Finite Group Databasebeta

← #41012424

Subgroups — #41012424

Order 27008.

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

normal subgroups only

This group has 236 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#27213771C211yesmaximal, characteristic
2#27213482C211yesmaximal, characteristic, complemented
3#27213479C211yesmaximal, characteristic, complemented
4#27213083C211yesmaximal, characteristic
5#27212882C211yesmaximal, characteristic, complemented
6#27212869C211yesmaximal, characteristic
7#328×C211C211yesmaximal, characteristic, complemented
8#20269957—21no—
9#20269956C2211yescharacteristic
10#20269955—21no—
11#20269955—21no—
12#20269954—21no—
13#20269953—21no—
14C2×C4⋊Dic211—21no—
15C2×C4⋊Dic211C2211yescharacteristic
16C2×(C2×D211)⋊C4—21no—
17C2×(C2×D211)⋊C4C2211yescharacteristic
18C22⋊C4×D211—21no—
19#20269911—21no—
20#20269910—21no—
21#20269902—21no—
22#20269901—21no—
23C22⋊(C211⋊C8)C2211yescharacteristic
24C22⋊Q8×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27C23×D211D411yescharacteristic
28C2×Dic422—21no—
29C2×Dic422D411yescharacteristic
30C22×Dic211—21no—
31C22×Dic211D411yescharacteristic
32C2×C4×D211—21no—
33C2×C4×D211—21no—
34C2×C211⋊C8—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211—21no—
37C4⋊Dic211—21no—
38C4⋊Dic211—42no—
39C4⋊Dic211D411yescharacteristic
40Dic211⋊C4—21no—
41Dic211⋊C4—21no—
42Dic211⋊C4—21no—
43C22⋊Dic211—21no—
44(C2×D211)⋊C4—21no—
45(C2×D211)⋊C4—21no—
46(C2×D211)⋊C4—21no—
47(C2×D211)⋊C4—21no—
48(C2×D211)⋊C4—42no—
49C2×Q8×C211D411yescharacteristic
50C22×C4×C211—21no—
51C22×C4×C211C2311yescharacteristic
52C2×C8×C211—21no—
53C4⋊C4×C211—21no—
54C4⋊C4×C211D411yescharacteristic
55C22⋊C4×C211—21no—
56C22×D211—21no—
57C22×D211—21no—
58C22×D211—41no—
59C22×D211—41no—
60C22×D211Q1612yes—
61Dic422—41no—
62Dic422—81no—
63C2×Dic211—21no—
64C2×Dic211—21no—
65C2×Dic211—21no—
66C2×Dic211—41no—
67C2×Dic211—41no—
68C2×Dic211—41no—
69C2×Dic211—41no—
70C4×D211—81no—
71C4×D211—81no—
72C211⋊C8—41no—
73C23×C211C2×D411yescharacteristic
74Q8×C211—41no—
75C2×C4×C211—22no—
76C2×C4×C211—21no—
77C2×C4×C211—21no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211—41no—
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C8×C211—41no—
84C22×C211—21no—
85C22×C211—21no—
86C22×C211#12011yescharacteristic
87C22×C211C2×Q1611yescharacteristic
88C22×C211#13711yescharacteristic
89C2×D211—22no—
90C2×D211—41no—
91C2×D211—41no—
92C2×D211—42no—
93C2×D211—81no—
94C4×C211—21no—
95C4×C211—21no—
96C4×C211—42no—
97C4×C211—41no—
98C4×C211—41no—
99Dic211—41no—
100Dic211—41no—
101Dic211—41no—
102Dic211—81no—
103C2×C211—21no—
104C2×C211—21no—
105C2×C211#45111yescharacteristic
106C2×C211#45311yescharacteristic
107C2×C211#45811yescharacteristic
108D211—42no—
109C211#161411yescharacteristic, complemented, Sylow
110#1614—2111nomaximal, Sylow
111#543—2111no—
112#484—2111no—
113#484—2111no—
114#399—2111no—
115#328—2111no—
116#328D21111yescharacteristic, complemented
117#323—2111no—
118C22⋊Q8—4221no—
119C22⋊Q8—4221no—
120C22⋊Q8—4221no—
121C22⋊Q8—4221no—
122C22⋊Q8—4221no—
123C22⋊Q8—2111no—
124C22⋊Q8C2×D21111yescharacteristic
125C2×C4⋊C4—4221no—
126C2×C4⋊C4—2111no—
127C2×C22⋊C4—4221no—
128C2×C22⋊C4—4221no—
129C2×C22⋊C4—2111no—
130#107—4221no—
131#107—4221no—
132#103—4221no—
133#103—4221no—
134C22⋊C8—2111no—
135C22⋊C8C2×D21111yescharacteristic
136#95C2×D21111yescharacteristic
137C24—2111no—
138C2×Q8—4221no—
139C2×Q8—2111no—
140C2×Q8D84411yescharacteristic
141C22×C4—21no—
142C22×C4—4221no—
143C22×C4—4221no—
144C22×C4—4221no—
145C22×C4—2111no—
146C22×C4C22×D21111yescharacteristic
147C2×C8—21no—
148C2×C8—4221no—
149C4⋊C4—21no—
150C4⋊C4—8442no—
151C4⋊C4—4221no—
152C4⋊C4—4221no—
153C4⋊C4—4221no—
154C4⋊C4—4221no—
155C4⋊C4—4221no—
156C4⋊C4—4221no—
157C4⋊C4—2111no—
158C4⋊C4D84411yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—8442no—
161C22⋊C4—4221no—
162C22⋊C4—4221no—
163C22⋊C4—4221no—
164C22⋊C4—4221no—
165C22⋊C4—4221no—
166C23—8441no—
167C23—8441no—
168C23—4221no—
169C23—4221no—
170C23—2112no—
171C23D4×D21111yescharacteristic
172Q8—41no—
173Q8—8441no—
174Q8—16881no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—8441no—
182C2×C4—8441no—
183C2×C4—8441no—
184C2×C4—8441no—
185C2×C4—16881no—
186C2×C4—16881no—
187C2×C4—4221no—
188C2×C4—4221no—
189C2×C4—4221no—
190C2×C4D4×D21111yescharacteristic
191C2×C4C2×D84411yescharacteristic
192C8—41no—
193C8—8441no—
194C22—21no—
195C22—21no—
196C22—8442no—
197C22—8441no—
198C22—8441no—
199C22—16881no—
200C22—4222no—
201C22#2026994711yescharacteristic
202C22Q16×D21111yescharacteristic
203C22#2027000611yescharacteristic
204C4—21no—
205C4—21no—
206C4—42no—
207C4—41no—
208C4—41no—
209C4—8441no—
210C4—8441no—
211C4—8441no—
212C4—16881no—
213C2—21no—
214C2—21no—
215C2—8442no—
216C2#2721330311yescharacteristic
217C2#2721331511yescharacteristic
218C2#2721333711yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#328×C211C2
socleC22×C211#120
prime radicalC23×D211D4
hypercenter#328D211
hypocenterC211#1614
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #27213482C2semidirect
3. #27213479C2semidirect
5. #27212882C2semidirect
7. #328×C211C2semidirect
109. C211#1614semidirect
116. #328D211semidirect

Maximal subgroups

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

subgroupindex
#272137712
#272134822
#272134792
#272130832
#272128822
#272128692
#328×C2112
#1614211