Finite Group Databasebeta

← #22572926

Subgroups — #22572926

Order 8576.

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

normal subgroups only

This group has 372 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#16864116C211yesmaximal, characteristic, complemented
2#16864061C211yesmaximal, characteristic, complemented
3#16863754C212yesmaximal, complemented
4#16863748C211yesmaximal, characteristic, complemented
5#16863690C212yesmaximal, complemented
6#16863610C212yesmaximal
7#16863602C211yesmaximal, characteristic, complemented
8#16863523C211yesmaximal, characteristic, complemented
9#16863495C212yesmaximal, complemented
10C4×#11761613C211yesmaximal, characteristic
11#474×C67C211yesmaximal, characteristic, complemented
12C2×Q8×D67—21no—
13C2×#11761617C2212yes—
14C2×#11761613C2211yescharacteristic
15#14524043C2211yescharacteristic
16#14524042C2212yes—
17#14524036—22no—
18#14524030C2212yes—
19#14524022—21no—
20#14524021—22no—
21#14524019—21no—
22C4×Dic134C2211yescharacteristic
23#14524008C2212yes—
24C4×D268C2211yescharacteristic
25#14524005C2212yes—
26#14524002—21no—
27#14523976C2211yescharacteristic
28Dic67⋊C8C2212yes—
29#14523968C2212yes—
30#14523966C2212yes—
31#14523966C2212yes—
32#14523964C2211yescharacteristic
33#14523963C2211yescharacteristic
34#14523953C2212yes—
35C4×C67⋊C8C2211yescharacteristic
36C4⋊Q8×C67C2211yescharacteristic
37C4×D4×C67C2211yescharacteristic
38#107×C67C2212yes—
39C4⋊C8×C67C2211yescharacteristic
40#102×C67C2212yes—
41C2×Dic134—22no—
42C2×Dic134D411yescharacteristic
43C2×Dic134C2312yes—
44Q8×D67—44no—
45C2×D268C2311yescharacteristic
46C2×C4×D67—21no—
47C2×C4×D67—22no—
48#11761617—24no—
49#11761613—22no—
50C2×C67⋊C8C2311yescharacteristic
51C2×C67⋊C8C2311yescharacteristic
52C4⋊Dic67—21no—
53C4⋊Dic67—21no—
54C4⋊Dic67D411yescharacteristic
55C4⋊Dic67C2311yescharacteristic
56Dic67⋊C4—22no—
57Dic67⋊C4—22no—
58Dic67⋊C4D412yes—
59C22⋊Dic67—22no—
60C22⋊Dic67—22no—
61(C2×D67)⋊C4—21no—
62(C2×D67)⋊C4—22no—
63C4×Dic67—22no—
64C4×Dic67C2312yes—
65C2×Q8×C67—21no—
66C2×D4×C67C2311yescharacteristic
67C22×C4×C67—21no—
68C2×C8×C67C2312yes—
69C4⋊C4×C67—21no—
70C4⋊C4×C67C2312yes—
71C4⋊C4×C67C2311yescharacteristic
72C22⋊C4×C67—21no—
73C42×C67C2311yescharacteristic
74C23×C67—21no—
75C22×D67—21no—
76Q8×C67—42no—
77D4×C67—21no—
78D4×C67—21no—
79C2×C4×C67—21no—
80C2×C4×C67—22no—
81C2×C4×C67—42no—
82C2×C4×C67C2×D411yescharacteristic
83C2×C4×C67C2×D411yescharacteristic
84C2×C4×C67C2411yescharacteristic
85C2×Dic67—22no—
86C2×Dic67—21no—
87C2×Dic67C2×D412yes—
88C2×Dic67C2×D412yes—
89D268—21no—
90D268#4112yes—
91C4×D67—44no—
92C4×D67—42no—
93Dic134—22no—
94Dic134—22no—
95Dic134—22no—
96Dic134—44no—
97C8×C67—22no—
98C67⋊C8—21no—
99C67⋊C8—21no—
100C22×C67—41no—
101C22×C67—41no—
102C22×C67—41no—
103C22×C67C22×D411yescharacteristic
104C2×D67—22no—
105C2×D67—41no—
106C2×D67—41no—
107C4×C67—21no—
108C4×C67—42no—
109C4×C67—41no—
110C4×C67AGL(1,ℤ/8)12yes—
111C4×C67C2×#4111yescharacteristic
