Finite Group Databasebeta

← #41012377

Subgroups — #41012377

Order 27008.

260 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 49 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#27213736C211yesmaximal, characteristic
2#27213285C211yesmaximal, characteristic, complemented
3#27213057C211yesmaximal, characteristic, complemented
4#27212883C211yesmaximal, characteristic, complemented
5#27212881C211yesmaximal, characteristic, complemented
6#27212869C211yesmaximal, characteristic, complemented
7#451×C211C211yesmaximal, characteristic, complemented
8C2×#16809949—21no—
9#20269972—21no—
10#20269966—21no—
11#20269962—21no—
12#20269952—21no—
13#20269951C2211yescharacteristic
14#20269947—21no—
15(C22⋊D211)⋊C4—21no—
16C2×(C2×D211)⋊C4—21no—
17C2×(C2×D211)⋊C4C2211yescharacteristic
18#20269901—21no—
19C22⋊(C211⋊C8)C2211yescharacteristic
20C211⋊#95—21no—
21C211⋊#95C2211yescharacteristic
22C22×Q8×C211C2211yescharacteristic
23C2×Q16×C211—21no—
24C22⋊Q8×C211C2211yescharacteristic
25#103×C211—21no—
26C22⋊C8×C211C2211yescharacteristic
27C23×D211D411yescharacteristic
28C2×D844—21no—
29C2×D844—21no—
30C2×D844D411yescharacteristic
31C2×C22⋊D211—21no—
32C22×Dic211—21no—
33C22×Dic211D411yescharacteristic
34C2×C4×D211—21no—
35#16809949—42no—
36C2×C211⋊C8—21no—
37C4⋊Dic211D411yescharacteristic
38Dic211⋊C4—21no—
39(C2×D211)⋊C4—21no—
40(C2×D211)⋊C4—21no—
41(C2×D211)⋊C4—21no—
42(C2×D211)⋊C4—21no—
43(C2×D211)⋊C4—21no—
44(C2×D211)⋊C4—21no—
45(C2×D211)⋊C4—42no—
46C4×Dic211—21no—
47C2×Q8×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211—42no—
50C2×Q8×C211D411yescharacteristic
51C22×C4×C211—21no—
52C22×C4×C211C2311yescharacteristic
53Q16×C211—42no—
54C2×C8×C211—21no—
55C4⋊C4×C211—21no—
56C4⋊C4×C211D411yescharacteristic
57C22⋊C4×C211—21no—
58C22×D211—21no—
59C22×D211—21no—
60C22×D211—21no—
61C22×D211—21no—
62C22×D211—41no—
63C22×D211—41no—
64D844—41no—
65D844—81no—
66D844—81no—
67C22⋊D211—41no—
68C22⋊D211—41no—
69C2×Dic211—21no—
70C2×Dic211—21no—
71C2×Dic211—41no—
72C2×Dic211—41no—
73C2×Dic211—41no—
74C2×Dic211—41no—
75C4×D211—81no—
76C211⋊C8—41no—
77C23×C211C2×D411yescharacteristic
78Q8×C211—22no—
79Q8×C211—22no—
80Q8×C211—41no—
81Q8×C211—41no—
82Q8×C211—41no—
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×C211C2×D411yescharacteristic
90C2×C4×C211C2×D411yescharacteristic
91C8×C211—41no—
92C22×C211—21no—
93C22×C211—21no—
94C22×C211#12011yescharacteristic
95C22×C211#13511yescharacteristic
96C22×C211#13711yescharacteristic
97C2×D211—42no—
98C2×D211—41no—
99C2×D211—41no—
100C2×D211—41no—
101C2×D211—41no—
102C2×D211—41no—
103C2×D211—41no—
104C2×D211—81no—
105C4×C211—21no—
106C4×C211—21no—
107C4×C211—41no—
108C4×C211—42no—
109C4×C211—41no—
110Dic211—41no—
111Dic211—41no—
112Dic211—81no—
113C2×C211—21no—
114C2×C211—21no—
115C2×C211#45211yescharacteristic
116C2×C211#45511yescharacteristic
