Finite Group Databasebeta

← #41014683

Subgroups — #41014683

Order 27008.

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

normal subgroups only

This group has 266 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
1#27213944C211yesmaximal, characteristic
2#27213169C211yesmaximal, characteristic, complemented
3C2×#20269904C211yesmaximal, characteristic, complemented
4C2×#20269903C211yesmaximal, characteristic, complemented
5#27213128C211yesmaximal, characteristic, complemented
6C211⋊#338C211yesmaximal, characteristic, complemented
7#413×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20270008—21no—
10#20270007—21no—
11#20270003—21no—
12#20270002—21no—
13#20269997—22no—
14C2×#16809946C2211yescharacteristic
15C2×#16809946C2211yescharacteristic
16C2×C4×Dic211C2211yescharacteristic
17Dic211⋊C8—22no—
18#20269904—21no—
19#20269904—21no—
20#20269903—21no—
21#20269903—21no—
22#20269895—21no—
23#20269893—21no—
24C2×#41×C211C2211yescharacteristic
25C2×M16×C211C2211yescharacteristic
26C2×M16×C211C2211yescharacteristic
27#101×C211—21no—
28#100×C211—21no—
29#16809962—21no—
30#16809962—21no—
31#16809962—21no—
32#16809962—21no—
33C2×Dic422D411yescharacteristic
34C2×D844D411yescharacteristic
35C2×C22⋊D211—21no—
36C22×Dic211—21no—
37C2×C4×D211—21no—
38#16809952—22no—
39#16809950—22no—
40#16809949—22no—
41#16809947—22no—
42#16809946—21no—
43#16809946—21no—
44#16809946—21no—
45#16809946—21no—
46C2×C211⋊C8—21no—
47C2×C211⋊C8D412yes—
48C4×Dic211—21no—
49C4×Dic211D411yescharacteristic
50C4×Dic211D411yescharacteristic
51#41×C211—21no—
52#41×C211—21no—
53#41×C211—21no—
54#41×C211—21no—
55C2×Q8×C211D411yescharacteristic
56C2×D4×C211—21no—
57C2×D4×C211D411yescharacteristic
58C22×C4×C211—21no—
59C22×C4×C211C2311yescharacteristic
60M16×C211—21no—
61M16×C211—21no—
62M16×C211—21no—
63M16×C211—21no—
64C2×C8×C211—21no—
65C2×C8×C211—21no—
66C22×D211—21no—
67Dic422—21no—
68Dic422—21no—
69D844—21no—
70D844—21no—
71C22⋊D211—41no—
72C22⋊D211—41no—
73C2×Dic211—21no—
74C2×Dic211—21no—
75C2×Dic211—21no—
76C2×Dic211—42no—
77C4×D211—41no—
78C4×D211—41no—
79C211⋊C8—22no—
80C211⋊C8—41no—
81C23×C211—21no—
82C23×C211#4111yescharacteristic
83Q8×C211—21no—
84Q8×C211—21no—
85D4×C211—21no—
86D4×C211—21no—
87D4×C211—41no—
88D4×C211—41no—
89C2×C4×C211—21no—
90C2×C4×C211—41no—
91C2×C4×C211—41no—
92C2×C4×C211C2×D411yescharacteristic
93C2×C4×C211C2×D411yescharacteristic
94C2×C4×C211C2×D411yescharacteristic
95C2×C4×C211C2×D411yescharacteristic
96C2×C4×C211#4111yescharacteristic
97C2×C4×C211#4111yescharacteristic
98C8×C211—41no—
99C8×C211—41no—
100C22×C211—41no—
101C22×C211—41no—
102C22×C211—81no—
103C22×C211#12111yescharacteristic
104C22×C211#12311yescharacteristic
105C22×C211#12311yescharacteristic
106C2×D211—41no—
107C2×D211—81no—
108C4×C211—41no—
109C4×C211#12011yescharacteristic
110C4×C211#12111yescharacteristic
111C4×C211#12111yescharacteristic
112C4×C211#12411yescharacteristic
113Dic211—22no—
114Dic211—41no—
115Dic211—41no—
116C2×C211—21no—
117C2×C211—21no—
118C2×C211—21no—
119C2×C211—81no—
