Finite Group Databasebeta

← #46839452

Subgroups — #46839452

Order 34688.

496 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 209 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×#22576158C211yesmaximal, characteristic
2#31773804C211yesmaximal, characteristic
3#31773801C212yesmaximal
4#31773794C212yesmaximal
5#31773792C212yesmaximal
6C4×Dic271⋊C4C212yesmaximal
7#31773748C212yesmaximal
8#31773748C212yesmaximal
9#391×C271C211yesmaximal, characteristic
10C22×Dic542C2211yescharacteristic
11#22576158C418yescomplemented
12C2×C4⋊Dic271C2211yescharacteristic
13C2×Dic271⋊C4C2212yes—
14C2×Dic271⋊C4C2212yes—
15C2×Dic271⋊C4C2212yes—
16C2×Dic271⋊C4C2212yes—
17C2×Dic271⋊C4C2212yes—
18C2×C4×Dic271C2212yes—
19C2×C4×Dic271C2212yes—
20C2×C4×Dic271C2211yescharacteristic
21C2×C4×Dic271C2212yes—
22C2×C4×Dic271C2211yescharacteristic
23C271⋊#95C2212yes—
24C271⋊#95C2212yes—
25C271⋊#95C2212yes—
26C271⋊#95C2212yes—
27C22×Q8×C271C2211yescharacteristic
28C2×C4⋊C4×C271C2211yescharacteristic
29C2×C42×C271C2211yescharacteristic
30#95×C271C2212yes—
31#95×C271C2212yes—
32C2×Dic542—41no—
33C2×Dic542—41no—
34C2×Dic542C2×C414yes—
35C22×Dic271C2312yes—
36C22×Dic271C2312yes—
37C22×Dic271C2312yes—
38C22×Dic271C2312yes—
39C4⋊Dic271—41no—
40Dic271⋊C4—24no—
41Dic271⋊C4—42no—
42Dic271⋊C4C2×C418yes—
43Dic271⋊C4C2×C418yes—
44Dic271⋊C4Q818yescomplemented
45C4×Dic271—24no—
46C4×Dic271—24no—
47C4×Dic271—42no—
48C4×Dic271—41no—
49C4×Dic271C2×C414yes—
50C2×Q8×C271—41no—
51C2×Q8×C271—41no—
52C2×Q8×C271C2×C414yes—
53C22×C4×C271C2312yes—
54C22×C4×C271C2311yescharacteristic
55C22×C4×C271C2311yescharacteristic
56C22×C4×C271C2311yescharacteristic
57C22×C4×C271C2312yes—
58C4⋊C4×C271—41no—
59C42×C271—41no—
60Dic542—44no—
61C2×Dic271—22no—
62C2×Dic271—22no—
63C2×Dic271—22no—
64C2×Dic271—22no—
65C2×Dic271—22no—
66C2×Dic271—22no—
67C2×Dic271—22no—
68C2×Dic271C22×C414yes—
69C2×Dic271C22×C414yes—
70C2×Dic271C2×Q814yes—
71C2×Dic271C2×Q814yes—
72C2×Dic271#4114yes—
73C23×C271C2411yescharacteristic
74Q8×C271—44no—
75C2×C4×C271—22no—
76C2×C4×C271—21no—
77C2×C4×C271—22no—
78C2×C4×C271—21no—
79C2×C4×C271—21no—
80C2×C4×C271—21no—
81C2×C4×C271—21no—
82C2×C4×C271—21no—
83C2×C4×C271—22no—
84C2×C4×C271—22no—
85C2×C4×C271—22no—
86C2×C4×C271C22×C412yes—
87C2×C4×C271C22×C412yes—
88C2×C4×C271C22×C412yes—
89C2×C4×C271C2×Q814yes—
90C22×C271C23×C411yescharacteristic
91C22×C271C22×Q812yes—
92C22×C271C2×#4112yes—
93C22×C271H(5,𝔽2)11yescharacteristic
94C22×C271H(5,𝔽2)11yescharacteristic
95C4×C271—22no—
96C4×C271—22no—
97C4×C271—22no—
98C4×C271—24no—
99C4×C271—42no—
100Dic271—24no—
101Dic271—24no—
102Dic271—42no—
103Dic271C4×Q818yes—
104C2×C271C2×C4×Q812yes—
105C2×C271#51811yescharacteristic
