Finite Group Databasebeta

← #41014695

Subgroups — #41014695

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×#20270007C211yesmaximal, characteristic
2C2×#20269912C211yesmaximal, characteristic, complemented
3C2×#20269900C211yesmaximal, characteristic, complemented
4C2×#20269895C211yesmaximal, characteristic, complemented
5#27213113C211yesmaximal, characteristic
6#27212915C211yesmaximal, characteristic, complemented
7#417×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9#20270007—22no—
10#20270007—22no—
11C2×#16809952—21no—
12C2×#16809950—21no—
13C2×#16809946C2211yescharacteristic
14C2×#16809946C2211yescharacteristic
15C22×C211⋊C8C2211yescharacteristic
16#20269912—22no—
17#20269900—22no—
18#20269895—22no—
19C22⋊(C211⋊C8)—22no—
20C2×#41×C211C2211yescharacteristic
21C2×M16×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23#103×C211—21no—
24#102×C211—21no—
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422D412yes—
31C22×Dic211—21no—
32#16809952—24no—
33#16809950—24no—
34#16809946—21no—
35#16809946—21no—
36#16809946—21no—
37#16809946D412yes—
38C2×C211⋊C8—21no—
39C2×C211⋊C8—21no—
40C2×C211⋊C8—21no—
41C2×C211⋊C8—21no—
42C2×C211⋊C8D412yes—
43#41×C211—21no—
44#41×C211—21no—
45#41×C211—21no—
46#41×C211—21no—
47C2×Q8×C211D411yescharacteristic
48C2×D4×C211—21no—
49C2×D4×C211D411yescharacteristic
50C22×C4×C211—21no—
51C22×C4×C211—21no—
52C22×C4×C211C2311yescharacteristic
53M16×C211—21no—
54M16×C211—21no—
55C2×C8×C211—21no—
56C4⋊C4×C211—21no—
57C4⋊C4×C211—21no—
58Dic422—22no—
59Dic422—22no—
60Dic422—41no—
61Dic422—41no—
62C2×Dic211—22no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C211⋊C8—22no—
66C211⋊C8—22no—
67C211⋊C8—41no—
68C23×C211—21no—
69C23×C211#4111yescharacteristic
70Q8×C211—21no—
71Q8×C211—21no—
72D4×C211—21no—
73D4×C211—21no—
74D4×C211—41no—
75D4×C211—41no—
76C2×C4×C211—21no—
77C2×C4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211—81no—
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—41no—
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—81no—
98C4×C211#12011yescharacteristic
99C4×C211#12111yescharacteristic
100C4×C211#12111yescharacteristic
101C4×C211#12311yescharacteristic
102Dic211—42no—
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×#137—2111no—
112C2×#108—2111no—
113#417D21111yescharacteristic
114C2×#103—2111no—
115C2×#101—2111no—
116#408—2111no—
117#343—2111no—
118C2×#41C2×D21111yescharacteristic
119C22×Q8—2111no—
120#137—4222no—
121#137—4222no—
122C2×Q16—4221no—
123C2×SD16—4221no—
124C2×M16—2111no—
125C2×M16—2111no—
126C2×M16C2×D21111yescharacteristic
127C22×C8—2111no—
128C2×C4⋊C4C2×D21111yescharacteristic
129#108—4222no—
130#103—21no—
131#103—4222no—
132#102—21no—
133#101—4222no—
134C22⋊C8—4222no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—4221no—
140C2×Q8—4221no—
141C2×Q8—4221no—
142C2×Q8—4221no—
143C2×Q8—4221no—
144C2×Q8—2112no—
145C2×Q8C22⋊D21111yescharacteristic
146C2×D4—21no—
147C2×D4C22⋊D21111yescharacteristic
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—4221no—
151C22×C4C22×D21111yescharacteristic
152Q16—4224no—
153SD16—4224no—
154M16—21no—
155M16—21no—
156M16—4221no—
157M16—4221no—
158M16—4221no—
159M16—2112no—
160C2×C8—21no—
161C2×C8—4221no—
162C2×C8—4221no—
163C2×C8—4221no—
164C2×C8—4221no—
165C2×C8—2112no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C23—21no—
169C23#1680996111yescharacteristic
170Q8—21no—
171Q8—21no—
172Q8—8441no—
173Q8—8441no—
174Q8—4222no—
175Q8—4222no—
176D4—21no—
177D4—21no—
178D4—41no—
179D4—41no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—81no—
185C2×C4—8441no—
186C2×C4—8441no—
187C2×C4—4222no—
188C2×C4D4×D21111yescharacteristic
189C2×C4D4×D21111yescharacteristic
190C2×C4D4×D21111yescharacteristic
191C2×C4C2×C22⋊D21111yescharacteristic
192C2×C4#1680996211yescharacteristic
193C2×C4#1680996211yescharacteristic
194C8—41no—
195C8—8441no—
196C8—4222no—
197C8—4222no—
198C22—21no—
199C22—21no—
200C22—41no—
201C22—41no—
202C22—41no—
203C22#2026994911yescharacteristic
204C22#2026995911yescharacteristic
205C22#2026996311yescharacteristic
206C4—41no—
207C4—81no—
208C4—8442no—
209C4#2026994511yescharacteristic
210C4#2026995011yescharacteristic
211C4#2026995011yescharacteristic
212C4#2026996011yescharacteristic
213C2—21no—
214C2—21no—
215C2—81no—
216C2#2721306111yescharacteristic
217C2#2721340711yescharacteristic
218C2#2721341911yescharacteristic
subgroupG/H
centerC22#20269949
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#417×C211C2
socleC22×C211#121
prime radicalC2×#41×C211C22
hypercenter#417D211
hypocenterC211#2052
LCSG ▷ C22×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×#20269900C2semidirect
4. C2×#20269895C2semidirect
6. #27212915C2semidirect
109. C211#2052semidirect

Maximal subgroups

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

subgroupindex
C2×#202700072
C2×#202699122
C2×#202699002
C2×#202698952
#272131132
#272129152
#417×C2112
#2052211