Finite Group Databasebeta

← #46846754

Subgroups — #46846754

Order 34688.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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#31774445C211yesmaximal, characteristic
2C4⋊Q8×D271C211yesmaximal, characteristic, complemented
3#31774255C214yesmaximal, complemented
4#31774249C211yesmaximal, characteristic, complemented
5#31774248C211yesmaximal, characteristic, complemented
6#31774243C211yesmaximal, characteristic, complemented
7#31773895C214yesmaximal, complemented
8C4×#16864637C211yesmaximal, characteristic, complemented
9#486×C271C211yesmaximal, characteristic, complemented
10C2×#16864653C2212yes—
11C2×Q8×D271—21no—
12C2×#16864637C2212yes—
13#22576160C2211yescharacteristic
14#22576160C2211yescharacteristic
15#22576159—21no—
16#22576159C2212yes—
17#22576158—21no—
18#22576147C2212yes—
19#22576131—21no—
20#22576131—21no—
21D4×Dic271—21no—
22D4×Dic271—21no—
23C4⋊C4×D271—21no—
24C4⋊C4×D271C2212yes—
25C42×D271C2211yescharacteristic
26#22576085C2214yes—
27#22576083C2214yes—
28#22576081C2214yes—
29#22576070C2212yes—
30C4×C271⋊C8C2211yescharacteristic
31C4⋊Q8×C271C2211yescharacteristic
32#127×C271C2211yescharacteristic
33#102×C271C2214yes—
34C4×C8×C271C2211yescharacteristic
35#16864653—28no—
36C2×Dic542—22no—
37C2×Dic542—21no—
38C2×Dic542C2312yes—
39Q8×D271—44no—
40C2×C22⋊D271—22no—
41C22×Dic271—22no—
42C2×C4×D271—22no—
43C2×C4×D271D411yescharacteristic
44C2×C4×D271D411yescharacteristic
45C2×C4×D271C2311yescharacteristic
46#16864637—24no—
47C2×C271⋊C8C2312yes—
48C4⋊Dic271—21no—
49C4⋊Dic271—21no—
50C4⋊Dic271—21no—
51C4⋊Dic271C2312yes—
52Dic271⋊C4—22no—
53Dic271⋊C4—22no—
54C22⋊Dic271—22no—
55C22⋊Dic271—22no—
56C4×Dic271D411yescharacteristic
57C4×Dic271D411yescharacteristic
58C4×Dic271C2311yescharacteristic
59C2×Q8×C271—21no—
60C2×D4×C271—21no—
61C2×D4×C271—21no—
62C2×D4×C271C2312yes—
63C2×C8×C271C2312yes—
64C4⋊C4×C271—21no—
65C4⋊C4×C271C2312yes—
66C42×C271C2311yescharacteristic
67C22×D271C2×D411yescharacteristic
68Dic542—24no—
69Dic542—44no—
70Dic542—42no—
71C22⋊D271—44no—
72C2×Dic271—22no—
73C2×Dic271—22no—
74C2×Dic271—44no—
75C2×Dic271C2×D411yescharacteristic
76C2×Dic271C2×D411yescharacteristic
77C2×Dic271C2×D411yescharacteristic
78C4×D271—22no—
79C4×D271—22no—
80C4×D271—44no—
81C4×D271#4114yes—
82C271⋊C8—22no—
83C23×C271—22no—
84Q8×C271—42no—
85D4×C271—24no—
86D4×C271—42no—
87D4×C271—42no—
88C2×C4×C271—22no—
89C2×C4×C271C2×D411yescharacteristic
90C2×C4×C271C2×D411yescharacteristic
91C2×C4×C271C2411yescharacteristic
92C8×C271—22no—
93C22×C271—42no—
94C22×C271—42no—
95C22×C271—42no—
96C22×C271C22×D411yescharacteristic
97C2×D271—21no—
98C2×D271—21no—
99C2×D271#12412yes—
100C4×C271—42no—
101C4×C271AGL(1,ℤ/8)12yes—
102C4×C271#13712yes—
103C4×C271C2×#4112yes—
104Dic271—21no—
105Dic271—21no—
106Dic271—42no—
107Dic271—42no—
108Dic271#12412yes—
109C2×C271—82no—
110C2×C271C2×#12411yescharacteristic
