Finite Group Databasebeta

← #46835385

Subgroups — #46835385

Order 34688.

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

normal subgroups only

This group has 288 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#31774455C211yesmaximal, characteristic
2#31774216C211yesmaximal, characteristic, complemented
3#31774213C211yesmaximal, characteristic, complemented
4#31773834C211yesmaximal, characteristic, complemented
5#31773630C211yesmaximal, characteristic, complemented
6#31773620C211yesmaximal, characteristic, complemented
7#327×C271C211yesmaximal, characteristic, complemented
8C2×D4×D271—21no—
9#22576145—21no—
10#22576142—21no—
11#22576134C2211yescharacteristic
12#22576132—21no—
13#22576130—21no—
14#22576128—21no—
15C2×C4⋊Dic271—21no—
16C2×C4⋊Dic271C2211yescharacteristic, complemented
17C2×(C2×D271)⋊C4—21no—
18C2×(C2×D271)⋊C4C2211yescharacteristic
19#22576092—21no—
20#22576091—21no—
21#22576082—21no—
22#22576081—21no—
23C22⋊(C271⋊C8)C2211yescharacteristic, complemented
24#121×C271C2211yescharacteristic
25C22⋊C8×C271C2211yescharacteristic, complemented
26#95×C271C2211yescharacteristic, complemented
27C23×D271—21no—
28C23×D271D411yescharacteristic
29C2×D1084—21no—
30C2×D1084D411yescharacteristic
31C2×C22⋊D271—21no—
32C2×C22⋊D271—21no—
33C2×C22⋊D271—21no—
34D4×D271—42no—
35D4×D271—42no—
36C22×Dic271—21no—
37C22×Dic271D411yescharacteristic
38C2×C4×D271—21no—
39C2×C271⋊C8—21no—
40C4⋊Dic271—21no—
41C4⋊Dic271—21no—
42C4⋊Dic271—42no—
43C4⋊Dic271D411yescharacteristic
44Dic271⋊C4—21no—
45C22⋊Dic271—21no—
46(C2×D271)⋊C4—21no—
47(C2×D271)⋊C4—21no—
48(C2×D271)⋊C4—21no—
49(C2×D271)⋊C4—21no—
50(C2×D271)⋊C4—42no—
51C2×D4×C271—21no—
52C2×D4×C271D411yescharacteristic
53C22×C4×C271—21no—
54C22×C4×C271C2311yescharacteristic
55C2×C8×C271—21no—
56C4⋊C4×C271D411yescharacteristic
57C22⋊C4×C271—21no—
58C22×D271—21no—
59C22×D271—21no—
60C22×D271—21no—
61C22×D271—42no—
62C22×D271—41no—
63C22×D271—41no—
64C22×D271—81no—
65C22×D271—81no—
66C22×D271SD1612yes—
67D1084—41no—
68D1084—42no—
69C22⋊D271—42no—
70C22⋊D271—41no—
71C22⋊D271—41no—
72C22⋊D271—81no—
73C2×Dic271—21no—
74C2×Dic271—21no—
75C2×Dic271—41no—
76C2×Dic271—41no—
77C2×Dic271—41no—
78C2×Dic271—41no—
79C4×D271—42no—
80C271⋊C8—41no—
81C23×C271—21no—
82C23×C271C2×D411yescharacteristic
83D4×C271—41no—
84D4×C271—41no—
85D4×C271—41no—
86C2×C4×C271—22no—
87C2×C4×C271—21no—
88C2×C4×C271—41no—
89C2×C4×C271—41no—
90C2×C4×C271—41no—
91C2×C4×C271C2×D411yescharacteristic
92C2×C4×C271C2×D411yescharacteristic
93C8×C271—41no—
94C22×C271—21no—
95C22×C271—21no—
96C22×C271—41no—
97C22×C271—41no—
98C22×C271—41no—
99C22×C271#12011yescharacteristic
100C22×C271C2×SD1611yescharacteristic
101C22×C271AGL(1,ℤ/8)11yescharacteristic
102C2×D271—22no—
103C2×D271—41no—
104C2×D271—41no—
105C2×D271—42no—
106C2×D271—41no—
107C2×D271—41no—
108C2×D271—41no—
109C2×D271—81no—
110C2×D271—84no—
111C4×C271—21no—
112C4×C271—21no—
113C4×C271—42no—
114C4×C271—41no—
115Dic271—41no—
116Dic271—41no—
117Dic271—81no—
118C2×C271—21no—
119C2×C271—21no—
120C2×C271—81no—
121C2×C271#45011yescharacteristic
122C2×C271#45311yescharacteristic
