Finite Group Databasebeta

← #46848019

Subgroups — #46848019

Order 34688.

476 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#22576189C212yesmaximal, complemented
2#31774555C212yesmaximal, complemented
3#31774140C212yesmaximal, complemented
4#31774116C212yesmaximal, complemented
5#31774051C212yesmaximal, complemented
6#31774039C212yesmaximal, complemented
7C4×Dic1084C211yesmaximal, characteristic, complemented
8#31773976C211yesmaximal, characteristic, complemented
9#495×C271C211yesmaximal, characteristic, complemented
10C2×#16864653C2212yes—
11C2×Q8×D271C2212yes—
12#22576189—28no—
13C2×Dic1084C2211yescharacteristic
14C2×#16864644C2212yes—
15C2×#16864642C2212yes—
16C2×#16864637C2212yes—
17#22576146—22no—
18#22576137—22no—
19#22576136C2212yes—
20#22576133—22no—
21#22576131C2212yes—
22C4×Dic542C2212yes—
23C4×D1084C2211yescharacteristic
24(C22⋊D271)⋊C4—22no—
25#22576094C2211yescharacteristic
26C4⋊(C271⋊C8)C2211yescharacteristic
27#22576085C2212yes—
28#22576084C2212yes—
29#22576081C2212yes—
30(C2×D271)⋊C8C2212yes—
31C2×Q16×C271C2211yescharacteristic
32C2×SD16×C271C2212yes—
33C2×D8×C271C2211yescharacteristic
34#124×C271C2212yes—
35C4×C8×C271C2211yescharacteristic
36#16864653—24no—
37#16864653—42no—
38C2×Dic542—22no—
39C2×Dic542C2312yes—
40Q8×D271—24no—
41Q8×D271—42no—
42C2×D1084D411yescharacteristic
43C2×C22⋊D271—22no—
44C2×C22⋊D271—22no—
45C22×Dic271—22no—
46C2×C4×D271—22no—
47C2×C4×D271C2312yes—
48Dic1084D414yescomplemented
49#16864644—24no—
50#16864642—24no—
51#16864637—24no—
52C2×C271⋊C8C2312yes—
53C4⋊Dic271D411yescharacteristic
54C4⋊Dic271C2312yes—
55Dic271⋊C4—22no—
56Dic271⋊C4—22no—
57C22⋊Dic271—22no—
58(C2×D271)⋊C4—22no—
59(C2×D271)⋊C4D412yes—
60C4×Dic271—22no—
61C2×Q8×C271C2312yes—
62C2×D4×C271C2312yes—
63Q16×C271—41no—
64SD16×C271—24no—
65D8×C271—41no—
66C2×C8×C271C2311yescharacteristic
67C2×C8×C271C2311yescharacteristic
68C22⋊C4×C271—22no—
69C22⋊C4×C271—22no—
70C42×C271C2311yescharacteristic
71C22×D271C2×D412yes—
72Dic542—22no—
73Dic542—44no—
74Dic542C2×D414yes—
75D1084—41no—
76C22⋊D271—44no—
77C22⋊D271—44no—
78C2×Dic271—22no—
79C2×Dic271—22no—
80C2×Dic271—44no—
81C2×Dic271C2×D412yes—
82C4×D271—22no—
83C4×D271—22no—
84C4×D271—44no—
85C271⋊C8—22no—
86C23×C271—22no—
87Q8×C271—22no—
88Q8×C271—22no—
89D4×C271—22no—
90D4×C271—22no—
91C2×C4×C271—22no—
92C2×C4×C271C2×D411yescharacteristic
93C2×C4×C271C2×D411yescharacteristic
94C2×C4×C271C2411yescharacteristic
95C8×C271—21no—
96C8×C271C2×D412yes—
97C22×C271—42no—
98C22×C271—42no—
99C22×C271—42no—
100C22×C271C22×D411yescharacteristic
101C2×D271—22no—
102C2×D271—22no—
103C2×D271—22no—
104C4×C271—21no—
105C4×C271—21no—
106C4×C271—42no—
107C4×C271C22×D411yescharacteristic
108C4×C271H(5,𝔽2)11yescharacteristic
109Dic271—22no—
110Dic271—24no—
111Dic271—42no—
112C2×C271—82no—
113C2×C271D4211yescharacteristic
114C2×C271#57511yescharacteristic
115C2×C271#57811yescharacteristic
