Finite Group Databasebeta

← #46835591

Subgroups — #46835591

Order 34688.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#31774612C211yesmaximal, characteristic
2#31774249C211yesmaximal, characteristic, complemented
3#31774099C211yesmaximal, characteristic, complemented
4#31773645C211yesmaximal, characteristic, complemented
5D4⋊(C271⋊C8)C211yesmaximal, characteristic, complemented
6#31773624C211yesmaximal, characteristic, complemented
7#474×C271C211yesmaximal, characteristic, complemented
8C2×#16864641—21no—
9#22576160C2211yescharacteristic
10#22576146—22no—
11#22576134—21no—
12#22576131—21no—
13C4×D1084C2211yescharacteristic
14#22576119—21no—
15#22576118—21no—
16C4⋊(C271⋊C8)C2211yescharacteristic
17#22576084—21no—
18#22576083—22no—
19C4×C271⋊C8C2211yescharacteristic
20C4⋊Q8×C271C2211yescharacteristic
21C4×D4×C271C2211yescharacteristic
22#107×C271—21no—
23C4⋊C8×C271C2211yescharacteristic
24#102×C271—21no—
25C2×Dic542D412yes—
26C2×D1084D411yescharacteristic
27C2×C22⋊D271—22no—
28C2×C4×D271—21no—
29#16864641—42no—
30C2×C271⋊C8—21no—
31C2×C271⋊C8—21no—
32C4⋊Dic271—21no—
33C4⋊Dic271—21no—
34C4⋊Dic271D411yescharacteristic
35C22⋊Dic271—22no—
36(C2×D271)⋊C4—22no—
37(C2×D271)⋊C4—21no—
38C4×Dic271—22no—
39C2×Q8×C271—21no—
40C2×D4×C271D411yescharacteristic
41C22×C4×C271—21no—
42C2×C8×C271—21no—
43C4⋊C4×C271—21no—
44C4⋊C4×C271—21no—
45C4⋊C4×C271D411yescharacteristic
46C22⋊C4×C271—21no—
47C42×C271C2311yescharacteristic
48C22×D271—21no—
49Dic542—42no—
50D1084—21no—
51D1084—21no—
52C22⋊D271—82no—
53C2×Dic271—21no—
54C2×Dic271—22no—
55C4×D271—41no—
56C4×D271—41no—
57C271⋊C8—41no—
58C271⋊C8—41no—
59C23×C271—21no—
60Q8×C271—41no—
61Q8×C271—41no—
62D4×C271—21no—
63D4×C271—21no—
64C2×C4×C271—21no—
65C2×C4×C271—41no—
66C2×C4×C271—41no—
67C2×C4×C271—41no—
68C2×C4×C271C2×D411yescharacteristic
69C2×C4×C271C2×D411yescharacteristic
70C2×C4×C271C2×D411yescharacteristic
71C8×C271—41no—
72C22×C271—41no—
73C22×C271—41no—
74C22×C271—41no—
75C22×C271#12011yescharacteristic
76C2×D271—41no—
77C2×D271—41no—
78C2×D271—41no—
79C4×C271—21no—
80C4×C271—41no—
81C4×C271—81no—
82C4×C271#13511yescharacteristic
83C4×C271#13511yescharacteristic
84C4×C271AGL(1,ℤ/8)11yescharacteristic
85C4×C271#13711yescharacteristic
86Dic271—41no—
87Dic271—42no—
88C2×C271—81no—
89C2×C271#44911yescharacteristic
90C2×C271#45211yescharacteristic
91C2×C271#45511yescharacteristic
92D271—81no—
93C271#164611yescharacteristic, complemented, Sylow
94#1646—2711nomaximal, Sylow
95#555—2711no—
96#487—2711no—
97#474D27111yescharacteristic, complemented
98#461—2711no—
99#333—2711no—
100D4⋊C8—2711no—
101D4⋊C8—2711no—
102C2×SD16—5421no—
103C4⋊Q8—2711no—
104C4⋊Q8C2×D27111yescharacteristic
105#124—5422no—
106#121—5421no—
107#121—5421no—
108C4×D4—2711no—
109C4×D4C2×D27111yescharacteristic
110#117—5421no—
111#117—5421no—
112#107—21no—
113C4⋊C8—2711no—
114C4⋊C8C2×D27111yescharacteristic
115#103—5422no—
116#103—5421no—
117#102—21no—
118C4×C8—2711no—
119C2×Q8—21no—
120C2×Q8—2712no—
121C2×D4—2711no—
122C2×D4—5422no—
123C2×D4C22⋊D27111yescharacteristic
124C22×C4—21no—
125C22×C4—5421no—
126SD16—10842no—
127C2×C8—21no—
128C2×C8—5421no—
129C2×C8—5421no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—2711no—
133C4⋊C4—5421no—
134C4⋊C4—5421no—
135C4⋊C4C22⋊D27111yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—5422no—
138C22⋊C4—5422no—
139C22⋊C4—5421no—
140C42—5422no—
141C42C22×D27111yescharacteristic
142C23—21no—
143C23—5421no—
144Q8—41no—
145Q8—41no—
146Q8—10842no—
147D4—21no—
148D4—21no—
149D4—5421no—
150D4—5421no—
151D4—21682no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—5421no—
157C2×C4—5422no—
158C2×C4—10841no—
159C2×C4—10841no—
160C2×C4D4×D27111yescharacteristic
161C2×C4D4×D27111yescharacteristic
162C2×C4C2×C22⋊D27111yescharacteristic
163C8—41no—
164C8—10841no—
165C8—10841no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—10841no—
170C22—10841no—
171C22—10841no—
172C22#2257612811yescharacteristic
173C4—21no—
174C4—41no—
175C4—81no—
176C4—10841no—
177C4—10842no—
178C4#2257617911yescharacteristic
179C4#2257618011yescharacteristic
180C4#2257618411yescharacteristic
181C4#2257619111yescharacteristic
182C2—81no—
183C2—21681no—
184C2#3177403911yescharacteristic
185C2#3177405711yescharacteristic
186C2#3177407311yescharacteristic
subgroupG/H
centerC22#22576128
commutator, cosocleC42×C271C23
FrattiniC42C22×D271
Fitting#474×C271C2
socleC22×C271#120
prime radical#31774612C2
hypercenter#474D271
hypocenterC271#1646
LCSG ▷ C42×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #31774249C2semidirect
3. #31774099C2semidirect
4. #31773645C2semidirect
5. D4⋊(C271⋊C8)C2semidirect
6. #31773624C2semidirect
7. #474×C271C2semidirect
93. C271#1646semidirect
97. #474D271semidirect

Maximal subgroups

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

subgroupindex
#317746122
#317742492
#317740992
#317736452
D4⋊(C271⋊C8)2
#317736242
#474×C2712
#1646271