Finite Group Databasebeta

← #41014715

Subgroups — #41014715

Order 27008.

324 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 89 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×#16809950C211yesmaximal, characteristic
2C2×C4⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
3C2×#20269900C211yesmaximal, characteristic, complemented
4C2×#20269898C211yesmaximal, characteristic
5#27213052C211yesmaximal, characteristic, complemented
6#27212908C211yesmaximal, characteristic, complemented
7#390×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809950—22no—
10C2×#16809950—22no—
11C2×#16809950—22no—
12C22×C211⋊C8C2211yescharacteristic
13C22×C211⋊C8C2211yescharacteristic
14C2×C4⋊Dic211C2211yescharacteristic
15C4⋊(C211⋊C8)—22no—
16#20269900—22no—
17#20269898—22no—
18C211⋊#95—21no—
19C211⋊#95—21no—
20C22×D4×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22C2×C22⋊C4×C211—21no—
23C2×C22⋊C4×C211—21no—
24C2×C42×C211C2211yescharacteristic
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422D412yes—
31C22×Dic211—21no—
32C22×Dic211—21no—
33#16809950—28no—
34C2×C211⋊C8—21no—
35C2×C211⋊C8—21no—
36C2×C211⋊C8—21no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39C2×C211⋊C8D412yes—
40C4⋊Dic211—21no—
41C4⋊Dic211—21no—
42C24×C211—21no—
43C2×D4×C211—21no—
44C2×D4×C211—21no—
45C2×D4×C211—21no—
46C2×D4×C211—21no—
47C2×D4×C211—21no—
48C2×D4×C211D412yes—
49C22×C4×C211—21no—
50C22×C4×C211D411yescharacteristic
51C22×C4×C211D411yescharacteristic
52C22×C4×C211C2311yescharacteristic
53C4⋊C4×C211—21no—
54C4⋊C4×C211—21no—
55C22⋊C4×C211—42no—
56C22⋊C4×C211—42no—
57C42×C211—21no—
58C42×C211—21no—
59Dic422—22no—
60Dic422—22no—
61Dic422—41no—
62Dic422—41no—
63C2×Dic211—22no—
64C2×Dic211—41no—
65C2×Dic211—41no—
66C2×Dic211—41no—
67C2×Dic211—41no—
68C2×Dic211—41no—
69C211⋊C8—22no—
70C211⋊C8—41no—
71C23×C211—22no—
72C23×C211—41no—
73C23×C211—41no—
74C23×C211—41no—
75C23×C211—41no—
76C23×C211—41no—
77C23×C211—41no—
78C23×C211C2×D411yescharacteristic
79D4×C211—22no—
80D4×C211—22no—
81D4×C211—41no—
82D4×C211—41no—
83C2×C4×C211—21no—
84C2×C4×C211—21no—
85C2×C4×C211—21no—
86C2×C4×C211—21no—
87C2×C4×C211—21no—
88C2×C4×C211—41no—
89C2×C4×C211—41no—
90C2×C4×C211—41no—
91C2×C4×C211SD1612yescomplemented
92C2×C4×C211C2×D411yescharacteristic
93C2×C4×C211C2×D411yescharacteristic
94C2×C4×C211C2×D411yescharacteristic
95C2×C4×C211#4111yescharacteristic
96C2×C4×C211#4111yescharacteristic
97C2×C4×C211#4111yescharacteristic
98C22×C211—42no—
99C22×C211—42no—
100C22×C211—42no—
101C22×C211—81no—
102C22×C211—81no—
103C22×C211—81no—
104C22×C211—81no—
105C22×C211#12111yescharacteristic
106C22×C211#12111yescharacteristic
107C22×C211#12111yescharacteristic
108C22×C211C2×SD1611yescharacteristic
109C22×C211#13511yescharacteristic
110C22×C211AGL(1,ℤ/8)11yescharacteristic
111C22×C211#13711yescharacteristic
112C4×C211—22no—
113C4×C211—41no—
114C4×C211—81no—
115C4×C211#12011yescharacteristic
116C4×C211#12311yescharacteristic
117C4×C211#12311yescharacteristic
118C4×C211#12411yescharacteristic
119Dic211—42no—
120Dic211—81no—
121C2×C211—82no—
122C2×C211#39411yescharacteristic
123C2×C211#46011yescharacteristic
124C2×C211#46111yescharacteristic
125C2×C211#46311yescharacteristic
126C2×C211#46411yescharacteristic
