Finite Group Databasebeta

← #22571379

Subgroups — #22571379

Order 8576.

334 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 121 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#16863930C211yesmaximal, characteristic
2C2×#14524035C211yesmaximal, characteristic, complemented
3#16863798C212yesmaximal, complemented
4#16863751C212yesmaximal, complemented
5#16863732C211yesmaximal, characteristic, complemented
6#16863727C211yesmaximal, characteristic, complemented
7#16863715C211yesmaximal, characteristic
8#16863710C211yesmaximal, characteristic
9#16863704C212yesmaximal, complemented
10#16863697C212yesmaximal, complemented
11#431×C67C211yesmaximal, characteristic, complemented
12#14524037—21no—
13#14524035—21no—
14#14524035—21no—
15#14524035C2212yes—
16#14524035C2211yescharacteristic
17#14524035C2212yes—
18#14524031C2211yescharacteristic
19#14524026—21no—
20#14524022C2211yescharacteristic
21#14524017C2211yescharacteristic
22C4×Dic134C2211yescharacteristic
23C4×D268C2211yescharacteristic
24C4×C22⋊D67—21no—
25#14524000—21no—
26C2×C4⋊Dic67C2211yescharacteristic
27C2×C4⋊Dic67C2211yescharacteristic
28C2×Dic67⋊C4—21no—
29C2×Dic67⋊C4—21no—
30#14523977C2212yes—
31#14523977C2212yes—
32#14523975C2212yes—
33#14523975C2212yes—
34#14523968C2212yes—
35#14523968C2212yes—
36#14523965C2212yes—
37#14523965C2212yes—
38C2×C42×C67C2211yescharacteristic
39C4⋊C8×C67C2212yes—
40C22⋊C8×C67C2211yescharacteristic
41C22⋊C8×C67C2211yescharacteristic
42C4×C8×C67C2212yes—
43C2×Dic134C2311yescharacteristic
44C2×D268C2311yescharacteristic
45C2×C22⋊D67—21no—
46C22×Dic67—21no—
47C22×Dic67—21no—
48C2×C4×D67—21no—
49C4⋊Dic67—21no—
50C4⋊Dic67—21no—
51C4⋊Dic67C2312yes—
52C4⋊Dic67C2312yes—
53C4⋊Dic67C2311yescharacteristic
54C4⋊Dic67C2311yescharacteristic
55Dic67⋊C4—22no—
56Dic67⋊C4—22no—
57Dic67⋊C4—22no—
58Dic67⋊C4—22no—
59Dic67⋊C4—21no—
60Dic67⋊C4—21no—
61Dic67⋊C4—21no—
62Dic67⋊C4—21no—
63C22⋊Dic67—21no—
64(C2×D67)⋊C4—21no—
65(C2×D67)⋊C4—21no—
66(C2×D67)⋊C4—21no—
67(C2×D67)⋊C4—21no—
68C4×Dic67—21no—
69C22×C4×C67D411yescharacteristic
70C22×C4×C67D411yescharacteristic
71C22×C4×C67C2311yescharacteristic
72C2×C8×C67C2312yes—
73C2×C8×C67C2312yes—
74C42×C67D411yescharacteristic
75C42×C67D411yescharacteristic
76C42×C67C2311yescharacteristic
77C42×C67C2311yescharacteristic
78C23×C67C2×D411yescharacteristic
79C22×D67—21no—
80C2×C4×C67—22no—
81C2×C4×C67—22no—
82C2×C4×C67—41no—
83C2×C4×C67C2×D411yescharacteristic
84C2×C4×C67C2×D411yescharacteristic
85C2×C4×C67C2×D411yescharacteristic
86C2×C4×C67C2×D411yescharacteristic
87C2×C4×C67C2×D411yescharacteristic
88C2×C4×C67C2411yescharacteristic
89C22⋊D67—42no—
90C2×Dic67—21no—
91C2×Dic67—21no—
92C2×Dic67—21no—
93C2×Dic67—22no—
94C2×Dic67—22no—
95C2×Dic67—42no—
96C2×Dic67—42no—
97D268—22no—
98C4×D67—42no—
99Dic134—22no—
100C8×C67—22no—
101C8×C67—22no—
102C22×C67—21no—
103C22×C67—21no—
104C22×C67#13512yes—
105C22×C67C22×D411yescharacteristic
106C2×D67—41no—
107C2×D67—41no—
108C2×D67—41no—
109C4×C67—21no—
110C4×C67—21no—
111C4×C67—21no—
112C4×C67—21no—
113C4×C67—21no—
114C4×C672−1+412yes—
115Dic67—41no—
