Finite Group Databasebeta

← #41014610

Subgroups — #41014610

Order 27008.

326 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 69 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#27213715C211yesmaximal, characteristic
2C2×#20269903C211yesmaximal, characteristic, complemented
3#27213145C211yesmaximal, characteristic, complemented
4C2×#20269893C211yesmaximal, characteristic, complemented
5#27213047C211yesmaximal, characteristic, complemented
6#27212905C211yesmaximal, characteristic, complemented
7#428×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C22×D844C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11#20269974—21no—
12#20269965—21no—
13#20269952—21no—
14#20269952—21no—
15#20269947—22no—
16C2×(C2×D211)⋊C4—22no—
17C2×C4×Dic211C2211yescharacteristic
18#20269903—22no—
19#20269900—21no—
20#20269897—21no—
21#20269893—22no—
22C2×M16×C211C2211yescharacteristic
23#117×C211C2211yescharacteristic
24C2×C4⋊C4×C211C2211yescharacteristic
25#107×C211—21no—
26#106×C211—21no—
27C23×D211D412yes—
28#16809962—21no—
29#16809962—21no—
30#16809962—21no—
31#16809962—21no—
32C2×Dic422D411yescharacteristic
33C2×D844—22no—
34C2×D844—22no—
35C2×D844—21no—
36C2×D844—21no—
37C2×D844—21no—
38C2×D844—21no—
39C2×D844—21no—
40C2×D844—21no—
41C2×D844D411yescharacteristic
42C2×C22⋊D211—21no—
43C22×Dic211—21no—
44C2×C4×D211—21no—
45#16809946—21no—
46#16809946—21no—
47C2×C211⋊C8—21no—
48(C2×D211)⋊C4—22no—
49(C2×D211)⋊C4—22no—
50(C2×D211)⋊C4—44no—
51C4×Dic211—21no—
52C4×Dic211—21no—
53C22×C4×C211—21no—
54C22×C4×C211C2311yescharacteristic
55M16×C211—21no—
56M16×C211D412yescomplemented
57C2×C8×C211—21no—
58C4⋊C4×C211—21no—
59C4⋊C4×C211—21no—
60C4⋊C4×C211D411yescharacteristic
61C4⋊C4×C211D411yescharacteristic
62C22⋊C4×C211—21no—
63C42×C211—21no—
64C22×D211—22no—
65C22×D211—21no—
66C22×D211—22no—
67C22×D211—22no—
68C22×D211—42no—
69C22×D211—42no—
70Dic422—21no—
71Dic422—21no—
72D844—21no—
73D844—21no—
74D844—41no—
75D844—41no—
76D844—41no—
77D844—41no—
78D844—82no—
79D844—82no—
80C22⋊D211—41no—
81C22⋊D211—41no—
82C2×Dic211—22no—
83C2×Dic211—21no—
84C2×Dic211—41no—
85C2×Dic211—41no—
86C4×D211—41no—
87C4×D211—41no—
88C211⋊C8—41no—
89C23×C211C2×D411yescharacteristic
90C2×C4×C211—21no—
91C2×C4×C211—21no—
92C2×C4×C211—42no—
93C2×C4×C211—41no—
94C2×C4×C211C2×D411yescharacteristic
95C2×C4×C211C2×D411yescharacteristic
96C2×C4×C211C2×D411yescharacteristic
97C2×C4×C211#4111yescharacteristic
98C2×C4×C211#4111yescharacteristic
99C2×C4×C211#4111yescharacteristic
100C8×C211—22no—
101C22×C211—21no—
102C22×C211—21no—
103C22×C211#12011yescharacteristic
104C22×C211#12111yescharacteristic
105C22×C211#12311yescharacteristic
106C2×D211—44no—
107C2×D211—41no—
108C2×D211—42no—
109C2×D211—41no—
110C2×D211—42no—
111C2×D211—41no—
112C2×D211—42no—
113C2×D211—82no—
114C4×C211—41no—
115C4×C211—41no—
116C4×C211—81no—
117C4×C211#12111yescharacteristic
118C4×C211#12111yescharacteristic
119C4×C211#12311yescharacteristic
120C4×C211#12411yescharacteristic
121Dic211—42no—
122Dic211—41no—
123C2×C211—21no—
124C2×C211—21no—
125C2×C211#39411yescharacteristic
126C2×C211#45311yescharacteristic
127C2×C211#45511yescharacteristic
128D211—81no—
129D211—82no—
