Finite Group Databasebeta

← #46835334

Subgroups — #46835334

Order 34688.

282 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 51 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#31774474C211yesmaximal, characteristic
2#31774038C211yesmaximal, characteristic, complemented
3#31774035C211yesmaximal, characteristic, complemented
4#31773818C211yesmaximal, characteristic, complemented
5#31773630C211yesmaximal, characteristic, complemented
6#31773622C211yesmaximal, characteristic, complemented
7#327×C271C211yesmaximal, characteristic, complemented
8C2×#16864653—21no—
9C22×Dic542C2211yescharacteristic
10C2×#16864643—21no—
11C2×#16864642—21no—
12#22576138—21no—
13#22576137—21no—
14#22576134C2211yescharacteristic
15#22576133—21no—
16#22576132—21no—
17#22576132—21no—
18C2×C4⋊Dic271—21no—
19C2×Dic271⋊C4C2211yescharacteristic
20#22576085—21no—
21#22576083—21no—
22C22⋊(C271⋊C8)C2211yescharacteristic
23C271⋊#95—21no—
24#121×C271C2211yescharacteristic
25C22⋊C8×C271C2211yescharacteristic
26#95×C271C2211yescharacteristic
27#16864653—42no—
28#16864653—42no—
29C2×Dic542—21no—
30C2×Dic542—21no—
31C2×Dic542—21no—
32C2×Dic542—42no—
33C2×D1084D411yescharacteristic
34C2×C22⋊D271—21no—
35C2×C22⋊D271—21no—
36C2×C22⋊D271—21no—
37C22×Dic271—21no—
38C22×Dic271—21no—
39C22×Dic271D411yescharacteristic
40C22×Dic271D411yescharacteristic
41C2×C4×D271—21no—
42#16864643—42no—
43#16864642—42no—
44C2×C271⋊C8—21no—
45C4⋊Dic271—42no—
46C4⋊Dic271D411yescharacteristic
47Dic271⋊C4—21no—
48Dic271⋊C4—21no—
49Dic271⋊C4—21no—
50C22⋊Dic271—21no—
51(C2×D271)⋊C4—21no—
52(C2×D271)⋊C4—21no—
53C2×D4×C271—21no—
54C2×D4×C271D411yescharacteristic
55C22×C4×C271—21no—
56C22×C4×C271C2311yescharacteristic
57C2×C8×C271—21no—
58C4⋊C4×C271D411yescharacteristic
59C22⋊C4×C271—21no—
60C22×D271—21no—
61Dic542—22no—
62Dic542—22no—
63Dic542—42no—
64Dic542—41no—
65Dic542—41no—
66D1084—41no—
67C22⋊D271—42no—
68C22⋊D271—41no—
69C22⋊D271—41no—
70C22⋊D271—81no—
71C2×Dic271—21no—
72C2×Dic271—21no—
73C2×Dic271—22no—
74C2×Dic271—21no—
75C2×Dic271—21no—
76C2×Dic271—42no—
77C2×Dic271—41no—
78C2×Dic271—41no—
79C2×Dic271—41no—
80C2×Dic271SD1612yescomplemented
81C4×D271—42no—
82C271⋊C8—41no—
83C23×C271—21no—
84C23×C271C2×D411yescharacteristic
85D4×C271—41no—
86D4×C271—41no—
87D4×C271—41no—
88C2×C4×C271—22no—
89C2×C4×C271—21no—
90C2×C4×C271—41no—
91C2×C4×C271—41no—
92C2×C4×C271—41no—
93C2×C4×C271C2×D411yescharacteristic
94C2×C4×C271C2×D411yescharacteristic
95C8×C271—41no—
96C22×C271—21no—
97C22×C271—21no—
98C22×C271—41no—
99C22×C271—41no—
100C22×C271—41no—
101C22×C271#12011yescharacteristic
102C22×C271C2×SD1611yescharacteristic
103C22×C271AGL(1,ℤ/8)11yescharacteristic
104C2×D271—41no—
105C2×D271—41no—
106C2×D271—41no—
107C4×C271—21no—
108C4×C271—21no—
109C4×C271—42no—
110C4×C271—41no—
111Dic271—22no—
112Dic271—41no—
113Dic271—42no—
114Dic271—41no—
115C2×C271—21no—
116C2×C271—21no—
117C2×C271—81no—
118C2×C271#45011yescharacteristic
