Finite Group Databasebeta

← #41014246

Subgroups — #41014246

Order 27008.

312 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 83 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×#20269966C211yesmaximal, characteristic
2#27213169C211yesmaximal, characteristic, complemented
3C2×#20269896C211yesmaximal, characteristic, complemented
4C2×#20269893C211yesmaximal, characteristic
5C2×#20269893C211yesmaximal, characteristic
6M16×Dic211C211yesmaximal, characteristic, complemented
7C2×#101×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11#20269966—21no—
12#20269966—21no—
13#20269966—22no—
14C2×(C2×D211)⋊C4C414yes—
15C2×C4×Dic211C2211yescharacteristic
16Dic211⋊C8—22no—
17#20269896—22no—
18#20269893—22no—
19#20269893—22no—
20#20269887—21no—
21C8×Dic211—21no—
22C22×Q8×C211C2211yescharacteristic
23C2×M16×C211C2211yescharacteristic
24C2×M16×C211C2211yescharacteristic
25#101×C211—22no—
26C23×D211C2×C412yes—
27C2×D844—21no—
28C2×D844—21no—
29C2×D844—21no—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844—21no—
33C22×Dic211C2×C412yes—
34#16809946—21no—
35#16809946—21no—
36#16809946—21no—
37#16809946D412yes—
38C2×C211⋊C8—21no—
39C2×C211⋊C8—21no—
40(C2×D211)⋊C4—24no—
41(C2×D211)⋊C4—24no—
42C4×Dic211D411yescharacteristic
43C4×Dic211D411yescharacteristic
44C4×Dic211D412yes—
45C2×Q8×C211—21no—
46C2×Q8×C211—21no—
47C2×Q8×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211—21no—
50C2×Q8×C211—21no—
51C22×C4×C211C2×C412yes—
52C22×C4×C211C2311yescharacteristic
53M16×C211—21no—
54M16×C211—21no—
55M16×C211—21no—
56M16×C211D412yes—
57C2×C8×C211—21no—
58C2×C8×C211—21no—
59C22×D211—22no—
60C22×D211—22no—
61C22×D211—22no—
62C22×D211—44no—
63D844—41no—
64D844—41no—
65D844—41no—
66D844—41no—
67C2×Dic211—42no—
68C2×Dic211C22⋊C412yes—
69C2×Dic211C22⋊C412yes—
70C211⋊C8—22no—
71C211⋊C8—41no—
72C23×C211C22×C411yescharacteristic
73Q8×C211—41no—
74Q8×C211—41no—
75Q8×C211—41no—
76Q8×C211—41no—
77C2×C4×C211—22no—
78C2×C4×C211—22no—
79C2×C4×C211—22no—
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—22no—
87C8×C211—41no—
88C22×C211—22no—
89C22×C211C2×C22⋊C411yescharacteristic
90C22×C211C4×D411yescharacteristic
91C22×C211C4×D411yescharacteristic
92C2×D211—42no—
93C2×D211—44no—
94C2×D211—82no—
95C2×D211—82no—
96C4×C211—42no—
97C4×C211#12111yescharacteristic
98C4×C211#12111yescharacteristic
99C4×C211#12411yescharacteristic
100C4×C211#12711yescharacteristic
101Dic211—22no—
102Dic211—22no—
103C2×C211—21no—
104C2×C211—21no—
105C2×C211#39011yescharacteristic
106C2×C211#41312yes—
107D211—82no—
108C211#196711yescharacteristic, complemented, Sylow
109#1967—2111nomaximal, Sylow
110C2×#124—2111no—
111#423—2111no—
112C2×#101—2111no—
113C2×#101D21111yescharacteristic, complemented
114C2×#100—2111no—
115C2×#100—2111no—
116C4×M16—2111no—
117C22×Q8C2×D21111yescharacteristic
118C22×D4—2111no—
119C2×M16—2111no—
120C2×M16—2111no—
121C2×M16C2×D21111yescharacteristic
122C2×M16C2×D21111yescharacteristic
123#124—4221no—
124#124—4221no—
125#124—4222no—
126C2×C22⋊C4—2114no—
127C2×C42—2111no—
128C4⋊C8—4222no—
129#101—22no—
130#101—4222no—
131#100—4222no—
132#100—4222no—
133#97—4221no—
134C4×C8—4221no—
135C24—2112no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×D4—4221no—
143C2×D4—4221no—
144C2×D4—4221no—
145C2×D4—4221no—
146C2×D4—4221no—
147C2×D4—4221no—
148C22×C4—2112no—
149C22×C4C4×D21112yes—
150C22×C4C22×D21111yescharacteristic
151M16—21no—
152M16—21no—
153M16—21no—
154M16—4221no—
155M16—4221no—
156M16—4221no—
157M16—2112no—
158M16C22⋊D21112yes—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—4221no—
162C2×C8—4221no—
163C22⋊C4—4224no—
164C22⋊C4—4224no—
165C42—2111no—
166C42—2111no—
167C42—2112no—
168C23—8444no—
169C23—4222no—
170C23—4222no—
171C23—4222no—
172C23C2×C4×D21111yescharacteristic
173Q8—41no—
174Q8—41no—
175Q8—41no—
176Q8—41no—
177D4—8441no—
178D4—8441no—
179D4—8441no—
180D4—8441no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—8442no—
185C2×C4—2112no—
186C2×C4—2112no—
187C2×C4D4×D21111yescharacteristic
188C2×C4D4×D21111yescharacteristic
189C2×C4D4×D21111yescharacteristic
190C2×C4C2×C22⋊D21111yescharacteristic
191C2×C4#1680996111yescharacteristic
192C2×C4#1680996211yescharacteristic
193C8—22no—
194C8—41no—
195C8—8441no—
196C8—4222no—
197C22—22no—
198C22—8444no—
199C22—8442no—
200C22—16882no—
201C22—16882no—
202C22C22⋊C4×D21111yescharacteristic
203C22(C22⋊D211)⋊C411yescharacteristic
204C22C4×C22⋊D21111yescharacteristic
205C4—42no—
206C4—4222no—
207C4—4222no—
208C4#2026994911yescharacteristic
209C4#2026995011yescharacteristic
210C4#2026996311yescharacteristic
211C4#2026997311yescharacteristic
212C2—21no—
213C2—21no—
214C2—16882no—
215C2#2721304611yescharacteristic
216C2#2721312912yes—
subgroupG/H
centerC22C22⋊C4×D211
commutatorC23×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#101×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC22×D844C22
cosocleC22×C4×C211C23
hypercenterC2×#101D211
hypocenterC211#1967
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. #27213169C2semidirect
3. C2×#20269896C2semidirect
6. M16×Dic211C2semidirect
7. C2×#101×C211C2semidirect
108. C211#1967semidirect
113. C2×#101D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699662
#272131692
C2×#202698962
C2×#202698932
M16×Dic2112
C2×#101×C2112
#1967211