Finite Group Databasebeta

← #41012419

Subgroups — #41012419

Order 27008.

288 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 51 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#27213704C211yesmaximal, characteristic
2#27213465C211yesmaximal, characteristic, complemented
3#27213462C211yesmaximal, characteristic, complemented
4#27213083C211yesmaximal, characteristic, complemented
5#27212879C211yesmaximal, characteristic, complemented
6#27212869C211yesmaximal, characteristic, complemented
7#327×C211C211yesmaximal, characteristic, complemented
8C2×D4×D211—21no—
9#20269962—21no—
10#20269959—21no—
11#20269951C2211yescharacteristic
12#20269949—21no—
13#20269947—21no—
14#20269945—21no—
15C2×C4⋊Dic211—21no—
16C2×C4⋊Dic211C2211yescharacteristic, complemented
17C2×(C2×D211)⋊C4—21no—
18C2×(C2×D211)⋊C4C2211yescharacteristic
19#20269909—21no—
20#20269908—21no—
21#20269899—21no—
22#20269898—21no—
23C22⋊(C211⋊C8)C2211yescharacteristic, complemented
24#121×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic, complemented
26#95×C211C2211yescharacteristic, complemented
27C23×D211—21no—
28C23×D211D411yescharacteristic
29C2×D844—21no—
30C2×D844D411yescharacteristic
31C2×C22⋊D211—21no—
32C2×C22⋊D211—21no—
33C2×C22⋊D211—21no—
34D4×D211—42no—
35D4×D211—42no—
36C22×Dic211—21no—
37C22×Dic211D411yescharacteristic
38C2×C4×D211—21no—
39C2×C211⋊C8—21no—
40C4⋊Dic211—21no—
41C4⋊Dic211—21no—
42C4⋊Dic211—42no—
43C4⋊Dic211D411yescharacteristic
44Dic211⋊C4—21no—
45C22⋊Dic211—21no—
46(C2×D211)⋊C4—21no—
47(C2×D211)⋊C4—21no—
48(C2×D211)⋊C4—21no—
49(C2×D211)⋊C4—21no—
50(C2×D211)⋊C4—42no—
51C2×D4×C211—21no—
52C2×D4×C211D411yescharacteristic
53C22×C4×C211—21no—
54C22×C4×C211C2311yescharacteristic
55C2×C8×C211—21no—
56C4⋊C4×C211D411yescharacteristic
57C22⋊C4×C211—21no—
58C22×D211—21no—
59C22×D211—21no—
60C22×D211—21no—
61C22×D211—42no—
62C22×D211—41no—
63C22×D211—41no—
64C22×D211—81no—
65C22×D211—81no—
66C22×D211SD1612yes—
67D844—41no—
68D844—42no—
69C22⋊D211—42no—
70C22⋊D211—41no—
71C22⋊D211—41no—
72C22⋊D211—81no—
73C2×Dic211—21no—
74C2×Dic211—21no—
75C2×Dic211—41no—
76C2×Dic211—41no—
77C2×Dic211—41no—
78C2×Dic211—41no—
79C4×D211—42no—
80C211⋊C8—41no—
81C23×C211—21no—
82C23×C211C2×D411yescharacteristic
83D4×C211—41no—
84D4×C211—41no—
85D4×C211—41no—
86C2×C4×C211—22no—
87C2×C4×C211—21no—
88C2×C4×C211—41no—
89C2×C4×C211—41no—
90C2×C4×C211—41no—
91C2×C4×C211C2×D411yescharacteristic
92C2×C4×C211C2×D411yescharacteristic
93C8×C211—41no—
94C22×C211—21no—
95C22×C211—21no—
96C22×C211—41no—
97C22×C211—41no—
98C22×C211—41no—
99C22×C211#12011yescharacteristic
100C22×C211C2×SD1611yescharacteristic
101C22×C211AGL(1,ℤ/8)11yescharacteristic
102C2×D211—22no—
103C2×D211—41no—
104C2×D211—41no—
105C2×D211—42no—
106C2×D211—41no—
107C2×D211—41no—
108C2×D211—41no—
109C2×D211—81no—
110C2×D211—84no—
111C4×C211—21no—
112C4×C211—21no—
113C4×C211—42no—
114C4×C211—41no—
115Dic211—41no—
116Dic211—41no—
117Dic211—81no—
118C2×C211—21no—
119C2×C211—21no—
120C2×C211—81no—
121C2×C211#45011yescharacteristic
122C2×C211#45311yescharacteristic
123C2×C211#45811yescharacteristic
