Finite Group Databasebeta

← #22573124

Subgroups — #22573124

Order 8576.

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

normal subgroups only

This group has 340 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#16864126C211yesmaximal, characteristic, complemented
2#16864107C211yesmaximal, characteristic
3#16863804C212yesmaximal, complemented
4#16863702C212yesmaximal, complemented
5#16863688C211yesmaximal, characteristic, complemented
6#16863683C211yesmaximal, characteristic
7#16863675C212yesmaximal
8#16863526C211yesmaximal, characteristic, complemented
9#16863495C212yesmaximal
10C4×#11761617C211yesmaximal, characteristic
11#500×C67C211yesmaximal, characteristic, complemented
12C2×#11761617C2211yescharacteristic
13#14524042—21no—
14#14524042C2212yes—
15#14524041—22no—
16#14524034C2212yes—
17#14524023—21no—
18#14524019—22no—
19C4×Dic134C2211yescharacteristic
20C4×Dic134C2211yescharacteristic
21#14524008—21no—
22#14524008C2212yes—
23C4×D268C2211yescharacteristic
24#14524005C2212yes—
25C4⋊C4×D67—21no—
26#14523976C2212yes—
27#14523975C2211yescharacteristic
28#14523974C2212yes—
29C4⋊(C67⋊C8)C2211yescharacteristic
30Dic67⋊C8C2212yes—
31#14523968C2211yescharacteristic
32#14523966C2212yes—
33#14523965C2211yescharacteristic
34#14523963C2212yes—
35#14523953C2212yes—
36C4⋊Q8×C67C2211yescharacteristic
37C4⋊Q8×C67C2211yescharacteristic
38#107×C67C2212yes—
39#107×C67C2212yes—
40C4×C8×C67C2211yescharacteristic
41C2×Dic134—22no—
42C2×Dic134D411yescharacteristic
43C2×Dic134C2311yescharacteristic
44C2×D268C2311yescharacteristic
45C2×C4×D67—21no—
46C2×C4×D67—22no—
47#11761617Q814yescomplemented
48C2×C67⋊C8C2312yes—
49C4⋊Dic67—22no—
50C4⋊Dic67D411yescharacteristic
51C4⋊Dic67C2311yescharacteristic
52C4⋊Dic67C2311yescharacteristic
53Dic67⋊C4—21no—
54Dic67⋊C4—22no—
55Dic67⋊C4—22no—
56Dic67⋊C4—22no—
57Dic67⋊C4D412yes—
58(C2×D67)⋊C4—21no—
59(C2×D67)⋊C4—22no—
60C4×Dic67—21no—
61C4×Dic67—22no—
62C4×Dic67C2312yes—
63C2×Q8×C67—21no—
64C2×Q8×C67—21no—
65C2×C8×C67C2311yescharacteristic
66C2×C8×C67C2311yescharacteristic
67C4⋊C4×C67—21no—
68C4⋊C4×C67—21no—
69C4⋊C4×C67C2312yes—
70C4⋊C4×C67C2312yes—
71C42×C67C2311yescharacteristic
72C22×D67—21no—
73Q8×C67—81no—
74Q8×C67—81no—
75C2×C4×C67—22no—
76C2×C4×C67—22no—
77C2×C4×C67C2×D411yescharacteristic
78C2×C4×C67C2×D411yescharacteristic
79C2×C4×C67C2411yescharacteristic
80C2×Dic67—21no—
81C2×Dic67—21no—
82C2×Dic67—21no—
83C2×Dic67C2×D412yes—
84C2×Dic67C2×D412yes—
85D268—21no—
86D268C2×Q812yes—
87C4×D67—42no—
88C4×D67—82no—
89Dic134—22no—
90Dic134—21no—
91Dic134—82no—
92Dic134C2×Q812yes—
93C8×C67—21no—
94C8×C67C2×Q812yes—
95C67⋊C8—22no—
96C22×C67C22×D411yescharacteristic
97C2×D67—22no—
98C2×D67—41no—
99C2×D67—41no—
100C4×C67—21no—
101C4×C67—42no—
102C4×C67—42no—
103C4×C67#13712yes—
104C4×C67C22×Q811yescharacteristic
