Finite Group Databasebeta

← #41014065

Subgroups — #41014065

Order 27008.

216 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 69 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#27213164C211yesmaximal, characteristic
3C2×#20269903C211yesmaximal, characteristic
4#27213128C211yesmaximal, characteristic
5M16×Dic211C211yesmaximal, characteristic, complemented
6#27212905C211yesmaximal, characteristic, complemented
7#344×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11C2×C4×Dic211C2211yescharacteristic
12#20269912—22no—
13#20269905—22no—
14#20269903—21no—
15#20269903—21no—
16#20269895C412yescomplemented
17#20269893C412yescomplemented
18#20269887—21no—
19C8×Dic211—21no—
20C2×M16×C211C2211yescharacteristic
21C2×M16×C211C2211yescharacteristic
22#117×C211C2211yescharacteristic
23#16809962—21no—
24#16809962—21no—
25#16809962—21no—
26#16809962—21no—
27C2×Dic422C2×C411yescharacteristic
28C2×D844C2×C411yescharacteristic
29C2×C22⋊D211—21no—
30C22×Dic211—21no—
31C2×C4×D211—21no—
32#16809946—21no—
33#16809946—21no—
34#16809946C2×C412yes—
35#16809946D412yes—
36C2×C211⋊C8—21no—
37C2×C211⋊C8—21no—
38C4×Dic211—21no—
39C4×Dic211D411yescharacteristic
40C4×Dic211D411yescharacteristic
41C22×C4×C211C2311yescharacteristic
42M16×C211—21no—
43M16×C211—21no—
44M16×C211—21no—
45M16×C211C2×C412yes—
46C2×C8×C211—21no—
47C2×C8×C211—21no—
48C4⋊C4×C211—21no—
49C22⋊C4×C211—21no—
50C42×C211—21no—
51C22×D211—21no—
52Dic422—21no—
53Dic422—21no—
54D844—21no—
55D844—21no—
56C22⋊D211—41no—
57C22⋊D211—41no—
58C2×Dic211—21no—
59C2×Dic211—21no—
60C2×Dic211—21no—
61C2×Dic211—42no—
62C4×D211—41no—
63C4×D211—41no—
64C211⋊C8—22no—
65C211⋊C8—22no—
66C23×C211#4111yescharacteristic
67C2×C4×C211—41no—
68C2×C4×C211C22×C411yescharacteristic
69C2×C4×C211C2×D411yescharacteristic
70C2×C4×C211C2×D411yescharacteristic
71C2×C4×C211#4111yescharacteristic
72C2×C4×C211#4111yescharacteristic
73C2×C4×C211#4111yescharacteristic
74C8×C211—22no—
75C8×C211—41no—
76C22×C211—41no—
77C22×C211#11711yescharacteristic
78C22×C211#12311yescharacteristic
79C22×C211#12411yescharacteristic
80C2×D211—41no—
81C2×D211—81no—
82C4×C211—81no—
83C4×C211C4×D411yescharacteristic
84C4×C211C4×D411yescharacteristic
85C4×C211#12111yescharacteristic
86C4×C211#12611yescharacteristic
87Dic211—22no—
88Dic211—41no—
89Dic211—41no—
90C2×C211—21no—
91C2×C211—21no—
92C2×C211—21no—
93C2×C211#38811yescharacteristic
94D211—81no—
95C211#193211yescharacteristic, complemented, Sylow
96#1932—2111nomaximal, Sylow
97#430—2111no—
98#421—2111no—
99C2×#104—2111no—
100#413—2111no—
101C4×M16—2111no—
102#344D21111yescharacteristic, complemented
103#339—2111no—
104C2×#41—2111no—
105C2×M16—2111no—
106C2×M16—2111no—
107C2×M16C2×D21111yescharacteristic
108C2×M16C2×D21111yescharacteristic
109#117C2×D21111yescharacteristic
110C2×C42—2111no—
111#108—4222no—
112#104—4222no—
113#104—4221no—
114#104—4221no—
115#101—2112no—
116#100—2112no—
117#97—4221no—
118C4×C8—4221no—
119#41—4221no—
120#41—4221no—
121#41—4221no—
122#41—4221no—
123C2×Q8—2111no—
124C2×D4—4221no—
125C2×D4—2111no—
126C22×C4—4221no—
127C22×C4—4221no—
128C22×C4C22×D21111yescharacteristic
129M16—21no—
130M16—21no—
131M16—21no—
132M16—4221no—
133M16—4221no—
134M16—2112no—
135M16—2112no—
136M16C4×D21112yes—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—4221no—
140C2×C8—4221no—
141C4⋊C4—21no—
142C22⋊C4—21no—
143C42—21no—
144C42—4221no—
145C42—2111no—
146C42—2111no—
147C23—4221no—
148C23#1680996211yescharacteristic
149Q8—4221no—
150Q8—4221no—
151D4—8441no—
152D4—8441no—
153D4—4221no—
154D4—4221no—
155C2×C4—41no—
156C2×C4—8442no—
157C2×C4—8441no—
158C2×C4—8441no—
159C2×C4—4221no—
160C2×C4—4221no—
161C2×C4—4221no—
162C2×C4C2×C4×D21111yescharacteristic
163C2×C4D4×D21111yescharacteristic
164C2×C4D4×D21111yescharacteristic
165C2×C4#1680996111yescharacteristic
166C2×C4#1680996211yescharacteristic
167C2×C4#1680996311yescharacteristic
168C8—22no—
169C8—41no—
170C8—4222no—
171C8—4222no—
172C22—41no—
173C22—8441no—
174C22—16881no—
175C22#2026993411yescharacteristic
176C22#2026996211yescharacteristic
177C22#2026996311yescharacteristic
178C4—81no—
179C4—8441no—
180C4—8441no—
181C4—4222no—
182C4(C22⋊D211)⋊C411yescharacteristic
183C4#2026993911yescharacteristic
184C4#2026995011yescharacteristic
185C4#2026997211yescharacteristic
186C2—21no—
187C2—21no—
188C2—21no—
189C2—16881no—
190C2#2721303511yescharacteristic
subgroupG/H
centerC4#20269950
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#344×C211C2
socleC2×C211#388
prime radicalC2×#16809962C22
cosocleC22×C4×C211C23
hypercenter#344D211
hypocenterC211#1932
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
1. #27213194C2semidirect
5. M16×Dic211C2semidirect
6. #27212905C2semidirect
7. #344×C211C2semidirect
16. #20269895C4semidirect
17. #20269893C4semidirect
95. C211#1932semidirect
102. #344D211semidirect

Maximal subgroups

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

subgroupindex
#272131942
#272131642
C2×#202699032
#272131282
M16×Dic2112
#272129052
#344×C2112
#1932211