Finite Group Databasebeta

← #22561405

Subgroups — #22561405

Order 8576.

316 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 87 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×#14524022C211yesmaximal, characteristic
2C2×#14523974C211yesmaximal, characteristic, complemented
3C2×#14523966C211yesmaximal, characteristic, complemented
4C2×#14523966C211yesmaximal, characteristic, complemented
5C2×C22⋊(C67⋊C8)C211yesmaximal, characteristic
6#16863158C211yesmaximal, characteristic, complemented
7#380×C67C211yesmaximal, characteristic
8C22×Dic134C2211yescharacteristic
9C22×C67⋊C8C2211yescharacteristic
10C22×C67⋊C8C2211yescharacteristic
11#14524022—22no—
12#14524022—22no—
13C2×C4⋊Dic67C2211yescharacteristic
14C2×Dic67⋊C4—21no—
15C2×C22⋊Dic67—21no—
16#14523974—22no—
17#14523966—22no—
18#14523966—22no—
19C22⋊(C67⋊C8)—22no—
20C23×C4×C67C2211yescharacteristic
21C2×C4⋊C4×C67C2211yescharacteristic
22C2×C4⋊C4×C67C2211yescharacteristic
23C2×C22⋊C4×C67—21no—
24#95×C67—21no—
25C2×Dic134—21no—
26C2×Dic134—21no—
27C2×Dic134—21no—
28C2×Dic134—21no—
29C2×Dic134—21no—
30C2×Dic134D412yes—
31C22×Dic67—21no—
32C22×Dic67—21no—
33C2×C67⋊C8—21no—
34C2×C67⋊C8—21no—
35C2×C67⋊C8—21no—
36C2×C67⋊C8—21no—
37C2×C67⋊C8—21no—
38C2×C67⋊C8D412yes—
39C4⋊Dic67—21no—
40C4⋊Dic67D412yes—
41Dic67⋊C4—24no—
42C22⋊Dic67—24no—
43C24×C67D411yescharacteristic
44C22×C4×C67—21no—
45C22×C4×C67—21no—
46C22×C4×C67—21no—
47C22×C4×C67—21no—
48C22×C4×C67—21no—
49C22×C4×C67—21no—
50C22×C4×C67—21no—
51C22×C4×C67—21no—
52C22×C4×C67D411yescharacteristic
53C22×C4×C67C2311yescharacteristic
54C4⋊C4×C67—21no—
55C4⋊C4×C67—21no—
56C4⋊C4×C67—21no—
57C4⋊C4×C67—21no—
58C22⋊C4×C67—42no—
59C23×C67—21no—
60C23×C67—21no—
61C23×C67—21no—
62C23×C67—21no—
63C23×C67—21no—
64C23×C67—21no—
65C23×C67SD1612yes—
66C23×C67C2×D411yescharacteristic
67C2×C4×C67—21no—
68C2×C4×C67—21no—
69C2×C4×C67—21no—
70C2×C4×C67—41no—
71C2×C4×C67—41no—
72C2×C4×C67—41no—
73C2×C4×C67—41no—
74C2×C4×C67—41no—
75C2×C4×C67—41no—
76C2×C4×C67—41no—
77C2×C4×C67—41no—
78C2×C4×C67—41no—
79C2×C4×C67—41no—
80C2×C4×C67C2×D411yescharacteristic
81C2×C4×C67C2×D411yescharacteristic
82C2×C4×C67C2×D411yescharacteristic
83C2×C4×C67#4111yescharacteristic
84C2×C4×C67#4111yescharacteristic
85C2×C4×C67#4111yescharacteristic
86C2×Dic67—22no—
87C2×Dic67—22no—
88C2×Dic67—41no—
89C2×Dic67—41no—
90C2×Dic67—41no—
91C2×Dic67—41no—
92Dic134—22no—
93Dic134—22no—
94Dic134—41no—
95Dic134—41no—
96C67⋊C8—22no—
97C67⋊C8—41no—
98C22×C67—22no—
99C22×C67—22no—
100C22×C67—22no—
101C22×C67—41no—
102C22×C67—41no—
103C22×C67—41no—
104C22×C67—41no—
105C22×C67#12011yescharacteristic
106C22×C67#12111yescharacteristic
107C22×C67#12311yescharacteristic
108C22×C67C2×SD1611yescharacteristic
109C22×C67#13511yescharacteristic
110C22×C67#13711yescharacteristic
111C22×C67#13711yescharacteristic
112C4×C67—41no—
113C4×C67—81no—
114C4×C67—81no—
115C4×C67#12111yescharacteristic
116C4×C67#12111yescharacteristic
117C4×C67#12311yescharacteristic
118C4×C67#12411yescharacteristic