116Dic67—42no—
117Dic67—41no—
118Dic67—41no—
119Dic67—42no—
120C2×C67—22no—
121C2×C67#53611yescharacteristic
122C2×C67C2×#13511yescharacteristic
123C2×C67#57511yescharacteristic
124D67—81no—
125#3232—671nomaximal, Sylow
126C67#323211yescharacteristic, complemented, Sylow
127#529—671no—
128C2×#125—671no—
129#499—672no—
130#488—672no—
131#484—671no—
132#483—671no—
133#481—671no—
134#480—671no—
135#479—672no—
136#478—672no—
137#431D6711yescharacteristic, complemented
138#126—1341no—
139#125—672no—
140#125—672no—
141#125—671no—
142#125—1341no—
143#125—1341no—
144#124—671no—
145#123—1341no—
146C22⋊Q8—671no—
147#121—671no—
148C4×Q8—671no—
149C4×D4—671no—
150C4×D4—1341no—
151#117—1341no—
152C2×C4⋊C4—671no—
153C2×C4⋊C4—671no—
154C2×C4⋊C4—1341no—
155C2×C4⋊C4—1341no—
156C2×C42C2×D6711yescharacteristic
157#107—672no—
158#107—672no—
159#106—672no—
160#106—672no—
161C4⋊C8C2×D6712yes—
162#103—672no—
163#103—672no—
164#102—672no—
165#102—672no—
166C22⋊C8C2×D6711yescharacteristic
167C22⋊C8C2×D6711yescharacteristic
168C4×C8C2×D6712yes—
169C2×Q8—671no—
170C2×D4—671no—
171C2×D4—1341no—
172C22×C4—1341no—
173C22×C4—1341no—
174C22×C4—1341no—
175C22×C4D26811yescharacteristic
176C22×C4D26811yescharacteristic
177C22×C4C22×D6711yescharacteristic
178C2×C8C22×D6712yes—
179C2×C8C22×D6712yes—
180C4⋊C4—672no—
181C4⋊C4—672no—
182C4⋊C4—671no—
183C4⋊C4—671no—
184C4⋊C4—1341no—
185C4⋊C4—1341no—
186C4⋊C4—1342no—
187C4⋊C4—1342no—
188C4⋊C4—1342no—
189C4⋊C4—1342no—
190C4⋊C4—1341no—
191C4⋊C4—1341no—
192C4⋊C4—1341no—
193C4⋊C4—1341no—
194C22⋊C4—1341no—
195C22⋊C4—1341no—
196C22⋊C4—1341no—
197C22⋊C4—1341no—
198C22⋊C4—1341no—
199C42—1341no—
200C42D26811yescharacteristic
201C42D26811yescharacteristic
202C42C22×D6711yescharacteristic
203C42C22×D6711yescharacteristic
204C23—1341no—
205C23C2×D26811yescharacteristic
206Q8—1342no—
207D4—2682no—
208D4—1342no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—2682no—
213C2×C4—2682no—
214C2×C4—2682no—
215C2×C4—1341no—
216C2×C4—1341no—
217C2×C4—1341no—
218C2×C4—1342no—
219C2×C4—1342no—
220C2×C4C2×D26811yescharacteristic
221C2×C4C2×D26811yescharacteristic
222C2×C4C2×D26811yescharacteristic
223C2×C4C2×D26811yescharacteristic
224C2×C4C2×D26811yescharacteristic
225C2×C4C23×D6711yescharacteristic
226C8—22no—
227C8—22no—
228C22—21no—
229C22—21no—
230C22—2681no—
231C22—2681no—
232C22—2681no—
233C22#1452406512yes—
234C22C22×D26811yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—2682no—
241C4—2681no—
242C4—2681no—
243C4—2682no—
244C4—2681no—
245C4#1452409012yes—
246C2—22no—
247C2—5361no—
248C2#1686397211yescharacteristic
249C2C2×#1452406511yescharacteristic
250C2#1686419611yescharacteristic
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#431×C67C2
socleC22×C67C22×D4
hypercenter#431D67
hypocenterC67#3232
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
2. C2×#14524035C2semidirect
3. #16863798C2semidirect
4. #16863751C2semidirect
5. #16863732C2semidirect
6. #16863727C2semidirect
9. #16863704C2semidirect
10. #16863697C2semidirect
11. #431×C67C2semidirect
126. C67#3232semidirect
137. #431D67semidirect