123C2×C271#45811yescharacteristic
124D271—42no—
125D271—81no—
126C271#161311yescharacteristic, complemented, Sylow
127#1613—2711nomaximal, Sylow
128#534—2711no—
129#481—2711no—
130#481—2711no—
131#399—2711no—
132#327—2711no—
133#327D27111yescharacteristic, complemented
134#323—2711no—
135C22×D4—5421no—
136#123—5421no—
137#123—5421no—
138#121—2711no—
139#121—5421no—
140#121C2×D27111yescharacteristic
141#120—5421no—
142#120—5421no—
143C2×C4⋊C4—2711no—
144C2×C4⋊C4—5421no—
145C2×C22⋊C4—2711no—
146C2×C22⋊C4—5421no—
147#106—5421no—
148#106—5421no—
149#102—5421no—
150#102—5421no—
151C22⋊C8—2711no—
152C22⋊C8C2×D27111yescharacteristic, complemented
153#95C2×D27111yescharacteristic, complemented
154C24—2711no—
155C24—5421no—
156C2×D4—21no—
157C2×D4—2711no—
158C2×D4—5421no—
159C2×D4—5421no—
160C2×D4—5421no—
161C2×D4—5421no—
162C2×D4—10842no—
163C2×D4—10842no—
164C2×D4D108411yescharacteristic
165C22×C4—21no—
166C22×C4—2711no—
167C22×C4—5421no—
168C22×C4—5421no—
169C22×C4C22×D27111yescharacteristic
170C2×C8—21no—
171C2×C8—5421no—
172C4⋊C4—2711no—
173C4⋊C4—5421no—
174C4⋊C4—5421no—
175C4⋊C4—5421no—
176C4⋊C4—10842no—
177C4⋊C4D108411yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—5421no—
180C22⋊C4—5421no—
181C22⋊C4—5421no—
182C22⋊C4—5421no—
183C22⋊C4—5421no—
184C22⋊C4—10842no—
185C23—21no—
186C23—2712no—
187C23—5421no—
188C23—5421no—
189C23—5421no—
190C23—10841no—
191C23—10841no—
192C23—10842no—
193C23—21681no—
194C23—21681no—
195C23D4×D27111yescharacteristic
196D4—41no—
197D4—41no—
198D4—41no—
199D4—10841no—
200D4—10842no—
201D4—10842no—
202D4—10841no—
203D4—10841no—
204D4—21681no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—5421no—
211C2×C4—5421no—
212C2×C4—10842no—
213C2×C4—10841no—
214C2×C4—10841no—
215C2×C4—10841no—
216C2×C4—10841no—
217C2×C4D4×D27111yescharacteristic
218C2×C4C2×D108411yescharacteristic
219C8—41no—
220C8—10841no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—5422no—
227C22—10842no—
228C22—10841no—
229C22—10841no—
230C22—10841no—
231C22—10841no—
232C22—10841no—
233C22—21681no—
234C22—21684no—
235C22#2257613011yescharacteristic
236C22SD16×D27111yescharacteristic
237C22#2257618411yescharacteristic
238C4—21no—
239C4—21no—
240C4—42no—
241C4—41no—
242C4—10841no—
243C4—10841no—
244C4—21681no—
245C2—21no—
246C2—21no—
247C2—81no—
248C2—10842no—
249C2—21681no—
250C2#3177404911yescharacteristic
251C2#3177406611yescharacteristic
252C2#3177408811yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#327×C271C2
socleC22×C271#120
prime radical#31774455C2
hypercenter#327D271
hypocenterC271#1613
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. #31774216C2semidirect
3. #31774213C2semidirect
4. #31773834C2semidirect
5. #31773630C2semidirect
6. #31773620C2semidirect
7. #327×C271C2semidirect
16. C2×C4⋊Dic271C22semidirect
23. C22⋊(C271⋊C8)C22semidirect
25. C22⋊C8×C271C22semidirect
26. #95×C271C22semidirect
126. C271#1613semidirect
133. #327D271semidirect
152. C22⋊C8C2×D271semidirect
153. #95C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317744552
#317742162
#317742132
#317738342
#317736302
#317736202
#327×C2712
#1613271