Finite Group Databasebeta

← #22561584

Subgroups — #22561584

Order 8576.

284 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#14523974C211yesmaximal, characteristic, complemented
2C2×Dic67⋊C8C211yesmaximal, characteristic, complemented
3C2×#14523965C211yesmaximal, characteristic
4C2×#14523963C211yesmaximal, characteristic
5#16863297C211yesmaximal, characteristic
6#16863292C211yesmaximal, characteristic, complemented
7#340×C67C211yesmaximal, characteristic, complemented
8C22×D268C2211yescharacteristic
9C22×C67⋊C8C2211yescharacteristic
10C2×C4⋊Dic67C2211yescharacteristic
11C2×Dic67⋊C4—21no—
12C2×(C2×D67)⋊C4—21no—
13C2×(C2×D67)⋊C4—21no—
14C2×C4×Dic67C2211yescharacteristic
15#14523974—22no—
16Dic67⋊C8—22no—
17#14523965—22no—
18#14523963—22no—
19C67⋊#95—21no—
20C22×C8×C67C2211yescharacteristic
21C2×C4⋊C4×C67C2211yescharacteristic
22C2×C4⋊C4×C67C2211yescharacteristic
23C23×D67—21no—
24C2×D268—21no—
25C2×D268—21no—
26C2×D268—21no—
27C2×D268—21no—
28C2×D268—21no—
29C2×D268Q812yes—
30C22×Dic67—21no—
31C22×Dic67D411yescharacteristic
32C22×Dic67D411yescharacteristic
33C2×C67⋊C8—21no—
34C2×C67⋊C8—21no—
35C2×C67⋊C8D412yes—
36C4⋊Dic67—21no—
37C4⋊Dic67D412yes—
38Dic67⋊C4—42no—
39(C2×D67)⋊C4—42no—
40(C2×D67)⋊C4—42no—
41C4×Dic67—21no—
42C4×Dic67—21no—
43C22×C4×C67—21no—
44C22×C4×C67—21no—
45C22×C4×C67C2311yescharacteristic
46C2×C8×C67—21no—
47C2×C8×C67—21no—
48C2×C8×C67—21no—
49C4⋊C4×C67—21no—
50C4⋊C4×C67—21no—
51C4⋊C4×C67—21no—
52C4⋊C4×C67—21no—
53C23×C67C2×D411yescharacteristic
54C22×D67—22no—
55C22×D67—41no—
56C22×D67—41no—
57C22×D67—41no—
58C22×D67—41no—
59C22×D67—41no—
60C22×D67—41no—
61C2×C4×C67—41no—
62C2×C4×C67—41no—
63C2×C4×C67—41no—
64C2×C4×C67—41no—
65C2×C4×C67—41no—
66C2×C4×C67—41no—
67C2×C4×C67C2×D411yescharacteristic
68C2×C4×C67C2×D411yescharacteristic
69C2×C4×C67C2×Q811yescharacteristic
70C2×C4×C67#4111yescharacteristic
71C2×C4×C67#4111yescharacteristic
72C2×C4×C67#4111yescharacteristic
73C2×Dic67—21no—
74C2×Dic67—21no—
75C2×Dic67—21no—
76C2×Dic67—22no—
77C2×Dic67—21no—
78C2×Dic67—21no—
79C2×Dic67—41no—
80C2×Dic67—41no—
81C2×Dic67SD1612yes—
82D268—22no—
83D268—22no—
84D268—41no—
85D268—41no—
86C8×C67—41no—
87C67⋊C8—22no—
88C22×C67#12111yescharacteristic
89C22×C67#12111yescharacteristic
90C22×C67C22⋊Q811yescharacteristic
91C22×C67C2×SD1611yescharacteristic
92C22×C67#13511yescharacteristic
93C22×C67AGL(1,ℤ/8)11yescharacteristic
94C22×C67AGL(1,ℤ/8)11yescharacteristic
95C2×D67—42no—
96C2×D67—42no—
97C2×D67—42no—
98C2×D67—81no—
99C2×D67—81no—
100C2×D67—81no—
101C2×D67—81no—
102C4×C67—81no—
103C4×C67—81no—
104C4×C67#12111yescharacteristic
105C4×C67C22⋊Q811yescharacteristic
106C4×C67C22⋊Q811yescharacteristic
107C4×C67#12611yescharacteristic
108Dic67—22no—
109Dic67—42no—
110Dic67—41no—
111C2×C67#39611yescharacteristic
112C2×C67#46011yescharacteristic
113C2×C67#46311yescharacteristic
114C2×C67#46511yescharacteristic
115C2×C67#46911yescharacteristic
