Finite Group Databasebeta

← #46839699

Subgroups — #46839699

Order 34688.

864 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C22×#16864654C211yesmaximal, characteristic
2C2×#22576143C212yesmaximal, complemented
3#31773817C212yesmaximal, complemented
4#31773814C212yesmaximal, complemented
5#31773776C212yesmaximal, complemented
6#31773768C214yesmaximal, complemented
7#31773760C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C271C211yesmaximal, characteristic, complemented
9C2×#16864654—28no—
10C2×#16864654—24no—
11C22×Dic542C2211yescharacteristic
12C22×D1084C2211yescharacteristic
13C22×C22⋊D271C2212yes—
14C22×C4×D271C2212yes—
15#22576143—28no—
16C2×Dic271⋊C4C2214yes—
17C2×C22⋊Dic271C2212yes—
18C2×(C2×D271)⋊C4C2214yes—
19C2×(C2×D271)⋊C4C2214yes—
20C271⋊#95C2212yes—
21C271⋊#95C2212yes—
22C271⋊#95C2214yes—
23C23×C4×C271C2211yescharacteristic
24C23×C4×C271C2211yescharacteristic
25C23×C4×C271C2211yescharacteristic
26C2×C4⋊C4×C271—24no—
27C2×C4⋊C4×C271C2212yes—
28C2×C4⋊C4×C271C2212yes—
29C23×D271C2312yes—
30#16864654—232no—
31C2×Dic542—24no—
32C2×Dic542—22no—
33C2×D1084—24no—
34C2×D1084—22no—
35C2×C22⋊D271—24no—
36C2×C22⋊D271—24no—
37C2×C22⋊D271D418yes—
38C22×Dic271C2314yes—
39C22×Dic271C2312yes—
40C2×C4×D271—24no—
41C2×C4×D271—24no—
42C2×C4×D271—24no—
43Dic271⋊C4—28no—
44C22⋊Dic271—24no—
45(C2×D271)⋊C4—28no—
46(C2×D271)⋊C4—28no—
47C24×C271C2311yescharacteristic
48C22×C4×C271—24no—
49C22×C4×C271—22no—
50C22×C4×C271—24no—
51C22×C4×C271—24no—
52C22×C4×C271—22no—
53C22×C4×C271D414yes—
54C22×C4×C271C2312yes—
55C22×C4×C271C2311yescharacteristic
56C22×C4×C271C2311yescharacteristic
57C22×C4×C271C2312yes—
58C4⋊C4×C271—24no—
59C4⋊C4×C271—24no—
60C22×D271—22no—
61C22×D271—22no—
62C22×D271—24no—
63C22×D271—24no—
64C22×D271C2×D414yes—
65Dic542—28no—
66D1084—28no—
67C22⋊D271—216no—
68C2×Dic271—22no—
69C2×Dic271—22no—
70C2×Dic271—24no—
71C2×Dic271—24no—
72C2×Dic271—24no—
73C2×Dic271C2×D414yes—
74C4×D271—216no—
75C23×C271—22no—
76C23×C271—22no—
77C23×C271C2×D414yes—
78C23×C271C2×D412yes—
79C23×C271C2411yescharacteristic
80C2×C4×C271—28no—
81C2×C4×C271—28no—
82C2×C4×C271—22no—
83C2×C4×C271—22no—
84C2×C4×C271—22no—
85C2×C4×C271—22no—
86C2×C4×C271—22no—
87C2×C4×C271—24no—
88C2×C4×C271—44no—
89C2×C4×C271C2×D412yes—
90C2×C4×C271C2×D412yes—
91C2×C4×C271#4114yescomplemented
92C22×C271—22no—
93C22×C271—22no—
94C22×C271—28no—
95C22×C271#12014yes—
96C22×C271C22×D412yes—
97C22×C271C22×D411yescharacteristic
98C22×C271C2×#4112yes—
99C22×C2712−1+412yes—
100C2×D271—24no—
101C2×D271—28no—
102C2×D271—42no—
103C2×D271—42no—
104C2×D271—44no—
105C4×C271—22no—
106C4×C271—22no—
107C4×C271—24no—
108C4×C271—42no—
109Dic271—24no—
110Dic271—44no—
111C2×C271—24no—
112C2×C271C2×#12011yescharacteristic
113C2×C271#52511yescharacteristic
114C2×C271#53611yescharacteristic
115C2×C271#54714yes—
116D271—44no—
117C271#240611yescharacteristic, complemented, Sylow
