Finite Group Databasebeta

← #41014461

Subgroups — #41014461

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#27213946C211yesmaximal, characteristic
2C2×#20270010C211yesmaximal, characteristic, complemented
3#27213722C211yesmaximal, characteristic, complemented
4#27213162C211yesmaximal, characteristic, complemented
5#27213156C211yesmaximal, characteristic, complemented
6#27213119C211yesmaximal, characteristic, complemented
7#344×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C2×#16809962C2211yescharacteristic
10C22×Dic422C2211yescharacteristic
11#20270010—22no—
12#20270010—21no—
13#20270010—21no—
14#20270010—22no—
15#20270005—21no—
16#20270005—21no—
17#20269999—22no—
18C2×Dic844—21no—
19C2×#16809951—21no—
20#20269977—21no—
21#20269965—21no—
22#20269961—22no—
23#20269957—22no—
24#20269936C2211yescharacteristic
25#20269903—22no—
26#20269902—22no—
27#20269891—22no—
28C2×M16×C211C2211yescharacteristic
29C2×M16×C211C2211yescharacteristic
30#117×C211C2211yescharacteristic
31#16809962—21no—
32#16809962—21no—
33#16809962—21no—
34#16809962—21no—
35#16809962—21no—
36#16809962—21no—
37#16809962—21no—
38#16809962—21no—
39C2×Dic422—21no—
40C2×Dic422—21no—
41C2×Dic422—21no—
42C2×Dic422—21no—
43C2×Dic422—21no—
44C2×Dic422D412yes—
45C2×Dic422D411yescharacteristic
46C2×Dic422D411yescharacteristic
47C2×D844D411yescharacteristic
48C2×D844D411yescharacteristic
49C2×C22⋊D211—21no—
50C2×C22⋊D211—21no—
51C22×Dic211—21no—
52C2×C4×D211—21no—
53C2×C4×D211—21no—
54Dic844—24no—
55Dic844—22no—
56#16809951—24no—
57#16809951—22no—
58#16809951—22no—
59D1688—22no—
60C4⋊Dic211—24no—
61C4⋊Dic211—21no—
62C22⋊Dic211—21no—
63(C2×D211)⋊C4—24no—
64C4×Dic211—21no—
65C22×C4×C211C2311yescharacteristic
66M16×C211—21no—
67M16×C211—21no—
68M16×C211—21no—
69M16×C211D412yes—
70C2×C8×C211—21no—
71C2×C8×C211D412yes—
72C4⋊C4×C211—21no—
73C22⋊C4×C211D412yes—
74C42×C211—21no—
75C22×D211—21no—
76C22×D211—21no—
77Dic422—22no—
78Dic422—21no—
79Dic422—21no—
80Dic422—21no—
81Dic422—22no—
82Dic422—21no—
83Dic422—41no—
84Dic422—41no—
85D844—21no—
86D844—21no—
87D844—21no—
88D844—21no—
89C22⋊D211—41no—
90C22⋊D211—41no—
91C22⋊D211—41no—
92C22⋊D211—41no—
93C2×Dic211—22no—
94C2×Dic211—21no—
95C2×Dic211—21no—
96C2×Dic211—41no—
97C2×Dic211—41no—
98C2×Dic211—41no—
99C4×D211—41no—
100C4×D211—41no—
101C4×D211—41no—
102C4×D211—41no—
103C23×C211C2×D411yescharacteristic
104C2×C4×C211—22no—
105C2×C4×C211C2×D411yescharacteristic
106C2×C4×C211C2×D411yescharacteristic
107C2×C4×C211C2×D411yescharacteristic
108C2×C4×C211C2×D411yescharacteristic
109C2×C4×C211C2×D411yescharacteristic
110C2×C4×C211#4111yescharacteristic
111C8×C211—22no—
112C8×C211—22no—
113C22×C211—41no—
114C22×C211#12011yescharacteristic
115C22×C211#12111yescharacteristic
116C22×C211#12711yescharacteristic
117C2×D211—42no—
118C2×D211—41no—
119C2×D211—41no—
120C2×D211—81no—
121C4×C211—42no—
122C4×C211#12011yescharacteristic
123C4×C211#12011yescharacteristic
124C4×C211#12111yescharacteristic
125C4×C211#12111yescharacteristic
126Dic211—41no—
127Dic211—41no—
128Dic211—42no—
129Dic211—81no—
130C2×C211—21no—
131C2×C211—21no—
132C2×C211—21no—
133C2×C211#39211yescharacteristic
134D211—81no—
135D211—81no—