112C4×C672−1+411yescharacteristic
113Dic67—22no—
114Dic67—22no—
115Dic67—42no—
116Dic67—41no—
117C2×C67—81no—
118C2×C67#54811yescharacteristic
119C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C67#57811yescharacteristic
121D67—42no—
122#3367—671nomaximal, Sylow
123C67#336711yescharacteristic, complemented, Sylow
124#556—671no—
125D4×Q8—671no—
126#489—672no—
127#487—671no—
128#477—672no—
129#474D6711yescharacteristic, complemented
130#463—672no—
131#461—671no—
132#446—671no—
133#440—672no—
134C4×D8—671no—
135C22×Q8—1341no—
136C2×SD16—672no—
137C2×D8—671no—
138C4⋊Q8—672no—
139C4⋊Q8—671no—
140C4⋊Q8C2×D6711yescharacteristic
141#126—1342no—
142#124—672no—
143C22⋊Q8—1342no—
144C22⋊Q8—1341no—
145C22⋊Q8—1341no—
146C4×Q8—671no—
147C4×Q8—672no—
148C4×D4—672no—
149C4×D4—671no—
150C4×D4C2×D6711yescharacteristic
151#117—1341no—
152#107—671no—
153#107C2×D6712yes—
154C4⋊C8—672no—
155C4⋊C8C2×D6711yescharacteristic
156#103—672no—
157#103—672no—
158#103—672no—
159#102—671no—
160#102—671no—
161#102C2×D6712yes—
162#97—672no—
163C4×C8—671no—
164C2×Q8—21no—
165C2×Q8—2684no—
166C2×Q8—671no—
167C2×Q8—672no—
168C2×Q8—1342no—
169C2×D4—671no—
170C2×D4C22×D6711yescharacteristic
171C22×C4—21no—
172C22×C4—1342no—
173C22×C4—1341no—
174SD16—1344no—
175D8—1342no—
176C2×C8—671no—
177C2×C8—671no—
178C2×C8C22×D6712yes—
179C4⋊C4—21no—
180C4⋊C4—671no—
181C4⋊C4—672no—
182C4⋊C4—671no—
183C4⋊C4—1342no—
184C4⋊C4—1342no—
185C4⋊C4—1341no—
186C4⋊C4—1341no—
187C4⋊C4C22×D6711yescharacteristic
188C4⋊C4C22×D6712yes—
189C22⋊C4—21no—
190C22⋊C4—1342no—
191C22⋊C4—1342no—
192C22⋊C4—1342no—
193C22⋊C4—1341no—
194C42—672no—
195C42—1342no—
196C42C22×D6711yescharacteristic
197C23—21no—
198C23—1341no—
199Q8—42no—
200Q8—2684no—
201Q8—1342no—
202Q8—1342no—
203Q8—1342no—
204D4—21no—
205D4—21no—
206D4—672no—
207D4—1341no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—2684no—
212C2×C4—2682no—
213C2×C4—672no—
214C2×C4—672no—
215C2×C4—1342no—
216C2×C4—1341no—
217C2×C4D4×D6711yescharacteristic
218C2×C4D4×D6711yescharacteristic
219C2×C4C23×D6711yescharacteristic
220C8—22no—
221C8—1341no—
222C8—1341no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—2681no—
227C22—2681no—
228C22—1342no—
229C22C2×D4×D6711yescharacteristic
230C4—21no—
231C4—41no—
232C4—42no—
233C4—2682no—
234C4—2681no—
235C4—1342no—
236C4—1342no—
237C4#1452406712yes—
238C4#41×D6711yescharacteristic
239C4#1452409111yescharacteristic
240C2—81no—
241C2—2682no—
242C2#1686405611yescharacteristic
243C2C2×#1452406711yescharacteristic
244C2#1686421211yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#474×C67C2
socleC22×C67C22×D4
hypercenter#474D67
hypocenterC67#3367
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #16864116C2semidirect
2. #16864061C2semidirect
3. #16863754C2semidirect
4. #16863748C2semidirect
5. #16863690C2semidirect
7. #16863602C2semidirect
8. #16863523C2semidirect
9. #16863495C2semidirect
11. #474×C67C2semidirect
123. C67#3367semidirect
129. #474D67semidirect