Finite Group Databasebeta

← #46846845

Subgroups — #46846845

Order 34688.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#22576186C212yesmaximal, complemented
2C2×#22576182C211yesmaximal, characteristic, complemented
3#31774464C212yesmaximal, complemented
4#31774285C214yesmaximal, complemented
5#31774280C212yesmaximal, complemented
6#31774272C212yesmaximal, complemented
7#31773852C211yesmaximal, characteristic, complemented
8C22×D8×C271C211yesmaximal, characteristic, complemented
9C2×#16864654C2212yes—
10#22576186—28no—
11#22576182—24no—
12C2×Dic1084C2211yescharacteristic, complemented
13C2×#16864643C2212yescomplemented
14C2×#16864642C2214yescomplemented
15C2×D2168C2211yescharacteristic, complemented
16C2×#16864639C2214yescomplemented
17C2×#16864638C2211yescharacteristic
18#22576156C2214yescomplemented
19#22576147C2214yescomplemented
20#22576131—24no—
21#22576129—24no—
22#22576119C2211yescharacteristic
23#22576070C2212yescomplemented
24C8×Dic271C2212yescomplemented
25C22×D4×C271C2212yes—
26C2×D8×C271—24no—
27C2×D8×C271C2212yes—
28C2×D8×C271C2212yes—
29C22×C8×C271C2211yescharacteristic
30#16864654—24no—
31#16864654—24no—
32C2×Dic542C2312yes—
33C2×D1084C2312yes—
34C2×C22⋊D271—28no—
35C2×C22⋊D271—22no—
36C2×C4×D271—22no—
37Dic1084—22no—
38#16864643—24no—
39#16864642—28no—
40D2168—22no—
41#16864639—28no—
42#16864638D414yescomplemented
43C2×C271⋊C8C2312yes—
44C4⋊Dic271D412yes—
45C22⋊Dic271—28no—
46C22⋊Dic271D412yes—
47C4×Dic271C2312yes—
48C24×C271—22no—
49C2×D4×C271—24no—
50C2×D4×C271—24no—
51C2×D4×C271—22no—
52C2×D4×C271C2314yes—
53C22×C4×C271C2311yescharacteristic
54D8×C271—24no—
55D8×C271—24no—
56C2×C8×C271D414yescomplemented
57C2×C8×C271C2311yescharacteristic
58C2×C8×C271C2311yescharacteristic
59C22×D271—22no—
60Dic542—22no—
61Dic542—22no—
62D1084—22no—
63D1084—22no—
64C22⋊D271—44no—
65C22⋊D271—88no—
66C2×Dic271—22no—
67C2×Dic271C2×D414yes—
68C4×D271—44no—
69C271⋊C8C2×D414yes—
70C23×C271—24no—
71C23×C271—44no—
72C23×C271—84no—
73C23×C271C2×D411yescharacteristic
74D4×C271—24no—
75D4×C271—24no—
76D4×C271—44no—
77C2×C4×C271C2×D411yescharacteristic
78C2×C4×C271C2×D412yes—
79C2×C4×C271C2×D412yes—
80C2×C4×C271C2411yescharacteristic
81C8×C271C2×D412yes—
82C8×C271C2×D412yes—
83C22×C271—22no—
84C22×C271—44no—
85C22×C271—48no—
86C22×C271—88no—
87C22×C271#12712yes—
88C22×C271C22×D411yescharacteristic
89C2×D271—42no—
90C2×D271—44no—
91C4×C271#12712yes—
92C4×C271C22×D411yescharacteristic
93C4×C271C22×D411yescharacteristic
94Dic271—24no—
95Dic271—42no—
96C2×C271—22no—
97C2×C271—84no—
98C2×C271C2×#12711yescharacteristic
99C2×C271#57612yes—
100D271—82no—
101C271#315511yescharacteristic, complemented, Sylow
102#3155—2711nomaximal, Sylow
103C2×AGL(1,ℤ/8)—2712no—
104C2×#135—2711no—
105C22×D8D27111yescharacteristic, complemented
