Finite Group Databasebeta

← #41014456

Subgroups — #41014456

Order 27008.

348 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 71 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#27213720C211yesmaximal, characteristic, complemented
3#27213162C211yesmaximal, characteristic, complemented
4#27213155C211yesmaximal, characteristic, complemented
5#27213120C211yesmaximal, characteristic, complemented
6#27212918C211yesmaximal, characteristic, complemented
7C2×#137×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20270008—21no—
10#20270007—21no—
11#20270003—21no—
12#20270002—21no—
13#20269997—22no—
14C2×#16809946C2211yescharacteristic
15#20269975—21no—
16#20269966—21no—
17#20269960—22no—
18#20269953—22no—
19#20269936C2211yescharacteristic
20#20269936C2211yescharacteristic
21#20269903—22no—
22#20269901—22no—
23#20269892—22no—
24C2×#41×C211C2211yescharacteristic
25C22×Q8×C211C2211yescharacteristic
26#137×C211—22no—
27#137×C211—22no—
28C2×Q16×C211—21no—
29C2×SD16×C211—21no—
30C2×M16×C211C2211yescharacteristic
31#16809962—21no—
32#16809962—21no—
33#16809962—21no—
34#16809962—21no—
35C2×Dic422D411yescharacteristic
36C2×D844D411yescharacteristic
37C2×C22⋊D211—21no—
38C2×C4×D211—21no—
39#16809952—22no—
40#16809950—22no—
41#16809949—22no—
42#16809947—22no—
43#16809946—21no—
44#16809946—21no—
45C2×C211⋊C8D412yes—
46C4⋊Dic211—21no—
47C4⋊Dic211—21no—
48Dic211⋊C4—24no—
49C22⋊Dic211—21no—
50C22⋊Dic211D412yes—
51(C2×D211)⋊C4—24no—
52C4×Dic211—21no—
53C4×Dic211—21no—
54#41×C211—21no—
55#41×C211—21no—
56#41×C211—21no—
57#41×C211—21no—
58C2×Q8×C211—21no—
59C2×Q8×C211—21no—
60C2×Q8×C211—21no—
61C2×Q8×C211—21no—
62C2×Q8×C211—21no—
63C2×Q8×C211D411yescharacteristic
64C2×Q8×C211D412yes—
65C2×D4×C211—21no—
66C2×D4×C211D411yescharacteristic
67C22×C4×C211—21no—
68C22×C4×C211—21no—
69C22×C4×C211C2311yescharacteristic
70Q16×C211—24no—
71SD16×C211—24no—
72M16×C211—21no—
73M16×C211D412yes—
74C2×C8×C211—21no—
75C22×D211—21no—
76Dic422—21no—
77Dic422—21no—
78D844—21no—
79D844—21no—
80C22⋊D211—41no—
81C22⋊D211—41no—
82C2×Dic211—22no—
83C2×Dic211—21no—
84C2×Dic211—41no—
85C4×D211—41no—
86C4×D211—41no—
87C211⋊C8—22no—
88C23×C211—21no—
89C23×C211C2×D411yescharacteristic
90Q8×C211—22no—
91Q8×C211—21no—
92Q8×C211—21no—
93Q8×C211—22no—
94Q8×C211—41no—
95Q8×C211—41no—
96D4×C211—21no—
97D4×C211—21no—
98D4×C211—41no—
99D4×C211—41no—
100C2×C4×C211—22no—
101C2×C4×C211—21no—
102C2×C4×C211—41no—
103C2×C4×C211—41no—
104C2×C4×C211—41no—
105C2×C4×C211—41no—
106C2×C4×C211C2×D411yescharacteristic
107C2×C4×C211C2×D411yescharacteristic
108C2×C4×C211C2×D411yescharacteristic
109C2×C4×C211C2×D411yescharacteristic
110C2×C4×C211C2×D411yescharacteristic
111C2×C4×C211#4111yescharacteristic
112C8×C211—22no—
113C22×C211—41no—
114C22×C211—41no—
115C22×C211—81no—
116C22×C211#12011yescharacteristic
117C22×C211#12111yescharacteristic
118C22×C211#12711yescharacteristic
119C2×D211—42no—
120C2×D211—41no—
121C4×C211—41no—
122C4×C211—42no—
123C4×C211#12011yescharacteristic
124C4×C211#12011yescharacteristic
125C4×C211#12111yescharacteristic
126C4×C211#12111yescharacteristic
127Dic211—42no—
128Dic211—41no—
129Dic211—81no—
130C2×C211—21no—
131C2×C211—21no—
132C2×C211—21no—
133C2×C211—81no—
