Finite Group Databasebeta

← #22571276

Subgroups — #22571276

Order 8576.

396 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 109 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#16864007C211yesmaximal, characteristic, complemented
2#16863964C211yesmaximal, characteristic
3#16863770C211yesmaximal, characteristic, complemented
4#16863761C211yesmaximal, characteristic, complemented
5#16863750C211yesmaximal, characteristic, complemented
6#16863620C212yesmaximal, complemented
7#16863594C212yesmaximal, complemented
8#16863557C212yesmaximal, complemented
9#16863545C212yesmaximal, complemented
10#16863458C211yesmaximal, characteristic, complemented
11#497×C67C211yesmaximal, characteristic, complemented
12C2×#11761629C2211yescharacteristic
13C2×D4×D67C2211yescharacteristic
14C2×#11761615C2212yes—
15C2×#11761615C2212yescomplemented
16#14524040C2211yescharacteristic
17#14524039C2211yescharacteristic
18#14524038—22no—
19#14524035C2211yescharacteristic, complemented
20#14524033—22no—
21#14524032C2211yescharacteristic
22#14524019C2212yes—
23#14524016—22no—
24#14524013—22no—
25Q8×Dic67C2212yescomplemented
26#14524000C2211yescharacteristic
27Dic67⋊C8C2212yescomplemented
28#14523967C2212yescomplemented
29#14523965C2212yes—
30#14523965C2212yescomplemented
31#14523964C2212yes—
32(C2×D67)⋊C8C2212yes—
33C4×C67⋊C8C2211yescharacteristic, complemented
34C2×Q16×C67C2212yescomplemented
35C2×SD16×C67C2212yes—
36C4⋊Q8×C67C2211yescharacteristic, complemented
37#124×C67C2211yescharacteristic
38#97×C67C2211yescharacteristic, complemented
39C23×D67—21no—
40#11761629—42no—
41C2×D268—21no—
42C2×D268—22no—
43C2×D268C2311yescharacteristic
44C2×D268C2311yescharacteristic
45C2×C22⋊D67—21no—
46C2×C22⋊D67—22no—
47D4×D67—42no—
48C2×C4×D67—21no—
49C2×C4×D67C2311yescharacteristic
50#11761615—42no—
51#11761615—42no—
52C2×C67⋊C8C2312yes—
53C4⋊Dic67—22no—
54C4⋊Dic67C2312yes—
55Dic67⋊C4—22no—
56Dic67⋊C4—22no—
57Dic67⋊C4D412yes—
58(C2×D67)⋊C4—22no—
59(C2×D67)⋊C4—22no—
60(C2×D67)⋊C4D412yes—
61C4×Dic67—22no—
62C4×Dic67C2311yescharacteristic
63C2×Q8×C67C2311yescharacteristic
64C2×Q8×C67C2312yes—
65C2×D4×C67C2311yescharacteristic
66Q16×C67—42no—
67SD16×C67—42no—
68C2×C8×C67C2312yes—
69C4⋊C4×C67—22no—
70C22⋊C4×C67—22no—
71C42×C67C2311yescharacteristic
72C23×C67—21no—
73C22×D67—21no—
74C22×D67—21no—
75C22×D67—42no—
76C22×D67—82no—
77C22×D67C2×D411yescharacteristic
78Q8×C67—22no—
79Q8×C67—22no—
80Q8×C67—22no—
81D4×C67—22no—
82C2×C4×C67—21no—
83C2×C4×C67—22no—
84C2×C4×C67C2×D412yes—
85C2×C4×C67C2411yescharacteristic
86C22⋊D67—42no—
87C22⋊D67—82no—
88C2×Dic67—22no—
89C2×Dic67C2×D412yes—
90C2×Dic67C2×D411yescharacteristic
91D268—22no—
92D268—22no—
93D268—42no—
94D268—82no—
95C4×D67—22no—
96C4×D67—42no—
97C8×C67—22no—
98C67⋊C8—22no—
99C22×C67—41no—
100C22×C67—42no—
101C22×C67C22×D411yescharacteristic
102C2×D67—21no—
103C2×D67—22no—
104C2×D67—41no—
105C2×D67—41no—
106C2×D67—42no—
107C2×D67—42no—
108C2×D67—84no—
109C4×C67—22no—
110C4×C67—41no—
111C4×C67—42no—
112C4×C67H(5,𝔽2)12yes—
113Dic67—21no—
114Dic67—22no—
115Dic67—42no—
116C2×C67—81no—
117C2×C67#53511yescharacteristic
