Finite Group Databasebeta

← #22570054

Subgroups — #22570054

Order 8576.

362 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 133 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#16864024C211yesmaximal, characteristic
2#16863873C211yesmaximal, characteristic
3#16863706C212yesmaximal, complemented
4#16863702C212yesmaximal, complemented
5#16863690C212yesmaximal, complemented
6#16863680C212yesmaximal, complemented
7#16863439C211yesmaximal, characteristic
8#16863422C211yesmaximal, characteristic
9#16863404C211yesmaximal, characteristic
10C2×#14523968C211yesmaximal, characteristic
11#427×C67C211yesmaximal, characteristic
12C22×Dic134C2211yescharacteristic
13C2×#11761612C2211yescharacteristic
14#14524043—21no—
15#14524042C2212yes—
16#14524035—21no—
17#14524034C2212yes—
18#14524020—22no—
19C4×Dic134—21no—
20C4×Dic134—21no—
21#14524008C2212yes—
22#14524008C2212yes—
23#14524002C2211yescharacteristic
24#14523999—22no—
25C2×C4⋊Dic67C2211yescharacteristic
26#14523976C2212yes—
27#14523974C2212yes—
28Dic67⋊C8C2212yes—
29Dic67⋊C8C2212yes—
30#14523968C2212yes—
31#14523968C2212yes—
32#14523966C2212yes—
33#14523966C2212yes—
34C22×C8×C67C2211yescharacteristic
35#117×C67C2211yescharacteristic
36#117×C67C2211yescharacteristic
37#107×C67C2211yescharacteristic
38#107×C67C2211yescharacteristic
39#106×C67C2211yescharacteristic
40#106×C67C2211yescharacteristic
41C2×Dic134—21no—
42C2×Dic134—22no—
43C2×Dic134—22no—
44C2×Dic134Q814yescomplemented
45C2×Dic134C2312yes—
46C22×Dic67—21no—
47C22×Dic67—21no—
48#11761612—22no—
49C2×C67⋊C8C2312yes—
50C4⋊Dic67—21no—
51C4⋊Dic67—22no—
52C4⋊Dic67D411yescharacteristic
53C4⋊Dic67D411yescharacteristic
54C4⋊Dic67C2312yes—
55Dic67⋊C4—22no—
56Dic67⋊C4—22no—
57Dic67⋊C4—22no—
58Dic67⋊C4—22no—
59C22⋊Dic67D412yes—
60C4×Dic67—22no—
61C4×Dic67—22no—
62C4×Dic67C2312yes—
63C22×C4×C67C2311yescharacteristic
64C2×C8×C67—22no—
65C2×C8×C67C2311yescharacteristic
66C2×C8×C67C2311yescharacteristic
67C4⋊C4×C67C2311yescharacteristic
68C4⋊C4×C67C2311yescharacteristic
69C4⋊C4×C67C2311yescharacteristic
70C4⋊C4×C67C2311yescharacteristic
71C22⋊C4×C67—21no—
72C22⋊C4×C67—21no—
73C42×C67—21no—
74C42×C67—21no—
75C23×C67C2×D411yescharacteristic
76C2×C4×C67—21no—
77C2×C4×C67—21no—
78C2×C4×C67—21no—
79C2×C4×C67—21no—
80C2×C4×C67C2×D411yescharacteristic
81C2×C4×C67C2×Q812yes—
82C2×C4×C67#4112yes—
83C2×C4×C67C2411yescharacteristic
84C2×Dic67—24no—
85C2×Dic67—22no—
86C2×Dic67—22no—
87C2×Dic67—81no—
88C2×Dic67C2×D412yes—
89C2×Dic67C2×D412yes—
90Dic134—24no—
91Dic134—22no—
92Dic134—82no—
93Dic134C2×Q814yes—
94C8×C67—21no—
95C8×C67—21no—
96C67⋊C8—22no—
97C22×C67—21no—
98C22×C67—21no—
99C22×C67C22⋊Q812yes—
100C22×C67C22×D411yescharacteristic
101C4×C67—41no—
102C4×C67—41no—
103C4×C67—41no—
104C4×C67—41no—
105C4×C67C22⋊Q812yes—
106C4×C67C22×Q811yescharacteristic
107C4×C67C2×#4111yescharacteristic