119C2×C271#45611yescharacteristic
120C2×C271B(4,𝔽2)11yescharacteristic
121D271—81no—
122C271#160411yescharacteristic, complemented, Sylow
123#1604—2711nomaximal, Sylow
124#537—2711no—
125#448—2711no—
126#448—2711no—
127#395—2711no—
128#327—2711no—
129#327D27111yescharacteristic, complemented
130#324—2711no—
131C2×#41—5421no—
132C22×Q8—2711no—
133C2×SD16—5421no—
134C2×SD16—5421no—
135C22⋊Q8—5421no—
136C22⋊Q8—5421no—
137#121—2711no—
138#121—5421no—
139#121—5421no—
140#121—5421no—
141#121C2×D27111yescharacteristic
142C2×C4⋊C4—2711no—
143C2×C4⋊C4—5421no—
144#103—5421no—
145#103—5421no—
146C22⋊C8—2711no—
147C22⋊C8C2×D27111yescharacteristic
148#95—5421no—
149#95C2×D27111yescharacteristic
150#41—10842no—
151#41—10842no—
152C2×Q8—5421no—
153C2×Q8—5421no—
154C2×Q8—5421no—
155C2×Q8—10842no—
156C2×D4—21no—
157C2×D4—2711no—
158C2×D4—5421no—
159C2×D4—5421no—
160C2×D4—5421no—
161C2×D4D108411yescharacteristic
162C22×C4—21no—
163C22×C4—2711no—
164C22×C4—2711no—
165C22×C4—5421no—
166C22×C4—5421no—
167C22×C4—5421no—
168C22×C4C22×D27111yescharacteristic
169SD16—10842no—
170SD16—10842no—
171C2×C8—21no—
172C2×C8—5421no—
173C4⋊C4—2711no—
174C4⋊C4—5421no—
175C4⋊C4—5421no—
176C4⋊C4—5421no—
177C4⋊C4—10842no—
178C4⋊C4D108411yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—5421no—
181C22⋊C4—5421no—
182C22⋊C4—5421no—
183C23—21no—
184C23—5421no—
185C23D4×D27111yescharacteristic
186Q8—5422no—
187Q8—5422no—
188Q8—10841no—
189Q8—10841no—
190Q8—10842no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—10841no—
195D4—10842no—
196D4—10841no—
197D4—10841no—
198D4—21681no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—2712no—
205C2×C4—5421no—
206C2×C4—5421no—
207C2×C4—5422no—
208C2×C4—5421no—
209C2×C4—5421no—
210C2×C4—10842no—
211C2×C4—10842no—
212C2×C4—10841no—
213C2×C4—10841no—
214C2×C4—10841no—
215C2×C4D4×D27111yescharacteristic
216C2×C4C2×D108411yescharacteristic
217C8—41no—
218C8—10841no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—10841no—
225C22—10841no—
226C22—10841no—
227C22#2257613011yescharacteristic
228C22SD16×D27111yescharacteristic
229C22#2257618411yescharacteristic
230C4—21no—
231C4—21no—
232C4—42no—
233C4—41no—
234C4—5422no—
235C4—10841no—
236C4—10842no—
237C4—10841no—
238C2—21no—
239C2—21no—
240C2—81no—
241C2—21681no—
242C2#3177404911yescharacteristic
243C2#3177408111yescharacteristic
244C2#3177408511yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#327×C271C2
socleC22×C271#120
prime radical#31774474C2
hypercenter#327D271
hypocenterC271#1604
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #31774038C2semidirect
3. #31774035C2semidirect
4. #31773818C2semidirect
5. #31773630C2semidirect
6. #31773622C2semidirect
7. #327×C271C2semidirect
80. C2×Dic271SD16semidirect
122. C271#1604semidirect
129. #327D271semidirect

Maximal subgroups

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

subgroupindex
#317744742
#317740382
#317740352
#317738182
#317736302
#317736222
#327×C2712
#1604271