Finite Group Databasebeta

← #41014537

Subgroups — #41014537

Order 27008.

290 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 290 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
1C2×#20269976C211yesmaximal, characteristic
2C2×#20269901C211yesmaximal, characteristic
3#27213053C211yesmaximal, characteristic
4#27212909C211yesmaximal, characteristic
5#27212908C211yesmaximal, characteristic
6#27212900C211yesmaximal, characteristic
7C2×#103×C211C211yesmaximal, characteristic
8C22×Dic422—21no—
9C22×C211⋊C8C2211yescharacteristic
10#20269976—22no—
11#20269976—22no—
12C2×C4⋊Dic211C2211yescharacteristic
13C2×C4⋊Dic211C2211yescharacteristic
14C2×Dic211⋊C4—21no—
15C2×C4×Dic211C2211yescharacteristic
16#20269901—22no—
17C211⋊#95—21no—
18C211⋊#95—21no—
19C22×Q8×C211C2211yescharacteristic
20C22×C8×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22#103×C211—22no—
23C2×Dic422—24no—
24C2×Dic422—42no—
25C2×Dic422—42no—
26C22×Dic211—21no—
27C22×Dic211—21no—
28C22×Dic211D411yescharacteristic
29C22×Dic211D411yescharacteristic
30C2×C211⋊C8—21no—
31C2×C211⋊C8—21no—
32C2×C211⋊C8—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211D412yes—
37Dic211⋊C4—24no—
38C4×Dic211—21no—
39C4×Dic211—21no—
40C2×Q8×C211—21no—
41C2×Q8×C211—21no—
42C2×Q8×C211—21no—
43C2×Q8×C211—21no—
44C2×Q8×C211—21no—
45C2×Q8×C211Q812yescomplemented
46C22×C4×C211—21no—
47C22×C4×C211—21no—
48C22×C4×C211C2311yescharacteristic
49C2×C8×C211—21no—
50C2×C8×C211—21no—
51C2×C8×C211—21no—
52C4⋊C4×C211—21no—
53C4⋊C4×C211D412yes—
54Dic422—48no—
55C2×Dic211—21no—
56C2×Dic211—21no—
57C2×Dic211—21no—
58C2×Dic211—21no—
59C2×Dic211—22no—
60C2×Dic211—41no—
61C2×Dic211—41no—
62C2×Dic211—41no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C2×Dic211SD1612yes—
66C2×Dic211Q1612yes—
67C211⋊C8—41no—
68C23×C211C2×D411yescharacteristic
69Q8×C211—22no—
70Q8×C211—22no—
71Q8×C211—41no—
72Q8×C211—41no—
73C2×C4×C211—22no—
74C2×C4×C211—22no—
75C2×C4×C211—41no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211C2×D411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×Q811yescharacteristic
82C2×C4×C211#4111yescharacteristic
83C2×C4×C211#4111yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C8×C211—41no—
86C22×C211#12011yescharacteristic
87C22×C211C22⋊Q811yescharacteristic
88C22×C211#12411yescharacteristic
89C22×C211C2×SD1611yescharacteristic
90C22×C211C2×Q1611yescharacteristic
91C22×C211AGL(1,ℤ/8)11yescharacteristic
92C22×C211#13711yescharacteristic
93C4×C211—42no—
94C4×C211—42no—
95C4×C211C22⋊Q811yescharacteristic
96C4×C211C22⋊Q811yescharacteristic
97C4×C211#12411yescharacteristic
98C4×C211#12411yescharacteristic
99Dic211—22no—
100Dic211—22no—
101Dic211—42no—
102Dic211—81no—
103C2×C211#39311yescharacteristic
104C2×C211#45011yescharacteristic
105C2×C211#45111yescharacteristic
106C2×C211#47511yescharacteristic
107C2×C211#47711yescharacteristic
108C2×C211#48711yescharacteristic
109C2×C211#48911yescharacteristic
110C211#202411yescharacteristic, complemented, Sylow
111#2024—2111nomaximal, Sylow
112C2×C4⋊Q8—2111no—
113C2×#103—2111no—
114C2×#103D21111yescharacteristic
115#391—2111no—
116#340—2111no—
117#340—2111no—
118#336—2111no—
119C22×Q8—4221no—
120C22×Q8C2×D21111yescharacteristic
121C22×C8—2111no—
122C22×C8C2×D21111yescharacteristic
123C4⋊Q8—4222no—
124C4⋊Q8—4222no—
125C2×C4⋊C4—4221no—
126C2×C4⋊C4—2111no—
127C2×C4⋊C4—2111no—
128C2×C4⋊C4C2×D21111yescharacteristic
129C2×C42—2111no—
130#103—22no—
131#103—4222no—
132#95—4221no—
133#95—4221no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—8442no—
140C2×Q8—8442no—
141C2×Q8—4224no—
142C2×Q8Dic42212yescomplemented
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—4221no—
146C22×C4—4221no—
147C22×C4—2111no—
148C22×C4—2111no—
149C22×C4C22×D21111yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—4221no—
154C2×C8—4221no—
155C2×C8—4221no—
156C4⋊C4—21no—
157C4⋊C4—4224no—
158C4⋊C4—4221no—
159C4⋊C4—4221no—
160C4⋊C4—4221no—
161C4⋊C4—2112no—
162C4⋊C4C22⋊D21112yes—
163C42—4221no—
164C42—4221no—
165C23D4×D21111yescharacteristic
166Q8—22no—
167Q8—22no—
168Q8—41no—
169Q8—41no—
170Q8—8448no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—8441no—
178C2×C4—8441no—
179C2×C4—8441no—
180C2×C4—8441no—
181C2×C4—8441no—
182C2×C4—4222no—
183C2×C4—4221no—
184C2×C4—4221no—
185C2×C4—4221no—
186C2×C4—4221no—
187C2×C4—2112no—
188C2×C4—2112no—
189C2×C4D4×D21111yescharacteristic
190C2×C4C2×C22⋊D21111yescharacteristic
191C2×C4C2×Dic42211yescharacteristic
192C2×C4#1680996111yescharacteristic
193C2×C4#1680996111yescharacteristic
194C2×C4#1680996211yescharacteristic
195C8—41no—
196C8—8441no—
197C22#2026994511yescharacteristic
198C22#2026995411yescharacteristic
199C22#2026996311yescharacteristic
200C22SD16×D21111yescharacteristic
201C22Q16×D21111yescharacteristic
202C22#2027000411yescharacteristic
203C22#2027000911yescharacteristic
204C4—42no—
205C4—42no—
206C4—8442no—
207C4—16881no—
208C4—4222no—
209C4—4222no—
210C4#2026995411yescharacteristic
211C4#2026995611yescharacteristic
212C4#2026996311yescharacteristic
213C4#2026996411yescharacteristic
214C2#2721305811yescharacteristic
215C2#2721329511yescharacteristic
216C2#2721330011yescharacteristic
217C2#2721342911yescharacteristic
218C2#2721343911yescharacteristic
219C2#2721349511yescharacteristic
220C2#2721350711yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#103×C211C2
socle, prime radicalC23×C211C2×D4
hypercenterC2×#103D211
hypocenterC211#2024
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
45. C2×Q8×C211Q8semidirect
110. C211#2024semidirect
142. C2×Q8Dic422semidirect

Maximal subgroups

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

subgroupindex
C2×#202699762
C2×#202699012
#272130532
#272129092
#272129082
#272129002
C2×#103×C2112
#2024211