105C4×C672−1+411yescharacteristic
106Dic67—22no—
107Dic67—22no—
108Dic67—22no—
109Dic67—41no—
110Dic67—41no—
111C2×C67D4×Q811yescharacteristic
112C2×C67C2×#13711yescharacteristic
113C2×C67#57611yescharacteristic
114D67—42no—
115#3385—671nomaximal, Sylow
116C67#338511yescharacteristic, complemented, Sylow
117Q82—671no—
118#554—671no—
119#501—672no—
120#500D6711yescharacteristic, complemented
121#479—672no—
122#477—671no—
123#476—671no—
124#474—672no—
125#446—671no—
126#440—672no—
127C4×SD16—671no—
128C2×SD16—671no—
129C4⋊Q8—672no—
130C4⋊Q8—1341no—
131C4⋊Q8—1342no—
132C4⋊Q8C2×D6711yescharacteristic
133C4⋊Q8C2×D6711yescharacteristic
134#125—672no—
135C22⋊Q8—1341no—
136C22⋊Q8—1342no—
137C4×Q8—671no—
138C4×Q8—672no—
139C4×Q8—671no—
140C4×Q8—1341no—
141C4×D4—672no—
142C4×D4—671no—
143C2×C4⋊C4—1341no—
144#107—672no—
145#107C2×D6712yes—
146#107C2×D6712yes—
147#106—671no—
148#106—672no—
149C4⋊C8—672no—
150C4⋊C8—671no—
151#103—671no—
152#103—672no—
153#102—671no—
154#102—672no—
155#97—672no—
156C4×C8C2×D6711yescharacteristic
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—671no—
160C2×Q8—671no—
161C2×Q8—1342no—
162C2×D4—671no—
163C22×C4—1342no—
164C22×C4—1341no—
165SD16—674no—
166C2×C8—672no—
167C2×C8C22×D6711yescharacteristic
168C2×C8C22×D6711yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—671no—
172C4⋊C4—672no—
173C4⋊C4—671no—
174C4⋊C4—671no—
175C4⋊C4—1342no—
176C4⋊C4—1342no—
177C4⋊C4—1342no—
178C4⋊C4—1342no—
179C4⋊C4—1341no—
180C4⋊C4C22×D6712yes—
181C4⋊C4C22×D6712yes—
182C22⋊C4—1342no—
183C22⋊C4—1341no—
184C42—672no—
185C42—1342no—
186C42—1341no—
187C42C22×D6711yescharacteristic
188C23—1341no—
189Q8—81no—
190Q8—81no—
191Q8—5362no—
192Q8—672no—
193Q8—1342no—
194Q8—1341no—
195D4—672no—
196D4—1341no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—2682no—
200C2×C4—5362no—
201C2×C4—672no—
202C2×C4—672no—
203C2×C4—1341no—
204C2×C4—1341no—
205C2×C4—1341no—
206C2×C4D4×D6711yescharacteristic
207C2×C4D4×D6711yescharacteristic
208C2×C4C23×D6711yescharacteristic
209C8—21no—
210C8—1342no—
211C8Q8×D6712yes—
212C22—2681no—
213C22—2681no—
214C22—1342no—
215C22C2×D4×D6711yescharacteristic
216C4—21no—
217C4—42no—
218C4—42no—
219C4—2681no—
220C4—2681no—
221C4—1342no—
222C4—1342no—
223C4—1342no—
224C4#1452407512yes—
225C4C2×Q8×D6711yescharacteristic
226C4#1452409111yescharacteristic
227C2—2682no—
228C2#1686406111yescharacteristic
229C2C2×#1452407511yescharacteristic
230C2#1686419711yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#500×C67C2
socleC22×C67C22×D4
hypercenter#500D67
hypocenterC67#3385
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
1. #16864126C2semidirect
3. #16863804C2semidirect
4. #16863702C2semidirect
5. #16863688C2semidirect
8. #16863526C2semidirect
11. #500×C67C2semidirect
47. #11761617Q8semidirect
116. C67#3385semidirect
120. #500D67semidirect