Finite Group Databasebeta

← #46846487

Subgroups — #46846487

Order 34688.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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#31774447C211yesmaximal, characteristic
2C2×C4×Dic542C211yesmaximal, characteristic, complemented
3#31774211C212yesmaximal, complemented
4#31774193C212yesmaximal, complemented
5#31774098C212yesmaximal, complemented
6#31774038C212yesmaximal, complemented
7#31773999C214yesmaximal, complemented
8#432×C271C211yesmaximal, characteristic, complemented
9C2×#16864654—21no—
10C22×Dic542C2211yescharacteristic
11C2×#16864643C2214yes—
12#22576160C2211yescharacteristic
13#22576157C2211yescharacteristic
14#22576148—21no—
15#22576134—21no—
16#22576134C2212yes—
17C4×Dic542—21no—
18C4×Dic542—21no—
19C4×Dic542C2212yes—
20C4×Dic542C2212yes—
21C4×D1084—21no—
22C4×D1084C2212yes—
23C2×C4⋊Dic271C2211yescharacteristic
24C2×Dic271⋊C4—21no—
25C2×C4×Dic271—21no—
26#22576094C2214yes—
27#22576085C2214yes—
28#22576082C2214yes—
29C2×C42×C271C2211yescharacteristic
30C4⋊C8×C271C2212yes—
31C22⋊C8×C271C2212yes—
32#97×C271C2212yes—
33#16864654—44no—
34C2×Dic542—24no—
35C2×Dic542—21no—
36C2×Dic542—21no—
37C2×Dic542C2312yes—
38C2×D1084—21no—
39C2×D1084—21no—
40C2×D1084C2312yes—
41C2×C22⋊D271—22no—
42C22×Dic271—21no—
43C22×Dic271—21no—
44C2×C4×D271—22no—
45#16864643—28no—
46C4⋊Dic271—21no—
47C4⋊Dic271—21no—
48C4⋊Dic271C2312yes—
49C4⋊Dic271C2312yes—
50Dic271⋊C4—22no—
51Dic271⋊C4—22no—
52(C2×D271)⋊C4—22no—
53(C2×D271)⋊C4—22no—
54C4×Dic271—22no—
55C4×Dic271—22no—
56C22×C4×C271D411yescharacteristic
57C22×C4×C271D411yescharacteristic
58C22×C4×C271C2311yescharacteristic
59C2×C8×C271C2314yes—
60C42×C271D411yescharacteristic
61C42×C271D411yescharacteristic
62C42×C271C2311yescharacteristic
63C42×C271C2311yescharacteristic
64C22×D271—22no—
65Dic542—24no—
66Dic542—22no—
67Dic542—42no—
68Dic542#4114yescomplemented
69D1084—24no—
70D1084—42no—
71D1084—42no—
72C22⋊D271—44no—
73C2×Dic271—24no—
74C2×Dic271—22no—
75C2×Dic271—22no—
76C2×Dic271—22no—
77C2×Dic271—42no—
78C4×D271—44no—
79C23×C271C2×D411yescharacteristic
80C2×C4×C271—22no—
81C2×C4×C271—22no—
82C2×C4×C271—22no—
83C2×C4×C271C2×D411yescharacteristic
84C2×C4×C271C2×D411yescharacteristic
85C2×C4×C271C2×D411yescharacteristic
86C2×C4×C271C2×D411yescharacteristic
87C2×C4×C271C2×D411yescharacteristic
88C2×C4×C271C2411yescharacteristic
89C8×C271—24no—
90C22×C271—21no—
91C22×C271—21no—
92C22×C271AGL(1,ℤ/8)12yes—
93C22×C271C22×D411yescharacteristic
94C2×D271—42no—
95C2×D271—42no—
96C2×D271—42no—
97C4×C271—21no—
98C4×C271—21no—
99C4×C271—21no—
100C4×C271—21no—
101C4×C271#13712yes—
102C4×C271C2×#4112yes—
103Dic271—24no—
104Dic271—42no—
105Dic271—42no—
106C2×C271—22no—
107C2×C271#52511yescharacteristic
108C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
109C2×C271C2×#13711yescharacteristic
110D271—82no—
