Finite Group Databasebeta

← #41014097

Subgroups — #41014097

Order 27008.

284 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#20269898C211yesmaximal, characteristic
2C2×#20269887C211yesmaximal, characteristic, complemented
3#27213049C211yesmaximal, characteristic
4#27213045C211yesmaximal, characteristic, complemented
5#27212909C211yesmaximal, characteristic, complemented
6#27212908C211yesmaximal, characteristic, complemented
7C2×#102×C211C211yesmaximal, characteristic
8C22×C211⋊C8C2211yescharacteristic
9C2×C4⋊Dic211—21no—
10C2×C4⋊Dic211C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12C2×C22⋊Dic211—21no—
13C2×C22⋊Dic211—21no—
14C2×C4×Dic211C2211yescharacteristic
15#20269898—22no—
16#20269887—22no—
17C211⋊#95—21no—
18C22×D4×C211C2211yescharacteristic
19C22×C8×C211C2211yescharacteristic
20C2×C4⋊C4×C211C2211yescharacteristic
21#102×C211C414yescomplemented
22C22×Dic211—21no—
23C22×Dic211—21no—
24C22×Dic211D411yescharacteristic
25C22×Dic211D411yescharacteristic
26C2×C211⋊C8—21no—
27C2×C211⋊C8—21no—
28C2×C211⋊C8D412yes—
29C4⋊Dic211—21no—
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32C4⋊Dic211—21no—
33C4⋊Dic211—42no—
34C22⋊Dic211—42no—
35C22⋊Dic211—42no—
36C4×Dic211—21no—
37C4×Dic211—21no—
38C24×C211—21no—
39C2×D4×C211—21no—
40C2×D4×C211—21no—
41C2×D4×C211—21no—
42C2×D4×C211—21no—
43C2×D4×C211—21no—
44C2×D4×C211C2×C412yes—
45C22×C4×C211—21no—
46C22×C4×C211C2311yescharacteristic
47C2×C8×C211—21no—
48C2×C8×C211—21no—
49C2×C8×C211C2×C412yes—
50C4⋊C4×C211—21no—
51C4⋊C4×C211C2×C412yes—
52C2×Dic211—21no—
53C2×Dic211—21no—
54C2×Dic211—21no—
55C2×Dic211—21no—
56C2×Dic211—21no—
57C2×Dic211—21no—
58C2×Dic211—41no—
59C2×Dic211—41no—
60C2×Dic211—41no—
61C2×Dic211—41no—
62C2×Dic211—41no—
63C2×Dic211—41no—
64C211⋊C8—22no—
65C23×C211—22no—
66C23×C211—41no—
67C23×C211—41no—
68C23×C211—41no—
69C23×C211—41no—
70C23×C211—41no—
71C23×C211—41no—
72C23×C211C2×D411yescharacteristic
73D4×C211—22no—
74D4×C211—22no—
75D4×C211—41no—
76D4×C211—41no—
77C2×C4×C211—22no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211C22×C411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211#4111yescharacteristic
83C2×C4×C211#4111yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C2×C4×C211#4111yescharacteristic
86C8×C211—22no—
87C22×C211—42no—
88C22×C211—42no—
89C22×C211—42no—
90C22×C211—81no—
91C22×C211—81no—
92C22×C211—81no—
93C22×C211—81no—
94C22×C211C4×D411yescharacteristic
95C22×C211#12111yescharacteristic
96C22×C211#12411yescharacteristic
97C22×C211AGL(1,ℤ/8)11yescharacteristic
98C22×C211AGL(1,ℤ/8)11yescharacteristic
99C22×C211AGL(1,ℤ/8)11yescharacteristic
100C22×C211#13711yescharacteristic
101C4×C211—42no—
102C4×C211#11711yescharacteristic
103C4×C211C4×D411yescharacteristic
104C4×C211#12311yescharacteristic
105C4×C211#12611yescharacteristic
106Dic211—41no—
107Dic211—41no—
108Dic211—81no—
109Dic211—81no—
110C2×C211—82no—
111C2×C211#38811yescharacteristic
112C2×C211#44011yescharacteristic
113C2×C211#44211yescharacteristic
114C2×C211#46811yescharacteristic
115C2×C211#46911yescharacteristic
116C2×C211#48911yescharacteristic
