Finite Group Databasebeta

← #46837408

Subgroups — #46837408

Order 34688.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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×#22576188C212yesmaximal, complemented
2#31774478C211yesmaximal, characteristic, complemented
3C2×#22576086C212yesmaximal, complemented
4#31773798C211yesmaximal, characteristic, complemented
5#343×C271C211yesmaximal, characteristic, complemented
6C2×#16864654C2212yes—
7C22×D1084C2211yescharacteristic
8#22576188—24no—
9#22576188—24no—
10C2×#16864643—22no—
11C2×D2168—22no—
12#22576140—22no—
13#22576135—22no—
14#22576134—22no—
15C2×(C2×D271)⋊C4—22no—
16C2×C4×Dic271C2211yescharacteristic
17#22576086—24no—
18C2×M16×C271C2212yes—
19C2×C4⋊C4×C271C2211yescharacteristic
20C23×D271—21no—
21#16864654—22no—
22#16864654—22no—
23#16864654—22no—
24#16864654—22no—
25C2×Dic542D412yes—
26C2×D1084—22no—
27C2×D1084—22no—
28C2×D1084—22no—
29C2×D1084—21no—
30C2×D1084D412yes—
31C2×D1084D412yescomplemented
32C2×C22⋊D271—22no—
33C22×Dic271—21no—
34C2×C4×D271—22no—
35#16864643—28no—
36D2168—28no—
37C4⋊Dic271—22no—
38(C2×D271)⋊C4—24no—
39(C2×D271)⋊C4—44no—
40C4×Dic271—22no—
41C22×C4×C271D412yes—
42C22×C4×C271C2311yescharacteristic
43M16×C271—22no—
44M16×C271D414yescomplemented
45C2×C8×C271—22no—
46C4⋊C4×C271—22no—
47C22×D271—22no—
48C22×D271—22no—
49C22×D271—42no—
50C22×D271—42no—
51C22×D271—41no—
52C22×D271—41no—
53Dic542—22no—
54Dic542—22no—
55D1084—22no—
56D1084—24no—
57D1084—22no—
58D1084—42no—
59D1084—82no—
60C22⋊D271—42no—
61C22⋊D271—42no—
62C2×Dic271—22no—
63C2×Dic271—22no—
64C2×Dic271—41no—
65C2×Dic271—41no—
66C4×D271—42no—
67C4×D271—42no—
68C23×C271C2×D411yescharacteristic
69C2×C4×C271—22no—
70C2×C4×C271—42no—
71C2×C4×C271C2×D411yescharacteristic
72C2×C4×C271C2×D412yes—
73C2×C4×C271C2×D412yes—
74C2×C4×C271#4111yescharacteristic
75C8×C271—24no—
76C22×C271—21no—
77C22×C271—21no—
78C22×C271#12011yescharacteristic
79C22×C271#12111yescharacteristic
80C22×C271#12711yescharacteristic
81C2×D271—44no—
82C2×D271—42no—
83C2×D271—42no—
84C2×D271—42no—
85C2×D271—42no—
86C2×D271—82no—
87C2×D271—81no—
88C2×D271—81no—
89C4×C271—42no—
90C4×C271#12012yes—
91C4×C271#12112yes—
92Dic271—42no—
93Dic271—42no—
94C2×C271—21no—
95C2×C271—21no—
96C2×C271#39211yescharacteristic
97C2×C271#45311yescharacteristic
98C2×C271#45411yescharacteristic
99D271—82no—
100D271—82no—
101C271#200611yescharacteristic, complemented, Sylow
102#2006—2711nomaximal, Sylow
103C2×AGL(1,ℤ/8)—2712no—
104#537—2711no—
105C2×#104—2712no—
106#390—2711no—
107#343D27111yescharacteristic, complemented
108C2×#41—2712no—
109C22×D4—2711no—
110AGL(1,ℤ/8)—5424no—
111AGL(1,ℤ/8)—5424no—
112C2×SD16—5422no—
113C2×D8—5422no—
114C2×M16C2×D27112yes—
115C22⋊Q8—5422no—
116#121—5422no—
117#121—5422no—
118C2×C4⋊C4C2×D27111yescharacteristic
119C2×C22⋊C4—5422no—
120C2×C42—2711no—
121#104—5424no—
122C24—5421no—
123#41—5422no—
124#41—5422no—
125#41—5422no—
126#41—5422no—
127C2×Q8—2712no—
128C2×D4—2712no—
129C2×D4—2712no—
130C2×D4—5422no—
131C2×D4—5422no—
132C2×D4—5422no—
133C2×D4—5422no—
134C2×D4—5421no—
135C22×C4—5421no—
136C22×C4—5422no—
137C22×C4D108412yes—
138C22×C4C22×D27111yescharacteristic
139SD16—5428no—
140D8—5428no—
141M16—22no—
142M16D108414yescomplemented
143C2×C8—22no—
144C4⋊C4—22no—
145C4⋊C4—5422no—
146C22⋊C4—5424no—
147C22⋊C4—10844no—
148C42—5422no—
149C23—5422no—
150C23—5422no—
151C23—10842no—
152C23—10842no—
153C23—10841no—
154C23—10841no—
155C23C2×D108411yescharacteristic
156Q8—5422no—
157Q8—5422no—
158D4—5422no—
159D4—5424no—
160D4—5422no—
161D4—10842no—
162D4—10842no—
163D4—10842no—
164D4—21682no—
165C2×C4—22no—
166C2×C4—42no—
167C2×C4—5422no—
168C2×C4—5422no—
169C2×C4—10841no—
170C2×C4—10841no—
171C2×C4—10842no—
172C2×C4—10842no—
173C2×C4D4×D27111yescharacteristic
174C2×C4D4×D27112yes—
175C2×C4C2×D108412yes—
176C2×C4#1686465511yescharacteristic
177C8—24no—
178C22—21no—
179C22—21no—
180C22—10844no—
181C22—10842no—
182C22—10842no—
183C22—10842no—
184C22—10842no—
185C22—21682no—
186C22—21681no—
187C22—21681no—
188C22#2257613011yescharacteristic
189C22#2257613511yescharacteristic
190C22#2257615711yescharacteristic
191C4—42no—
192C4—10842no—
193C4—10842no—
194C4#2257613012yes—
195C4#2257613512yes—
196C2—21no—
197C2—21no—
198C2—21682no—
199C2—21682no—
200C2#3177380711yescharacteristic
201C2#3177406711yescharacteristic
202C2#3177407211yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#343×C271C2
socleC22×C271#120
hypercenter#343D271
hypocenterC271#2006
LCSG ▷ C22×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#22576188C2semidirect
2. #31774478C2semidirect
3. C2×#22576086C2semidirect
4. #31773798C2semidirect
5. #343×C271C2semidirect
31. C2×D1084D4semidirect
44. M16×C271D4semidirect
101. C271#2006semidirect
107. #343D271semidirect
142. M16D1084semidirect

Maximal subgroups

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

subgroupindex
C2×#225761882
#317744782
C2×#225760862
#317737982
#343×C2712
#2006271