111C271#311211yescharacteristic, complemented, Sylow
112#3112—2711nomaximal, Sylow
113#532—2711no—
114C2×C4×Q8—2711no—
115#480—2712no—
116#477—2712no—
117#460—2712no—
118#448—2712no—
119#440—2714no—
120#432D27111yescharacteristic, complemented
121C2×#41—5421no—
122C22×Q8—2711no—
123C2×SD16—2714no—
124C4⋊Q8—2711no—
125#127—2711no—
126#124—5421no—
127#121—2712no—
128#121—5421no—
129C4×Q8—2712no—
130C4×Q8—2712no—
131C4×Q8—5421no—
132C4×Q8—5421no—
133C4×D4—2712no—
134C4×D4—5421no—
135C2×C4⋊C4—2711no—
136C2×C4⋊C4—5421no—
137C2×C42—5421no—
138C2×C42C2×D27111yescharacteristic
139#107—2714no—
140C4⋊C8C2×D27112yes—
141#103—2714no—
142#102—2714no—
143C22⋊C8C2×D27112yes—
144#97C2×D27112yes—
145#41—10844no—
146C2×Q8—2712no—
147C2×Q8—5424no—
148C2×Q8—5421no—
149C2×Q8—5421no—
150C2×D4—2712no—
151C2×D4—5422no—
152C2×D4—5421no—
153C2×D4—5421no—
154C22×C4—5421no—
155C22×C4—5421no—
156C22×C4—5422no—
157C22×C4D108411yescharacteristic
158C22×C4D108411yescharacteristic
159C22×C4C22×D27111yescharacteristic
160SD16—5428no—
161C2×C8C22×D27114yes—
162C4⋊C4—2712no—
163C4⋊C4—2712no—
164C4⋊C4—5421no—
165C4⋊C4—5422no—
166C4⋊C4—5422no—
167C4⋊C4—5421no—
168C22⋊C4—5422no—
169C22⋊C4—5422no—
170C42—5422no—
171C42—5422no—
172C42D108411yescharacteristic
173C42D108411yescharacteristic
174C42C22×D27111yescharacteristic
175C42C22×D27111yescharacteristic
176C23—5422no—
177C23C2×D108411yescharacteristic
178Q8—2714no—
179Q8—5424no—
180Q8—5422no—
181Q8—10842no—
182D4—5424no—
183D4—10842no—
184D4—10844no—
185D4—10842no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—5424no—
190C2×C4—5422no—
191C2×C4—5422no—
192C2×C4—5422no—
193C2×C4—10844no—
194C2×C4—10842no—
195C2×C4C2×D108411yescharacteristic
196C2×C4C2×D108411yescharacteristic
197C2×C4C2×D108411yescharacteristic
198C2×C4C2×D108411yescharacteristic
199C2×C4C2×D108411yescharacteristic
200C2×C4C23×D27111yescharacteristic
201C8—24no—
202C22—21no—
203C22—21no—
204C22—10842no—
205C22—10842no—
206C22—10842no—
207C22#2257618812yes—
208C22C22×D108411yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—5424no—
214C4—10842no—
215C4—10842no—
216C4#2257619312yes—
217C4#41×D27112yes—
218C2—22no—
219C2—21682no—
220C2#3177440911yescharacteristic
221C2C2×#2257618811yescharacteristic
222C2C2×#2257619311yescharacteristic
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#432×C271C2
socleC22×C271C22×D4
prime radical#31774447C2
hypercenter#432D271
hypocenterC271#3112
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic542C2semidirect
3. #31774211C2semidirect
4. #31774193C2semidirect
5. #31774098C2semidirect
6. #31774038C2semidirect
7. #31773999C2semidirect
8. #432×C271C2semidirect
68. Dic542#41semidirect
111. C271#3112semidirect
120. #432D271semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#317744472
C2×C4×Dic5422
#317742112
#317741932
#317740982
#317740382
#317739992
#432×C2712
#3112271