130C211#203711yescharacteristic, complemented, Sylow
131#2037—2111nomaximal, Sylow
132#535—2111no—
133#428D21111yescharacteristic, complemented
134C2×#104—2111no—
135#417—2111no—
136C2×#100—2111no—
137#390—2111no—
138#339—2111no—
139C2×#41—2111no—
140C22×D4—2111no—
141C2×M16—2111no—
142C2×M16C2×D21111yescharacteristic
143#127—4221no—
144#124—4221no—
145#121—4221no—
146#121—4221no—
147#120—4222no—
148#117C2×D21111yescharacteristic
149C2×C4⋊C4C2×D21111yescharacteristic
150C2×C22⋊C4—4222no—
151C2×C42—2111no—
152#107—21no—
153#106—21no—
154#104—4222no—
155#103—4221no—
156#102—4221no—
157#100—4222no—
158C24—2112no—
159#41—4221no—
160#41—4221no—
161#41—4221no—
162#41—4221no—
163C2×Q8—2111no—
164C2×D4—4222no—
165C2×D4—4222no—
166C2×D4—4221no—
167C2×D4—4221no—
168C2×D4—4221no—
169C2×D4—4221no—
170C2×D4—4221no—
171C2×D4—4221no—
172C2×D4—4221no—
173C2×D4—2111no—
174C22×C4—21no—
175C22×C4—4221no—
176C22×C4—4221no—
177C22×C4C22×D21111yescharacteristic
178M16—21no—
179M16—4221no—
180M16—4221no—
181M16D84412yescomplemented
182C2×C8—21no—
183C2×C8—4221no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4C22⋊D21111yescharacteristic
187C4⋊C4C22⋊D21111yescharacteristic
188C22⋊C4—21no—
189C22⋊C4—8444no—
190C22⋊C4—4222no—
191C22⋊C4—4222no—
192C42—21no—
193C42—4221no—
194C42—4221no—
195C23—8442no—
196C23—8442no—
197C23—4222no—
198C23—4221no—
199C23—4222no—
200C23—4222no—
201C23D4×D21111yescharacteristic
202Q8—4221no—
203Q8—4221no—
204D4—8441no—
205D4—8441no—
206D4—8441no—
207D4—8441no—
208D4—8441no—
209D4—8441no—
210D4—16882no—
211D4—16882no—
212D4—4221no—
213D4—4221no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—8441no—
219C2×C4—8441no—
220C2×C4—8441no—
221C2×C4—8441no—
222C2×C4—4222no—
223C2×C4—4221no—
224C2×C4D4×D21111yescharacteristic
225C2×C4C2×C22⋊D21111yescharacteristic
226C2×C4C2×D84411yescharacteristic
227C2×C4#1680996211yescharacteristic
228C2×C4#1680996311yescharacteristic
229C2×C4#1680996311yescharacteristic
230C8—22no—
231C8—8441no—
232C22—21no—
233C22—21no—
234C22—8444no—
235C22—8441no—
236C22—8442no—
237C22—8441no—
238C22—8442no—
239C22—8441no—
240C22—8442no—
241C22—16882no—
242C22#2026994511yescharacteristic
243C22#2026995211yescharacteristic
244C22#2026996211yescharacteristic
245C4—41no—
246C4—41no—
247C4—81no—
248C4—8442no—
249C4—8441no—
250C4#2026995111yescharacteristic
251C4#2026995211yescharacteristic
252C4#2026996211yescharacteristic
253C4#2026996611yescharacteristic
254C2—21no—
255C2—21no—
256C2—16882no—
257C2—16881no—
258C2#2721306311yescharacteristic
259C2#2721331211yescharacteristic
260C2#2721332211yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#428×C211C2
socleC22×C211#120
prime radical#27213715C2
hypercenter#428D211
hypocenterC211#2037
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. C2×#20269903C2semidirect
3. #27213145C2semidirect
4. C2×#20269893C2semidirect
5. #27213047C2semidirect
6. #27212905C2semidirect
7. #428×C211C2semidirect
56. M16×C211D4semidirect
130. C211#2037semidirect
133. #428D211semidirect
181. M16D844semidirect

Maximal subgroups

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

subgroupindex
#272137152
C2×#202699032
#272131452
C2×#202698932
#272130472
#272129052
#428×C2112
#2037211