116D271—42no—
117C271#327711yescharacteristic, complemented, Sylow
118#3277—2711nomaximal, Sylow
119C2×#137—2712no—
120#548—2712no—
121#495D27111yescharacteristic, complemented
122#468—2712no—
123#464—2712no—
124#451—2712no—
125#449—2712no—
126C4×Q16—2711no—
127#436—2711no—
128C2×#41—2712no—
129C22×Q8—2712no—
130#137—5428no—
131C2×Q16—2712no—
132C2×Q16—2711no—
133C2×Q16C2×D27111yescharacteristic
134C2×SD16—2712no—
135C2×SD16C2×D27112yes—
136C2×D8C2×D27111yescharacteristic
137C2×M16—2712no—
138#124—5422no—
139#124C2×D27112yes—
140C22⋊Q8—2712no—
141C22⋊Q8—5422no—
142#121—2712no—
143#121—5422no—
144C4×Q8—2712no—
145C4×D4—2711no—
146C4×D4—5422no—
147#107—2711no—
148C4⋊C8—2711no—
149#103—2712no—
150#103—2712no—
151#102—2712no—
152C22⋊C8—2712no—
153C4×C8C2×D27111yescharacteristic
154#41—5424no—
155#41—10842no—
156C2×Q8—2712no—
157C2×Q8—5422no—
158C2×Q8—5424no—
159C2×Q8—10842no—
160C2×Q8C22×D27112yes—
161C2×D4—2711no—
162C2×D4—5422no—
163C2×D4—5422no—
164C2×D4C22×D27112yes—
165C22×C4—2712no—
166C22×C4—5422no—
167C22×C4—5422no—
168Q16—41no—
169Q16—2714no—
170Q16—5424no—
171SD16—24no—
172SD16—5424no—
173D8—41no—
174M16—5424no—
175C2×C8—2712no—
176C2×C8C22×D27111yescharacteristic
177C2×C8C22×D27111yescharacteristic
178C4⋊C4—2711no—
179C4⋊C4—2712no—
180C4⋊C4—5422no—
181C4⋊C4—5422no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—2712no—
185C22⋊C4—5422no—
186C22⋊C4—5422no—
187C42—5422no—
188C42C22×D27111yescharacteristic
189C23—22no—
190C23—2712no—
191Q8—22no—
192Q8—22no—
193Q8—2714no—
194Q8—5422no—
195Q8—10844no—
196D4—22no—
197D4—22no—
198D4—10841no—
199D4—10844no—
200D4—10844no—
201C2×C4—22no—
202C2×C4—2712no—
203C2×C4—5422no—
204C2×C4—5422no—
205C2×C4—5422no—
206C2×C4—5422no—
207C2×C4—10844no—
208C2×C4—10844no—
209C2×C4D4×D27111yescharacteristic
210C2×C4D4×D27111yescharacteristic
211C2×C4C23×D27111yescharacteristic
212C8—21no—
213C8—5422no—
214C8D4×D27112yes—
215C22—42no—
216C22—42no—
217C22—42no—
218C22—5422no—
219C22—5422no—
220C22—5422no—
221C22C2×D4×D27111yescharacteristic
222C4—21no—
223C4—21no—
224C4—42no—
225C4—5422no—
226C4—5424no—
227C4—10842no—
228C4C2×D4×D27111yescharacteristic
229C4#2257620511yescharacteristic
230C2—82no—
231C2—10842no—
232C2#3177453211yescharacteristic
233C2#3177469411yescharacteristic
234C2#3177471111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#495×C271C2
socleC22×C271C22×D4
hypercenter#495D271
hypocenterC271#3277
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#22576189C2semidirect
2. #31774555C2semidirect
3. #31774140C2semidirect
4. #31774116C2semidirect
5. #31774051C2semidirect
6. #31774039C2semidirect
7. C4×Dic1084C2semidirect
8. #31773976C2semidirect
9. #495×C271C2semidirect
48. Dic1084D4semidirect
117. C271#3277semidirect
121. #495D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761892
#317745552
#317741402
#317741162
#317740512
#317740392
C4×Dic10842
#317739762
#495×C2712
#3277271