Finite Group Databasebeta

← #22572929

Subgroups — #22572929

Order 8576.

372 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 123 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#16864119C211yesmaximal, characteristic, complemented
2Q8×D268C211yesmaximal, characteristic, complemented
3#16863756C212yesmaximal, complemented
4#16863747C211yesmaximal, characteristic, complemented
5#16863692C212yesmaximal, complemented
6#16863677C211yesmaximal, characteristic, complemented
7#16863612C212yesmaximal
8#16863602C211yesmaximal, characteristic, complemented
9#16863498C212yesmaximal, complemented
10C4×D536C211yesmaximal, characteristic
11#446×C67C211yesmaximal, characteristic, complemented
12C2×Q8×D67—21no—
13C2×#11761617C2212yes—
14C2×D536C2211yescharacteristic
15#14524043C2211yescharacteristic
16#14524043C2211yescharacteristic
17#14524043C2212yes—
18#14524038—22no—
19#14524031C2212yes—
20#14524023—21no—
21#14524023—22no—
22#14524019—21no—
23C4×Dic134C2212yes—
24C4×D268C2212yes—
25C4×D268C2211yescharacteristic
26C4×D268C2211yescharacteristic
27#14524001—21no—
28#14523977C2212yes—
29#14523977C2211yescharacteristic
30#14523968C2212yes—
31#14523968C2212yes—
32#14523968C2212yes—
33#14523965C2211yescharacteristic
34#14523965C2211yescharacteristic
35#14523965C2212yes—
36C4×Q8×C67C2211yescharacteristic
37C4⋊C8×C67C2211yescharacteristic
38C4⋊C8×C67C2212yes—
39#97×C67C2212yes—
40C4×C8×C67C2211yescharacteristic
41C2×Dic134—22no—
42C2×Dic134—21no—
43C2×Dic134C2312yes—
44Q8×D67—44no—
45C2×D268C2311yescharacteristic
46C2×D268C2311yescharacteristic
47C2×C4×D67—21no—
48C2×C4×D67—22no—
49C2×C4×D67—21no—
50#11761617—24no—
51D536—22no—
52C4⋊Dic67—21no—
53C4⋊Dic67—22no—
54C4⋊Dic67—21no—
55C4⋊Dic67—21no—
56C4⋊Dic67C2311yescharacteristic
57C4⋊Dic67C2312yes—
58C4⋊Dic67C2311yescharacteristic
59Dic67⋊C4—22no—
60(C2×D67)⋊C4—22no—
61(C2×D67)⋊C4—22no—
62(C2×D67)⋊C4—21no—
63(C2×D67)⋊C4—21no—
64(C2×D67)⋊C4—22no—
65C4×Dic67—22no—
66C2×Q8×C67D411yescharacteristic
67C2×C8×C67C2312yes—
68C2×C8×C67C2311yescharacteristic
69C2×C8×C67C2311yescharacteristic
70C4⋊C4×C67D411yescharacteristic
71C4⋊C4×C67D412yes—
72C42×C67C2311yescharacteristic
73C42×C67C2312yes—
74C22×D67—21no—
75C22×D67—21no—
76Q8×C67—22no—
77C2×C4×C67C2×D412yes—
78C2×C4×C67C2×D411yescharacteristic
79C2×C4×C67C2×D412yes—
80C2×C4×C67C2×D411yescharacteristic
81C2×C4×C67C2411yescharacteristic
82C2×Dic67—21no—
83C2×Dic67—22no—
84C2×Dic67—21no—
85C2×Dic67—22no—
86D268—21no—
87D268—21no—
88D268—21no—
89D268#4112yes—
90C4×D67—44no—
91C4×D67—42no—
92C4×D67—42no—
93Dic134—22no—
94Dic134—22no—
95Dic134—44no—
96Dic134—42no—
97C8×C67—21no—
98C8×C67—22no—
99C8×C67—21no—
100C22×C67C22×D411yescharacteristic
101C2×D67—22no—
102C2×D67—41no—
103C2×D67—41no—
104C2×D67—41no—
105C2×D67—41no—
106C2×D67—41no—
107C4×C67—22no—
108C4×C67—22no—
109C4×C67—21no—
110C4×C67AGL(1,ℤ/8)12yes—
111C4×C67C2×#4111yescharacteristic
112C4×C672−1+411yescharacteristic
113Dic67—42no—
114Dic67—41no—
115Dic67—42no—