120C2×C211#39411yescharacteristic
121D211—81no—
122C211#205011yescharacteristic, complemented, Sylow
123#2050—2111nomaximal, Sylow
124#575—2111no—
125#423—2111no—
126C2×#104—2111no—
127C2×#104—2111no—
128#413—2111no—
129#413D21111yescharacteristic, complemented
130#338—2111no—
131C2×#41—2111no—
132C2×#41C2×D21111yescharacteristic
133#137—4221no—
134#137—4221no—
135AGL(1,ℤ/8)—4221no—
136AGL(1,ℤ/8)—4221no—
137#135—4222no—
138C2×M16—2111no—
139C2×M16—2111no—
140C2×M16C2×D21111yescharacteristic
141C2×M16C2×D21111yescharacteristic
142C2×C42—2111no—
143C4⋊C8—4222no—
144#104—4221no—
145#104—4221no—
146#104—4221no—
147#104—4221no—
148#101—21no—
149#101—4221no—
150#100—21no—
151#100—4221no—
152#41—21no—
153#41—21no—
154#41—21no—
155#41—21no—
156#41—4221no—
157#41—4221no—
158#41—4221no—
159#41—4221no—
160C2×Q8—2111no—
161C2×Q8C22⋊D21111yescharacteristic
162C2×D4—21no—
163C2×D4—4221no—
164C2×D4—2111no—
165C2×D4C22⋊D21111yescharacteristic
166C22×C4—21no—
167C22×C4—4221no—
168C22×C4—4221no—
169C22×C4C22×D21111yescharacteristic
170Q16—4222no—
171SD16—4222no—
172SD16—4222no—
173D8—4222no—
174M16—21no—
175M16—21no—
176M16—21no—
177M16—21no—
178M16—4221no—
179M16—4221no—
180M16—4221no—
181M16—4221no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—4221no—
185C2×C8—2112no—
186C42—4221no—
187C42—2111no—
188C42—2111no—
189C23—21no—
190C23—4221no—
191C23#1680996211yescharacteristic
192Q8—21no—
193Q8—21no—
194Q8—4221no—
195Q8—4221no—
196D4—21no—
197D4—21no—
198D4—41no—
199D4—41no—
200D4—8441no—
201D4—8441no—
202D4—4221no—
203D4—4221no—
204C2×C4—21no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—8442no—
208C2×C4—8441no—
209C2×C4—8441no—
210C2×C4—4221no—
211C2×C4—4221no—
212C2×C4—4221no—
213C2×C4D4×D21111yescharacteristic
214C2×C4D4×D21111yescharacteristic
215C2×C4D4×D21111yescharacteristic
216C2×C4C2×C22⋊D21111yescharacteristic
217C2×C4#1680996111yescharacteristic
218C2×C4#1680996211yescharacteristic
219C8—41no—
220C8—41no—
221C8—8441no—
222C8—4222no—
223C22—41no—
224C22—41no—
225C22—81no—
226C22—8441no—
227C22—16881no—
228C22#2026995011yescharacteristic
229C22#2026995911yescharacteristic
230C22#2026996011yescharacteristic
231C4—41no—
232C4—8441no—
233C4—8441no—
234C4—4222no—
235C4#2026994511yescharacteristic
236C4#2026994911yescharacteristic
237C4#2026995011yescharacteristic
238C4#2026996311yescharacteristic
239C2—21no—
240C2—21no—
241C2—21no—
242C2—81no—
243C2—16881no—
244C2#2721306111yescharacteristic
subgroupG/H
centerC4#20269963
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#413×C211C2
socleC2×C211#394
prime radical#27213944C2
hypercenter#413D211
hypocenterC211#2050
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #27213169C2semidirect
3. C2×#20269904C2semidirect
4. C2×#20269903C2semidirect
5. #27213128C2semidirect
6. C211⋊#338C2semidirect
7. #413×C211C2semidirect
122. C211#2050semidirect
129. #413D211semidirect

Maximal subgroups

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

subgroupindex
#272139442
#272131692
C2×#202699042
C2×#202699032
#272131282
C211⋊#3382
#413×C2112
#2050211