118C2×C67#57712yes—
119D67—41no—
120D67—81no—
121D67—81no—
122#3222—671nomaximal, Sylow
123C67#322211yescharacteristic, complemented, Sylow
124#541—671no—
125#535—671no—
126#497D6711yescharacteristic, complemented
127#492—671no—
128#490—671no—
129#488—671no—
130#464—672no—
131#460—672no—
132#452—672no—
133#450—672no—
134#433—671no—
135C2×#41—671no—
136C22×D4—671no—
137C2×Q16C2×D6712yescomplemented
138C2×SD16—672no—
139C2×SD16—672no—
140C2×SD16C2×D6712yes—
141C4⋊Q8C2×D6711yescharacteristic, complemented
142#127—671no—
143#127—671no—
144#126—1342no—
145#125—671no—
146#125—1342no—
147#124—671no—
148#124C2×D6711yescharacteristic
149C22⋊Q8—672no—
150#121—1342no—
151#120—1342no—
152C4×Q8—672no—
153#117—671no—
154C4⋊C8—672no—
155#103—672no—
156#102—672no—
157#102—672no—
158#102—672no—
159C22⋊C8—672no—
160#97C2×D6711yescharacteristic, complemented
161C4×C8—671no—
162C24—1341no—
163#41—2682no—
164C2×Q8C22×D6711yescharacteristic
165C2×Q8C22×D6712yes—
166C2×D4—2682no—
167C2×D4—671no—
168C2×D4—671no—
169C2×D4—1341no—
170C2×D4—1341no—
171C2×D4—1342no—
172C2×D4—1342no—
173C2×D4C22×D6711yescharacteristic
174C22×C4—671no—
175C22×C4—1341no—
176Q16—42no—
177SD16—42no—
178SD16—2682no—
179SD16—2682no—
180C2×C8—672no—
181C2×C8C22×D6712yes—
182C4⋊C4—22no—
183C4⋊C4—672no—
184C4⋊C4—672no—
185C4⋊C4—1342no—
186C4⋊C4—1342no—
187C4⋊C4—1342no—
188C22⋊C4—22no—
189C22⋊C4—672no—
190C22⋊C4—1342no—
191C22⋊C4—1342no—
192C42—671no—
193C42—1342no—
194C42C22×D6711yescharacteristic
195C23—21no—
196C23—2682no—
197C23—5362no—
198C23—671no—
199C23—1341no—
200C23—1341no—
201Q8—22no—
202Q8—22no—
203Q8—22no—
204D4—22no—
205D4—2682no—
206D4—2682no—
207D4—5362no—
208D4—5362no—
209D4—1342no—
210D4—1342no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—2682no—
214C2×C4—672no—
215C2×C4—671no—
216C2×C4—1342no—
217C2×C4—1342no—
218C2×C4D4×D6712yes—
219C2×C4C23×D6711yescharacteristic
220C8—22no—
221C8—1342no—
222C22—42no—
223C22—41no—
224C22—2682no—
225C22—2682no—
226C22—2681no—
227C22—2681no—
228C22—5364no—
229C22—1342no—
230C22—1341no—
231C22C2×D4×D6711yescharacteristic
232C4—22no—
233C4—41no—
234C4—42no—
235C4—2682no—
236C4—1342no—
237C4—1341no—
238C4#1452408812yes—
239C2—81no—
240C2—2681no—
241C2—5361no—
242C2—5361no—
243C2#1686395911yescharacteristic
244C2#1686420512yes—
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#497×C67C2
socleC22×C67C22×D4
hypercenter#497D67
hypocenterC67#3222
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #16864007C2semidirect
3. #16863770C2semidirect
4. #16863761C2semidirect
5. #16863750C2semidirect
6. #16863620C2semidirect
7. #16863594C2semidirect
8. #16863557C2semidirect
9. #16863545C2semidirect
10. #16863458C2semidirect
11. #497×C67C2semidirect
15. C2×#11761615C22semidirect
19. #14524035C22semidirect
25. Q8×Dic67C22semidirect
27. Dic67⋊C8C22semidirect
28. #14523967C22semidirect
30. #14523965C22semidirect
33. C4×C67⋊C8C22semidirect
34. C2×Q16×C67C22semidirect
36. C4⋊Q8×C67C22semidirect
38. #97×C67C22semidirect
123. C67#3222semidirect
126. #497D67semidirect
137. C2×Q16C2×D67semidirect
141. C4⋊Q8C2×D67semidirect
160. #97C2×D67semidirect