Finite Group Databasebeta

← #46835331

Subgroups — #46835331

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#31774036C211yesmaximal, characteristic, complemented
3#31773816C211yesmaximal, characteristic, complemented
4#31773632C211yesmaximal, characteristic, complemented
5#31773631C211yesmaximal, characteristic, complemented
6#31773622C211yesmaximal, characteristic, complemented
7#448×C271C211yesmaximal, characteristic, complemented
8C2×#16864653—21no—
9C2×#16864641—21no—
10#22576138—21no—
11#22576137—21no—
12#22576134C2211yescharacteristic
13#22576133—21no—
14#22576132—21no—
15#22576132—21no—
16C2×Dic271⋊C4—21no—
17C2×Dic271⋊C4C2211yescharacteristic
18#22576084—21no—
19C22⋊(C271⋊C8)C2211yescharacteristic
20C271⋊#95—21no—
21C271⋊#95C2211yescharacteristic
22C22×Q8×C271C2211yescharacteristic
23C2×SD16×C271—21no—
24#121×C271C2211yescharacteristic
25#103×C271—21no—
26C22⋊C8×C271C2211yescharacteristic
27#16864653—42no—
28#16864653—42no—
29C2×Dic542—21no—
30C2×D1084D411yescharacteristic
31C2×C22⋊D271—21no—
32C2×C22⋊D271—21no—
33C2×C22⋊D271—21no—
34C22×Dic271—21no—
35C22×Dic271—21no—
36C22×Dic271D411yescharacteristic
37C22×Dic271D411yescharacteristic
38C2×C4×D271—21no—
39#16864641—42no—
40C2×C271⋊C8—21no—
41C4⋊Dic271D411yescharacteristic
42Dic271⋊C4—21no—
43Dic271⋊C4—21no—
44Dic271⋊C4—21no—
45Dic271⋊C4—42no—
46C22⋊Dic271—21no—
47(C2×D271)⋊C4—21no—
48(C2×D271)⋊C4—21no—
49C2×Q8×C271—21no—
50C2×Q8×C271—21no—
51C2×Q8×C271—42no—
52C2×D4×C271—21no—
53C2×D4×C271D411yescharacteristic
54C22×C4×C271—21no—
55C22×C4×C271C2311yescharacteristic
56SD16×C271—42no—
57C2×C8×C271—21no—
58C4⋊C4×C271D411yescharacteristic
59C22⋊C4×C271—21no—
60C22×D271—21no—
61Dic542—42no—
62D1084—41no—
63C22⋊D271—42no—
64C22⋊D271—41no—
65C22⋊D271—41no—
66C22⋊D271—81no—
67C2×Dic271—21no—
68C2×Dic271—21no—
69C2×Dic271—22no—
70C2×Dic271—21no—
71C2×Dic271—21no—
72C2×Dic271—42no—
73C2×Dic271—41no—
74C2×Dic271—41no—
75C2×Dic271—41no—
76C2×Dic271SD1612yescomplemented
77C4×D271—42no—
78C271⋊C8—41no—
79C23×C271—21no—
80C23×C271C2×D411yescharacteristic
81Q8×C271—22no—
82Q8×C271—22no—
83Q8×C271—41no—
84Q8×C271—41no—
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—41no—
110C4×C271—42no—
111Dic271—22no—
112Dic271—42no—
113Dic271—41no—
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#448D27111yescharacteristic, complemented
127#395—2711no—
128#327—2711no—
129#327—2711no—
130#324—2711no—
131C2×#41—5421no—
132C22×Q8C2×D27111yescharacteristic
133C2×SD16—21no—
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—21no—
145#103—5421no—
146C22⋊C8—2711no—
147C22⋊C8C2×D27111yescharacteristic
148#95—2711no—
149#95—5421no—
150#41—10842no—
151#41—10842no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—42no—
155C2×Q8—5421no—
156C2×D4—21no—
157C2×D4—2711no—
158C2×D4—5421no—
159C2×D4—5421no—
160C2×D4—5421no—
161C2×D4C22⋊D27111yescharacteristic
162C22×C4—21no—
163C22×C4—2711no—
164C22×C4—2711no—
165C22×C4—5421no—
166C22×C4—5421no—
167C22×C4—5421no—
168C22×C4C22×D27111yescharacteristic
169SD16—42no—
170SD16—10842no—
171C2×C8—21no—
172C2×C8—5421no—
173C4⋊C4—2711no—
174C4⋊C4—5421no—
175C4⋊C4—5421no—
176C4⋊C4—5421no—
177C4⋊C4—10842no—
178C4⋊C4C22⋊D27111yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—5421no—
181C22⋊C4—5421no—
182C22⋊C4—5421no—
183C23—21no—
184C23—5421no—
185C23D4×D27111yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
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×C22⋊D27111yescharacteristic
217C8—41no—
218C8—10841no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—10841no—
225C22—10841no—
226C22—10841no—
227C22#2257612811yescharacteristic
228C22SD16×D27111yescharacteristic
229C22#2257618711yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—42no—
234C4—5422no—
235C4—10842no—
236C4—10841no—
237C4—10841no—
238C2—21no—
239C2—21no—
240C2—81no—
241C2—21681no—
242C2#3177404611yescharacteristic
243C2#3177407811yescharacteristic
244C2#3177408311yescharacteristic
subgroupG/H
centerC22#22576128
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#448×C271C2
socleC22×C271#120
prime radical#31774474C2
hypercenter#448D271
hypocenterC271#1604
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #31774036C2semidirect
3. #31773816C2semidirect
4. #31773632C2semidirect
5. #31773631C2semidirect
6. #31773622C2semidirect
7. #448×C271C2semidirect
76. C2×Dic271SD16semidirect
122. C271#1604semidirect
126. #448D271semidirect

Maximal subgroups

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

subgroupindex
#317744742
#317740362
#317738162
#317736322
#317736312
#317736222
#448×C2712
#1604271