Finite Group Databasebeta

← #22573088

Subgroups — #22573088

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#16864103C211yesmaximal, characteristic
3#16863797C212yesmaximal, complemented
4#16863792C211yesmaximal, characteristic, complemented
5#16863702C212yesmaximal, complemented
6#16863694C212yesmaximal
7#16863692C211yesmaximal, characteristic, complemented
8#16863520C211yesmaximal, characteristic, complemented
9#16863487C212yesmaximal
10C4×#11761616C211yesmaximal, characteristic
11#474×C67C211yesmaximal, characteristic
12C2×#11761616C2211yescharacteristic
13#14524043C2211yescharacteristic
14#14524041—21no—
15#14524034C2212yes—
16#14524034C2212yes—
17#14524033—22no—
18#14524022—21no—
19#14524020—22no—
20Q8×Dic67—21no—
21C4×Dic134C2211yescharacteristic
22C4×Dic134C2211yescharacteristic
23#14524008C2212yes—
24D4×Dic67C2212yes—
25C2×C4⋊Dic67—21no—
26#14523977C2212yes—
27#14523974C2212yes—
28#14523974C2212yes—
29#14523974C2211yescharacteristic
30Dic67⋊C8C2212yes—
31#14523968C2212yes—
32#14523966C2211yescharacteristic
33#14523964C2211yescharacteristic
34C4×C67⋊C8C2211yescharacteristic
35C8×Dic67C2212yes—
36C4⋊Q8×C67C2211yescharacteristic
37C4×D4×C67C2211yescharacteristic
38#107×C67C2212yes—
39C4⋊C8×C67C2211yescharacteristic
40#102×C67C2212yes—
41C2×Dic134—21no—
42C2×Dic134D411yescharacteristic
43C2×Dic134C2311yescharacteristic
44C22×Dic67—22no—
45#11761616Q814yescomplemented
46C2×C67⋊C8C2311yescharacteristic
47C2×C67⋊C8C2311yescharacteristic
48C4⋊Dic67—21no—
49C4⋊Dic67—22no—
50C4⋊Dic67—22no—
51C4⋊Dic67D411yescharacteristic
52C4⋊Dic67C2311yescharacteristic
53C4⋊Dic67C2312yes—
54Dic67⋊C4—21no—
55Dic67⋊C4—22no—
56Dic67⋊C4—22no—
57Dic67⋊C4—22no—
58Dic67⋊C4D412yes—
59C22⋊Dic67—22no—
60C4×Dic67—21no—
61C4×Dic67—22no—
62C4×Dic67C2312yes—
63C2×Q8×C67—21no—
64C2×D4×C67C2311yescharacteristic
65C22×C4×C67—21no—
66C2×C8×C67C2312yes—
67C4⋊C4×C67—21no—
68C4⋊C4×C67C2312yes—
69C4⋊C4×C67C2311yescharacteristic
70C22⋊C4×C67—21no—
71C42×C67C2311yescharacteristic
72C23×C67—21no—
73Q8×C67—42no—
74D4×C67—21no—
75D4×C67C2×Q812yes—
76C2×C4×C67—21no—
77C2×C4×C67—22no—
78C2×C4×C67—42no—
79C2×C4×C67C2×D411yescharacteristic
80C2×C4×C67C2×D411yescharacteristic
81C2×C4×C67C2411yescharacteristic
82C2×Dic67—21no—
83C2×Dic67—21no—
84C2×Dic67—22no—
85C2×Dic67—82no—
86C2×Dic67C2×D412yes—
87C2×Dic67C2×D412yes—
88Dic134—22no—
89Dic134—21no—
90Dic134—81no—
91Dic134C2×Q812yes—
92C8×C67—22no—
93C67⋊C8—21no—
94C67⋊C8C2×Q812yes—
95C22×C67—22no—
96C22×C67—41no—
97C22×C67—41no—
98C22×C67C22×D411yescharacteristic
99C4×C67—21no—
100C4×C67—41no—
101C4×C67—42no—
102C4×C67#13512yes—
103C4×C67C22×Q811yescharacteristic
104C4×C672−1+411yescharacteristic
105Dic67—22no—
106Dic67—22no—