134C2×C211#39211yescharacteristic
135D211—81no—
136C211#200911yescharacteristic, complemented, Sylow
137#2009—2111nomaximal, Sylow
138#575—2111no—
139C2×#137D21111yescharacteristic, complemented
140#536—2111no—
141#421—2111no—
142#419—2111no—
143#410—2111no—
144#344—2111no—
145C2×#41—2111no—
146C2×#41C2×D21111yescharacteristic
147C22×Q8C2×D21111yescharacteristic
148#137—22no—
149#137—22no—
150#137—4221no—
151#137—4221no—
152AGL(1,ℤ/8)—4221no—
153AGL(1,ℤ/8)—4221no—
154#135—4222no—
155C2×Q16—21no—
156C2×SD16—21no—
157C2×M16—2111no—
158C2×M16C2×D21111yescharacteristic
159C4⋊Q8—4221no—
160#124—4221no—
161#123—4222no—
162C22⋊Q8—4222no—
163#117—2111no—
164#117—2111no—
165#104—4222no—
166#103—4222no—
167#99—4222no—
168#41—21no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—4221no—
173#41—4221no—
174#41—4221no—
175#41—4221no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—2111no—
182C2×Q8C22⋊D21112yes—
183C2×Q8C22⋊D21111yescharacteristic
184C2×D4—21no—
185C2×D4—4221no—
186C2×D4—2111no—
187C2×D4C22⋊D21111yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—4221no—
191C22×C4C22×D21111yescharacteristic
192Q16—24no—
193Q16—4222no—
194SD16—24no—
195SD16—4222no—
196SD16—4222no—
197D8—4222no—
198M16—21no—
199M16—4221no—
200M16—4221no—
201M16C22⋊D21112yes—
202C2×C8—21no—
203C2×C8—2112no—
204C4⋊C4—4224no—
205C4⋊C4—4221no—
206C4⋊C4—4221no—
207C22⋊C4—4224no—
208C22⋊C4—4221no—
209C22⋊C4—2112no—
210C42—4221no—
211C42—4221no—
212C23—21no—
213C23—4221no—
214C23D4×D21111yescharacteristic
215Q8—21no—
216Q8—22no—
217Q8—21no—
218Q8—22no—
219Q8—41no—
220Q8—41no—
221Q8—4221no—
222Q8—4221no—
223D4—21no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—8441no—
228D4—8441no—
229D4—4221no—
230D4—4221no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—8441no—
238C2×C4—8441no—
239C2×C4—8441no—
240C2×C4—4222no—
241C2×C4—4221no—
242C2×C4D4×D21111yescharacteristic
243C2×C4D4×D21111yescharacteristic
244C2×C4C2×C22⋊D21111yescharacteristic
245C2×C4C2×C22⋊D21111yescharacteristic
246C2×C4C2×C22⋊D21111yescharacteristic
247C2×C4#1680996111yescharacteristic
248C8—22no—
249C8—4222no—
250C22—41no—
251C22—41no—
252C22—81no—
253C22—8442no—
254C22—8441no—
255C22#2026994511yescharacteristic
256C22#2026994911yescharacteristic
257C22#2026997311yescharacteristic
258C4—41no—
259C4—42no—
260C4—8442no—
261C4—8441no—
262C4—16881no—
263C4#2026994511yescharacteristic
264C4#2026994611yescharacteristic
265C4#2026994811yescharacteristic
266C4#2026994911yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—81no—
271C2—16881no—
272C2#2721305411yescharacteristic
subgroupG/H
centerC2#27213054
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#137×C211C2
socleC2×C211#392
prime radical#27213944C2
hypercenterC2×#137D211
hypocenterC211#2009
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. #27213720C2semidirect
3. #27213162C2semidirect
4. #27213155C2semidirect
5. #27213120C2semidirect
6. #27212918C2semidirect
7. C2×#137×C211C2semidirect
136. C211#2009semidirect
139. C2×#137D211semidirect

Maximal subgroups

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

subgroupindex
#272139442
#272137202
#272131622
#272131552
#272131202
#272129182
C2×#137×C2112
#2009211