Finite Group Databasebeta

← #41014713

Subgroups — #41014713

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×#16809949C211yesmaximal, characteristic
2C2×C4⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
3C2×#20269901C211yesmaximal, characteristic, complemented
4C2×#20269897C211yesmaximal, characteristic
5#27213048C211yesmaximal, characteristic
6#27212908C211yesmaximal, characteristic, complemented
7#391×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C2×#16809949—22no—
10C2×#16809949—22no—
11C2×#16809949—22no—
12C22×C211⋊C8C2211yescharacteristic
13C22×C211⋊C8C2211yescharacteristic
14C2×C4⋊Dic211C2211yescharacteristic
15C2×(C2×D211)⋊C4—21no—
16C2×(C2×D211)⋊C4—21no—
17C4⋊(C211⋊C8)—22no—
18#20269901—22no—
19#20269897—22no—
20C22×Q8×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22C2×C42×C211C2211yescharacteristic
23#95×C211—21no—
24#95×C211—21no—
25C23×D211—21no—
26C2×D844—21no—
27C2×D844—21no—
28C2×D844—21no—
29C2×D844—21no—
30C2×D844—21no—
31C2×D844D412yes—
32C22×Dic211—21no—
33#16809949—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—
42(C2×D211)⋊C4—42no—
43(C2×D211)⋊C4—42no—
44C2×Q8×C211—21no—
45C2×Q8×C211—21no—
46C2×Q8×C211—21no—
47C2×Q8×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211D412yes—
50C22×C4×C211—21no—
51C22×C4×C211—21no—
52C22×C4×C211D411yescharacteristic
53C22×C4×C211D411yescharacteristic
54C22×C4×C211C2311yescharacteristic
55C4⋊C4×C211—21no—
56C4⋊C4×C211—21no—
57C42×C211—21no—
58C42×C211—21no—
59C22×D211—22no—
60C22×D211—41no—
61C22×D211—41no—
62C22×D211—41no—
63C22×D211—41no—
64C22×D211—41no—
65C22×D211—41no—
66D844—22no—
67D844—22no—
68D844—41no—
69D844—41no—
70C2×Dic211—41no—
71C2×Dic211—41no—
72C2×Dic211—41no—
73C211⋊C8—22no—
74C211⋊C8—41no—
75C23×C211C2×D411yescharacteristic
76Q8×C211—22no—
77Q8×C211—22no—
78Q8×C211—41no—
79Q8×C211—41no—
80C2×C4×C211—21no—
81C2×C4×C211—21no—
82C2×C4×C211—21no—
83C2×C4×C211—22no—
84C2×C4×C211—21no—
85C2×C4×C211—21no—
86C2×C4×C211—41no—
87C2×C4×C211—41no—
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#12111yescharacteristic
99C22×C211#12111yescharacteristic
100C22×C211#12111yescharacteristic
101C22×C211C2×SD1611yescharacteristic
102C22×C211#13511yescharacteristic
103C22×C211AGL(1,ℤ/8)11yescharacteristic
104C22×C211#13711yescharacteristic
105C2×D211—42no—
106C2×D211—42no—
107C2×D211—42no—
108C2×D211—81no—
109C2×D211—81no—
110C2×D211—81no—
111C2×D211—81no—
112C4×C211—22no—
113C4×C211—42no—
114C4×C211—41no—
115C4×C211—81no—
116C4×C211#12011yescharacteristic
117C4×C211#12311yescharacteristic
118C4×C211#12311yescharacteristic
119C4×C211#12411yescharacteristic
120Dic211—81no—
121C2×C211#39411yescharacteristic
122C2×C211#46011yescharacteristic
123C2×C211#46111yescharacteristic
124C2×C211#46311yescharacteristic
125C2×C211#46411yescharacteristic
126C2×C211#46511yescharacteristic
127C2×C211#46811yescharacteristic
128D211—82no—
129C211#205511yescharacteristic, complemented, Sylow
130#2055—2111nomaximal, Sylow
131C22×SD16—2111no—
132C2×C4⋊C8—2111no—
133C2×#103—2111no—
134C2×#102—2111no—
135#391D21111yescharacteristic, complemented
136#390—2111no—
137#340—2111no—
138C22×Q8C2×D21111yescharacteristic
139C22×D4—2111no—
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—4221no—
148C2×C22⋊C4—4221no—
149C2×C42C2×D21111yescharacteristic
150C4⋊C8—4222no—
151#103—4222no—
152#102—4222no—
153#95—21no—
154#95—21no—
155C24—4221no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8D84412yes—
162C2×D4—4221no—
163C2×D4—4221no—
164C2×D4—4221no—
165C2×D4—4221no—
166C2×D4—4221no—
167C2×D4—2112no—
168C22×C4—21no—
169C22×C4—21no—
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—8442no—
186C22⋊C4—8442no—
187C42—21no—
188C42—21no—
189C23—8441no—
190C23—8441no—
191C23—8441no—
192C23—8441no—
193C23—8441no—
194C23—8441no—
195C23—4222no—
196C23C2×C22⋊D21111yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201D4—8441no—
202D4—8441no—
203D4—4222no—
204D4—4222no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—8441no—
217C2×C4—8441no—
218C2×C4—8441no—
219C2×C4#1680994912yescomplemented
220C2×C4D4×D21111yescharacteristic
221C2×C4D4×D21111yescharacteristic
222C2×C4C2×D84411yescharacteristic
223C2×C4#1680996111yescharacteristic
224C2×C4#1680996111yescharacteristic
225C2×C4#1680996211yescharacteristic
226C8—8441no—
227C8—4222no—
228C22—8442no—
229C22—8442no—
230C22—8442no—
231C22—16881no—
232C22—16881no—
233C22—16881no—
234C22—16881no—
235C22#2026994811yescharacteristic
236C22#2026994911yescharacteristic
237C22#2026995111yescharacteristic
238C22C2×#1680994911yescharacteristic
239C22#2026999711yescharacteristic
240C22#2027000211yescharacteristic
241C22#2027000811yescharacteristic
242C4—22no—
243C4—42no—
244C4—41no—
245C4—81no—
246C4—16881no—
247C4#2026994711yescharacteristic
248C4#2026995911yescharacteristic
249C4#2026996111yescharacteristic
250C4#2026996311yescharacteristic
251C2—16882no—
252C2#2721306211yescharacteristic
253C2#2721334611yescharacteristic
254C2#2721334811yescharacteristic
255C2#2721335811yescharacteristic
256C2#2721336811yescharacteristic
257C2#2721337711yescharacteristic
258C2#2721339411yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#391×C211C2
socleC23×C211C2×D4
prime radicalC22×D844C22
hypercenter#391D211
hypocenterC211#2055
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊(C211⋊C8)C2semidirect
3. C2×#20269901C2semidirect
6. #27212908C2semidirect
7. #391×C211C2semidirect
91. C2×C4×C211SD16semidirect
129. C211#2055semidirect
135. #391D211semidirect
219. C2×C4#16809949semidirect

Maximal subgroups

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

subgroupindex
C22×#168099492
C2×C4⋊(C211⋊C8)2
C2×#202699012
C2×#202698972
#272130482
#272129082
#391×C2112
#2055211