Finite Group Databasebeta

← #22571597

Subgroups — #22571597

Order 8576.

386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 125 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#16863953C211yesmaximal, characteristic, complemented
2#16863946C211yesmaximal, characteristic
3#16863692C212yesmaximal, complemented
4#16863677C212yesmaximal, complemented
5#16863642C214yesmaximal, complemented
6#16863602C212yesmaximal, complemented
7#16863597C212yesmaximal, complemented
8#423×C67C211yesmaximal, characteristic, complemented
9C2×#11761628—21no—
10C2×#11761617C2214yes—
11#14524043C2211yescharacteristic
12#14524043C2211yescharacteristic
13#14524043C2211yescharacteristic
14#14524040C2211yescharacteristic
15#14524035—21no—
16#14524031—21no—
17#14524022—21no—
18#14524022C2212yes—
19#14524017—21no—
20#14524017C2212yes—
21C4×Dic134—21no—
22C4×Dic134C2212yes—
23C4×D268—21no—
24C4×D268C2212yes—
25#14524002—21no—
26#14523977C2214yes—
27#14523968C2214yes—
28#14523965C2214yes—
29C2×M16×C67C2212yes—
30C2×C42×C67C2211yescharacteristic
31C4⋊C8×C67C2212yes—
32C4⋊C8×C67C2212yes—
33#11761628—44no—
34C2×Dic134—21no—
35C2×Dic134—21no—
36C2×Dic134C2312yes—
37C2×D268—21no—
38C2×D268—21no—
39C2×D268C2312yes—
40C2×C22⋊D67—22no—
41C2×C4×D67—22no—
42#11761617—44no—
43C4⋊Dic67—21no—
44C4⋊Dic67—21no—
45C4⋊Dic67—21no—
46C4⋊Dic67—21no—
47C4⋊Dic67C2312yes—
48C4⋊Dic67C2312yes—
49Dic67⋊C4—22no—
50Dic67⋊C4—22no—
51C22⋊Dic67—22no—
52(C2×D67)⋊C4—22no—
53(C2×D67)⋊C4—22no—
54C4×Dic67—22no—
55C22×C4×C67D411yescharacteristic
56C22×C4×C67D411yescharacteristic
57C22×C4×C67C2311yescharacteristic
58M16×C67—42no—
59C2×C8×C67C2314yes—
60C42×C67D411yescharacteristic
61C42×C67D411yescharacteristic
62C42×C67C2311yescharacteristic
63C42×C67C2311yescharacteristic
64C23×C67C2×D411yescharacteristic
65C22×D67—22no—
66C2×C4×C67—22no—
67C2×C4×C67—22no—
68C2×C4×C67—21no—
69C2×C4×C67—21no—
70C2×C4×C67C2×D411yescharacteristic
71C2×C4×C67C2×D411yescharacteristic
72C2×C4×C67C2×D411yescharacteristic
73C2×C4×C67C2×D411yescharacteristic
74C2×C4×C67C2×D411yescharacteristic
75C2×C4×C67C2411yescharacteristic
76C22⋊D67—44no—
77C2×Dic67—22no—
78C2×Dic67—22no—
79C2×Dic67—22no—
80D268—22no—
81D268—22no—
82D268—42no—
83D268—42no—
84C4×D67—44no—
85Dic134—22no—
86Dic134—22no—
87Dic134—42no—
88Dic134—42no—
89C8×C67—24no—
90C22×C67—21no—
91C22×C67—21no—
92C22×C67—21no—
93C22×C67C22×D411yescharacteristic
94C2×D67—42no—
95C2×D67—42no—
96C2×D67—42no—
97C4×C67—21no—
98C4×C67—21no—
99C4×C67—21no—
100C4×C67AGL(1,ℤ/8)12yes—
101C4×C67#13712yes—
102C4×C67H(5,𝔽2)11yescharacteristic
103C4×C672−1+411yescharacteristic
104Dic67—42no—
105Dic67—42no—
106Dic67—42no—
107C2×C67—41no—
108C2×C67#53711yescharacteristic
109C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
