Finite Group Databasebeta

← #46835593

Subgroups — #46835593

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#31774613C211yesmaximal, characteristic
2#31774247C211yesmaximal, characteristic, complemented
3#31774178C211yesmaximal, characteristic, complemented
4#31774103C211yesmaximal, characteristic, complemented
5#31773624C211yesmaximal, characteristic, complemented
6#31773624C211yesmaximal, characteristic, complemented
7#333×C271C211yesmaximal, characteristic, complemented
8C2×#16864643—21no—
9#22576160C2211yescharacteristic
10#22576149—22no—
11#22576135—21no—
12#22576135—21no—
13C4×D1084C2211yescharacteristic
14C4×D1084C2211yescharacteristic
15#22576118—21no—
16#22576118—21no—
17#22576094—21no—
18#22576085—21no—
19#22576084—22no—
20#22576082—21no—
21C4×C271⋊C8C2211yescharacteristic
22C4⋊Q8×C271C2211yescharacteristic
23C4⋊C8×C271C2211yescharacteristic
24C4⋊C8×C271C2211yescharacteristic
25C2×Dic542—21no—
26C2×D1084—22no—
27C2×D1084D411yescharacteristic
28C2×D1084D411yescharacteristic
29C2×C4×D271—21no—
30C2×C4×D271—21no—
31#16864643—42no—
32C2×C271⋊C8—21no—
33C4⋊Dic271—21no—
34C4⋊Dic271—21no—
35C4⋊Dic271D411yescharacteristic
36C4⋊Dic271D411yescharacteristic
37(C2×D271)⋊C4—22no—
38(C2×D271)⋊C4—22no—
39(C2×D271)⋊C4—21no—
40(C2×D271)⋊C4—21no—
41C4×Dic271—22no—
42C2×Q8×C271D412yes—
43C2×C8×C271—21no—
44C2×C8×C271—21no—
45C4⋊C4×C271—21no—
46C4⋊C4×C271—21no—
47C42×C271C2311yescharacteristic
48C22×D271—21no—
49C22×D271—21no—
50Dic542—41no—
51Dic542—41no—
52D1084—21no—
53D1084—21no—
54D1084—21no—
55D1084—21no—
56D1084—82no—
57C2×Dic271—21no—
58C2×Dic271—21no—
59C2×Dic271—41no—
60C4×D271—41no—
61C4×D271—41no—
62C4×D271—41no—
63C4×D271—41no—
64C271⋊C8—41no—
65Q8×C271—42no—
66C2×C4×C271—22no—
67C2×C4×C271C2×D411yescharacteristic
68C2×C4×C271C2×D411yescharacteristic
69C2×C4×C271C2×D411yescharacteristic
70C8×C271—41no—
71C8×C271—41no—
72C22×C271#12011yescharacteristic
73C2×D271—41no—
74C2×D271—41no—
75C2×D271—41no—
76C2×D271—41no—
77C2×D271—41no—
78C2×D271—41no—
79C4×C271—21no—
80C4×C271—42no—
81C4×C271#13511yescharacteristic
82C4×C271#13511yescharacteristic
83C4×C271AGL(1,ℤ/8)11yescharacteristic
84C4×C271#13711yescharacteristic
85Dic271—41no—
86Dic271—41no—
87Dic271—81no—
88C2×C271#44911yescharacteristic
89C2×C271#45211yescharacteristic
90C2×C271#45511yescharacteristic
91D271—81no—
92D271—81no—
93C271#164611yescharacteristic, complemented, Sylow
94#1646—2711nomaximal, Sylow
95#555—2711no—
96#487—2711no—
97#474—2711no—
98#461—2711no—
99#333D27111yescharacteristic, complemented
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×D4—2711no—
110#117—5421no—
111#117—5421no—
112#107—5421no—
113C4⋊C8C2×D27111yescharacteristic
114C4⋊C8C2×D27111yescharacteristic
115#103—5421no—
116#103—5422no—
117#102—5421no—
118C4×C8—2711no—
119C2×Q8—5421no—
120C2×Q8D108412yes—
121C2×D4—2711no—
122C2×D4—2711no—
123C2×D4—5422no—
124C22×C4—5421no—
125C22×C4—5421no—
126SD16—10842no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—5421no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—2711no—
133C4⋊C4—2711no—
134C4⋊C4—5421no—
135C4⋊C4—5421no—
136C22⋊C4—5422no—
137C22⋊C4—5422no—
138C22⋊C4—5421no—
139C22⋊C4—5421no—
140C42—5422no—
141C42C22×D27111yescharacteristic
142C23—5421no—
143C23—5421no—
144Q8—42no—
145Q8—10841no—
146Q8—10841no—
147D4—5421no—
148D4—5421no—
149D4—5421no—
150D4—5421no—
151D4—21682no—
152C2×C4—22no—
153C2×C4—5421no—
154C2×C4—5421no—
155C2×C4—10841no—
156C2×C4—10841no—
157C2×C4—10841no—
158C2×C4—10841no—
159C2×C4—10841no—
160C2×C4D4×D27111yescharacteristic
161C2×C4D4×D27111yescharacteristic
162C2×C4C2×D108411yescharacteristic
163C8—41no—
164C8—41no—
165C8—10841no—
166C22—10841no—
167C22—10841no—
168C22—10841no—
169C22—10841no—
170C22—10841no—
171C22—10841no—
172C22#2257613011yescharacteristic
173C4—21no—
174C4—42no—
175C4—10841no—
176C4—10841no—
177C4—21681no—
178C4#2257618111yescharacteristic
179C4#2257618311yescharacteristic
180C4#2257618711yescharacteristic
181C4#2257619211yescharacteristic
182C2—21681no—
183C2—21681no—
184C2#3177404511yescharacteristic
185C2#3177406211yescharacteristic
186C2#3177407711yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC42×C271C23
FrattiniC42C22×D271
Fitting#333×C271C2
socleC22×C271#120
prime radical#31774613C2
hypercenter#333D271
hypocenterC271#1646
LCSG ▷ C42×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #31774247C2semidirect
3. #31774178C2semidirect
4. #31774103C2semidirect
5. #31773624C2semidirect
6. #31773624C2semidirect
7. #333×C271C2semidirect
93. C271#1646semidirect
99. #333D271semidirect

Maximal subgroups

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

subgroupindex
#317746132
#317742472
#317741782
#317741032
#317736242
#333×C2712
#1646271