117C2×C211#49011yescharacteristic
118C211#193611yescharacteristic, complemented, Sylow
119#1936—2111nomaximal, Sylow
120C2×#102—2111no—
121C2×#102D21111yescharacteristic
122C2×#97—2111no—
123#390—2111no—
124#389—2111no—
125#340—2111no—
126#340—2111no—
127C22×D4C2×D21111yescharacteristic
128C22×C8—2111no—
129C22×C8C2×D21111yescharacteristic
130C2×C4⋊C4—4221no—
131C2×C4⋊C4—2111no—
132C2×C4⋊C4—2111no—
133C2×C4⋊C4C2×D21111yescharacteristic
134C2×C22⋊C4—4221no—
135C2×C22⋊C4—4221no—
136C2×C42—2111no—
137#102—4222no—
138#102Dic21114yescomplemented
139#97—4222no—
140#95—4221no—
141C24—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4C2×Dic21112yes—
148C22×C4—21no—
149C22×C4—4221no—
150C22×C4—4221no—
151C22×C4—2111no—
152C22×C4—2111no—
153C22×C4C22×D21111yescharacteristic
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—4221no—
157C2×C8—4221no—
158C2×C8—2112no—
159C2×C8C2×Dic21112yes—
160C4⋊C4—21no—
161C4⋊C4—8442no—
162C4⋊C4—4221no—
163C4⋊C4—4221no—
164C4⋊C4—4221no—
165C4⋊C4—4221no—
166C4⋊C4C2×Dic21112yes—
167C22⋊C4—8442no—
168C22⋊C4—8442no—
169C42—4221no—
170C42—4221no—
171C23—22no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23D4×D21111yescharacteristic
179D4—22no—
180D4—22no—
181D4—41no—
182D4—41no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—8441no—
187C2×C4—8441no—
188C2×C4—8441no—
189C2×C4—8441no—
190C2×C4—8441no—
191C2×C4—8441no—
192C2×C4—4221no—
193C2×C4—4221no—
194C2×C4—4221no—
195C2×C4—4221no—
196C2×C4—4221no—
197C2×C4—4221no—
198C2×C4C22×Dic21111yescharacteristic
199C2×C4D4×D21111yescharacteristic
200C2×C4#1680996111yescharacteristic
201C2×C4#1680996111yescharacteristic
202C2×C4#1680996211yescharacteristic
203C2×C4#1680996211yescharacteristic
204C8—22no—
205C8—4222no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22—81no—
213C22D4×Dic21111yescharacteristic
214C22#2026995011yescharacteristic
215C22#2026996311yescharacteristic
216C22#2027000111yescharacteristic
217C22#2027000111yescharacteristic
218C22#2027000411yescharacteristic
219C22#2027000611yescharacteristic
220C4—42no—
221C4—8441no—
222C4—8441no—
223C4—16881no—
224C4—16881no—
225C4#2026993611yescharacteristic
226C4D4×Dic21111yescharacteristic
227C4#2026995911yescharacteristic
228C4#2026997011yescharacteristic
229C2—82no—
230C2#2721303911yescharacteristic
231C2#2721324711yescharacteristic
232C2#2721325811yescharacteristic
233C2#2721339111yescharacteristic
234C2#2721339811yescharacteristic
235C2#2721350411yescharacteristic
236C2#2721350911yescharacteristic
subgroupG/H
centerC23D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#102×C211C2
socleC23×C211C2×D4
prime radicalC22×D4×C211C22
cosocleC22×C4×C211C23
hypercenterC2×#102D211
hypocenterC211#1936
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#20269887C2semidirect
4. #27213045C2semidirect
5. #27212909C2semidirect
6. #27212908C2semidirect
21. #102×C211C4semidirect
118. C211#1936semidirect
138. #102Dic211semidirect

Maximal subgroups

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

subgroupindex
C2×#202698982
C2×#202698872
#272130492
#272130452
#272129092
#272129082
C2×#102×C2112
#1936211