Finite Group Databasebeta

← #46846894

Subgroups — #46846894

Order 34688.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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#31774528C211yesmaximal, characteristic
2#31774525C211yesmaximal, characteristic
3#31774305C212yesmaximal, complemented
4#31774304C212yesmaximal, complemented
5#31774298C212yesmaximal, complemented
6#31774294C212yesmaximal, complemented
7#31773939C211yesmaximal, characteristic
8#31773933C211yesmaximal, characteristic
9C2×#22576091C211yesmaximal, characteristic
10#31773855C211yesmaximal, characteristic
11#428×C271C211yesmaximal, characteristic
12C22×C271⋊C8C2211yescharacteristic
13#22576160—21no—
14#22576159—21no—
15#22576159C2212yes—
16#22576159C2212yes—
17#22576152—21no—
18#22576151—21no—
19#22576151C2212yes—
20#22576151C2212yes—
21#22576139—22no—
22#22576137—22no—
23#22576119C2211yescharacteristic
24#22576119C2211yescharacteristic
25C2×C4⋊Dic271C2211yescharacteristic
26#22576094C2212yes—
27#22576093C2211yescharacteristic
28#22576093C2211yescharacteristic
29#22576092C2212yes—
30#22576091C2211yescharacteristic
31#22576091C2212yes—
32#22576091C2211yescharacteristic
33#22576091C2212yes—
34#22576070C2212yes—
35C8×Dic271C2212yes—
36C2×M16×C271C2211yescharacteristic
37#117×C271C2211yescharacteristic
38C2×C4⋊C4×C271C2211yescharacteristic
39#107×C271C2212yes—
40#106×C271C2212yes—
41C2×Dic542—22no—
42C2×Dic542—22no—
43C22×Dic271—21no—
44C2×C271⋊C8Q814yescomplemented
45C2×C271⋊C8C2311yescharacteristic
46C2×C271⋊C8C2311yescharacteristic
47C4⋊Dic271—22no—
48C4⋊Dic271—22no—
49C4⋊Dic271—21no—
50C4⋊Dic271D411yescharacteristic
51C4⋊Dic271D411yescharacteristic
52C4⋊Dic271C2311yescharacteristic
53C4⋊Dic271C2312yes—
54C4⋊Dic271C2311yescharacteristic
55Dic271⋊C4—22no—
56Dic271⋊C4—22no—
57Dic271⋊C4—22no—
58Dic271⋊C4—22no—
59C22⋊Dic271—21no—
60C22⋊Dic271D412yes—
61C4×Dic271—21no—
62C4×Dic271C2312yes—
63C22×C4×C271—21no—
64C22×C4×C271C2311yescharacteristic
65M16×C271Q814yescomplemented
66C2×C8×C271C2312yes—
67C4⋊C4×C271—21no—
68C4⋊C4×C271C2311yescharacteristic
69C4⋊C4×C271C2312yes—
70C4⋊C4×C271C2311yescharacteristic
71C22⋊C4×C271—21no—
72C42×C271—21no—
73Dic542—82no—
74Dic542—82no—
75C2×Dic271—21no—
76C2×Dic271—21no—
77C2×Dic271—22no—
78C2×Dic271—81no—
79C2×Dic271C2×D412yes—
80C2×Dic271C2×D412yes—
81C271⋊C8C2×Q812yes—
82C271⋊C8C2×Q812yes—
83C23×C271C2×D411yescharacteristic
84C2×C4×C271—21no—
85C2×C4×C271—21no—
86C2×C4×C271—22no—
87C2×C4×C271—81no—
88C2×C4×C271C2×D411yescharacteristic
89C2×C4×C271C2×Q812yes—
90C2×C4×C271C2×Q812yes—
91C2×C4×C271C2411yescharacteristic
92C8×C271C2×Q814yes—
93C22×C271—21no—
94C22×C271—21no—
95C22×C271C4⋊Q812yes—
96C22×C271C22×D411yescharacteristic
97C4×C271—41no—
98C4×C271—42no—
99C4×C271—41no—
100C4×C271C4⋊Q812yes—
101C4×C271C22×Q811yescharacteristic
102C4×C271C22×Q811yescharacteristic
103Dic271—22no—
104Dic271—22no—
105Dic271—41no—
106Dic271—42no—
107Dic271—41no—
108C2×C271—22no—
