Finite Group Databasebeta

← #44400818

Subgroups — #44400818

Order 29056.

406 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 93 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×#16813873C211yesmaximal, characteristic
2C22×#16813871C211yesmaximal, characteristic
3C2×#20300286C211yesmaximal, characteristic, complemented
4C2×#20300211C211yesmaximal, characteristic, complemented
5#28408933C211yesmaximal, characteristic
6#28408786C211yesmaximal, characteristic, complemented
7C2×#103×C227C211yesmaximal, characteristic, complemented
8C22×Dic454C2211yescharacteristic
9C22×D908C2211yescharacteristic
10C2×#16813873—22no—
11C2×#16813873—22no—
12C2×#16813873—22no—
13C2×#16813871—22no—
14C2×#16813871—22no—
15C2×#16813871—22no—
16C22×C227⋊C8C2211yescharacteristic
17#20300286—22no—
18#20300286—22no—
19C2×Dic227⋊C4—22no—
20C2×(C2×D227)⋊C4—22no—
21C2×C4×Dic227C2211yescharacteristic
22#20300211—22no—
23C22×Q8×C227C2211yescharacteristic
24C22×C8×C227C2211yescharacteristic
25C2×C4⋊C4×C227C2211yescharacteristic
26#103×C227—22no—
27C23×D227—21no—
28C2×Dic454—21no—
29C2×Dic454—21no—
30C2×Dic454—21no—
31C2×Dic454—21no—
32C2×Dic454—21no—
33C2×Dic454D412yes—
34C2×D908—21no—
35C2×D908—21no—
36C2×D908—21no—
37C2×D908—21no—
38C2×D908—21no—
39C2×D908D412yes—
40C22×Dic227—21no—
41C22×Dic227D412yes—
42#16813873—28no—
43#16813871—28no—
44C2×C227⋊C8—21no—
45C2×C227⋊C8—21no—
46C2×C227⋊C8D412yes—
47Dic227⋊C4—28no—
48(C2×D227)⋊C4—44no—
49C4×Dic227—21no—
50C4×Dic227—21no—
51C2×Q8×C227—21no—
52C2×Q8×C227—21no—
53C2×Q8×C227—21no—
54C2×Q8×C227—21no—
55C2×Q8×C227—21no—
56C2×Q8×C227D412yes—
57C22×C4×C227—21no—
58C22×C4×C227—21no—
59C22×C4×C227C2311yescharacteristic
60C2×C8×C227—21no—
61C2×C8×C227—21no—
62C2×C8×C227D412yes—
63C4⋊C4×C227—21no—
64C4⋊C4×C227—21no—
65C22×D227—22no—
66C22×D227—41no—
67C22×D227—41no—
68C22×D227—42no—
69C22×D227—42no—
70Dic454—22no—
71Dic454—22no—
72Dic454—41no—
73Dic454—41no—
74D908—22no—
75D908—22no—
76D908—41no—
77D908—41no—
78C2×Dic227—22no—
79C2×Dic227—22no—
80C2×Dic227—22no—
81C2×Dic227—41no—
82C2×Dic227—41no—
83C2×Dic227SD1614yescomplemented
84C227⋊C8—22no—
85C23×C227C2×D411yescharacteristic
86Q8×C227—22no—
87Q8×C227—22no—
88Q8×C227—41no—
89Q8×C227—41no—
90C2×C4×C227—22no—
91C2×C4×C227—41no—
92C2×C4×C227—41no—
93C2×C4×C227—41no—
94C2×C4×C227—42no—
95C2×C4×C227C2×D411yescharacteristic
96C2×C4×C227C2×D411yescharacteristic
97C2×C4×C227C2×D411yescharacteristic
98C2×C4×C227C2×D411yescharacteristic
99C2×C4×C227C2×D411yescharacteristic
100C2×C4×C227#4111yescharacteristic
101C8×C227—22no—
102C22×C227#12011yescharacteristic
103C22×C227#12111yescharacteristic
104C22×C227#12711yescharacteristic
105C22×C227C2×SD1612yes—
106C22×C227#13712yes—
107C2×D227—42no—
108C2×D227—44no—
109C2×D227—82no—
110C2×D227—81no—
111C2×D227—81no—
112C4×C227—42no—
113C4×C227—81no—
114C4×C227#12011yescharacteristic
115C4×C227#12011yescharacteristic
116C4×C227#12111yescharacteristic
117C4×C227#12111yescharacteristic
118Dic227—24no—
119Dic227—42no—
120C2×C227#39211yescharacteristic
121C2×C227#44812yes—
122C2×C227#46112yes—
123C2×C227#49211yescharacteristic
124C2×C227#49711yescharacteristic
125D227—82no—
