Finite Group Databasebeta

← #46845873

Subgroups — #46845873

Order 34688.

494 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 141 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×#22576186C211yesmaximal, characteristic
2#31774471C211yesmaximal, characteristic, complemented
3#31774372C211yesmaximal, characteristic, complemented
4#31774114C214yesmaximal, complemented
5#31774100C214yesmaximal, complemented
6#31773923C211yesmaximal, characteristic, complemented
7#31773906C211yesmaximal, characteristic, complemented
8#31773894C211yesmaximal, characteristic, complemented
9C22×SD16×C271C211yesmaximal, characteristic, complemented
10C2×#16864654C2211yescharacteristic
11#22576186—24no—
12C2×#16864642C2212yes—
13C2×#16864639C2212yes—
14C2×#16864638C2211yescharacteristic
15#22576158C2212yescomplemented
16#22576149C2212yescomplemented
17#22576143—22no—
18#22576136—22no—
19D4×Dic271—22no—
20D4×Dic271C2214yes—
21#22576119C2211yescharacteristic
22#22576119C2211yescharacteristic
23C2×C22⋊Dic271—22no—
24Dic271⋊C8C2214yescomplemented
25#22576085C2212yescomplemented
26#22576084C2214yescomplemented
27#22576082C2212yescomplemented
28C22×Q8×C271C2211yescharacteristic
29C22×D4×C271C2211yescharacteristic
30C2×SD16×C271—24no—
31C2×SD16×C271C2214yes—
32C22×C8×C271C2211yescharacteristic
33#16864654—22no—
34#16864654—22no—
35C2×Dic542C2311yescharacteristic
36C2×D1084C2311yescharacteristic
37C2×C22⋊D271—21no—
38C22×Dic271—24no—
39C2×C4×D271—21no—
40#16864642—24no—
41#16864639—24no—
42#16864638—22no—
43C2×C271⋊C8C2312yes—
44C4⋊Dic271D412yes—
45C4⋊Dic271C2312yes—
46Dic271⋊C4—24no—
47C22⋊Dic271—24no—
48C22⋊Dic271—21no—
49C22⋊Dic271D412yes—
50(C2×D271)⋊C4—24no—
51C4×Dic271—21no—
52C4×Dic271C2312yes—
53C24×C271—21no—
54C2×Q8×C271—21no—
55C2×Q8×C271—22no—
56C2×Q8×C271—22no—
57C2×Q8×C271C2312yes—
58C2×D4×C271—21no—
59C2×D4×C271—22no—
60C2×D4×C271D414yes—
61C2×D4×C271C2312yes—
62C22×C4×C271—21no—
63C22×C4×C271C2311yescharacteristic
64SD16×C271—28no—
65C2×C8×C271—22no—
66C2×C8×C271C2312yes—
67C22×D271—21no—
68Dic542—21no—
69Dic542—21no—
70D1084—21no—
71D1084—21no—
72C22⋊D271—42no—
73C2×Dic271—22no—
74C2×Dic271—21no—
75C2×Dic271—84no—
76C2×Dic271C2×D414yes—
77C4×D271—42no—
78C271⋊C8—22no—
79C23×C271—22no—
80C23×C271—24no—
81C23×C271—82no—
82C23×C271C2×D411yescharacteristic
83Q8×C271—22no—
84Q8×C271—22no—
85Q8×C271—42no—
86D4×C271—24no—
87D4×C271—22no—
88D4×C271C2×D414yes—
89C2×C4×C271—22no—
90C2×C4×C271—42no—
91C2×C4×C271C2×D411yescharacteristic
92C2×C4×C271C2×D412yes—
93C2×C4×C271#4112yes—
94C2×C4×C271C2411yescharacteristic
95C8×C271—22no—
96C22×C271—22no—
97C22×C271—24no—
98C22×C271—44no—
99C22×C271—84no—
100C22×C271#12112yes—
101C22×C271C22×D411yescharacteristic
102C2×D271—41no—
103C2×D271—42no—
104C4×C271—42no—
105C4×C271#12112yes—
106C4×C271C22×D411yescharacteristic
107C4×C271C2×#4111yescharacteristic
108Dic271—24no—
109Dic271—41no—
110Dic271—42no—
111C2×C271—22no—
112C2×C271—44no—
113C2×C271C2×#12111yescharacteristic
114C2×C271#57712yes—
115D271—81no—