116C2×C67#47611yescharacteristic
117C2×C67#47811yescharacteristic
118D67—82no—
119#2065—671nomaximal, Sylow
120C67#206511yescharacteristic, complemented, Sylow
121C2×#106—671no—
122C2×C4⋊C8—671no—
123C2×#102—671no—
124C2×#102—671no—
125#390—671no—
126#389—671no—
127#340D6711yescharacteristic, complemented
128C22×D4—671no—
129C22×C8—671no—
130C22×C8C2×D6711yescharacteristic
131C2×C4⋊C4—671no—
132C2×C4⋊C4—1341no—
133C2×C4⋊C4C2×D6711yescharacteristic
134C2×C4⋊C4C2×D6711yescharacteristic
135C2×C22⋊C4—1341no—
136C2×C22⋊C4—1341no—
137C2×C42—671no—
138#106—1342no—
139C4⋊C8—1342no—
140#102—1342no—
141#102—1342no—
142#95—1341no—
143C24—1341no—
144C2×D4—672no—
145C2×D4—1341no—
146C2×D4—1341no—
147C2×D4—1341no—
148C2×D4—1341no—
149C2×D4—1341no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—671no—
153C22×C4—671no—
154C22×C4—1341no—
155C22×C4C22×D6711yescharacteristic
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—672no—
160C2×C8—1341no—
161C2×C8—1341no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—2682no—
167C4⋊C4—672no—
168C4⋊C4—1341no—
169C22⋊C4—2682no—
170C22⋊C4—2682no—
171C42—1341no—
172C42—1341no—
173C23—2681no—
174C23—2681no—
175C23—2681no—
176C23—2681no—
177C23—2681no—
178C23—2681no—
179C23—1342no—
180C23D4×D6711yescharacteristic
181D4—2681no—
182D4—2681no—
183D4—1342no—
184D4—1342no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—2681no—
192C2×C4—2681no—
193C2×C4—672no—
194C2×C4—1341no—
195C2×C4—1342no—
196C2×C4—1341no—
197C2×C4—1341no—
198C2×C4—1341no—
199C2×C4—1341no—
200C2×C4D4×D6711yescharacteristic
201C2×C4D4×D6711yescharacteristic
202C2×C4Q8×D6711yescharacteristic
203C2×C4#1176162811yescharacteristic
204C2×C4#1176162811yescharacteristic
205C2×C4#1176162811yescharacteristic
206C8—41no—
207C8—1342no—
208C22—2682no—
209C22—2682no—
210C22—2682no—
211C22—5361no—
212C22—5361no—
213C22—5361no—
214C22—5361no—
215C22#1452401611yescharacteristic
216C22#1452401611yescharacteristic
217C22#1452402111yescharacteristic
218C22SD16×D6711yescharacteristic
219C22#1452406611yescharacteristic
220C22#1452406711yescharacteristic
221C22#1452407011yescharacteristic
222C4—81no—
223C4—81no—
224C4—2682no—
225C4—2681no—
226C4—1342no—
227C4#1452401611yescharacteristic
228C4#1452402111yescharacteristic
229C4#1452402111yescharacteristic
230C4#1452403711yescharacteristic
231C2—5362no—
232C2#1686332011yescharacteristic
233C2#1686359511yescharacteristic
234C2#1686361411yescharacteristic
235C2#1686362811yescharacteristic
236C2#1686364811yescharacteristic
237C2#1686368511yescharacteristic
238C2#1686369911yescharacteristic
subgroupG/H
centerC23D4×D67
commutatorC22×C4×C67C23
FrattiniC22×C4C22×D67
Fitting#340×C67C2
socleC23×C67C2×D4
hypercenter#340D67
hypocenterC67#2065
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
1. C2×#14523974C2semidirect
2. C2×Dic67⋊C8C2semidirect
6. #16863292C2semidirect
7. #340×C67C2semidirect
120. C67#2065semidirect
127. #340D67semidirect