Finite Group Databasebeta

← #41014697

Subgroups — #41014697

Order 27008.

262 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 67 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
2C2×#20269912C211yesmaximal, characteristic, complemented
3#27213148C211yesmaximal, characteristic
4C2×#20269901C211yesmaximal, characteristic, complemented
5#27213114C211yesmaximal, characteristic
6#27212914C211yesmaximal, characteristic, complemented
7C2×#101×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20270008—22no—
10#20270008—22no—
11C2×#16809952—21no—
12C2×#16809949—21no—
13C2×#16809946C2211yescharacteristic
14C22×C211⋊C8C2211yescharacteristic
15C2×C4⋊Dic211C2211yescharacteristic
16#20269912—22no—
17#20269902—21no—
18#20269901—22no—
19#20269899—21no—
20(C2×D211)⋊C8—22no—
21C22×Q8×C211C2211yescharacteristic
22C2×M16×C211C2211yescharacteristic
23C2×M16×C211C2211yescharacteristic
24#101×C211—22no—
25#16809962—21no—
26#16809962—21no—
27#16809962—21no—
28#16809962—21no—
29C2×Dic422D411yescharacteristic
30C2×D844D411yescharacteristic
31C2×C22⋊D211—21no—
32C22×Dic211—21no—
33C2×C4×D211—21no—
34#16809952—24no—
35#16809949—24no—
36#16809946—21no—
37#16809946D412yes—
38C2×C211⋊C8—21no—
39C2×C211⋊C8—21no—
40C2×C211⋊C8—21no—
41C2×C211⋊C8D412yes—
42C4⋊Dic211—21no—
43C4⋊Dic211—21no—
44C2×Q8×C211—21no—
45C2×Q8×C211—21no—
46C2×Q8×C211—21no—
47C2×Q8×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211D412yes—
50C22×C4×C211—21no—
51C22×C4×C211C2311yescharacteristic
52M16×C211—21no—
53M16×C211—21no—
54M16×C211—21no—
55M16×C211—21no—
56C2×C8×C211—21no—
57C2×C8×C211—21no—
58C22×D211—21no—
59Dic422—21no—
60Dic422—21no—
61D844—21no—
62D844—21no—
63C22⋊D211—41no—
64C22⋊D211—41no—
65C2×Dic211—21no—
66C2×Dic211—41no—
67C2×Dic211—81no—
68C4×D211—41no—
69C4×D211—41no—
70C211⋊C8—22no—
71C211⋊C8—22no—
72C23×C211#4111yescharacteristic
73Q8×C211—22no—
74Q8×C211—22no—
75Q8×C211—41no—
76Q8×C211—41no—
77C2×C4×C211—22no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C2×C4×C211#4111yescharacteristic
86C8×C211—41no—
87C8×C211—41no—
88C22×C211—21no—
89C22×C211—21no—
90C22×C211#12111yescharacteristic
91C22×C211#12311yescharacteristic
92C22×C211#12411yescharacteristic
93C2×D211—41no—
94C2×D211—41no—
95C2×D211—41no—
96C4×C211—42no—
97C4×C211#12011yescharacteristic
98C4×C211#12111yescharacteristic
99C4×C211#12111yescharacteristic
100C4×C211#12311yescharacteristic
101Dic211—41no—
102Dic211—81no—
103C2×C211—21no—
104C2×C211—21no—
105C2×C211#39411yescharacteristic
106C2×C211#47111yescharacteristic
107C2×C211#47311yescharacteristic
108D211—81no—
109C211#205211yescharacteristic, complemented, Sylow
110#2052—2111nomaximal, Sylow
111C2×#137—2111no—
112C2×#108—2111no—
113#417—2111no—
114C2×#103—2111no—
115C2×#101D21111yescharacteristic, complemented
116#408—2111no—
117#343—2111no—
118C2×#41—2111no—
119C22×Q8C2×D21111yescharacteristic
120#137—4222no—
121#137—4222no—
122C2×Q16—4221no—
123C2×SD16—4221no—
124C2×M16—2111no—
125C2×M16C2×D21111yescharacteristic
126C2×M16C2×D21111yescharacteristic
127C22×C8—2111no—
128C2×C4⋊C4—2111no—
129#108—4222no—
130#103—4221no—
131#103—4222no—
132#102—4221no—
133#101—22no—
134C22⋊C8—4222no—
135#41—4221no—
136#41—4221no—
137#41—4221no—
138#41—4221no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—2111no—
145C2×Q8C22⋊D21112yes—
146C2×D4—4221no—
147C2×D4—2111no—
148C22×C4—21no—
149C22×C4—4221no—
150C22×C4—4221no—
151C22×C4C22×D21111yescharacteristic
152Q16—4224no—
153SD16—4224no—
154M16—21no—
155M16—21no—
156M16—21no—
157M16—21no—
158M16—4221no—
159M16—2112no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—4221no—
163C2×C8—4221no—
164C2×C8—4221no—
165C2×C8—2112no—
166C4⋊C4—4221no—
167C4⋊C4—4221no—
168C23—4221no—
169C23#1680996211yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—41no—
173Q8—41no—
174Q8—4221no—
175Q8—4221no—
176D4—8441no—
177D4—8441no—
178D4—4221no—
179D4—4221no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—8441no—
184C2×C4—8441no—
185C2×C4—8441no—
186C2×C4—16881no—
187C2×C4—4221no—
188C2×C4D4×D21111yescharacteristic
189C2×C4D4×D21111yescharacteristic
190C2×C4D4×D21111yescharacteristic
191C2×C4C2×C22⋊D21111yescharacteristic
192C2×C4#1680996111yescharacteristic
193C2×C4#1680996211yescharacteristic
194C8—41no—
195C8—41no—
196C8—4222no—
197C8—4222no—
198C22—21no—
199C22—21no—
200C22—8441no—
201C22—8441no—
202C22—8441no—
203C22#2026995011yescharacteristic
204C22#2026996011yescharacteristic
205C22#2026996311yescharacteristic
206C4—42no—
207C4—8441no—
208C4—16881no—
209C4#2026994511yescharacteristic
210C4#2026994911yescharacteristic
211C4#2026995011yescharacteristic
212C4#2026995911yescharacteristic
213C2—21no—
214C2—21no—
215C2—16881no—
216C2#2721306111yescharacteristic
217C2#2721341211yescharacteristic
218C2#2721342011yescharacteristic
subgroupG/H
centerC22#20269950
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#101×C211C2
socleC22×C211#121
prime radicalC2×#16809962C22
hypercenterC2×#101D211
hypocenterC211#2052
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×#20269912C2semidirect
4. C2×#20269901C2semidirect
6. #27212914C2semidirect
7. C2×#101×C211C2semidirect
109. C211#2052semidirect
115. C2×#101D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700082
C2×#202699122
#272131482
C2×#202699012
#272131142
#272129142
C2×#101×C2112
#2052211