescharacteristic
59C2×C8×C67C2312yes—
60C4⋊C4×C67D411yescharacteristic
61C4⋊C4×C67D411yescharacteristic
62C22⋊C4×C67D412yes—
63C42×C67C2312yes—
64C23×C67C2×D411yescharacteristic
65C22×D67—22no—
66C2×C4×C67—21no—
67C2×C4×C67—21no—
68C2×C4×C67C2×D412yes—
69C2×C4×C67C2×D412yes—
70C2×C4×C67C2×D411yescharacteristic
71C2×C4×C67C2411yescharacteristic
72C22⋊D67—82no—
73C2×Dic67—22no—
74C2×Dic67—22no—
75C2×Dic67—22no—
76D268—22no—
77D268—22no—
78C4×D67—82no—
79Dic134—22no—
80Dic134—22no—
81Dic134—82no—
82C8×C67—21no—
83C8×C67—21no—
84C8×C67—22no—
85C22×C67—21no—
86C22×C67—21no—
87C22×C67—21no—
88C22×C67C22×D411yescharacteristic
89C2×D67—42no—
90C2×D67—42no—
91C2×D67—42no—
92C4×C67—21no—
93C4×C67—22no—
94C4×C67—22no—
95C4×C67H(5,𝔽2)11yescharacteristic
96C4×C672−1+411yescharacteristic
97Dic67—42no—
98Dic67—42no—
99Dic67—42no—
100C2×C67—41no—
101C2×C67#53711yescharacteristic
102C2×C67#57611yescharacteristic
103C2×C67#57811yescharacteristic
104D67—82no—
105#3262—671nomaximal, Sylow
106C67#326211yescharacteristic, complemented, Sylow
107#542—672no—
108#477—672no—
109#474—672no—
110#468—672no—
111#467—671no—
112#466—671no—
113#464—672no—
114#463—672no—
115#424D6711yescharacteristic, complemented
116C2×Q16—671no—
117C2×SD16—672no—
118C2×D8—671no—
119C2×M16C2×D6711yescharacteristic
120C22×C8C2×D6711yescharacteristic
121C4⋊Q8—672no—
122#126—1342no—
123#124—672no—
124#124—1342no—
125#123—1342no—
126C22⋊Q8—672no—
127C22⋊Q8—1342no—
128#121—672no—
129C4×Q8—672no—
130C4×D4—672no—
131#117C2×D6711yescharacteristic
132#107—671no—
133#107—671no—
134#107—672no—
135C4⋊C8C2×D6712yes—
136C4⋊C8C2×D6712yes—
137#103—672no—
138#103—672no—
139#102—672no—
140#102—672no—
141C2×Q8—672no—
142C2×Q8—1342no—
143C2×D4—672no—
144C2×D4—1342no—
145C22×C4—1342no—
146C22×C4C22×D6711yescharacteristic
147Q16—2681no—
148SD16—2682no—
149D8—2681no—
150M16—41no—
151C2×C8—41no—
152C2×C8C22×D6711yescharacteristic
153C2×C8C22×D6712yes—
154C2×C8C22×D6711yescharacteristic
155C4⋊C4—672no—
156C4⋊C4—672no—
157C4⋊C4—1342no—
158C4⋊C4—1342no—
159C4⋊C4—1342no—
160C4⋊C4D26811yescharacteristic
161C4⋊C4D26811yescharacteristic
162C22⋊C4—1342no—
163C22⋊C4—1342no—
164C22⋊C4—1342no—
165C22⋊C4—1342no—
166C22⋊C4—1342no—
167C22⋊C4D26812yes—
168C42—1342no—
169C42C22×D6712yes—
170C23—1342no—
171C23C2×D26811yescharacteristic
172Q8—5362no—
173Q8—1342no—
174Q8—1342no—
175D4—5362no—
176D4—1342no—
177D4—1342no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—5362no—
181C2×C4—1342no—
182C2×C4—1342no—
183C2×C4—1342no—
184C2×C4C2×D26812yes—
185C2×C4C2×D26812yes—
186C2×C4C2×D26811yescharacteristic
187C2×C4C23×D6711yescharacteristic
188C8—21no—
189C8—21no—
190C8—22no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—2682no—
195C22—2682no—
196C22—2682no—
197C22C22×D26811yescharacteristic
198C4—22no—
199C4—22no—
200C4—21no—
201C4—2682no—
202C4—2682no—
203C4—2682no—
204C4#1452408811yescharacteristic
205C4#1452409111yescharacteristic
206C2—41no—
207C2—5362no—
208C2#1686397711yescharacteristic
209C2#1686420111yescharacteristic
210C2#1686421311yescharacteristic
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#424×C67C2
socleC22×C67C22×D4
hypercenter#424D67
hypocenterC67#3262
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
1. #16864017C2semidirect
2. #16863692C2semidirect
3. #16863677C2semidirect
4. #16863642C2semidirect
5. #16863637C2semidirect
7. #16863619C2semidirect
8. #16863612C2semidirect
9. #424×C67C2semidirect
106. C67#3262semidirect
115. #424D67semidirect