109C2×C271C2×C4⋊Q811yescharacteristic
110C2×C271#57711yescharacteristic
111C2×C271#57711yescharacteristic
112C271#316311yescharacteristic, complemented, Sylow
113#3163—2711nomaximal, Sylow
114#544—2711no—
115#544—2711no—
116#501—2712no—
117#501—2712no—
118#499—2712no—
119#498—2712no—
120#428—2711no—
121#428D27111yescharacteristic
122#427—2711no—
123C2×#106—2711no—
124#405—2711no—
125C2×M16C2×D27111yescharacteristic
126C22×C8—2711no—
127C4⋊Q8—2712no—
128C4⋊Q8—2712no—
129C4⋊Q8—5421no—
130C4⋊Q8—5421no—
131#125—2712no—
132#125—2712no—
133#125—5421no—
134#125—5421no—
135C22⋊Q8—5422no—
136C22⋊Q8—5422no—
137#117—2711no—
138#117—2711no—
139#117C2×D27111yescharacteristic
140C2×C4⋊C4—2711no—
141C2×C4⋊C4C2×D27111yescharacteristic
142#107—2712no—
143#107—2711no—
144#107—2711no—
145#107C2×D27112yes—
146#106—2712no—
147#106—2712no—
148#106—2712no—
149#106—2711no—
150#106—2711no—
151#106C2×D27112yes—
152#97—2712no—
153C4×C8—2712no—
154C2×Q8—5422no—
155C2×Q8—5422no—
156C22×C4—21no—
157C22×C4—5421no—
158C22×C4C22×D27111yescharacteristic
159M16Dic54214yescomplemented
160C2×C8—2714no—
161C2×C8—2711no—
162C2×C8—2711no—
163C2×C8C22×D27112yes—
164C4⋊C4—21no—
165C4⋊C4—2711no—
166C4⋊C4—2711no—
167C4⋊C4—2712no—
168C4⋊C4—2711no—
169C4⋊C4—2711no—
170C4⋊C4—5422no—
171C4⋊C4—5422no—
172C4⋊C4—5422no—
173C4⋊C4—5422no—
174C4⋊C4—5422no—
175C4⋊C4—5422no—
176C4⋊C4—5421no—
177C4⋊C4C22×D27112yes—
178C4⋊C4C22×D27111yescharacteristic
179C4⋊C4C22×D27111yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—2712no—
182C22⋊C4—5421no—
183C42—21no—
184C42—2712no—
185C42—5421no—
186C23D4×D27111yescharacteristic
187Q8—21682no—
188Q8—21682no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—81no—
193C2×C4—2712no—
194C2×C4—2712no—
195C2×C4—5421no—
196C2×C4—5421no—
197C2×C4—5422no—
198C2×C4—21681no—
199C2×C4D4×D27111yescharacteristic
200C2×C4Q8×D27112yes—
201C2×C4C2×Dic54212yes—
202C2×C4C23×D27111yescharacteristic
203C8—2712no—
204C8—2712no—
205C8C2×Dic54214yes—
206C22—21no—
207C22—21no—
208C22#2257615912yes—
209C22C2×D4×D27111yescharacteristic
210C4—41no—
211C4—42no—
212C4—41no—
213C4—5422no—
214C4—5422no—
215C4—10842no—
216C4—10841no—
217C4—10841no—
218C4#2257615912yes—
219C4C2×Q8×D27111yescharacteristic
220C4C22×Dic54211yescharacteristic
221C2—22no—
222C2C2×#2257615911yescharacteristic
223C2#3177470611yescharacteristic
224C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#428×C271C2
socleC22×C271C22×D4
prime radicalC23×C271C2×D4
hypercenter#428D271
hypocenterC271#3163
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #31774305C2semidirect
4. #31774304C2semidirect
5. #31774298C2semidirect
6. #31774294C2semidirect
44. C2×C271⋊C8Q8semidirect
65. M16×C271Q8semidirect
112. C271#3163semidirect
159. M16Dic542semidirect

Maximal subgroups

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

subgroupindex
#317745282
#317745252
#317743052
#317743042
#317742982
#317742942
#317739392
#317739332
C2×#225760912
#317738552
#428×C2712
#3163271