136C211#200911yescharacteristic, complemented, Sylow
137#2009—2111nomaximal, Sylow
138#575—2111no—
139C2×#137—2111no—
140#536—2111no—
141#421—2111no—
142#419—2111no—
143#410—2111no—
144#344D21111yescharacteristic, complemented
145C2×#41—2111no—
146C2×#41—2111no—
147C22×Q8—2111no—
148#137—4222no—
149#137—4221no—
150#137—4221no—
151#137—4222no—
152AGL(1,ℤ/8)—4221no—
153AGL(1,ℤ/8)—4221no—
154#135—4222no—
155C2×Q16—4221no—
156C2×SD16—4221no—
157C2×M16C2×D21111yescharacteristic
158C2×M16C2×D21111yescharacteristic
159C4⋊Q8—4221no—
160#124—4221no—
161#123—4222no—
162C22⋊Q8—4222no—
163#117—2111no—
164#117C2×D21111yescharacteristic
165#104—4222no—
166#103—4222no—
167#99—4222no—
168#41—4221no—
169#41—4221no—
170#41—4221no—
171#41—4221no—
172#41—4221no—
173#41—4221no—
174#41—4221no—
175#41—4221no—
176C2×Q8—4221no—
177C2×Q8—4221no—
178C2×Q8—4221no—
179C2×Q8—4221no—
180C2×Q8—4221no—
181C2×Q8—2112no—
182C2×Q8—2111no—
183C2×Q8—2111no—
184C2×D4—4221no—
185C2×D4—4221no—
186C2×D4—2111no—
187C2×D4—2111no—
188C22×C4—4221no—
189C22×C4—4221no—
190C22×C4—4221no—
191C22×C4C22×D21111yescharacteristic
192Q16—4224no—
193Q16—4222no—
194SD16—4224no—
195SD16—4222no—
196SD16—4222no—
197D8—4222no—
198M16—21no—
199M16—21no—
200M16—21no—
201M16D84412yes—
202C2×C8—21no—
203C2×C8D84412yes—
204C4⋊C4—21no—
205C4⋊C4—4224no—
206C4⋊C4—4221no—
207C22⋊C4—4224no—
208C22⋊C4—4221no—
209C22⋊C4D84412yes—
210C42—21no—
211C42—4221no—
212C23—4221no—
213C23—4221no—
214C23C2×D84411yescharacteristic
215Q8—8441no—
216Q8—8441no—
217Q8—4222no—
218Q8—4221no—
219Q8—4221no—
220Q8—4221no—
221Q8—4222no—
222Q8—4221no—
223D4—8441no—
224D4—8441no—
225D4—8441no—
226D4—8441no—
227D4—4221no—
228D4—4221no—
229D4—4221no—
230D4—4221no—
231C2×C4—22no—
232C2×C4—8441no—
233C2×C4—8441no—
234C2×C4—8441no—
235C2×C4—8441no—
236C2×C4—8441no—
237C2×C4—8441no—
238C2×C4—8441no—
239C2×C4—4222no—
240C2×C4—4221no—
241C2×C4—4221no—
242C2×C4D4×D21111yescharacteristic
243C2×C4D4×D21111yescharacteristic
244C2×C4D4×D21111yescharacteristic
245C2×C4C2×D84411yescharacteristic
246C2×C4C2×D84411yescharacteristic
247C2×C4#1680996311yescharacteristic
248C8—22no—
249C8—22no—
250C22—41no—
251C22—8442no—
252C22—8441no—
253C22—8441no—
254C22—16881no—
255C22#2026994711yescharacteristic
256C22#2026995211yescharacteristic
257C22#2026997411yescharacteristic
258C4—42no—
259C4—8441no—
260C4—8441no—
261C4—8442no—
262C4—16881no—
263C4#2026994711yescharacteristic
264C4#2026994711yescharacteristic
265C4#2026995211yescharacteristic
266C4#2026995211yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—16881no—
271C2—16881no—
272C2#2721305611yescharacteristic
subgroupG/H
centerC2#27213056
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#344×C211C2
socleC2×C211#392
prime radical#27213946C2
hypercenter#344D211
hypocenterC211#2009
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. C2×#20270010C2semidirect
3. #27213722C2semidirect
4. #27213162C2semidirect
5. #27213156C2semidirect
6. #27213119C2semidirect
7. #344×C211C2semidirect
136. C211#2009semidirect
144. #344D211semidirect

Maximal subgroups

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

subgroupindex
#272139462
C2×#202700102
#272137222
#272131622
#272131562
#272131192
#344×C2112
#2009211