111C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
112C2×C271C2×#13711yescharacteristic
113D271—22no—
114C271#314111yescharacteristic, complemented, Sylow
115#3141—2711nomaximal, Sylow
116#532—2711no—
117C2×C4⋊Q8—2711no—
118#489—2714no—
119#487—2711no—
120#487—2711no—
121#486—2711no—
122#486D27111yescharacteristic, complemented
123#417—2714no—
124C4×M16—2711no—
125C2×#41—2712no—
126C22×Q8—5421no—
127C2×M16—2712no—
128C4⋊Q8—2712no—
129C4⋊Q8—2711no—
130C4⋊Q8—2711no—
131C4⋊Q8—5421no—
132C4⋊Q8—5421no—
133C4⋊Q8C2×D27111yescharacteristic
134#127C2×D27111yescharacteristic
135#124—2712no—
136#121—5421no—
137#121—5421no—
138C4×D4—5421no—
139C4×D4—5421no—
140C2×C4⋊C4—2712no—
141C2×C4⋊C4—5421no—
142C2×C42—2711no—
143#103—2714no—
144#103—2714no—
145#102—2714no—
146#102C2×D27114yes—
147#97—2712no—
148C4×C8—2711no—
149C4×C8C2×D27111yescharacteristic
150#41—5428no—
151C2×Q8—21no—
152C2×Q8—2712no—
153C2×Q8—5422no—
154C2×Q8—5421no—
155C2×Q8—10844no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—5422no—
159C2×D4C22×D27112yes—
160C22×C4—2711no—
161C22×C4—2711no—
162C22×C4—2711no—
163C22×C4—5422no—
164C22×C4—5422no—
165M16—5424no—
166C2×C8—2712no—
167C2×C8C22×D27112yes—
168C4⋊C4—21no—
169C4⋊C4—2712no—
170C4⋊C4—5422no—
171C4⋊C4—5422no—
172C4⋊C4—5421no—
173C4⋊C4—5421no—
174C4⋊C4—5421no—
175C4⋊C4C22×D27112yes—
176C22⋊C4—5422no—
177C22⋊C4—5422no—
178C42—2711no—
179C42—2711no—
180C42—2711no—
181C42C22×D27111yescharacteristic
182C23—22no—
183C23—2711no—
184Q8—42no—
185Q8—5424no—
186Q8—10844no—
187Q8—10842no—
188D4—24no—
189D4—42no—
190D4—42no—
191D4—10844no—
192C2×C4—22no—
193C2×C4—2711no—
194C2×C4—2714no—
195C2×C4—2711no—
196C2×C4—2711no—
197C2×C4—5422no—
198C2×C4—5422no—
199C2×C4—5422no—
200C2×C4—5422no—
201C2×C4—10844no—
202C2×C4—10844no—
203C2×C4D4×D27111yescharacteristic
204C2×C4D4×D27111yescharacteristic
205C2×C4C23×D27111yescharacteristic
206C8—22no—
207C8—5422no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—2712no—
212C22—5421no—
213C22—5421no—
214C22C2×D4×D27111yescharacteristic
215C4—42no—
216C4—2712no—
217C4—5421no—
218C4—5421no—
219C4—10842no—
220C4—10842no—
221C4#2257618412yes—
222C4#2257618912yes—
223C4#41×D27112yes—
224C2—82no—
225C2—5422no—
226C2#124×D27111yescharacteristic
227C2C2×#2257618411yescharacteristic
228C2C2×#2257618911yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#486×C271C2
socleC22×C271C22×D4
prime radical#31774445C2
hypercenter#486D271
hypocenterC271#3141
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D271C2semidirect
3. #31774255C2semidirect
4. #31774249C2semidirect
5. #31774248C2semidirect
6. #31774243C2semidirect
7. #31773895C2semidirect
8. C4×#16864637C2semidirect
9. #486×C271C2semidirect
114. C271#3141semidirect
122. #486D271semidirect

Maximal subgroups

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

subgroupindex
#317744452
C4⋊Q8×D2712
#317742552
#317742492
#317742482
#317742432
#317738952
C4×#168646372
#486×C2712
#3141271