Finite Group Databasebeta

← #41013460

Subgroups — #41013460

Order 27008.

264 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 93 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×#20269984C211yesmaximal, characteristic
2C2×#20269912C211yesmaximal, characteristic, complemented
3C2×#20269910C211yesmaximal, characteristic, complemented
4#27213147C211yesmaximal, characteristic
5#27212915C211yesmaximal, characteristic, complemented
6#27212914C211yesmaximal, characteristic, complemented
7#417×C211C211yesmaximal, characteristic
8#20269984—22no—
9#20269984C414yescomplemented
10C2×#16809946—21no—
11C2×#16809946C2211yescharacteristic
12C2×#16809946C2211yescharacteristic
13C22×C211⋊C8—21no—
14C22×C211⋊C8C2211yescharacteristic
15C2×C4⋊Dic211C2211yescharacteristic
16#20269912—22no—
17#20269910—21no—
18#20269910—21no—
19#20269901—21no—
20#20269898—21no—
21C2×#41×C211C2211yescharacteristic
22C2×M16×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24#103×C211—21no—
25#102×C211—21no—
26C22×Dic211—21no—
27#16809946—24no—
28#16809946—21no—
29#16809946—21no—
30#16809946—21no—
31#16809946C2×C412yes—
32C2×C211⋊C8—24no—
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C2×C211⋊C8C2×C412yes—
36C2×C211⋊C8D411yescharacteristic
37C2×C211⋊C8D411yescharacteristic
38C2×C211⋊C8D412yes—
39C4⋊Dic211—21no—
40C4⋊Dic211—21no—
41#41×C211—21no—
42#41×C211—21no—
43#41×C211C2×C412yes—
44#41×C211Q812yes—
45C2×Q8×C211D411yescharacteristic
46C2×D4×C211—21no—
47C2×D4×C211D411yescharacteristic
48C22×C4×C211—21no—
49C22×C4×C211—21no—
50C22×C4×C211C2311yescharacteristic
51M16×C211—21no—
52M16×C211—21no—
53C2×C8×C211—21no—
54C4⋊C4×C211—21no—
55C4⋊C4×C211—21no—
56C2×Dic211—41no—
57C2×Dic211—81no—
58C211⋊C8—22no—
59C211⋊C8—41no—
60C211⋊C8C22⋊C412yes—
61C211⋊C8C22⋊C412yes—
62C23×C211—21no—
63C23×C211#4111yescharacteristic
64Q8×C211—21no—
65Q8×C211C4⋊C412yes—
66D4×C211—22no—
67D4×C211—22no—
68D4×C211—21no—
69D4×C211C4⋊C412yes—
70C2×C4×C211—21no—
71C2×C4×C211—22no—
72C2×C4×C211—22no—
73C2×C4×C211—41no—
74C2×C4×C211—81no—
75C2×C4×C211C22×C411yescharacteristic
76C2×C4×C211C2×D411yescharacteristic
77C2×C4×C211C2×D411yescharacteristic
78C2×C4×C211C2×D411yescharacteristic
79C2×C4×C211C2×Q811yescharacteristic
80C2×C4×C211#4111yescharacteristic
81C8×C211—41no—
82C22×C211—21no—
83C22×C211—21no—
84C22×C211—22no—
85C22×C211—41no—
86C22×C211—41no—
87C22×C211#11711yescharacteristic
88C22×C211#12111yescharacteristic
89C22×C211C22⋊Q811yescharacteristic
90C4×C211—22no—
91C4×C211—81no—
92C4×C211C2×C22⋊C411yescharacteristic
93C4×C211C2×C4⋊C411yescharacteristic
94C4×C211#12111yescharacteristic
95C4×C211C22⋊Q811yescharacteristic
96Dic211—81no—
97C2×C211—21no—
98C2×C211—21no—
99C2×C211—42no—
100C2×C211#38011yescharacteristic
101C2×C211#47211yescharacteristic
102C2×C211#47311yescharacteristic
103C211#181611yescharacteristic, complemented, Sylow
104#1816—2111nomaximal, Sylow
105C2×#131—2111no—
106C2×#108—2111no—
107C2×#107—2111no—
108#417—2111no—
109#417D21111yescharacteristic
110#343—2111no—
111#343—2111no—
112C2×#41C2×D21111yescharacteristic
113#131—4222no—
114#131—2114no—
115C2×M16—4221no—
116C2×M16—2111no—
117C2×M16—2111no—
118C2×M16C2×D21111yescharacteristic
119C22×C8—4221no—
120C22×C8—2111no—
121C2×C4⋊C4—2111no—
122C2×C4⋊C4C2×D21111yescharacteristic
123#108—4222no—
124#107—4221no—
125#107—4221no—
126#103—21no—
127#103—4221no—
128#102—21no—
129#102—4221no—
130#41—21no—
131#41—21no—
132#41C4×D21112yes—
133#41Dic42212yes—
134C2×Q8C22⋊D21111yescharacteristic
135C2×D4—21no—
136C2×D4C22⋊D21111yescharacteristic
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—4221no—
140C22×C4C22×D21111yescharacteristic
141M16—21no—
142M16—21no—
143M16—4221no—
144M16—4221no—
145M16—4221no—
146M16—4224no—
147M16—2112no—
148C2×C8—21no—
149C2×C8—4221no—
150C2×C8—4221no—
151C2×C8—4224no—
152C2×C8—2112no—
153C2×C8—2111no—
154C2×C8—2111no—
155C2×C8—2112no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—4221no—
159C4⋊C4—4221no—
160C23—21no—
161C23#1680996111yescharacteristic
162Q8—21no—
163Q8Dic211⋊C412yes—
164D4—22no—
165D4—22no—
166D4—21no—
167D4Dic211⋊C412yes—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—41no—
172C2×C4—81no—
173C2×C4—8441no—
174C2×C4—16881no—
175C2×C4C2×C4×D21111yescharacteristic
176C2×C4D4×D21111yescharacteristic
177C2×C4D4×D21111yescharacteristic
178C2×C4C2×C22⋊D21111yescharacteristic
179C2×C4C2×Dic42211yescharacteristic
180C2×C4#1680996111yescharacteristic
181C8—41no—
182C8—8441no—
183C8—4222no—
184C8—2112no—
185C8—2112no—
186C22—21no—
187C22—21no—
188C22—22no—
189C22—41no—
190C22—41no—
191C22#2026993311yescharacteristic
192C22#2026994911yescharacteristic
193C22#2026995411yescharacteristic
194C4—22no—
195C4—81no—
196C4—16881no—
197C4C22⋊C4×D21111yescharacteristic
198C4C2×Dic211⋊C411yescharacteristic
199C4#2026994911yescharacteristic
200C4#2026995411yescharacteristic
201C2—21no—
202C2—21no—
203C2—42no—
204C2#2721300411yescharacteristic
205C2#2721341411yescharacteristic
206C2#2721341911yescharacteristic
subgroupG/H
centerC22#20269949
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#417×C211C2
socleC22×C211#121
prime radicalC2×#41×C211C22
cosocleC22×C4×C211C23
hypercenter#417D211
hypocenterC211#1816
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#20269912C2semidirect
3. C2×#20269910C2semidirect
5. #27212915C2semidirect
6. #27212914C2semidirect
9. #20269984C4semidirect
103. C211#1816semidirect

Maximal subgroups

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

subgroupindex
C2×#202699842
C2×#202699122
C2×#202699102
#272131472
#272129152
#272129142
#417×C2112
#1816211