Finite Group Databasebeta

← #22573242

Subgroups — #22573242

Order 8576.

350 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 125 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#16864125C211yesmaximal, characteristic, complemented
2#16864119C211yesmaximal, characteristic
3#16863790C212yesmaximal
4#16863776C211yesmaximal, characteristic, complemented
5#16863619C212yesmaximal
6#16863607C211yesmaximal, characteristic, complemented
7#16863607C212yesmaximal, complemented
8#16863602C211yesmaximal, characteristic
9#16863504C212yesmaximal, complemented
10C4×#11761617C211yesmaximal, characteristic
11#446×C67C211yesmaximal, characteristic, complemented
12C2×Dic268C2212yes—
13C2×Dic268C2212yes—
14C2×Dic268C2212yes—
15C2×#11761617C2211yescharacteristic
16C2×#11761617C2212yes—
17#14524043C2211yescharacteristic
18#14524043C2211yescharacteristic
19#14524043C2212yes—
20#14524038—22no—
21#14524034—21no—
22#14524034—22no—
23#14524031C2212yes—
24#14524019—21no—
25Q8×Dic67—21no—
26C4×Dic134C2212yes—
27C4×Dic134C2212yes—
28C4×Dic134C2211yescharacteristic
29C4×D268C2211yescharacteristic
30#14524001—21no—
31#14523975C2211yescharacteristic
32#14523968C2212yes—
33#14523968C2212yes—
34#14523968C2211yescharacteristic
35#14523965C2211yescharacteristic
36C4×Q8×C67C2211yescharacteristic
37C4⋊C8×C67C2211yescharacteristic
38C4⋊C8×C67C2212yes—
39#97×C67C2212yes—
40C4×C8×C67C2211yescharacteristic
41C2×Dic134C2312yes—
42C2×Dic134C2311yescharacteristic
43C2×Dic134C2312yes—
44C2×D268C2311yescharacteristic
45C2×C4×D67—21no—
46Dic268—24no—
47Dic268—42no—
48Dic268—42no—
49#11761617—22no—
50#11761617—42no—
51C4⋊Dic67—21no—
52C4⋊Dic67—22no—
53C4⋊Dic67—21no—
54C4⋊Dic67—22no—
55C4⋊Dic67—21no—
56C4⋊Dic67—21no—
57C4⋊Dic67C2311yescharacteristic
58C4⋊Dic67C2311yescharacteristic
59Dic67⋊C4—22no—
60Dic67⋊C4—22no—
61Dic67⋊C4—21no—
62(C2×D67)⋊C4—22no—
63(C2×D67)⋊C4—21no—
64(C2×D67)⋊C4—22no—
65C4×Dic67—22no—
66C4×Dic67—22no—
67C4×Dic67—21no—
68C2×Q8×C67D411yescharacteristic
69C2×C8×C67C2311yescharacteristic
70C2×C8×C67C2312yes—
71C2×C8×C67C2311yescharacteristic
72C4⋊C4×C67D411yescharacteristic
73C4⋊C4×C67D412yes—
74C42×C67C2311yescharacteristic
75C42×C67C2312yes—
76C22×D67—21no—
77Q8×C67—22no—
78C2×C4×C67C2×D412yes—
79C2×C4×C67C2×D411yescharacteristic
80C2×C4×C67C2×D412yes—
81C2×C4×C67C2×D411yescharacteristic
82C2×C4×C67C2411yescharacteristic
83C2×Dic67—21no—
84C2×Dic67—21no—
85C2×Dic67—22no—
86C2×Dic67—21no—
87C2×Dic67—22no—
88D268—21no—
89D268—21no—
90C4×D67—42no—
91Dic134—21no—
92Dic134—22no—
93Dic134—22no—
94Dic134—22no—
95Dic134—22no—
96Dic134—21no—
97C8×C67—22no—
98C8×C67—21no—
99C8×C67#4112yes—
100C22×C67C22×D411yescharacteristic
101C2×D67—41no—
102C2×D67—41no—
103C2×D67—41no—
104C4×C67—22no—
105C4×C67—22no—
106C4×C67—21no—
107C4×C67#13712yes—
108C4×C67C2×#4111yescharacteristic
109C4×C67H(5,𝔽2)11yescharacteristic
110Dic67—41no—
