Finite Group Databasebeta

← #22565220

Subgroups — #22565220

Order 8576.

544 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 173 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×#14524022C212yesmaximal
2#16863330C212yesmaximal, complemented
3#16863317C212yesmaximal, complemented
4#16863300C212yesmaximal, complemented
5#16863293C212yesmaximal, complemented
6#16863269C212yesmaximal
7#16863256C211yesmaximal, characteristic
8C4×C22⋊Dic67C211yesmaximal, characteristic
9C2×#95×C67C211yesmaximal, characteristic
10C22×Dic134C2212yes—
11#14524022—28no—
12C2×C4⋊Dic67C2212yes—
13C2×C4⋊Dic67C2212yes—
14C2×Dic67⋊C4C2212yes—
15C2×Dic67⋊C4C2212yes—
16C2×Dic67⋊C4C2212yes—
17C2×Dic67⋊C4C2212yes—
18C2×C22⋊Dic67C2212yes—
19C2×C22⋊Dic67C2211yescharacteristic
20C2×C22⋊Dic67C2211yescharacteristic
21C2×C4×Dic67C2212yes—
22C67⋊#95C2212yes—
23C67⋊#95C2211yescharacteristic
24C67⋊#95C2211yescharacteristic
25C67⋊#95C2212yes—
26C67⋊#95C2212yes—
27C23×C4×C67C2212yes—
28C23×C4×C67C2211yescharacteristic
29#95×C67C2212yes—
30#95×C67C2211yescharacteristic
31#95×C67C2211yescharacteristic
32C2×Dic134—24no—
33C2×Dic134—24no—
34C2×Dic134—42no—
35C22×Dic67C2312yes—
36C22×Dic67C2312yes—
37C22×Dic67C2312yes—
38C22×Dic67C2312yes—
39C4⋊Dic67—24no—
40C4⋊Dic67—42no—
41Dic67⋊C4—24no—
42Dic67⋊C4—24no—
43Dic67⋊C4—24no—
44Dic67⋊C4—42no—
45C22⋊Dic67—24no—
46C22⋊Dic67D414yes—
47C22⋊Dic67Q814yescomplemented
48C4×Dic67—42no—
49C24×C67C2311yescharacteristic
50C22×C4×C67—24no—
51C22×C4×C67—24no—
52C22×C4×C67—42no—
53C22×C4×C67—42no—
54C22×C4×C67Q814yescomplemented
55C22×C4×C67C2312yes—
56C22×C4×C67C2311yescharacteristic
57C22×C4×C67C2311yescharacteristic
58C22×C4×C67C2312yes—
59C23×C67—22no—
60C23×C67—22no—
61C23×C67C2×D412yes—
62C23×C67C2×Q812yes—
63C23×C67C2×Q812yes—
64C23×C67C2411yescharacteristic
65C2×C4×C67—22no—
66C2×C4×C67—22no—
67C2×C4×C67—22no—
68C2×C4×C67—22no—
69C2×C4×C67—22no—
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67—44no—
73C2×C4×C67—48no—
74C2×C4×C67C2×Q812yes—
75C2×C4×C67C2×Q812yes—
76C2×C4×C67#4114yes—
77C2×Dic67—22no—
78C2×Dic67—22no—
79C2×Dic67—22no—
80C2×Dic67—22no—
81C2×Dic67—22no—
82C2×Dic67—22no—
83C2×Dic67—22no—
84C2×Dic67—22no—
85C2×Dic67—22no—
86C2×Dic67—22no—
87C2×Dic67C2×D414yes—
88C2×Dic67C2×Q814yes—
89Dic134—48no—
90C22×C67—22no—
91C22×C67—22no—
92C22×C67—24no—
93C22×C67—24no—
94C22×C67C4⋊Q814yes—
95C22×C67C22×D411yescharacteristic
96C22×C67C22×Q811yescharacteristic
97C22×C67C22×Q811yescharacteristic
98C22×C67C2×#4112yes—
99C22×C67H(5,𝔽2)12yes—
100C4×C67—22no—
101C4×C67—22no—
102C4×C67—24no—
103C4×C67—42no—
104Dic67—24no—
105Dic67—24no—
106Dic67—42no—
107Dic67—42no—
108C2×C67—24no—
