Finite Group Databasebeta

← #41013868

Subgroups — #41013868

Order 27008.

304 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 79 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×#20270008C211yesmaximal, characteristic
2#27213912C211yesmaximal, characteristic
3#27213164C211yesmaximal, characteristic
4C2×#20269896C211yesmaximal, characteristic, complemented
5#27212916C211yesmaximal, characteristic, complemented
6C211⋊#338C211yesmaximal, characteristic, complemented
7#419×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20270008—22no—
10#20270008—22no—
11C2×#16809952C412yescomplemented
12C2×#16809949C412yescomplemented
13#20269985—22no—
14#20269983—22no—
15C2×#16809946C2211yescharacteristic
16C2×#16809946C2211yescharacteristic
17C2×#16809946C2211yescharacteristic
18M16×D211—22no—
19#20269905—22no—
20#20269896—22no—
21C22×Q8×C211C2211yescharacteristic
22C2×M16×C211C2211yescharacteristic
23#117×C211C2211yescharacteristic
24#103×C211—22no—
25#16809962—21no—
26#16809962—21no—
27#16809962D412yes—
28#16809962D412yes—
29C2×Dic422C2×C411yescharacteristic
30C2×D844C2×C411yescharacteristic
31C2×C22⋊D211—21no—
32C2×C4×D211—21no—
33#16809952—24no—
34#16809949—24no—
35#16809946—21no—
36#16809946—21no—
37#16809946—21no—
38#16809946—21no—
39#16809946—21no—
40#16809946D412yes—
41#16809945—24no—
42C2×C211⋊C8—21no—
43C2×C211⋊C8—21no—
44C2×C211⋊C8C2×C412yes—
45C8×D211—24no—
46C2×Q8×C211—21no—
47C2×Q8×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211—21no—
50C2×Q8×C211—21no—
51C2×Q8×C211C2×C412yes—
52C22×C4×C211—21no—
53C22×C4×C211C2311yescharacteristic
54M16×C211—21no—
55M16×C211—21no—
56C2×C8×C211D412yes—
57C4⋊C4×C211—21no—
58C22⋊C4×C211—21no—
59C42×C211—21no—
60C22×D211—21no—
61Dic422—21no—
62Dic422C22⋊C412yes—
63D844—21no—
64D844C22⋊C412yes—
65C22⋊D211—22no—
66C22⋊D211—22no—
67C2×Dic211—21no—
68C4×D211—22no—
69C4×D211—22no—
70C211⋊C8—22no—
71C211⋊C8—22no—
72C211⋊C8—41no—
73C23×C211#4111yescharacteristic
74Q8×C211—22no—
75Q8×C211—22no—
76Q8×C211—41no—
77Q8×C211—41no—
78C2×C4×C211—22no—
79C2×C4×C211—41no—
80C2×C4×C211—41no—
81C2×C4×C211—41no—
82C2×C4×C211C22×C411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×D411yescharacteristic
85C2×C4×C211C2×D411yescharacteristic
86C2×C4×C211C2×D411yescharacteristic
87C2×C4×C211#4111yescharacteristic
88C8×C211—22no—
89C22×C211—41no—
90C22×C211C4×D411yescharacteristic
91C22×C211#12111yescharacteristic
92C22×C211#12311yescharacteristic
93C2×D211—22no—
94C2×D211—81no—
95C4×C211—42no—
96C4×C211—81no—
97C4×C211C2×C22⋊C411yescharacteristic
98C4×C211C4×D411yescharacteristic
99C4×C211#12011yescharacteristic
100C4×C211#12111yescharacteristic
101Dic211—22no—
102C2×C211—21no—
103C2×C211—21no—
104C2×C211—21no—
105C2×C211#38611yescharacteristic
106D211—42no—
107C211#190211yescharacteristic, complemented, Sylow
108#1902—2111nomaximal, Sylow
109C2×#137—2111no—
110#568—2111no—
111#421—2111no—
112#419D21111yescharacteristic, complemented
113C2×#101—2111no—
114#344—2111no—
115#338—2111no—
116C2×#41—2111no—
117C22×Q8C2×D21111yescharacteristic
118#137—4222no—
119#137—4222no—
120C2×Q16—2112no—
121C2×SD16—2112no—
122#131—4222no—
123#131—4222no—
124C2×M16—4222no—
125C2×M16—2111no—
126C2×M16—2111no—
127C2×M16—2111no—
128C2×M16C2×D21111yescharacteristic
129#117C2×D21111yescharacteristic
130#104—4222no—
131#103—22no—
132#101—4222no—
133#41—4221no—
134#41—4221no—
135#41—2112no—
136#41—2112no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—2111no—
143C2×Q8C4×D21112yes—
144C2×D4—4221no—
145C2×D4—2111no—
146C22×C4—21no—
147C22×C4—4221no—
148C22×C4C22×D21111yescharacteristic
149Q16—4224no—
150SD16—4224no—
151M16—21no—
152M16—21no—
153M16—4224no—
154M16—4221no—
155M16—4221no—
156M16—4221no—
157M16—4221no—
158M16—4221no—
159M16—2112no—
160C2×C8—4224no—
161C2×C8—4221no—
162C2×C8—4221no—
163C2×C8—2112no—
164C2×C8C22⋊D21112yes—
165C4⋊C4—21no—
166C22⋊C4—21no—
167C42—21no—
168C23—4221no—
169C23#1680996111yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—41no—
173Q8—41no—
174Q8—4221no—
175Q8—2112no—
176D4—4222no—
177D4—4222no—
178D4—4221no—
179D4—2112no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—4221no—
185C2×C4—4222no—
186C2×C4—4222no—
187C2×C4C2×C4×D21111yescharacteristic
188C2×C4D4×D21111yescharacteristic
189C2×C4D4×D21111yescharacteristic
190C2×C4D4×D21111yescharacteristic
191C2×C4C2×C22⋊D21111yescharacteristic
192C2×C4#1680996211yescharacteristic
193C8—22no—
194C8—8441no—
195C8—4222no—
196C8—4222no—
197C22—41no—
198C22—16881no—
199C22—4222no—
200C22(C22⋊D211)⋊C411yescharacteristic
201C22#2026994811yescharacteristic
202C22#2026995911yescharacteristic
203C4—42no—
204C4—81no—
205C4—4222no—
206C4C22⋊C4×D21111yescharacteristic
207C4C4×C22⋊D21111yescharacteristic
208C4#2026994511yescharacteristic
209C4#2026995011yescharacteristic
210C2—21no—
211C2—21no—
212C2—21no—
213C2—8442no—
214C2#2721302211yescharacteristic
subgroupG/H
centerC2#27213022
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#419×C211C2
socleC2×C211#386
prime radicalC2×#16809962C22
cosocleC22×C4×C211C23
hypercenter#419D211
hypocenterC211#1902
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
4. C2×#20269896C2semidirect
5. #27212916C2semidirect
6. C211⋊#338C2semidirect
7. #419×C211C2semidirect
11. C2×#16809952C4semidirect
12. C2×#16809949C4semidirect
107. C211#1902semidirect
112. #419D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700082
#272139122
#272131642
C2×#202698962
#272129162
C211⋊#3382
#419×C2112
#1902211