Finite Group Databasebeta

← #41014669

Subgroups — #41014669

Order 27008.

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

normal subgroups only

This group has 274 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×#20270003C211yesmaximal, characteristic
2#27213194C211yesmaximal, characteristic, complemented
3#27213164C211yesmaximal, characteristic, complemented
4#27213128C211yesmaximal, characteristic, complemented
5C2×#20269894C211yesmaximal, characteristic, complemented
6#27212916C211yesmaximal, characteristic, complemented
7#418×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20270003—22no—
10#20270003—22no—
11C2×#16809950—21no—
12C2×#16809947—21no—
13C2×#16809946C2211yescharacteristic
14C2×#16809946C2211yescharacteristic
15C2×#16809946C2211yescharacteristic
16#20269912—22no—
17#20269905—22no—
18#20269895—21no—
19#20269894—22no—
20#20269893—21no—
21C22×D4×C211C2211yescharacteristic
22C2×M16×C211C2211yescharacteristic
23#117×C211C2211yescharacteristic
24#102×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#16809950—24no—
34#16809947—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—
44C24×C211—21no—
45C2×D4×C211—21no—
46C2×D4×C211—21no—
47C2×D4×C211—21no—
48C2×D4×C211—21no—
49C2×D4×C211—21no—
50C2×D4×C211D412yes—
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—22no—
72C23×C211—41no—
73C23×C211—41no—
74C23×C211—82no—
75C23×C211#4111yescharacteristic
76D4×C211—22no—
77D4×C211—22no—
78D4×C211—41no—
79D4×C211—41no—
80C2×C4×C211—41no—
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—41no—
89C22×C211—42no—
90C22×C211—82no—
91C22×C211—82no—
92C22×C211—82no—
93C22×C211#12111yescharacteristic
94C22×C211#12311yescharacteristic
95C22×C211#12411yescharacteristic
96C2×D211—41no—
97C2×D211—81no—
98C4×C211—81no—
99C4×C211#12011yescharacteristic
100C4×C211#12111yescharacteristic
101C4×C211#12111yescharacteristic
102C4×C211#12311yescharacteristic
103Dic211—41no—
104C2×C211—21no—
105C2×C211—21no—
106C2×C211—21no—
107C2×C211—82no—
108C2×C211#39411yescharacteristic
109D211—81no—
110C211#204811yescharacteristic, complemented, Sylow
111#2048—2111nomaximal, Sylow
112C2×AGL(1,ℤ/8)—2111no—
113#430—2111no—
114#421—2111no—
115#418D21111yescharacteristic, complemented
116#413—2111no—
117C2×#100—2111no—
118#344—2111no—
119C2×#41—2111no—
120C22×D4C2×D21111yescharacteristic
121AGL(1,ℤ/8)—4222no—
122AGL(1,ℤ/8)—4222no—
123C2×SD16—4221no—
124C2×D8—4221no—
125C2×M16—2111no—
126C2×M16—2111no—
127C2×M16—2111no—
128C2×M16C2×D21111yescharacteristic
129#117C2×D21111yescharacteristic
130#108—4222no—
131#104—4222no—
132#102—22no—
133#101—4221no—
134#100—4221no—
135#100—4222no—
136C24—21no—
137#41—4221no—
138#41—4221no—
139#41—4221no—
140#41—4221no—
141C2×Q8—2111no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—4221no—
148C2×D4—2111no—
149C2×D4C22⋊D21112yes—
150C22×C4—4221no—
151C22×C4C22×D21111yescharacteristic
152SD16—4224no—
153D8—4224no—
154M16—21no—
155M16—21no—
156M16—4221no—
157M16—4221no—
158M16—4221no—
159M16—4221no—
160M16—2112no—
161M16—2112no—
162C2×C8—21no—
163C2×C8—4221no—
164C2×C8—4221no—
165C2×C8—4221no—
166C4⋊C4—21no—
167C22⋊C4—21no—
168C42—21no—
169C23—22no—
170C23—41no—
171C23—41no—
172C23—82no—
173C23—4221no—
174C23#1680996111yescharacteristic
175Q8—4221no—
176Q8—4221no—
177D4—22no—
178D4—22no—
179D4—41no—
180D4—41no—
181D4—8441no—
182D4—8441no—
183D4—4221no—
184D4—4221no—
185C2×C4—41no—
186C2×C4—8441no—
187C2×C4—8441no—
188C2×C4—4221no—
189C2×C4D4×D21111yescharacteristic
190C2×C4D4×D21111yescharacteristic
191C2×C4D4×D21111yescharacteristic
192C2×C4C2×C22⋊D21111yescharacteristic
193C2×C4#1680996211yescharacteristic
194C2×C4#1680996211yescharacteristic
195C8—41no—
196C8—8441no—
197C8—4222no—
198C8—4222no—
199C22—41no—
200C22—42no—
201C22—82no—
202C22—82no—
203C22—82no—
204C22—8441no—
205C22—16881no—
206C22#2026994911yescharacteristic
207C22#2026995911yescharacteristic
208C22#2026996311yescharacteristic
209C4—81no—
210C4—8441no—
211C4#2026994511yescharacteristic
212C4#2026995011yescharacteristic
213C4#2026995011yescharacteristic
214C4#2026996011yescharacteristic
215C2—21no—
216C2—21no—
217C2—21no—
218C2—82no—
219C2—16881no—
220C2#2721306111yescharacteristic
subgroupG/H
centerC2#27213061
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#418×C211C2
socleC2×C211#394
prime radicalC2×#20270003C2
hypercenter#418D211
hypocenterC211#2048
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #27213194C2semidirect
3. #27213164C2semidirect
4. #27213128C2semidirect
5. C2×#20269894C2semidirect
6. #27212916C2semidirect
7. #418×C211C2semidirect
110. C211#2048semidirect
115. #418D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700032
#272131942
#272131642
#272131282
C2×#202698942
#272129162
#418×C2112
#2048211