106C2×C271#55412yes—
107C2×C271#55511yescharacteristic
108C2×C271Q8211yescharacteristic
109C271#236911yescharacteristic, complemented, Sylow
110#2369—2711nomaximal, Sylow
111C2×C4⋊Q8—2711no—
112#391—2711no—
113#391—2712no—
114#391D27111yescharacteristic
115#389—2712no—
116#389—2712no—
117C4×C4⋊C4—2712no—
118C4×C4⋊C4—2712no—
119C4×C4⋊C4—2712no—
120C22×Q8—2711no—
121C22×Q8C2×D27111yescharacteristic
122C4⋊Q8—2718no—
123C2×C4⋊C4—2712no—
124C2×C4⋊C4—2712no—
125C2×C4⋊C4—2712no—
126C2×C4⋊C4—2712no—
127C2×C4⋊C4—2712no—
128C2×C4⋊C4—2711no—
129C2×C4⋊C4C2×D27111yescharacteristic
130C2×C42—2711no—
131C2×C42—2712no—
132C2×C42—2712no—
133C2×C42—2711no—
134C2×C42—2712no—
135C2×C42C2×D27111yescharacteristic
136#95—2712no—
137#95—2712no—
138#95—2712no—
139#95—2712no—
140#95C2×D27112yes—
141#95C2×D27112yes—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—2714no—
145C2×Q8—10841no—
146C2×Q8—10841no—
147C2×Q8C4×D27114yes—
148C22×C4—2712no—
149C22×C4—2712no—
150C22×C4—2712no—
151C22×C4—2712no—
152C22×C4C22×D27112yes—
153C22×C4C22×D27111yescharacteristic
154C22×C4C22×D27111yescharacteristic
155C22×C4C22×D27111yescharacteristic
156C22×C4C22×D27112yes—
157C4⋊C4—41no—
158C4⋊C4—2718no—
159C4⋊C4—2718no—
160C4⋊C4—2718no—
161C4⋊C4—5424no—
162C4⋊C4—10842no—
163C4⋊C4—10841no—
164C42—41no—
165C42—2714no—
166C42—5424no—
167C42—5424no—
168C42—10842no—
169C42—10841no—
170C23C23×D27111yescharacteristic
171Q8—44no—
172Q8—10844no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—2714no—
185C2×C4—2714no—
186C2×C4—2714no—
187C2×C4—2714no—
188C2×C4—2714no—
189C2×C4—5422no—
190C2×C4—5422no—
191C2×C4—5422no—
192C2×C4—5422no—
193C2×C4—5422no—
194C2×C4—5422no—
195C2×C4—5422no—
196C2×C4C2×C4×D27112yes—
197C2×C4C2×C4×D27112yes—
198C2×C4C2×C4×D27112yes—
199C2×C4Q8×D27114yes—
200C22C22×C4×D27111yescharacteristic
201C22C2×Q8×D27112yes—
202C22#41×D27112yes—
203C22#2257620511yescharacteristic
204C22#2257620511yescharacteristic
205C4—24no—
206C4—22no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—2718no—
211C4—5424no—
212C4—5424no—
213C4—10842no—
214C2C4×Q8×D27112yes—
215C2#3177437011yescharacteristic
216C2#3177460112yes—
217C2#3177460911yescharacteristic
218C2#3177462711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutatorC22×C271C23×C4
Fitting#391×C271C2
socle, prime radical, cosocleC23×C271C24
hypercenter#391D271
hypocenterC271#2369
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #22576158C4semidirect
44. Dic271⋊C4Q8semidirect
109. C271#2369semidirect

Maximal subgroups

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

subgroupindex
C2×#225761582
#317738042
#317738012
#317737942
#317737922
C4×Dic271⋊C42
#317737482
#391×C2712
#2369271