109C2×C67C2×C4⋊Q811yescharacteristic
110C2×C67#53311yescharacteristic
111C2×C67#54311yescharacteristic
112C2×C67#54612yes—
113C2×C67#55412yes—
114#2549—671nomaximal, Sylow
115C67#254911yescharacteristic, complemented, Sylow
116C2×C22⋊Q8—672no—
117#398—672no—
118#395—672no—
119#391—672no—
120#389—672no—
121#385—672no—
122#380—671no—
123C4×C22⋊C4—671no—
124C2×#95D6711yescharacteristic
125C22×Q8—672no—
126C23×C4C2×D6712yes—
127C23×C4C2×D6711yescharacteristic
128C22⋊Q8—1348no—
129C2×C4⋊C4—672no—
130C2×C4⋊C4—672no—
131C2×C4⋊C4—672no—
132C2×C4⋊C4—672no—
133C2×C4⋊C4—672no—
134C2×C4⋊C4—672no—
135C2×C22⋊C4—671no—
136C2×C22⋊C4—671no—
137C2×C22⋊C4—672no—
138C2×C42—672no—
139#95—672no—
140#95—671no—
141#95—671no—
142#95—672no—
143#95—672no—
144#95C2×D6712yes—
145#95C2×D6711yescharacteristic
146#95C2×D6711yescharacteristic
147C24C22×D6711yescharacteristic
148C2×Q8—2682no—
149C2×Q8—1344no—
150C2×Q8—1344no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—672no—
156C22×C4—672no—
157C22×C4—672no—
158C22×C4—672no—
159C22×C4Dic13414yescomplemented
160C22×C4C22×D6712yes—
161C22×C4C22×D6711yescharacteristic
162C22×C4C22×D6711yescharacteristic
163C22×C4C22×D6712yes—
164C4⋊C4—2682no—
165C4⋊C4—2682no—
166C4⋊C4—1344no—
167C4⋊C4—1344no—
168C4⋊C4—1344no—
169C4⋊C4—1344no—
170C22⋊C4—674no—
171C22⋊C4—674no—
172C22⋊C4—1344no—
173C42—2682no—
174C23—22no—
175C23—22no—
176C23D4×D6712yes—
177C23Q8×D6712yes—
178C23C2×Dic13412yes—
179C23C23×D6711yescharacteristic
180Q8—2688no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—44no—
189C2×C4—48no—
190C2×C4—674no—
191C2×C4—674no—
192C2×C4—1342no—
193C2×C4—1342no—
194C2×C4—1342no—
195C2×C4—1342no—
196C2×C4—1342no—
197C2×C4—1342no—
198C2×C4—1342no—
199C2×C4—1342no—
200C2×C4—1342no—
201C2×C4—1342no—
202C2×C4C2×Dic13412yes—
203C2×C4C2×Dic13412yes—
204C2×C4#1176162814yes—
205C22—22no—
206C22—24no—
207C22—24no—
208C22—22no—
209C22#1452404214yes—
210C22C2×D4×D6711yescharacteristic
211C22C2×Q8×D6711yescharacteristic
212C22C22×Dic13411yescharacteristic
213C22C2×#1176162812yes—
214C22#1452408812yes—
215C4—22no—
216C4—22no—
217C4—24no—
218C4—42no—
219C4—2682no—
220C4—2682no—
221C4—1344no—
222C4—1344no—
223C2—24no—
224C2C2×#1452404211yescharacteristic
225C2#1686395311yescharacteristic
226C2#1686402211yescharacteristic
227C2#1686403612yes—
228C2#1686410112yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC2×#95×C67C2
hypercenterC2×#95D67
hypocenterC67#2549
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
2. #16863330C2semidirect
3. #16863317C2semidirect
4. #16863300C2semidirect
5. #16863293C2semidirect
47. C22⋊Dic67Q8semidirect
54. C22×C4×C67Q8semidirect
115. C67#2549semidirect
159. C22×C4Dic134semidirect