119Dic67—42no—
120Dic67—42no—
121C2×C67—42no—
122C2×C67#39411yescharacteristic
123C2×C67#44811yescharacteristic
124C2×C67#45211yescharacteristic
125C2×C67#46511yescharacteristic
126C2×C67#47011yescharacteristic
127C2×C67#48311yescharacteristic
128C2×C67#48511yescharacteristic
129#2035—671nomaximal, Sylow
130C67#203511yescharacteristic, complemented, Sylow
131C2×C22⋊Q8—671no—
132C2×#106—671no—
133C2×#103—671no—
134C2×#103—671no—
135C2×C22⋊C8—671no—
136#380D6711yescharacteristic
137#340—671no—
138C22×Q8—671no—
139C23×C4C2×D6711yescharacteristic
140C22×C8—671no—
141C22×C8—671no—
142C22⋊Q8—1342no—
143C22⋊Q8—1342no—
144C2×C4⋊C4—671no—
145C2×C4⋊C4—1341no—
146C2×C4⋊C4C2×D6711yescharacteristic
147C2×C4⋊C4C2×D6711yescharacteristic
148C2×C22⋊C4—21no—
149C2×C22⋊C4—1341no—
150#106—1342no—
151#103—1342no—
152#103—1342no—
153C22⋊C8—1342no—
154#95—21no—
155C24C22⋊D6711yescharacteristic
156C2×Q8—672no—
157C2×Q8—1341no—
158C2×Q8—1341no—
159C2×Q8—1341no—
160C2×Q8—1341no—
161C2×Q8—1341no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—1341no—
171C22×C4—1341no—
172C22×C4C22⋊D6711yescharacteristic
173C22×C4C22×D6711yescharacteristic
174C2×C8—672no—
175C2×C8—1341no—
176C2×C8—1341no—
177C2×C8—1341no—
178C2×C8—1341no—
179C2×C8—1341no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—672no—
185C4⋊C4—1344no—
186C4⋊C4—1341no—
187C22⋊C4—42no—
188C22⋊C4—1344no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23#1176161612yes—
196C23C2×C22⋊D6711yescharacteristic
197Q8—2681no—
198Q8—2681no—
199Q8—1342no—
200Q8—1342no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—2681no—
215C2×C4—2681no—
216C2×C4—2681no—
217C2×C4—2681no—
218C2×C4—1342no—
219C2×C4—1342no—
220C2×C4D4×D6711yescharacteristic
221C2×C4D4×D6711yescharacteristic
222C2×C4D4×D6711yescharacteristic
223C2×C4#1176162711yescharacteristic
224C2×C4#1176162811yescharacteristic
225C2×C4#1176162811yescharacteristic
226C8—2681no—
227C8—1342no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22#1452401111yescharacteristic
236C22#1452401511yescharacteristic
237C22#1452402611yescharacteristic
238C22C2×#1176161611yescharacteristic
239C22#1452406311yescharacteristic
240C22#1452407311yescharacteristic
241C22#1452407411yescharacteristic
242C4—41no—
243C4—81no—
244C4—81no—
245C4—2682no—
246C4—2682no—
247C4#1452401611yescharacteristic
248C4#1452401611yescharacteristic
249C4#1452402511yescharacteristic
250C4#1452402911yescharacteristic
251C2—42no—
252C2#1686331111yescharacteristic
253C2#1686353411yescharacteristic
254C2#1686355611yescharacteristic
255C2#1686362311yescharacteristic
256C2#1686365211yescharacteristic
257C2#1686372611yescharacteristic
258C2#1686373711yescharacteristic
subgroupG/H
centerC23C2×C22⋊D67
commutatorC22×C4×C67C23
FrattiniC22×C4C22×D67
Fitting#380×C67C2
socleC23×C67C2×D4
hypercenter#380D67
hypocenterC67#2035
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#14523974C2semidirect
3. C2×#14523966C2semidirect
4. C2×#14523966C2semidirect
6. #16863158C2semidirect
130. C67#2035semidirect