Finite Group Databasebeta

← #22573089

Subgroups — #22573089

Order 8576.

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

normal subgroups only

This group has 338 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#16864121C211yesmaximal, characteristic, complemented
2#16864107C211yesmaximal, characteristic
3#16863796C212yesmaximal, complemented
4#16863706C212yesmaximal, complemented
5#16863695C212yesmaximal
6#16863688C211yesmaximal, characteristic, complemented
7#16863673C211yesmaximal, characteristic
8C8×Dic134C211yesmaximal, characteristic, complemented
9C4×#11761617C211yesmaximal, characteristic
10#16863483C212yesmaximal
11#498×C67C211yesmaximal, characteristic, complemented
12C2×#11761617C2211yescharacteristic
13#14524041—21no—
14#14524034C2212yes—
15#14524034C2212yes—
16#14524033—22no—
17#14524023—21no—
18#14524019—22no—
19Q8×Dic67—21no—
20C4×Dic134C2211yescharacteristic
21C4×Dic134C2211yescharacteristic
22#14524008C2212yes—
23C4×D268C2211yescharacteristic
24#14524005C2212yes—
25C4⋊C4×D67—21no—
26#14523976C2212yes—
27#14523976C2212yes—
28#14523975C2211yescharacteristic
29C4⋊(C67⋊C8)C2211yescharacteristic
30Dic67⋊C8C2212yes—
31#14523968C2211yescharacteristic
32#14523966C2212yes—
33#14523965C2211yescharacteristic
34#14523963C2212yes—
35C8×Dic67C2212yes—
36C4⋊Q8×C67C2211yescharacteristic
37C4⋊Q8×C67C2211yescharacteristic
38#106×C67C2212yes—
39#106×C67C2212yes—
40C4×C8×C67C2211yescharacteristic
41C2×Dic134D411yescharacteristic
42C2×Dic134C2311yescharacteristic
43C2×D268C2311yescharacteristic
44C2×C4×D67—21no—
45C2×C4×D67—22no—
46#11761617Q814yescomplemented
47C2×C67⋊C8C2312yes—
48C4⋊Dic67—22no—
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—42no—
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—
91Dic134C2×Q812yes—
92C8×C67—21no—
93C8×C67C2×Q812yes—
94C67⋊C8—22no—
95C22×C67C22×D411yescharacteristic
96C2×D67—22no—
97C2×D67—41no—
98C2×D67—41no—
99C4×C67—21no—
100C4×C67—42no—
101C4×C67—42no—
102C4×C67#13512yes—
103C4×C67C22×Q811yescharacteristic
104C4×C672−1+411yescharacteristic
105Dic67—22no—
106Dic67—22no—
107Dic67—22no—
108Dic67—41no—
109Dic67—41no—
110C2×C67D4×Q811yescharacteristic
111C2×C67C2×#13511yescharacteristic
112C2×C67#57711yescharacteristic
113D67—42no—
114#3381—671nomaximal, Sylow
115C67#338111yescharacteristic, complemented, Sylow
116#557—671no—
117#554—671no—
118#499—672no—
119#498D6711yescharacteristic, complemented
120#479—672no—
121#478—672no—
122#477—671no—
123#474—671no—
124C8×Q8—671no—
125C4×SD16—672no—
126C4×SD16—671no—
127C2×SD16—671no—
128C4⋊Q8—1341no—
129C4⋊Q8C2×D6711yescharacteristic
130C4⋊Q8C2×D6711yescharacteristic
131#125—672no—
132#125—672no—
133#125—1342no—
134C22⋊Q8—1341no—
135C22⋊Q8—1342no—
136C4×Q8—671no—
137C4×Q8—672no—
138C4×Q8—671no—
139C4×Q8—1341no—
140C4×D4—672no—
141C4×D4—671no—
142C2×C4⋊C4—1341no—
143#107—672no—
144#107—672no—
145#106—671no—
146#106C2×D6712yes—
147#106C2×D6712yes—
148C4⋊C8—672no—
149C4⋊C8—671no—
150#103—671no—
151#103—672no—
152#102—671no—
153#102—672no—
154C4×C8—672no—
155C4×C8C2×D6711yescharacteristic
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—671no—
159C2×Q8—671no—
160C2×D4—671no—
161C22×C4—1342no—
162C22×C4—1341no—
163SD16—674no—
164C2×C8—672no—
165C2×C8C22×D6711yescharacteristic
166C2×C8C22×D6711yescharacteristic
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—671no—
170C4⋊C4—672no—
171C4⋊C4—671no—
172C4⋊C4—671no—
173C4⋊C4—1342no—
174C4⋊C4—1342no—
175C4⋊C4—1342no—
176C4⋊C4—1342no—
177C4⋊C4—1342no—
178C4⋊C4—1341no—
179C4⋊C4C22×D6712yes—
180C4⋊C4C22×D6712yes—
181C22⋊C4—1342no—
182C22⋊C4—1341no—
183C42—672no—
184C42—1342no—
185C42—1341no—
186C42C22×D6711yescharacteristic
187C23—1341no—
188Q8—42no—
189Q8—81no—
190Q8—672no—
191Q8—1342no—
192Q8—1341no—
193D4—672no—
194D4—1341no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—2682no—
198C2×C4—5362no—
199C2×C4—672no—
200C2×C4—672no—
201C2×C4—1341no—
202C2×C4—1341no—
203C2×C4—1341no—
204C2×C4D4×D6711yescharacteristic
205C2×C4D4×D6711yescharacteristic
206C2×C4C23×D6711yescharacteristic
207C8—21no—
208C8—1342no—
209C8Q8×D6712yes—
210C22—2681no—
211C22—2681no—
212C22—1342no—
213C22C2×D4×D6711yescharacteristic
214C4—21no—
215C4—42no—
216C4—42no—
217C4—2681no—
218C4—2681no—
219C4—1342no—
220C4—1342no—
221C4—1342no—
222C4#1452406412yes—
223C4C2×Q8×D6711yescharacteristic
224C4#1452409111yescharacteristic
225C2—2682no—
226C2#1686406111yescharacteristic
227C2C2×#1452406411yescharacteristic
228C2#1686420211yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#498×C67C2
socleC22×C67C22×D4
hypercenter#498D67
hypocenterC67#3381
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
1. #16864121C2semidirect
3. #16863796C2semidirect
4. #16863706C2semidirect
6. #16863688C2semidirect
8. C8×Dic134C2semidirect
11. #498×C67C2semidirect
46. #11761617Q8semidirect
115. C67#3381semidirect
119. #498D67semidirect