Finite Group Databasebeta

← #41014673

Subgroups — #41014673

Order 27008.

274 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 65 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×#20270005C211yesmaximal, characteristic
2#27213162C211yesmaximal, characteristic, complemented
3#27213152C211yesmaximal, characteristic, complemented
4#27213128C211yesmaximal, characteristic, complemented
5C2×#20269893C211yesmaximal, characteristic, complemented
6#27212918C211yesmaximal, characteristic, complemented
7#430×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C22×D844C2211yescharacteristic
10#20270005—22no—
11#20270005—22no—
12C2×#16809951—21no—
13C2×D1688—21no—
14C2×#16809946C2211yescharacteristic
15#20269936C2211yescharacteristic
16#20269903—22no—
17#20269899—22no—
18#20269895—21no—
19#20269893—21no—
20#20269893—22no—
21C2×M16×C211C2211yescharacteristic
22C2×M16×C211C2211yescharacteristic
23C2×M16×C211C2211yescharacteristic
24#108×C211—22no—
25C23×D211—21no—
26#16809962—21no—
27#16809962—21no—
28#16809962—21no—
29#16809962—21no—
30C2×Dic422D411yescharacteristic
31C2×D844—21no—
32C2×D844—21no—
33C2×D844—21no—
34C2×D844—21no—
35C2×D844—21no—
36C2×D844D411yescharacteristic
37C2×D844D412yes—
38C2×C22⋊D211—21no—
39C2×C4×D211—21no—
40#16809951—24no—
41D1688—24no—
42#16809946—21no—
43#16809946—21no—
44C2×C211⋊C8—21no—
45C4⋊Dic211—21no—
46C22⋊Dic211—21no—
47C4×Dic211—21no—
48C22×C4×C211C2311yescharacteristic
49M16×C211—21no—
50M16×C211—21no—
51M16×C211—21no—
52M16×C211—21no—
53M16×C211D412yes—
54M16×C211D412yes—
55C2×C8×C211—21no—
56C2×C8×C211—21no—
57C2×C8×C211—21no—
58C22×D211—22no—
59C22×D211—21no—
60C22×D211—41no—
61C22×D211—41no—
62C22×D211—82no—
63Dic422—21no—
64Dic422—21no—
65D844—21no—
66D844—22no—
67D844—21no—
68D844—22no—
69D844—41no—
70D844—41no—
71C22⋊D211—41no—
72C22⋊D211—41no—
73C2×Dic211—21no—
74C2×Dic211—41no—
75C4×D211—41no—
76C4×D211—41no—
77C211⋊C8—41no—
78C23×C211#4111yescharacteristic
79C2×C4×C211C2×D411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211#4111yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C8×C211—22no—
86C8×C211—22no—
87C8×C211—41no—
88C22×C211—41no—
89C22×C211#12111yescharacteristic
90C22×C211#12311yescharacteristic
91C22×C211#12411yescharacteristic
92C2×D211—41no—
93C2×D211—42no—
94C2×D211—82no—
95C2×D211—82no—
96C2×D211—81no—
97C2×D211—82no—
98C4×C211#12011yescharacteristic
99C4×C211#12111yescharacteristic
100C4×C211#12111yescharacteristic
101C4×C211#12311yescharacteristic
102Dic211—41no—
103Dic211—81no—
104C2×C211—21no—
105C2×C211—21no—
106C2×C211—21no—
107C2×C211#39411yescharacteristic
108D211—81no—
109D211—82no—
110C211#204811yescharacteristic, complemented, Sylow
111#2048—2111nomaximal, Sylow
112C2×AGL(1,ℤ/8)—2111no—
113#430D21111yescharacteristic, complemented
114#421—2111no—
115#418—2111no—
116#413—2111no—
117C2×#100—2111no—
118#344—2111no—
119C2×#41—2111no—
120C22×D4—2111no—
121AGL(1,ℤ/8)—4222no—
122AGL(1,ℤ/8)—4222no—
123C2×SD16—4221no—
124C2×D8—4221no—
125C2×M16—2111no—
126C2×M16C2×D21111yescharacteristic
127C2×M16C2×D21111yescharacteristic
128C2×M16C2×D21111yescharacteristic
129#117—2111no—
130#108—22no—
131#104—4222no—
132#102—4222no—
133#101—4221no—
134#100—4221no—
135#100—4222no—
136C24—4221no—
137#41—4221no—
138#41—4221no—
139#41—4221no—
140#41—4221no—
141C2×Q8—2111no—
142C2×D4—4221no—
143C2×D4—4221no—
144C2×D4—4221no—
145C2×D4—4221no—
146C2×D4—4221no—
147C2×D4—4221no—
148C2×D4—2111no—
149C2×D4—2112no—
150C22×C4—4221no—
151C22×C4C22×D21111yescharacteristic
152SD16—4224no—
153D8—4224no—
154M16—21no—
155M16—21no—
156M16—21no—
157M16—21no—
158M16—4221no—
159M16—4221no—
160M16C22⋊D21112yes—
161M16D84412yes—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—4221no—
166C4⋊C4—4221no—
167C22⋊C4—4221no—
168C42—4221no—
169C23—8441no—
170C23—8441no—
171C23—16882no—
172C23—4222no—
173C23—4221no—
174C23#1680996311yescharacteristic
175Q8—4221no—
176Q8—4221no—
177D4—8441no—
178D4—8441no—
179D4—8441no—
180D4—8441no—
181D4—4221no—
182D4—4222no—
183D4—4221no—
184D4—4222no—
185C2×C4—8441no—
186C2×C4—8441no—
187C2×C4—8441no—
188C2×C4—4221no—
189C2×C4D4×D21111yescharacteristic
190C2×C4D4×D21111yescharacteristic
191C2×C4C2×C22⋊D21111yescharacteristic
192C2×C4C2×D84411yescharacteristic
193C2×C4#1680996211yescharacteristic
194C2×C4#1680996311yescharacteristic
195C8—22no—
196C8—22no—
197C8—41no—
198C8—8441no—
199C22—41no—
200C22—8441no—
201C22—8442no—
202C22—16882no—
203C22—16881no—
204C22—16882no—
205C22—16882no—
206C22#2026995211yescharacteristic
207C22#2026996211yescharacteristic
208C22#2026996611yescharacteristic
209C4—8441no—
210C4—16881no—
211C4#2026994511yescharacteristic
212C4#2026995111yescharacteristic
213C4#2026995211yescharacteristic
214C4#2026996211yescharacteristic
215C2—21no—
216C2—21no—
217C2—21no—
218C2—16882no—
219C2—16881no—
220C2#2721306311yescharacteristic
subgroupG/H
centerC2#27213063
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#430×C211C2
socleC2×C211#394
prime radicalC2×#20270005C2
hypercenter#430D211
hypocenterC211#2048
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C22×C4 ◁ #430

Complements

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

NH ≅ G/Ntype
2. #27213162C2semidirect
3. #27213152C2semidirect
4. #27213128C2semidirect
5. C2×#20269893C2semidirect
6. #27212918C2semidirect
7. #430×C211C2semidirect
110. C211#2048semidirect
113. #430D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700052
#272131622
#272131522
#272131282
C2×#202698932
#272129182
#430×C2112
#2048211