Finite Group Databasebeta

← #46837405

Subgroups — #46837405

Order 34688.

406 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 73 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
2#31774478C211yesmaximal, characteristic, complemented
3C2×#22576088C212yesmaximal, complemented
4#31773666C211yesmaximal, characteristic, complemented
5#390×C271C211yesmaximal, characteristic, complemented
6C2×#16864654C2212yes—
7#22576186—24no—
8#22576186—24no—
9C2×#16864642—22no—
10C2×#16864639—22no—
11C2×#16864638C2212yes—
12#22576140—22no—
13#22576135—22no—
14#22576134—22no—
15#22576088—24no—
16C22×D4×C271C2211yescharacteristic
17C2×C4⋊C4×C271C2211yescharacteristic
18C2×C22⋊C4×C271—22no—
19C2×C42×C271C2211yescharacteristic
20#16864654—22no—
21#16864654—22no—
22#16864654—22no—
23#16864654—22no—
24C2×Dic542D412yes—
25C2×D1084—22no—
26C2×D1084D412yes—
27C2×C22⋊D271—22no—
28C2×C4×D271—22no—
29#16864642—28no—
30#16864639—28no—
31#16864638—22no—
32#16864638D414yescomplemented
33C2×C271⋊C8—22no—
34C4⋊Dic271—22no—
35(C2×D271)⋊C4—24no—
36C24×C271—21no—
37C2×D4×C271—22no—
38C2×D4×C271—22no—
39C2×D4×C271—21no—
40C2×D4×C271D412yescomplemented
41C22×C4×C271—21no—
42C22×C4×C271D412yes—
43C22×C4×C271C2311yescharacteristic
44C4⋊C4×C271—22no—
45C22⋊C4×C271—44no—
46C42×C271—22no—
47C22×D271—22no—
48Dic542—22no—
49Dic542—22no—
50D1084—22no—
51D1084—22no—
52D1084—82no—
53C22⋊D271—42no—
54C22⋊D271—42no—
55C2×Dic271—22no—
56C4×D271—42no—
57C4×D271—42no—
58C271⋊C8—24no—
59C23×C271—22no—
60C23×C271—42no—
61C23×C271—42no—
62C23×C271—41no—
63C23×C271—41no—
64C23×C271C2×D411yescharacteristic
65D4×C271—24no—
66D4×C271—42no—
67C2×C4×C271—22no—
68C2×C4×C271—22no—
69C2×C4×C271—41no—
70C2×C4×C271—41no—
71C2×C4×C271—42no—
72C2×C4×C271C2×D412yes—
73C2×C4×C271C2×D412yes—
74C2×C4×C271C2×D411yescharacteristic
75C2×C4×C271#4111yescharacteristic
76C22×C271—21no—
77C22×C271—21no—
78C22×C271—44no—
79C22×C271—42no—
80C22×C271—82no—
81C22×C271—81no—
82C22×C271—81no—
83C22×C271#12011yescharacteristic
84C22×C271#12111yescharacteristic
85C22×C271#12711yescharacteristic
86C2×D271—42no—
87C2×D271—42no—
88C2×D271—42no—
89C4×C271—42no—
90C4×C271—42no—
91C4×C271#12012yes—
92C4×C271#12112yes—
93Dic271—42no—
94C2×C271—21no—
95C2×C271—21no—
96C2×C271—82no—
97C2×C271#39211yescharacteristic
98C2×C271#45311yescharacteristic
99C2×C271#45411yescharacteristic
100D271—82no—
101C271#200611yescharacteristic, complemented, Sylow
102#2006—2711nomaximal, Sylow
103C2×AGL(1,ℤ/8)—2712no—
104#537—2711no—
105C2×#104—2712no—
106#390D27111yescharacteristic, complemented
107#343—2711no—
108C2×#41—2712no—
109C22×D4C2×D27111yescharacteristic
110AGL(1,ℤ/8)—5424no—
111AGL(1,ℤ/8)—5424no—
112C2×SD16—5422no—
113C2×D8—5422no—
114C2×M16—2712no—
115C22⋊Q8—5422no—
116#121—5422no—
117#121—5422no—
118C2×C4⋊C4C2×D27111yescharacteristic
119C2×C22⋊C4—22no—
120C2×C42C2×D27111yescharacteristic
121#104—5424no—
122C24—21no—
123#41—5422no—
124#41—5422no—
125#41—5422no—
126#41—5422no—
127C2×Q8—2712no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—2712no—
132C2×D4—5422no—
133C2×D4—5422no—
134C2×D4D108412yescomplemented
135C22×C4—21no—
136C22×C4—5422no—
137C22×C4C22⋊D27112yes—
138C22×C4C22×D27111yescharacteristic
139SD16—5428no—
140D8—5428no—
141M16—2714no—
142M16—5422no—
143C2×C8—5422no—
144C4⋊C4—22no—
145C4⋊C4—5422no—
146C22⋊C4—44no—
147C22⋊C4—5424no—
148C42—22no—
149C23—22no—
150C23—42no—
151C23—42no—
152C23—41no—
153C23—41no—
154C23—5422no—
155C23C2×C22⋊D27111yescharacteristic
156Q8—5422no—
157Q8—5422no—
158D4—24no—
159D4—42no—
160D4—5422no—
161D4—5422no—
162D4—10842no—
163D4—10842no—
164D4—21682no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—5422no—
171C2×C4—10842no—
172C2×C4—10842no—
173C2×C4D4×D27112yes—
174C2×C4D4×D27112yes—
175C2×C4C2×D108411yescharacteristic
176C2×C4#1686465411yescharacteristic
177C8—5424no—
178C22—21no—
179C22—21no—
180C22—44no—
181C22—42no—
182C22—82no—
183C22—81no—
184C22—81no—
185C22—10842no—
186C22—10842no—
187C22—10842no—
188C22#2257612811yescharacteristic
189C22#2257613411yescharacteristic
190C22#2257615611yescharacteristic
191C4—42no—
192C4—42no—
193C4—10842no—
194C4#2257613012yes—
195C4#2257613312yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—21682no—
200C2#3177380611yescharacteristic
201C2#3177406411yescharacteristic
202C2#3177406911yescharacteristic
subgroupG/H
centerC22#22576128
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#390×C271C2
socleC22×C271#120
hypercenter#390D271
hypocenterC271#2006
LCSG ▷ C22×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#22576186C2semidirect
2. #31774478C2semidirect
3. C2×#22576088C2semidirect
4. #31773666C2semidirect
5. #390×C271C2semidirect
32. #16864638D4semidirect
40. C2×D4×C271D4semidirect
101. C271#2006semidirect
106. #390D271semidirect
134. C2×D4D1084semidirect

Maximal subgroups

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

subgroupindex
C2×#225761862
#317744782
C2×#225760882
#317736662
#390×C2712
#2006271