106#535—2712no—
107#496—2714no—
108#495—2712no—
109#493—2712no—
110#405—2711no—
111C2×#41—2712no—
112C22×D4C2×D27112yes—
113AGL(1,ℤ/8)—5428no—
114#135—5424no—
115C2×Q16—2711no—
116C2×SD16—2714no—
117C2×SD16—2712no—
118C2×D8—24no—
119C2×D8—2714no—
120C2×D8—2711no—
121C2×D8C2×D27112yes—
122C2×D8C2×D27112yes—
123C2×M16—2711no—
124C22×C8C2×D27111yescharacteristic
125#127—2714no—
126#124—2714no—
127#121—5424no—
128#120—5424no—
129#117—2711no—
130#97—2712no—
131C4×C8—2712no—
132C24—22no—
133#41—5424no—
134#41—5424no—
135C2×Q8—2712no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—24no—
139C2×D4—2712no—
140C2×D4—5428no—
141C2×D4—5422no—
142C2×D4C22×D27114yes—
143C22×C4—5422no—
144C22×C4C22×D27111yescharacteristic
145Q16—5422no—
146SD16—5428no—
147SD16—5424no—
148D8—24no—
149D8—24no—
150D8—5428no—
151D8—5422no—
152M16—2714no—
153C2×C8—2712no—
154C2×C8C22⋊D27114yescomplemented
155C2×C8C22×D27111yescharacteristic
156C2×C8C22×D27111yescharacteristic
157C4⋊C4—2712no—
158C22⋊C4—2712no—
159C22⋊C4—5428no—
160C42—2712no—
161C23—24no—
162C23—44no—
163C23—84no—
164C23—5422no—
165C23D4×D27111yescharacteristic
166Q8—5422no—
167Q8—5422no—
168D4—24no—
169D4—24no—
170D4—44no—
171D4—5422no—
172D4—5422no—
173D4—10844no—
174D4—21688no—
175C2×C4—2714no—
176C2×C4—5422no—
177C2×C4—10844no—
178C2×C4D4×D27111yescharacteristic
179C2×C4D4×D27112yes—
180C2×C4C2×C22⋊D27112yes—
181C2×C4C23×D27111yescharacteristic
182C8—2714no—
183C8C2×C22⋊D27112yes—
184C8C2×C22⋊D27112yes—
185C22—22no—
186C22—44no—
187C22—48no—
188C22—88no—
189C22—10842no—
190C22—10844no—
191C22#2257615612yes—
192C22C2×D4×D27111yescharacteristic
193C4—5424no—
194C4—10842no—
195C4#2257615612yes—
196C4C2×D4×D27111yescharacteristic
197C4C22×C22⋊D27111yescharacteristic
198C2—22no—
199C2—84no—
200C2—21682no—
201C2C2×#2257615611yescharacteristic
202C2#3177469812yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC22×D8×C271C2
socleC22×C271C22×D4
hypercenterC22×D8D271
hypocenterC271#3155
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#22576186C2semidirect
2. C2×#22576182C2semidirect
3. #31774464C2semidirect
4. #31774285C2semidirect
5. #31774280C2semidirect
6. #31774272C2semidirect
7. #31773852C2semidirect
8. C22×D8×C271C2semidirect
12. C2×Dic1084C22semidirect
13. C2×#16864643C22semidirect
14. C2×#16864642C22semidirect
15. C2×D2168C22semidirect
16. C2×#16864639C22semidirect
18. #22576156C22semidirect
19. #22576147C22semidirect
23. #22576070C22semidirect
24. C8×Dic271C22semidirect
42. #16864638D4semidirect
56. C2×C8×C271D4semidirect
101. C271#3155semidirect
105. C22×D8D271semidirect
154. C2×C8C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761862
C2×#225761822
#317744642
#317742852
#317742802
#317742722
#317738522
C22×D8×C2712
#3155271