124D211—42no—
125D211—81no—
126C211#161311yescharacteristic, complemented, Sylow
127#1613—2111nomaximal, Sylow
128#534—2111no—
129#481—2111no—
130#481—2111no—
131#399—2111no—
132#327—2111no—
133#327D21111yescharacteristic, complemented
134#323—2111no—
135C22×D4—4221no—
136#123—4221no—
137#123—4221no—
138#121—4221no—
139#121—2111no—
140#121C2×D21111yescharacteristic
141#120—4221no—
142#120—4221no—
143C2×C4⋊C4—4221no—
144C2×C4⋊C4—2111no—
145C2×C22⋊C4—4221no—
146C2×C22⋊C4—2111no—
147#106—4221no—
148#106—4221no—
149#102—4221no—
150#102—4221no—
151C22⋊C8—2111no—
152C22⋊C8C2×D21111yescharacteristic, complemented
153#95C2×D21111yescharacteristic, complemented
154C24—4221no—
155C24—2111no—
156C2×D4—21no—
157C2×D4—8442no—
158C2×D4—8442no—
159C2×D4—4221no—
160C2×D4—4221no—
161C2×D4—4221no—
162C2×D4—4221no—
163C2×D4—2111no—
164C2×D4D84411yescharacteristic
165C22×C4—21no—
166C22×C4—4221no—
167C22×C4—4221no—
168C22×C4—2111no—
169C22×C4C22×D21111yescharacteristic
170C2×C8—21no—
171C2×C8—4221no—
172C4⋊C4—8442no—
173C4⋊C4—4221no—
174C4⋊C4—4221no—
175C4⋊C4—4221no—
176C4⋊C4—2111no—
177C4⋊C4D84411yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—8442no—
180C22⋊C4—4221no—
181C22⋊C4—4221no—
182C22⋊C4—4221no—
183C22⋊C4—4221no—
184C22⋊C4—4221no—
185C23—21no—
186C23—8441no—
187C23—8441no—
188C23—8442no—
189C23—16881no—
190C23—16881no—
191C23—4221no—
192C23—4221no—
193C23—4221no—
194C23—2112no—
195C23D4×D21111yescharacteristic
196D4—41no—
197D4—41no—
198D4—41no—
199D4—8441no—
200D4—8442no—
201D4—8442no—
202D4—8441no—
203D4—8441no—
204D4—16881no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—8442no—
211C2×C4—8441no—
212C2×C4—8441no—
213C2×C4—8441no—
214C2×C4—8441no—
215C2×C4—4221no—
216C2×C4—4221no—
217C2×C4D4×D21111yescharacteristic
218C2×C4C2×D84411yescharacteristic
219C8—41no—
220C8—8441no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—8442no—
227C22—8441no—
228C22—8441no—
229C22—8441no—
230C22—8441no—
231C22—8441no—
232C22—16881no—
233C22—16884no—
234C22—4222no—
235C22#2026994711yescharacteristic
236C22SD16×D21111yescharacteristic
237C22#2027000111yescharacteristic
238C4—21no—
239C4—21no—
240C4—42no—
241C4—41no—
242C4—8441no—
243C4—8441no—
244C4—16881no—
245C2—21no—
246C2—21no—
247C2—81no—
248C2—8442no—
249C2—16881no—
250C2#2721329811yescharacteristic
251C2#2721331511yescharacteristic
252C2#2721333711yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#327×C211C2
socleC22×C211#120
prime radical#27213704C2
hypercenter#327D211
hypocenterC211#1613
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #27213465C2semidirect
3. #27213462C2semidirect
4. #27213083C2semidirect
5. #27212879C2semidirect
6. #27212869C2semidirect
7. #327×C211C2semidirect
16. C2×C4⋊Dic211C22semidirect
23. C22⋊(C211⋊C8)C22semidirect
25. C22⋊C8×C211C22semidirect
26. #95×C211C22semidirect
126. C211#1613semidirect
133. #327D211semidirect
152. C22⋊C8C2×D211semidirect
153. #95C2×D211semidirect

Maximal subgroups

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

subgroupindex
#272137042
#272134652
#272134622
#272130832
#272128792
#272128692
#327×C2112
#1613211