117C2×C211B(4,𝔽2)11yescharacteristic
118D211—81no—
119D211—81no—
120C211#160611yescharacteristic, complemented, Sylow
121#1606—2111nomaximal, Sylow
122#538—2111no—
123#451D21111yescharacteristic, complemented
124#448—2111no—
125#393—2111no—
126#328—2111no—
127#327—2111no—
128#323—2111no—
129C22×Q8C2×D21111yescharacteristic
130C2×Q16—21no—
131C2×SD16—4221no—
132#126—4221no—
133#124—4221no—
134#123—4221no—
135C22⋊Q8C2×D21111yescharacteristic
136#121—4221no—
137#121—2111no—
138#120—4221no—
139C4×D4—4221no—
140C2×C22⋊C4—4221no—
141C2×C22⋊C4—2111no—
142#103—21no—
143#103—4221no—
144C22⋊C8—2111no—
145C22⋊C8C2×D21111yescharacteristic
146#95—4221no—
147#95—2111no—
148C24—2111no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—42no—
152C2×Q8C22⋊D21111yescharacteristic
153C2×D4—4221no—
154C2×D4—4221no—
155C2×D4—4221no—
156C2×D4—2111no—
157C22×C4—21no—
158C22×C4—4221no—
159C22×C4—4221no—
160C22×C4—2111no—
161C22×C4C22×D21111yescharacteristic
162Q16—42no—
163SD16—8442no—
164C2×C8—21no—
165C2×C8—4221no—
166C4⋊C4—21no—
167C4⋊C4—4221no—
168C4⋊C4—2111no—
169C4⋊C4C22⋊D21111yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—8442no—
172C22⋊C4—4221no—
173C22⋊C4—4221no—
174C22⋊C4—4221no—
175C22⋊C4—4221no—
176C22⋊C4—4221no—
177C22⋊C4—4221no—
178C42—4221no—
179C23—8441no—
180C23—8441no—
181C23—4221no—
182C23—4221no—
183C23—4221no—
184C23—4221no—
185C23D4×D21111yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—41no—
191D4—8441no—
192D4—8441no—
193D4—8441no—
194D4—16881no—
195D4—16881no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—8441no—
203C2×C4—8441no—
204C2×C4—8441no—
205C2×C4—8441no—
206C2×C4—16881no—
207C2×C4—4221no—
208C2×C4—4221no—
209C2×C4D4×D21111yescharacteristic
210C2×C4C2×C22⋊D21111yescharacteristic
211C8—41no—
212C8—8441no—
213C22—21no—
214C22—21no—
215C22—8442no—
216C22—8441no—
217C22—8441no—
218C22—8441no—
219C22—8441no—
220C22—8441no—
221C22—8441no—
222C22—16881no—
223C22#2026994511yescharacteristic
224C22#2027000011yescharacteristic
225C22#2027000911yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—42no—
230C4—41no—
231C4—8441no—
232C4—8441no—
233C4—16881no—
234C2—21no—
235C2—21no—
236C2—16881no—
237C2—16881no—
238C2#2721330711yescharacteristic
239C2#2721332211yescharacteristic
240C2#2721333211yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#451×C211C2
socleC22×C211#120
prime radical#27213736C2
hypercenter#451D211
hypocenterC211#1606
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #27213285C2semidirect
3. #27213057C2semidirect
4. #27212883C2semidirect
5. #27212881C2semidirect
6. #27212869C2semidirect
7. #451×C211C2semidirect
120. C211#1606semidirect
123. #451D211semidirect

Maximal subgroups

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

subgroupindex
#272137362
#272132852
#272130572
#272128832
#272128812
#272128692
#451×C2112
#1606211