116C271#304211yescharacteristic, complemented, Sylow
117#3042—2711nomaximal, Sylow
118C2×AGL(1,ℤ/8)—2711no—
119C22×SD16D27111yescharacteristic, complemented
120#536—2711no—
121#518—2711no—
122#463—2714no—
123#461—2714no—
124#424—2711no—
125#419—2711no—
126#416—2711no—
127C2×#41—2711no—
128C22×Q8C2×D27111yescharacteristic
129C22×D4C2×D27111yescharacteristic
130AGL(1,ℤ/8)—5424no—
131C2×SD16—24no—
132C2×SD16—2712no—
133C2×SD16C2×D27114yes—
134C2×D8—2712no—
135C2×M16—2711no—
136C22×C8C2×D27111yescharacteristic
137C4⋊Q8—2712no—
138#124—2712no—
139#123—5422no—
140C22⋊Q8—5422no—
141C4×D4—2714no—
142C4×D4—5422no—
143#117—2711no—
144#117—2711no—
145C2×C22⋊C4—5422no—
146C4⋊C8—2714no—
147#103—2714no—
148#103—2712no—
149#102—2712no—
150C24—21no—
151#41—5422no—
152#41—5422no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—2711no—
157C2×Q8C22×D27112yes—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—2711no—
161C2×D4—5421no—
162C2×D4C22⋊D27114yes—
163C2×D4C22×D27112yes—
164C22×C4—21no—
165C22×C4—5424no—
166C22×C4—5421no—
167C22×C4C22×D27111yescharacteristic
168SD16—28no—
169SD16—5424no—
170D8—5424no—
171M16—5422no—
172C2×C8—22no—
173C2×C8—2712no—
174C2×C8C22×D27112yes—
175C4⋊C4—2712no—
176C4⋊C4—2712no—
177C4⋊C4—5424no—
178C22⋊C4—2712no—
179C22⋊C4—5424no—
180C22⋊C4—5424no—
181C22⋊C4—5421no—
182C42—2712no—
183C42—5421no—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—5421no—
188C23D4×D27111yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—5421no—
193Q8—5421no—
194D4—24no—
195D4—22no—
196D4—5421no—
197D4—5421no—
198D4—10842no—
199D4C2×C22⋊D27114yes—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—2714no—
203C2×C4—5422no—
204C2×C4—5421no—
205C2×C4—10842no—
206C2×C4—21684no—
207C2×C4D4×D27111yescharacteristic
208C2×C4C2×C22⋊D27112yes—
209C2×C4#1686465312yes—
210C2×C4C23×D27111yescharacteristic
211C8—22no—
212C8—5422no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—10841no—
218C22—10842no—
219C22#2257613212yes—
220C22C2×D4×D27111yescharacteristic
221C4—42no—
222C4—5424no—
223C4—10841no—
224C4—10842no—
225C4#2257613212yes—
226C4C22×C22⋊D27111yescharacteristic
227C4C2×#1686465311yescharacteristic
228C2—22no—
229C2—44no—
230C2—21681no—
231C2C2×#2257613211yescharacteristic
232C2#3177470312yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC22×SD16×C271C2
socleC22×C271C22×D4
prime radicalC2×#22576186C2
hypercenterC22×SD16D271
hypocenterC271#3042
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. #31774471C2semidirect
3. #31774372C2semidirect
4. #31774114C2semidirect
5. #31774100C2semidirect
6. #31773923C2semidirect
7. #31773906C2semidirect
8. #31773894C2semidirect
9. C22×SD16×C271C2semidirect
15. #22576158C22semidirect
16. #22576149C22semidirect
24. Dic271⋊C8C22semidirect
25. #22576085C22semidirect
26. #22576084C22semidirect
27. #22576082C22semidirect
116. C271#3042semidirect
119. C22×SD16D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761862
#317744712
#317743722
#317741142
#317741002
#317739232
#317739062
#317738942
C22×SD16×C2712
#3042271