116Dic67—41no—
117C2×C67#54811yescharacteristic
118C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
119C2×C67#57811yescharacteristic
120D67—42no—
121D67—81no—
122#3367—671nomaximal, Sylow
123C67#336711yescharacteristic, complemented, Sylow
124#556—671no—
125D4×Q8—671no—
126#489—672no—
127#487—671no—
128#477—672no—
129#474—671no—
130#463—672no—
131#461—671no—
132#446D6711yescharacteristic, complemented
133#440—672no—
134C4×D8—671no—
135C22×Q8—1341no—
136C2×SD16—672no—
137C2×D8—671no—
138C4⋊Q8—671no—
139C4⋊Q8—672no—
140C4⋊Q8—671no—
141#126—1342no—
142#124—672no—
143C22⋊Q8—1341no—
144C22⋊Q8—1341no—
145C22⋊Q8—1342no—
146C4×Q8—672no—
147C4×Q8C2×D6711yescharacteristic
148C4×D4—671no—
149C4×D4—672no—
150C4×D4—671no—
151#117—1341no—
152#107—671no—
153#107—672no—
154C4⋊C8C2×D6711yescharacteristic
155C4⋊C8C2×D6712yes—
156#103—672no—
157#103—672no—
158#103—672no—
159#102—671no—
160#102—671no—
161#102—672no—
162#97C2×D6712yes—
163C4×C8C2×D6711yescharacteristic
164C2×Q8—2684no—
165C2×Q8—672no—
166C2×Q8—1342no—
167C2×Q8—1341no—
168C2×Q8D26811yescharacteristic
169C2×D4—671no—
170C2×D4—671no—
171C22×C4—1341no—
172C22×C4—1342no—
173C22×C4—1341no—
174SD16—1344no—
175D8—1342no—
176C2×C8C22×D6711yescharacteristic
177C2×C8C22×D6711yescharacteristic
178C2×C8C22×D6712yes—
179C4⋊C4—671no—
180C4⋊C4—672no—
181C4⋊C4—671no—
182C4⋊C4—1341no—
183C4⋊C4—1342no—
184C4⋊C4—1342no—
185C4⋊C4—1341no—
186C4⋊C4—1341no—
187C4⋊C4D26811yescharacteristic
188C4⋊C4D26812yes—
189C22⋊C4—1342no—
190C22⋊C4—1342no—
191C22⋊C4—1341no—
192C22⋊C4—1341no—
193C22⋊C4—1342no—
194C42—1342no—
195C42C22×D6711yescharacteristic
196C42C22×D6712yes—
197C23—1341no—
198C23—1341no—
199Q8—22no—
200Q8—2684no—
201Q8—2682no—
202Q8—1342no—
203Q8—1342no—
204D4—672no—
205D4—1341no—
206D4—1341no—
207D4—1341no—
208C2×C4—2684no—
209C2×C4—2682no—
210C2×C4—2682no—
211C2×C4—1341no—
212C2×C4—1342no—
213C2×C4—1342no—
214C2×C4—1341no—
215C2×C4C2×D26812yes—
216C2×C4C2×D26811yescharacteristic
217C2×C4C2×D26812yes—
218C2×C4C2×D26811yescharacteristic
219C2×C4C23×D6711yescharacteristic
220C8—21no—
221C8—22no—
222C8—21no—
223C22—2681no—
224C22—2681no—
225C22—2681no—
226C22—2681no—
227C22—2681no—
228C22—1342no—
229C22C22×D26811yescharacteristic
230C4—22no—
231C4—22no—
232C4—21no—
233C4—2682no—
234C4—2682no—
235C4—2681no—
236C4—2681no—
237C4#1452407112yes—
238C4#41×D6711yescharacteristic
239C4#1452409111yescharacteristic
240C2—2682no—
241C2—5361no—
242C2#1686405811yescharacteristic
243C2C2×#1452407111yescharacteristic
244C2#1686421311yescharacteristic
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#446×C67C2
socleC22×C67C22×D4
hypercenter#446D67
hypocenterC67#3367
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. #16864119C2semidirect
2. Q8×D268C2semidirect
3. #16863756C2semidirect
4. #16863747C2semidirect
5. #16863692C2semidirect
6. #16863677C2semidirect
8. #16863602C2semidirect
9. #16863498C2semidirect
11. #446×C67C2semidirect
123. C67#3367semidirect
132. #446D67semidirect