110C2×C67C2×#13711yescharacteristic
111D67—82no—
112#3254—671nomaximal, Sylow
113C67#325411yescharacteristic, complemented, Sylow
114#533—671no—
115#532—671no—
116#477—672no—
117#474—672no—
118#468—674no—
119#461—672no—
120#460—672no—
121#423D6711yescharacteristic, complemented
122C2×#41—1341no—
123C2×SD16—674no—
124C2×M16C2×D6712yes—
125C4⋊Q8—671no—
126C4⋊Q8—671no—
127C4⋊Q8—671no—
128#127—671no—
129#125—1341no—
130#124—1341no—
131C22⋊Q8—672no—
132C22⋊Q8—1341no—
133#121—672no—
134#121—1341no—
135C4×Q8—672no—
136C4×Q8—1341no—
137C4×D4—672no—
138C4×D4—1341no—
139#117—1341no—
140C2×C42C2×D6711yescharacteristic
141#107—674no—
142C4⋊C8C2×D6712yes—
143C4⋊C8C2×D6712yes—
144#103—674no—
145#102—674no—
146#41—2684no—
147C2×Q8—672no—
148C2×Q8—1341no—
149C2×Q8—1341no—
150C2×D4—672no—
151C2×D4—1342no—
152C2×D4—1341no—
153C2×D4—1341no—
154C22×C4—1342no—
155C22×C4D26811yescharacteristic
156C22×C4D26811yescharacteristic
157C22×C4C22×D6711yescharacteristic
158SD16—2684no—
159M16—42no—
160C2×C8C22×D6714yes—
161C4⋊C4—672no—
162C4⋊C4—672no—
163C4⋊C4—1341no—
164C4⋊C4—1341no—
165C4⋊C4—1341no—
166C4⋊C4—1342no—
167C4⋊C4—1342no—
168C4⋊C4—1341no—
169C22⋊C4—1342no—
170C22⋊C4—1342no—
171C22⋊C4—1342no—
172C42—1342no—
173C42D26811yescharacteristic
174C42D26811yescharacteristic
175C42C22×D6711yescharacteristic
176C42C22×D6711yescharacteristic
177C23—1342no—
178C23C2×D26811yescharacteristic
179Q8—2682no—
180Q8—2682no—
181Q8—1342no—
182Q8—1342no—
183D4—2682no—
184D4—2684no—
185D4—2682no—
186D4—1342no—
187D4—1342no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—2684no—
193C2×C4—1342no—
194C2×C4—1342no—
195C2×C4—1342no—
196C2×C4C2×D26811yescharacteristic
197C2×C4C2×D26811yescharacteristic
198C2×C4C2×D26811yescharacteristic
199C2×C4C2×D26811yescharacteristic
200C2×C4C2×D26811yescharacteristic
201C2×C4C23×D6711yescharacteristic
202C8—24no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—2682no—
207C22—2682no—
208C22—2682no—
209C22C22×D26811yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—2682no—
214C4—2682no—
215C4—2682no—
216C4#1452407112yes—
217C4#1452407612yes—
218C4#1452408811yescharacteristic
219C4#1452409111yescharacteristic
220C2—41no—
221C2—5362no—
222C2#1686397711yescharacteristic
223C2C2×#1452407111yescharacteristic
224C2C2×#1452407611yescharacteristic
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#423×C67C2
socleC22×C67C22×D4
hypercenter#423D67
hypocenterC67#3254
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
1. #16863953C2semidirect
3. #16863692C2semidirect
4. #16863677C2semidirect
5. #16863642C2semidirect
6. #16863602C2semidirect
7. #16863597C2semidirect
8. #423×C67C2semidirect
113. C67#3254semidirect
121. #423D67semidirect