107Dic67—22no—
108Dic67—42no—
109Dic67—41no—
110C2×C67—42no—
111C2×C67D4×Q811yescharacteristic
112C2×C67C2×#13511yescharacteristic
113C2×C67#57711yescharacteristic
114#3381—671nomaximal, Sylow
115C67#338111yescharacteristic, complemented, Sylow
116#557—671no—
117#554—671no—
118#499—672no—
119#498—671no—
120#479—672no—
121#478—672no—
122#477—671no—
123#474D6711yescharacteristic
124C8×Q8—671no—
125C4×SD16—671no—
126C4×SD16—672no—
127C2×SD16—671no—
128C4⋊Q8—671no—
129C4⋊Q8—1341no—
130C4⋊Q8C2×D6711yescharacteristic
131#125—672no—
132#125—672no—
133#125—1342no—
134C22⋊Q8—1342no—
135C22⋊Q8—1341no—
136C4×Q8—671no—
137C4×Q8—672no—
138C4×Q8—671no—
139C4×Q8—1341no—
140C4×D4—672no—
141C4×D4C2×D6711yescharacteristic
142C2×C4⋊C4—1341no—
143#107—672no—
144#107C2×D6712yes—
145#106—672no—
146#106—671no—
147#106—672no—
148C4⋊C8—672no—
149C4⋊C8C2×D6711yescharacteristic
150#103—672no—
151#103—671no—
152#102—671no—
153#102C2×D6712yes—
154C4×C8—672no—
155C4×C8—671no—
156C2×Q8—21no—
157C2×Q8—671no—
158C2×Q8—671no—
159C2×Q8—1341no—
160C2×D4C22×D6711yescharacteristic
161C22×C4—21no—
162C22×C4—1342no—
163SD16—674no—
164C2×C8—671no—
165C2×C8—671no—
166C2×C8C22×D6712yes—
167C4⋊C4—21no—
168C4⋊C4—671no—
169C4⋊C4—672no—
170C4⋊C4—671no—
171C4⋊C4—672no—
172C4⋊C4—1342no—
173C4⋊C4—1342no—
174C4⋊C4—1342no—
175C4⋊C4—1342no—
176C4⋊C4—1342no—
177C4⋊C4—1341no—
178C4⋊C4—1341no—
179C4⋊C4C22×D6712yes—
180C4⋊C4C22×D6711yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—1342no—
183C42—672no—
184C42—1342no—
185C42—1341no—
186C42C22×D6711yescharacteristic
187C23—21no—
188Q8—42no—
189Q8—5361no—
190Q8—672no—
191Q8—1342no—
192Q8—1341no—
193D4—21no—
194D4Q8×D6712yes—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—42no—
198C2×C4—5362no—
199C2×C4—672no—
200C2×C4—672no—
201C2×C4—1341no—
202C2×C4—1342no—
203C2×C4—1341no—
204C2×C4D4×D6711yescharacteristic
205C2×C4D4×D6711yescharacteristic
206C2×C4C23×D6711yescharacteristic
207C8—22no—
208C8—672no—
209C8—1341no—
210C22—22no—
211C22—41no—
212C22—41no—
213C22C2×D4×D6711yescharacteristic
214C4—21no—
215C4—41no—
216C4—42no—
217C4—2682no—
218C4—2681no—
219C4—1342no—
220C4—1342no—
221C4—1342no—
222C4#1452406212yes—
223C4C2×Q8×D6711yescharacteristic
224C4#1452409011yescharacteristic
225C2—42no—
226C2#1686405911yescharacteristic
227C2C2×#1452406211yescharacteristic
228C2#1686420511yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#474×C67C2
socleC22×C67C22×D4
hypercenter#474D67
hypocenterC67#3381
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. #16864121C2semidirect
3. #16863797C2semidirect
4. #16863792C2semidirect
5. #16863702C2semidirect
7. #16863692C2semidirect
8. #16863520C2semidirect
45. #11761616Q8semidirect
115. C67#3381semidirect