111Dic67—42no—
112Dic67—42no—
113Dic67—41no—
114Dic67—41no—
115C2×C67#55011yescharacteristic
116C2×C67C2×#13711yescharacteristic
117C2×C67#57811yescharacteristic
118D67—81no—
119#3398—671nomaximal, Sylow
120C67#339811yescharacteristic, complemented, Sylow
121#557—671no—
122#556—671no—
123#497—672no—
124#494—671no—
125#464—672no—
126#462—672no—
127#462—671no—
128#461—671no—
129#446D6711yescharacteristic, complemented
130#441—672no—
131C4×SD16—671no—
132C2×Q16—672no—
133C2×Q16—672no—
134C2×Q16—672no—
135C2×SD16—672no—
136C2×SD16—671no—
137C4⋊Q8—671no—
138C4⋊Q8—671no—
139C4⋊Q8—672no—
140#126—1342no—
141#125—1341no—
142#125—1342no—
143#124—672no—
144C22⋊Q8—1341no—
145C4×Q8—672no—
146C4×Q8—671no—
147C4×Q8—672no—
148C4×Q8—1341no—
149C4×Q8C2×D6711yescharacteristic
150C4×D4—671no—
151#117—1341no—
152#106—671no—
153C4⋊C8C2×D6711yescharacteristic
154C4⋊C8C2×D6712yes—
155#103—671no—
156#103—672no—
157#103—672no—
158#102—671no—
159#97C2×D6712yes—
160C4×C8C2×D6711yescharacteristic
161C2×Q8—672no—
162C2×Q8—671no—
163C2×Q8—672no—
164C2×Q8D26811yescharacteristic
165C2×D4—671no—
166C22×C4—1341no—
167Q16—2682no—
168Q16—2682no—
169Q16—1344no—
170SD16—2682no—
171SD16—1342no—
172C2×C8C22×D6711yescharacteristic
173C2×C8C22×D6712yes—
174C2×C8C22×D6711yescharacteristic
175C4⋊C4—671no—
176C4⋊C4—671no—
177C4⋊C4—1341no—
178C4⋊C4—1342no—
179C4⋊C4—1342no—
180C4⋊C4—1341no—
181C4⋊C4—1342no—
182C4⋊C4—1341no—
183C4⋊C4—1342no—
184C4⋊C4—1341no—
185C4⋊C4—1341no—
186C4⋊C4D26811yescharacteristic
187C4⋊C4D26812yes—
188C22⋊C4—1342no—
189C22⋊C4—1341no—
190C22⋊C4—1342no—
191C42—1342no—
192C42—1342no—
193C42—1341no—
194C42C22×D6711yescharacteristic
195C42C22×D6712yes—
196C23—1341no—
197Q8—22no—
198Q8—1341no—
199Q8—1342no—
200Q8—1342no—
201Q8—1342no—
202Q8—1342no—
203Q8—1341no—
204D4—1341no—
205D4—1341no—
206C2×C4—2682no—
207C2×C4—1341no—
208C2×C4—1341no—
209C2×C4—1342no—
210C2×C4—1341no—
211C2×C4—1342no—
212C2×C4C2×D26812yes—
213C2×C4C2×D26812yes—
214C2×C4C2×D26811yescharacteristic
215C2×C4C2×D26811yescharacteristic
216C2×C4C23×D6711yescharacteristic
217C8—22no—
218C8—21no—
219C8#1176162912yes—
220C22—2681no—
221C22—2681no—
222C22—2681no—
223C22C22×D26811yescharacteristic
224C4—22no—
225C4—22no—
226C4—21no—
227C4—2682no—
228C4—2681no—
229C4—2682no—
230C4—2681no—
231C4—2681no—
232C4#1452407612yes—
233C4C2×#1176162911yescharacteristic
234C4#1452408911yescharacteristic
235C2—5361no—
236C2#1686406811yescharacteristic
237C2C2×#1452407611yescharacteristic
238C2#1686421311yescharacteristic
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#446×C67C2
socleC22×C67C22×D4
hypercenter#446D67
hypocenterC67#3398
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
1. #16864125C2semidirect
4. #16863776C2semidirect
6. #16863607C2semidirect
7. #16863607C2semidirect
9. #16863504C2semidirect
11. #446×C67C2semidirect
120. C67#3398semidirect
129. #446D67semidirect