Finite Group Databasebeta

← #46835381

Subgroups — #46835381

Order 34688.

232 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 49 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#31774487C211yesmaximal, characteristic
2#31774225C211yesmaximal, characteristic, complemented
3#31774211C211yesmaximal, characteristic, complemented
4#31773824C211yesmaximal, characteristic, complemented
5#31773630C211yesmaximal, characteristic, complemented
6#31773620C211yesmaximal, characteristic, complemented
7#328×C271C211yesmaximal, characteristic, complemented
8#22576155—21no—
9#22576149—21no—
10#22576145—21no—
11#22576135—21no—
12#22576134C2211yescharacteristic
13#22576130—21no—
14(C22⋊D271)⋊C4—21no—
15C2×C4⋊Dic271C2211yescharacteristic
16C2×(C2×D271)⋊C4—21no—
17C2×(C2×D271)⋊C4C2211yescharacteristic
18#22576094—21no—
19#22576091—21no—
20#22576084—21no—
21#22576082—21no—
22C22⋊(C271⋊C8)C2211yescharacteristic
23C271⋊#95—21no—
24C22⋊Q8×C271C2211yescharacteristic
25C22⋊C8×C271C2211yescharacteristic
26#95×C271C2211yescharacteristic
27C23×D271D411yescharacteristic
28C2×D1084—21no—
29C2×D1084—21no—
30C2×D1084D411yescharacteristic
31C2×C22⋊D271—21no—
32C22×Dic271—21no—
33C22×Dic271D411yescharacteristic
34C2×C4×D271—21no—
35C2×C271⋊C8—21no—
36C4⋊Dic271—21no—
37C4⋊Dic271—21no—
38C4⋊Dic271D411yescharacteristic
39Dic271⋊C4—21no—
40(C2×D271)⋊C4—21no—
41(C2×D271)⋊C4—21no—
42(C2×D271)⋊C4—21no—
43(C2×D271)⋊C4—21no—
44(C2×D271)⋊C4—21no—
45(C2×D271)⋊C4—21no—
46(C2×D271)⋊C4—42no—
47C4×Dic271—21no—
48C2×Q8×C271D411yescharacteristic
49C22×C4×C271—21no—
50C22×C4×C271C2311yescharacteristic
51C2×C8×C271—21no—
52C4⋊C4×C271—21no—
53C4⋊C4×C271D411yescharacteristic
54C22⋊C4×C271—21no—
55C22×D271—21no—
56C22×D271—21no—
57C22×D271—21no—
58C22×D271—21no—
59C22×D271—41no—
60C22×D271—41no—
61D1084—41no—
62D1084—81no—
63D1084—81no—
64C22⋊D271—41no—
65C22⋊D271—41no—
66C2×Dic271—21no—
67C2×Dic271—21no—
68C2×Dic271—41no—
69C2×Dic271—41no—
70C2×Dic271—41no—
71C2×Dic271—41no—
72C4×D271—81no—
73C271⋊C8—41no—
74C23×C271C2×D411yescharacteristic
75Q8×C271—41no—
76C2×C4×C271—21no—
77C2×C4×C271—21no—
78C2×C4×C271—41no—
79C2×C4×C271—41no—
80C2×C4×C271—41no—
81C2×C4×C271—41no—
82C2×C4×C271C2×D411yescharacteristic
83C2×C4×C271C2×D411yescharacteristic
84C8×C271—41no—
85C22×C271—21no—
86C22×C271—21no—
87C22×C271#12011yescharacteristic
88C22×C271#13511yescharacteristic
89C22×C271AGL(1,ℤ/8)11yescharacteristic
90C2×D271—42no—
91C2×D271—41no—
92C2×D271—41no—
93C2×D271—41no—
94C2×D271—41no—
95C2×D271—41no—
96C2×D271—41no—
97C2×D271—81no—
98C4×C271—21no—
99C4×C271—21no—
100C4×C271—41no—
101C4×C271—41no—
102C4×C271—81no—
103Dic271—41no—
104Dic271—41no—
105Dic271—81no—
106C2×C271—21no—
107C2×C271—21no—
108C2×C271#44911yescharacteristic
109C2×C271#45511yescharacteristic