118#2406—2711nomaximal, Sylow
119C22×#41—2711no—
120C2×#123—2712no—
121C22×C4⋊C4D27111yescharacteristic, complemented
122#395—2712no—
123#394—2712no—
124#386—2712no—
125#385—2714no—
126#381—2711no—
127C2×#41—5428no—
128C2×#41—5424no—
129C22×Q8—2711no—
130C22×D4—2712no—
131C22×D4—2711no—
132C23×C4—2712no—
133C23×C4C2×D27111yescharacteristic
134C23×C4C2×D27111yescharacteristic
135C23×C4C2×D27111yescharacteristic
136#123—5428no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—2714no—
139C2×C4⋊C4C2×D27112yes—
140C2×C4⋊C4C2×D27112yes—
141C2×C22⋊C4—2712no—
142C2×C22⋊C4—2714no—
143C2×C22⋊C4—2714no—
144#95—2712no—
145#95—2712no—
146#95—2714no—
147C24—2712no—
148C24C22×D27111yescharacteristic
149#41—54232no—
150C2×Q8—5424no—
151C2×Q8—5422no—
152C2×D4—2718no—
153C2×D4—5424no—
154C2×D4—5424no—
155C2×D4—5424no—
156C2×D4—5422no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—2712no—
163C22×C4—2714no—
164C22×C4—5424no—
165C22×C4—5424no—
166C22×C4—5424no—
167C22×C4C22⋊D27114yes—
168C22×C4C22×D27112yes—
169C22×C4C22×D27111yescharacteristic
170C22×C4C22×D27111yescharacteristic
171C22×C4C22×D27112yes—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—5428no—
175C22⋊C4—5428no—
176C22⋊C4—5424no—
177C22⋊C4—5428no—
178C23—22no—
179C23—22no—
180C23—2714no—
181C23—5422no—
182C23—5422no—
183C23—5424no—
184C23—5424no—
185C23D4×D27114yes—
186C23C2×C22⋊D27112yes—
187C23C23×D27111yescharacteristic
188Q8—5428no—
189D4—5428no—
190D4—54216no—
191C2×C4—28no—
192C2×C4—28no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—44no—
200C2×C4—2714no—
201C2×C4—54216no—
202C2×C4—5422no—
203C2×C4—5422no—
204C2×C4—5424no—
205C2×C4—5424no—
206C2×C4—5424no—
207C2×C4C2×C22⋊D27112yes—
208C2×C4C2×C22⋊D27112yes—
209C2×C4#1686465414yescomplemented
210C22—22no—
211C22—28no—
212C22—22no—
213C22—5428no—
214C22—5424no—
215C22—10842no—
216C22—10844no—
217C22—10842no—
218C22#2257612814yes—
219C22C2×D4×D27112yes—
220C22C22×C22⋊D27111yescharacteristic
221C22C2×#1686465412yes—
222C22#2257620812yes—
223C4—22no—
224C4—22no—
225C4—24no—
226C4—42no—
227C4—5424no—
228C4—10844no—
229C2—24no—
230C2—10844no—
231C2C2×#2257612811yescharacteristic
232C2#3177440811yescharacteristic
233C2#3177447111yescharacteristic
234C2#3177454814yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC22×C4⋊C4×C271C2
prime radicalC22×#16864654C2
hypercenterC22×C4⋊C4D271
hypocenterC271#2406
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#22576143C2semidirect
3. #31773817C2semidirect
4. #31773814C2semidirect
5. #31773776C2semidirect
6. #31773768C2semidirect
7. #31773760C2semidirect
8. C22×C4⋊C4×C271C2semidirect
91. C2×C4×C271#41semidirect
117. C271#2406semidirect
121. C22×C4⋊C4D271semidirect
209. C2×C4#16864654semidirect

Maximal subgroups

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

subgroupindex
C22×#168646542
C2×#225761432
#317738172
#317738142
#317737762
#317737682
#317737602
C22×C4⋊C4×C2712
#2406271