108Dic67—24no—
109Dic67—22no—
110Dic67—22no—
111Dic67—42no—
112C2×C67—22no—
113C2×C67C2×C22⋊Q811yescharacteristic
114C2×C67#57711yescharacteristic
115C2×C67#57811yescharacteristic
116#3082—671nomaximal, Sylow
117C67#308211yescharacteristic, complemented, Sylow
118#544—671no—
119#519—671no—
120#479—672no—
121#479—672no—
122#477—672no—
123#475—672no—
124#428—671no—
125#427D6711yescharacteristic
126#424—671no—
127#419—671no—
128C2×#103—671no—
129C22×Q8—671no—
130C2×M16—671no—
131C22×C8C2×D6711yescharacteristic
132C4⋊Q8—672no—
133C4⋊Q8—1341no—
134#125—672no—
135#125—1341no—
136C22⋊Q8—1342no—
137C4×Q8—672no—
138C4×Q8—672no—
139C4×Q8—1341no—
140C4×Q8—1341no—
141#117—671no—
142#117—1342no—
143#117C2×D6711yescharacteristic
144#117C2×D6711yescharacteristic
145C2×C4⋊C4—671no—
146#107—672no—
147#107C2×D6711yescharacteristic
148#107C2×D6711yescharacteristic
149#106—672no—
150#106C2×D6711yescharacteristic
151#106C2×D6711yescharacteristic
152C4⋊C8—672no—
153C4⋊C8—672no—
154#103—672no—
155#103—672no—
156#103—672no—
157#103—672no—
158C2×Q8—674no—
159C2×Q8—672no—
160C2×Q8—1342no—
161C2×Q8—1341no—
162C2×Q8—1342no—
163C22×C4—1341no—
164C22×C4—1341no—
165C22×C4C22×D6711yescharacteristic
166M16—1342no—
167C2×C8—22no—
168C2×C8—672no—
169C2×C8C22×D6711yescharacteristic
170C2×C8C22×D6711yescharacteristic
171C4⋊C4—671no—
172C4⋊C4—671no—
173C4⋊C4—672no—
174C4⋊C4—1342no—
175C4⋊C4—1342no—
176C4⋊C4—1342no—
177C4⋊C4—1342no—
178C4⋊C4—1342no—
179C4⋊C4—1341no—
180C4⋊C4C22×D6711yescharacteristic
181C4⋊C4C22×D6711yescharacteristic
182C4⋊C4C22×D6711yescharacteristic
183C4⋊C4C22×D6711yescharacteristic
184C22⋊C4—21no—
185C22⋊C4—21no—
186C22⋊C4—672no—
187C42—21no—
188C42—21no—
189C42—672no—
190C42—1342no—
191C42—1342no—
192C23D4×D6711yescharacteristic
193Q8—5362no—
194Q8—674no—
195Q8—1344no—
196Q8—1342no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—5361no—
202C2×C4—672no—
203C2×C4—672no—
204C2×C4—1342no—
205C2×C4—1344no—
206C2×C4—1342no—
207C2×C4D4×D6711yescharacteristic
208C2×C4Q8×D6712yes—
209C2×C4#1176162812yes—
210C2×C4C23×D6711yescharacteristic
211C8—21no—
212C8—21no—
213C8—1342no—
214C22—21no—
215C22—21no—
216C22#1452402112yes—
217C22C2×D4×D6711yescharacteristic
218C4—41no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—2682no—
223C4—1344no—
224C4—1342no—
225C4—1342no—
226C4#1452402112yes—
227C4C2×Q8×D6711yescharacteristic
228C4C2×#1176162811yescharacteristic
229C2—22no—
230C2C2×#1452402111yescharacteristic
231C2#1686420311yescharacteristic
232C2#1686421011yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#427×C67C2
socleC22×C67C22×D4
hypercenter#427D67
hypocenterC67#3082
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
3. #16863706C2semidirect
4. #16863702C2semidirect
5. #16863690C2semidirect
6. #16863680C2semidirect
44. C2×Dic134Q8semidirect
117. C67#3082semidirect