126C227#201311yescharacteristic, complemented, Sylow
127#2013—2271nomaximal, Sylow
128C22×SD16—2271no—
129C22×SD16—2271no—
130C2×C4⋊Q8—2271no—
131C2×#103—2271no—
132C2×#103D22711yescharacteristic, complemented
133#390—2271no—
134#336—2271no—
135C22×Q8—2271no—
136C22×Q8C2×D22711yescharacteristic
137C22×D4—2271no—
138C2×SD16—4542no—
139C2×SD16—4542no—
140C2×SD16—4542no—
141C2×SD16—4542no—
142C2×SD16—4542no—
143C2×SD16—4542no—
144C22×C8—2271no—
145C22×C8C2×D22711yescharacteristic
146C4⋊Q8—4542no—
147C4⋊Q8—4542no—
148C2×C4⋊C4—4542no—
149C2×C4⋊C4C2×D22711yescharacteristic
150C2×C22⋊C4—4542no—
151C2×C42—2271no—
152#103—22no—
153#103—4542no—
154C24—4541no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—4541no—
161C2×Q8—4541no—
162C2×Q8—4541no—
163C2×Q8—4541no—
164C2×Q8—4541no—
165C2×Q8—2272no—
166C2×Q8D90812yes—
167C2×D4—4541no—
168C2×D4—4541no—
169C2×D4—4541no—
170C2×D4—4541no—
171C2×D4—4541no—
172C2×D4—2272no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—4541no—
176C22×C4—2272no—
177C22×C4C22×D22711yescharacteristic
178SD16—4548no—
179SD16—4548no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—4541no—
183C2×C8—4541no—
184C2×C8—2272no—
185C2×C8C22⋊D22712yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—4548no—
189C22⋊C4—9084no—
190C42—4541no—
191C42—4541no—
192C23—9082no—
193C23—9082no—
194C23—9081no—
195C23—9081no—
196C23—4542no—
197C23D4×D22711yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—41no—
201Q8—41no—
202Q8—9081no—
203Q8—9081no—
204Q8—4542no—
205Q8—4542no—
206D4—9081no—
207D4—9081no—
208D4—4542no—
209D4—4542no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—9081no—
216C2×C4—9081no—
217C2×C4—4542no—
218C2×C4—4542no—
219C2×C4—4542no—
220C2×C4—2274no—
221C2×C4D4×D22711yescharacteristic
222C2×C4D4×D22711yescharacteristic
223C2×C4D4×D22711yescharacteristic
224C2×C4C2×C22⋊D22711yescharacteristic
225C2×C4C2×D90811yescharacteristic
226C2×C4#1681388411yescharacteristic
227C8—22no—
228C8—4542no—
229C22—18161no—
230C22—18161no—
231C22—18162no—
232C22—9082no—
233C22—9084no—
234C22#2030025811yescharacteristic
235C22#2030026111yescharacteristic
236C22#2030028411yescharacteristic
237C22SD16×D22712yes—
238C22#2030032012yes—
239C4—42no—
240C4—81no—
241C4—9082no—
242C4—4544no—
243C4#2030025611yescharacteristic
244C4#2030025811yescharacteristic
245C4#2030026111yescharacteristic
246C4#2030026211yescharacteristic
247C2—18162no—
248C2#2840894111yescharacteristic
249C2#2840917212yes—
250C2#2840923612yes—
251C2#2840940311yescharacteristic
252C2#2840942511yescharacteristic
subgroupG/H
centerC23D4×D227
commutator, cosocleC22×C4×C227C23
FrattiniC22×C4C22×D227
FittingC2×#103×C227C2
socleC23×C227C2×D4
prime radicalC22×D908C22
hypercenterC2×#103D227
hypocenterC227#2013
LCSG ▷ C22×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. C2×#20300286C2semidirect
4. C2×#20300211C2semidirect
6. #28408786C2semidirect
7. C2×#103×C227C2semidirect
83. C2×Dic227SD16semidirect
126. C227#2013semidirect
132. C2×#103D227semidirect

Maximal subgroups

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

subgroupindex
C22×#168138732
C22×#168138712
C2×#203002862
C2×#203002112
#284089332
#284087862
C2×#103×C2272
#2013227