127C2×C211#46511yescharacteristic
128C2×C211#46811yescharacteristic
129C211#205511yescharacteristic, complemented, Sylow
130#2055—2111nomaximal, Sylow
131C22×SD16—2111no—
132C2×C4⋊C8—2111no—
133C2×#103—2111no—
134C2×#102—2111no—
135#391—2111no—
136#390D21111yescharacteristic
137#340—2111no—
138C22×Q8—2111no—
139C22×D4C2×D21111yescharacteristic
140C2×SD16—4222no—
141C2×SD16—4222no—
142C2×SD16—4222no—
143C22×C8—2111no—
144C22×C8—2111no—
145C2×C4⋊C4—2111no—
146C2×C4⋊C4C2×D21111yescharacteristic
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C42C2×D21111yescharacteristic
150C4⋊C8—4222no—
151#103—4222no—
152#102—4222no—
153#95—4221no—
154#95—4221no—
155C24—21no—
156C2×Q8—4221no—
157C2×Q8—4221no—
158C2×Q8—4221no—
159C2×Q8—4221no—
160C2×Q8—4221no—
161C2×Q8—2112no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4D84412yes—
168C22×C4—21no—
169C22×C4—4221no—
170C22×C4—4221no—
171C22×C4C22⋊D21111yescharacteristic
172C22×C4C22⋊D21111yescharacteristic
173C22×C4C22×D21111yescharacteristic
174SD16—4228no—
175C2×C8—4221no—
176C2×C8—4221no—
177C2×C8—4221no—
178C2×C8—4221no—
179C2×C8—4221no—
180C2×C8—2112no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—4221no—
184C4⋊C4—4221no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C42—21no—
188C42—21no—
189C23—22no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23C2×C22⋊D21111yescharacteristic
197Q8—8441no—
198Q8—8441no—
199Q8—4222no—
200Q8—4222no—
201D4—22no—
202D4—22no—
203D4—41no—
204D4—41no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—8441no—
214C2×C4—8441no—
215C2×C4—8441no—
216C2×C4—8441no—
217C2×C4—8441no—
218C2×C4—4222no—
219C2×C4#1680995012yescomplemented
220C2×C4D4×D21111yescharacteristic
221C2×C4D4×D21111yescharacteristic
222C2×C4C2×D84411yescharacteristic
223C2×C4#1680996111yescharacteristic
224C2×C4#1680996111yescharacteristic
225C2×C4#1680996211yescharacteristic
226C8—8441no—
227C8—4222no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22#2026994811yescharacteristic
236C22#2026994911yescharacteristic
237C22#2026995111yescharacteristic
238C22C2×#1680995011yescharacteristic
239C22#2026999711yescharacteristic
240C22#2027000311yescharacteristic
241C22#2027000711yescharacteristic
242C4—22no—
243C4—41no—
244C4—81no—
245C4—8442no—
246C4—16881no—
247C4#2026994711yescharacteristic
248C4#2026995911yescharacteristic
249C4#2026996111yescharacteristic
250C4#2026996311yescharacteristic
251C2—82no—
252C2#2721306211yescharacteristic
253C2#2721334311yescharacteristic
254C2#2721335111yescharacteristic
255C2#2721336111yescharacteristic
256C2#2721336511yescharacteristic
257C2#2721337311yescharacteristic
258C2#2721339011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#390×C211C2
socleC23×C211C2×D4
prime radicalC22×D4×C211C22
hypercenter#390D211
hypocenterC211#2055
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊(C211⋊C8)C2semidirect
3. C2×#20269900C2semidirect
5. #27213052C2semidirect
6. #27212908C2semidirect
91. C2×C4×C211SD16semidirect
129. C211#2055semidirect
219. C2×C4#16809950semidirect

Maximal subgroups

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

subgroupindex
C22×#168099502
C2×C4⋊(C211⋊C8)2
C2×#202699002
C2×#202698982
#272130522
#272129082
#390×C2112
#2055211