Finite Group Databasebeta

← #41014694

Subgroups — #41014694

Order 27008.

262 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 71 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×#20269912C211yesmaximal, characteristic, complemented
2#27213147C211yesmaximal, characteristic
3C2×#20269901C211yesmaximal, characteristic, complemented
4C2×#20269896C211yesmaximal, characteristic, complemented
5#27213113C211yesmaximal, characteristic
6#27212914C211yesmaximal, characteristic, complemented
7C2×#137×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C22×C211⋊C8C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12#20269912—22no—
13#20269901—22no—
14#20269901—21no—
15#20269898—21no—
16#20269896—22no—
17C22⋊(C211⋊C8)—22no—
18C2×#41×C211C2211yescharacteristic
19C22×Q8×C211C2211yescharacteristic
20#137×C211—22no—
21#137×C211—22no—
22C2×Q16×C211—21no—
23C2×SD16×C211—21no—
24C2×M16×C211C2211yescharacteristic
25C22×Dic211—21no—
26#16809946—21no—
27#16809946—21no—
28#16809946—21no—
29#16809946—21no—
30C2×C211⋊C8—21no—
31C2×C211⋊C8—21no—
32C2×C211⋊C8—21no—
33C2×C211⋊C8—21no—
34C2×C211⋊C8D412yes—
35C4⋊Dic211—21no—
36C4⋊Dic211—21no—
37#41×C211—21no—
38#41×C211—21no—
39#41×C211—21no—
40#41×C211—21no—
41C2×Q8×C211—21no—
42C2×Q8×C211—21no—
43C2×Q8×C211—21no—
44C2×Q8×C211—21no—
45C2×Q8×C211—21no—
46C2×Q8×C211D411yescharacteristic
47C2×Q8×C211D412yes—
48C2×D4×C211—21no—
49C2×D4×C211D411yescharacteristic
50C22×C4×C211—21no—
51C22×C4×C211—21no—
52C22×C4×C211C2311yescharacteristic
53Q16×C211—24no—
54SD16×C211—24no—
55M16×C211—21no—
56M16×C211D412yes—
57C2×C8×C211—21no—
58C2×Dic211—41no—
59C2×Dic211—81no—
60C211⋊C8—22no—
61C211⋊C8—41no—
62C211⋊C8—41no—
63C23×C211—21no—
64C23×C211#4111yescharacteristic
65Q8×C211—22no—
66Q8×C211—21no—
67Q8×C211—22no—
68Q8×C211—21no—
69Q8×C211—41no—
70Q8×C211—41no—
71D4×C211—21no—
72D4×C211—21no—
73D4×C211—41no—
74D4×C211—41no—
75C2×C4×C211—21no—
76C2×C4×C211—22no—
77C2×C4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211—41no—
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×D411yescharacteristic
85C2×C4×C211#4111yescharacteristic
86C2×C4×C211#4111yescharacteristic
87C8×C211—22no—
88C22×C211—21no—
89C22×C211—21no—
90C22×C211—41no—
91C22×C211—41no—
92C22×C211—41no—
93C22×C211#12111yescharacteristic
94C22×C211#12311yescharacteristic
95C22×C211#12411yescharacteristic
96C4×C211—41no—
97C4×C211—42no—
98C4×C211#12011yescharacteristic
99C4×C211#12111yescharacteristic
100C4×C211#12111yescharacteristic
101C4×C211#12311yescharacteristic
102Dic211—81no—
103C2×C211—21no—
104C2×C211—21no—
105C2×C211—81no—
106C2×C211#39411yescharacteristic
107C2×C211#47111yescharacteristic
108C2×C211#47311yescharacteristic
109C211#205211yescharacteristic, complemented, Sylow
110#2052—2111nomaximal, Sylow
111C2×#137D21111yescharacteristic
112C2×#108—2111no—
113#417—2111no—
114C2×#103—2111no—
115C2×#101—2111no—
116#408—2111no—
117#343—2111no—
118C2×#41C2×D21111yescharacteristic
119C22×Q8C2×D21111yescharacteristic
120#137—22no—
121#137—22no—
122C2×Q16—21no—
123C2×SD16—21no—
124C2×M16—2111no—
125C2×M16—2111no—
126C2×M16C2×D21111yescharacteristic
127C22×C8—2111no—
128C2×C4⋊C4—2111no—
129#108—4222no—
130#103—4221no—
131#103—4222no—
132#102—4221no—
133#101—4222no—
134C22⋊C8—4222no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8C22⋊D21112yes—
145C2×Q8C22⋊D21111yescharacteristic
146C2×D4—21no—
147C2×D4C22⋊D21111yescharacteristic
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—4221no—
151C22×C4C22×D21111yescharacteristic
152Q16—24no—
153SD16—24no—
154M16—21no—
155M16—4221no—
156M16—4221no—
157M16—4221no—
158M16—4221no—
159M16C22⋊D21112yes—
160C2×C8—21no—
161C2×C8—4221no—
162C2×C8—4221no—
163C2×C8—4221no—
164C2×C8—4221no—
165C2×C8—2112no—
166C4⋊C4—4221no—
167C4⋊C4—4221no—
168C23—21no—
169C23#1680996111yescharacteristic
170Q8—22no—
171Q8—21no—
172Q8—22no—
173Q8—21no—
174Q8—41no—
175Q8—41no—
176D4—21no—
177D4—21no—
178D4—41no—
179D4—41no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—8441no—
187C2×C4—16881no—
188C2×C4D4×D21111yescharacteristic
189C2×C4C2×C22⋊D21111yescharacteristic
190C2×C4C2×C22⋊D21111yescharacteristic
191C2×C4C2×C22⋊D21111yescharacteristic
192C2×C4#1680996111yescharacteristic
193C2×C4#1680996111yescharacteristic
194C8—22no—
195C8—8441no—
196C8—8441no—
197C8—4222no—
198C22—21no—
199C22—21no—
200C22—41no—
201C22—41no—
202C22—41no—
203C22#2026994911yescharacteristic
204C22#2026995811yescharacteristic
205C22#2026996411yescharacteristic
206C4—41no—
207C4—42no—
208C4—16881no—
209C4#2026994611yescharacteristic
210C4#2026994811yescharacteristic
211C4#2026994911yescharacteristic
212C4#2026995811yescharacteristic
213C2—21no—
214C2—21no—
215C2—81no—
216C2#2721306011yescharacteristic
217C2#2721341111yescharacteristic
218C2#2721341911yescharacteristic
subgroupG/H
centerC22#20269949
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#137×C211C2
socleC22×C211#121
prime radicalC2×#41×C211C22
hypercenterC2×#137D211
hypocenterC211#2052
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#20269912C2semidirect
3. C2×#20269901C2semidirect
4. C2×#20269896C2semidirect
6. #27212914C2semidirect
109. C211#2052semidirect

Maximal subgroups

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

subgroupindex
C2×#202699122
#272131472
C2×#202699012
C2×#202698962
#272131132
#272129142
C2×#137×C2112
#2052211