Finite Group Databasebeta

← #41014675

Subgroups — #41014675

Order 27008.

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

normal subgroups only

This group has 254 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
1#27213194C211yesmaximal, characteristic, complemented
2#27213163C211yesmaximal, characteristic
3#27213155C211yesmaximal, characteristic, complemented
4#27213130C211yesmaximal, characteristic
5C2×#20269896C211yesmaximal, characteristic, complemented
6#27212918C211yesmaximal, characteristic, complemented
7C2×#137×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11#20269936C2211yescharacteristic
12#20269912—22no—
13#20269904—22no—
14#20269901—22no—
15#20269896—22no—
16#20269896—21no—
17#20269894—21no—
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
25#16809946—21no—
26#16809946—21no—
27#16809946—21no—
28#16809946—21no—
29#16809946—21no—
30#16809946D412yes—
31C2×C211⋊C8—21no—
32C2×C211⋊C8—21no—
33C2×C211⋊C8—21no—
34C4⋊Dic211—21no—
35C22⋊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—
59C211⋊C8—22no—
60C211⋊C8—41no—
61C211⋊C8—41no—
62C23×C211—21no—
63C23×C211#4111yescharacteristic
64Q8×C211—22no—
65Q8×C211—21no—
66Q8×C211—21no—
67Q8×C211—22no—
68Q8×C211—41no—
69Q8×C211—41no—
70D4×C211—21no—
71D4×C211—21no—
72D4×C211—41no—
73D4×C211—41no—
74C2×C4×C211—22no—
75C2×C4×C211—21no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
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—22no—
87C22×C211—41no—
88C22×C211—41no—
89C22×C211—81no—
90C22×C211#12111yescharacteristic
91C22×C211#12311yescharacteristic
92C22×C211#12411yescharacteristic
93C4×C211—41no—
94C4×C211—42no—
95C4×C211#12011yescharacteristic
96C4×C211#12111yescharacteristic
97C4×C211#12111yescharacteristic
98C4×C211#12311yescharacteristic
99Dic211—81no—
100C2×C211—21no—
101C2×C211—21no—
102C2×C211—21no—
103C2×C211—81no—
104C2×C211#39411yescharacteristic
105C211#204911yescharacteristic, complemented, Sylow
106#2049—2111nomaximal, Sylow
107C2×#137D21111yescharacteristic
108#430—2111no—
109#421—2111no—
110#419—2111no—
111#413—2111no—
112C2×#101—2111no—
113#344—2111no—
114C2×#41C2×D21111yescharacteristic
115C22×Q8C2×D21111yescharacteristic
116#137—22no—
117#137—22no—
118C2×Q16—21no—
119C2×SD16—21no—
120C2×M16—2111no—
121C2×M16—2111no—
122C2×M16—2111no—
123C2×M16C2×D21111yescharacteristic
124#117—2111no—
125#108—4222no—
126#104—4222no—
127#103—4222no—
128#101—4221no—
129#101—4222no—
130#100—4221no—
131#41—21no—
132#41—21no—
133#41—21no—
134#41—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8C22⋊D21111yescharacteristic
141C2×Q8C22⋊D21112yes—
142C2×D4—21no—
143C2×D4C22⋊D21111yescharacteristic
144C22×C4—21no—
145C22×C4—21no—
146C22×C4C22×D21111yescharacteristic
147Q16—24no—
148SD16—24no—
149M16—21no—
150M16—4221no—
151M16—4221no—
152M16—4221no—
153M16—4221no—
154M16—4221no—
155M16—2112no—
156M16C22⋊D21112yes—
157C2×C8—21no—
158C2×C8—4221no—
159C2×C8—4221no—
160C2×C8—4221no—
161C4⋊C4—4221no—
162C22⋊C4—4221no—
163C42—4221no—
164C23—21no—
165C23#1680996111yescharacteristic
166Q8—22no—
167Q8—21no—
168Q8—21no—
169Q8—22no—
170Q8—41no—
171Q8—41no—
172D4—21no—
173D4—21no—
174D4—41no—
175D4—41no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—8441no—
183C2×C4D4×D21111yescharacteristic
184C2×C4C2×C22⋊D21111yescharacteristic
185C2×C4C2×C22⋊D21111yescharacteristic
186C2×C4C2×C22⋊D21111yescharacteristic
187C2×C4#1680996111yescharacteristic
188C2×C4#1680996111yescharacteristic
189C8—22no—
190C8—8441no—
191C8—8441no—
192C8—4222no—
193C22—41no—
194C22—41no—
195C22—81no—
196C22#2026994911yescharacteristic
197C22#2026995811yescharacteristic
198C22#2026996411yescharacteristic
199C4—41no—
200C4—42no—
201C4—16881no—
202C4#2026994611yescharacteristic
203C4#2026994811yescharacteristic
204C4#2026994911yescharacteristic
205C4#2026995811yescharacteristic
206C2—21no—
207C2—21no—
208C2—21no—
209C2—81no—
210C2#2721306011yescharacteristic
subgroupG/H
centerC2#27213060
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#137×C211C2
socleC2×C211#394
prime radicalC2×#41×C211C22
hypercenterC2×#137D211
hypocenterC211#2049
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. #27213194C2semidirect
3. #27213155C2semidirect
5. C2×#20269896C2semidirect
6. #27212918C2semidirect
105. C211#2049semidirect

Maximal subgroups

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

subgroupindex
#272131942
#272131632
#272131552
#272131302
C2×#202698962
#272129182
C2×#137×C2112
#2049211