Finite Group Databasebeta

← #41014676

Subgroups — #41014676

Order 27008.

254 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 63 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×#20270008C211yesmaximal, characteristic
2#27213194C211yesmaximal, characteristic, complemented
3#27213164C211yesmaximal, characteristic
4#27213128C211yesmaximal, characteristic
5C2×#20269896C211yesmaximal, characteristic, complemented
6#27212916C211yesmaximal, characteristic, complemented
7#419×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20270008—22no—
10#20270008—22no—
11C2×#16809952—21no—
12C2×#16809949—21no—
13C2×#16809946C2211yescharacteristic
14C2×#16809946C2211yescharacteristic
15C2×#16809946C2211yescharacteristic
16#20269912—22no—
17#20269905—22no—
18#20269896—22no—
19#20269895—21no—
20#20269893—21no—
21C22×Q8×C211C2211yescharacteristic
22C2×M16×C211C2211yescharacteristic
23#117×C211C2211yescharacteristic
24#103×C211—22no—
25#16809962—21no—
26#16809962—21no—
27#16809962—21no—
28#16809962—21no—
29C2×Dic422D411yescharacteristic
30C2×D844D411yescharacteristic
31C2×C22⋊D211—21no—
32C2×C4×D211—21no—
33#16809952—24no—
34#16809949—24no—
35#16809946—21no—
36#16809946—21no—
37#16809946—21no—
38#16809946—21no—
39#16809946D412yes—
40#16809946D412yes—
41C2×C211⋊C8—21no—
42C2×C211⋊C8—21no—
43C2×C211⋊C8—21no—
44C2×Q8×C211—21no—
45C2×Q8×C211—21no—
46C2×Q8×C211—21no—
47C2×Q8×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211D412yes—
50C22×C4×C211—21no—
51C22×C4×C211C2311yescharacteristic
52M16×C211—21no—
53M16×C211—21no—
54C2×C8×C211—21no—
55C4⋊C4×C211—21no—
56C22⋊C4×C211—21no—
57C42×C211—21no—
58C22×D211—21no—
59Dic422—21no—
60Dic422—21no—
61D844—21no—
62D844—21no—
63C22⋊D211—41no—
64C22⋊D211—41no—
65C2×Dic211—21no—
66C4×D211—41no—
67C4×D211—41no—
68C211⋊C8—22no—
69C211⋊C8—22no—
70C211⋊C8—41no—
71C23×C211#4111yescharacteristic
72Q8×C211—22no—
73Q8×C211—22no—
74Q8×C211—41no—
75Q8×C211—41no—
76C2×C4×C211—22no—
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—41no—
87C22×C211—41no—
88C22×C211#12111yescharacteristic
89C22×C211#12311yescharacteristic
90C22×C211#12411yescharacteristic
91C2×D211—41no—
92C2×D211—81no—
93C4×C211—42no—
94C4×C211—81no—
95C4×C211#12011yescharacteristic
96C4×C211#12111yescharacteristic
97C4×C211#12111yescharacteristic
98C4×C211#12311yescharacteristic
99Dic211—41no—
100C2×C211—21no—
101C2×C211—21no—
102C2×C211—21no—
103C2×C211#39411yescharacteristic
104D211—81no—
105C211#204911yescharacteristic, complemented, Sylow
106#2049—2111nomaximal, Sylow
107C2×#137—2111no—
108#430—2111no—
109#421—2111no—
110#419D21111yescharacteristic, complemented
111#413—2111no—
112C2×#101—2111no—
113#344—2111no—
114C2×#41—2111no—
115C22×Q8C2×D21111yescharacteristic
116#137—4222no—
117#137—4222no—
118C2×Q16—4221no—
119C2×SD16—4221no—
120C2×M16—2111no—
121C2×M16—2111no—
122C2×M16—2111no—
123C2×M16C2×D21111yescharacteristic
124#117C2×D21111yescharacteristic
125#108—4222no—
126#104—4222no—
127#103—22no—
128#101—4221no—
129#101—4222no—
130#100—4221no—
131#41—4221no—
132#41—4221no—
133#41—4221no—
134#41—4221no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—2111no—
141C2×Q8C22⋊D21112yes—
142C2×D4—4221no—
143C2×D4—2111no—
144C22×C4—21no—
145C22×C4—4221no—
146C22×C4C22×D21111yescharacteristic
147Q16—4224no—
148SD16—4224no—
149M16—21no—
150M16—21no—
151M16—4221no—
152M16—4221no—
153M16—4221no—
154M16—4221no—
155M16—2112no—
156M16—2112no—
157C2×C8—21no—
158C2×C8—4221no—
159C2×C8—4221no—
160C2×C8—4221no—
161C4⋊C4—21no—
162C22⋊C4—21no—
163C42—21no—
164C23—4221no—
165C23#1680996111yescharacteristic
166Q8—22no—
167Q8—22no—
168Q8—41no—
169Q8—41no—
170Q8—4221no—
171Q8—4221no—
172D4—8441no—
173D4—8441no—
174D4—4221no—
175D4—4221no—
176C2×C4—22no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—8441no—
181C2×C4—8441no—
182C2×C4—4221no—
183C2×C4D4×D21111yescharacteristic
184C2×C4D4×D21111yescharacteristic
185C2×C4D4×D21111yescharacteristic
186C2×C4C2×C22⋊D21111yescharacteristic
187C2×C4#1680996211yescharacteristic
188C2×C4#1680996211yescharacteristic
189C8—41no—
190C8—8441no—
191C8—4222no—
192C8—4222no—
193C22—41no—
194C22—8441no—
195C22—16881no—
196C22#2026994911yescharacteristic
197C22#2026995911yescharacteristic
198C22#2026996311yescharacteristic
199C4—42no—
200C4—81no—
201C4—8441no—
202C4#2026994511yescharacteristic
203C4#2026995011yescharacteristic
204C4#2026995011yescharacteristic
205C4#2026996011yescharacteristic
206C2—21no—
207C2—21no—
208C2—21no—
209C2—16881no—
210C2#2721306111yescharacteristic
subgroupG/H
centerC2#27213061
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#419×C211C2
socleC2×C211#394
prime radicalC2×#16809962C22
hypercenter#419D211
hypocenterC211#2049
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. #27213194C2semidirect
5. C2×#20269896C2semidirect
6. #27212916C2semidirect
7. #419×C211C2semidirect
105. C211#2049semidirect
110. #419D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700082
#272131942
#272131642
#272131282
C2×#202698962
#272129162
#419×C2112
#2049211