110C2×C271#45811yescharacteristic
111D271—81no—
112D271—81no—
113C271#161211yescharacteristic, complemented, Sylow
114#1612—2711nomaximal, Sylow
115#538—2711no—
116#483—2711no—
117#480—2711no—
118#397—2711no—
119#328D27111yescharacteristic, complemented
120#327—2711no—
121#323—2711no—
122#126—5421no—
123#124—5421no—
124#123—5421no—
125C22⋊Q8C2×D27111yescharacteristic
126#121—2711no—
127#121—5421no—
128#120—5421no—
129C4×D4—5421no—
130C2×C4⋊C4—2711no—
131C2×C22⋊C4—2711no—
132C2×C22⋊C4—5421no—
133#107—5421no—
134#106—5421no—
135#103—5421no—
136#102—5421no—
137C22⋊C8—2711no—
138C22⋊C8C2×D27111yescharacteristic
139#95—5421no—
140#95C2×D27111yescharacteristic
141C24—2711no—
142C2×Q8D108411yescharacteristic
143C2×D4—2711no—
144C2×D4—5421no—
145C2×D4—5421no—
146C2×D4—5421no—
147C22×C4—21no—
148C22×C4—2711no—
149C22×C4—5421no—
150C22×C4—5421no—
151C22×C4C22×D27111yescharacteristic
152C2×C8—21no—
153C2×C8—5421no—
154C4⋊C4—21no—
155C4⋊C4—2711no—
156C4⋊C4—5421no—
157C4⋊C4—5421no—
158C4⋊C4—5421no—
159C4⋊C4D108411yescharacteristic
160C22⋊C4—21no—
161C22⋊C4—5421no—
162C22⋊C4—5421no—
163C22⋊C4—5421no—
164C22⋊C4—5421no—
165C22⋊C4—5421no—
166C22⋊C4—5421no—
167C22⋊C4—10842no—
168C42—5421no—
169C23—5421no—
170C23—5421no—
171C23—5421no—
172C23—5421no—
173C23—10841no—
174C23—10841no—
175C23D4×D27111yescharacteristic
176Q8—41no—
177D4—10841no—
178D4—10841no—
179D4—10841no—
180D4—21681no—
181D4—21681no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—5421no—
189C2×C4—5421no—
190C2×C4—10841no—
191C2×C4—10841no—
192C2×C4—10841no—
193C2×C4—10841no—
194C2×C4—21681no—
195C2×C4D4×D27111yescharacteristic
196C2×C4C2×D108411yescharacteristic
197C8—41no—
198C8—10841no—
199C22—21no—
200C22—21no—
201C22—10842no—
202C22—10841no—
203C22—10841no—
204C22—10841no—
205C22—10841no—
206C22—10841no—
207C22—10841no—
208C22—21681no—
209C22#2257613011yescharacteristic
210C22#2257618311yescharacteristic
211C22#2257618711yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—41no—
216C4—81no—
217C4—10841no—
218C4—10841no—
219C4—21681no—
220C2—21no—
221C2—21no—
222C2—21681no—
223C2—21681no—
224C2#3177404511yescharacteristic
225C2#3177407611yescharacteristic
226C2#3177408811yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#328×C271C2
socleC22×C271#120
prime radical#31774487C2
hypercenter#328D271
hypocenterC271#1612
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #31774225C2semidirect
3. #31774211C2semidirect
4. #31773824C2semidirect
5. #31773630C2semidirect
6. #31773620C2semidirect
7. #328×C271C2semidirect
113. C271#1612semidirect
119. #328D271semidirect

Maximal subgroups

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

subgroupindex
#317744872
#317742252
#317742112
#317738242
#317736302
#317736202
#328×C2712
#1612271