Finite Group Databasebeta

← #41014455

Subgroups — #41014455

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
1C2×#20270007C211yesmaximal, characteristic, complemented
2#27213717C211yesmaximal, characteristic, complemented
3#27213163C211yesmaximal, characteristic, complemented
4#27213156C211yesmaximal, characteristic, complemented
5#27213120C211yesmaximal, characteristic, complemented
6#27212917C211yesmaximal, characteristic, complemented
7#575×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9#20270007—22no—
10#20270007—22no—
11C2×#16809952—21no—
12C2×#16809950—21no—
13C2×#16809946C2211yescharacteristic
14#20269975—21no—
15#20269964—21no—
16#20269958—22no—
17#20269956—22no—
18#20269936C2211yescharacteristic
19#20269936C2211yescharacteristic
20#20269904—22no—
21#20269902—22no—
22#20269892—22no—
23C2×#41×C211C2211yescharacteristic
24C2×#41×C211C2211yescharacteristic
25#137×C211—21no—
26#137×C211—21no—
27AGL(1,ℤ/8)×C211—21no—
28AGL(1,ℤ/8)×C211—21no—
29#135×C211—22no—
30C2×M16×C211C2211yescharacteristic
31C2×Dic422—21no—
32C2×Dic422—21no—
33C2×Dic422—21no—
34C2×Dic422—21no—
35C2×Dic422—21no—
36C2×Dic422D412yes—
37C22×Dic211—21no—
38#16809952—24no—
39#16809950—24no—
40#16809946—21no—
41#16809946D412yes—
42C2×C211⋊C8—21no—
43C4⋊Dic211—21no—
44C4⋊Dic211—21no—
45Dic211⋊C4—24no—
46C22⋊Dic211—24no—
47C22⋊Dic211—21no—
48C22⋊Dic211D412yes—
49C4×Dic211—21no—
50C4×Dic211—21no—
51#41×C211—21no—
52#41×C211—21no—
53#41×C211—21no—
54#41×C211—21no—
55#41×C211—21no—
56#41×C211—21no—
57#41×C211—21no—
58#41×C211—21no—
59C2×Q8×C211D411yescharacteristic
60C2×Q8×C211D411yescharacteristic
61C2×D4×C211—21no—
62C2×D4×C211—21no—
63C2×D4×C211D411yescharacteristic
64C2×D4×C211D411yescharacteristic
65C22×C4×C211—21no—
66C22×C4×C211—21no—
67C22×C4×C211C2311yescharacteristic
68Q16×C211—22no—
69SD16×C211—22no—
70SD16×C211—22no—
71D8×C211—22no—
72M16×C211—21no—
73M16×C211—21no—
74C2×C8×C211D412yes—
75Dic422—22no—
76Dic422—22no—
77Dic422—41no—
78Dic422—41no—
79C2×Dic211—22no—
80C2×Dic211—22no—
81C2×Dic211—41no—
82C2×Dic211—41no—
83C2×Dic211—41no—
84C211⋊C8—22no—
85C23×C211—21no—
86C23×C211—21no—
87C23×C211C2×D411yescharacteristic
88Q8×C211—21no—
89Q8×C211—21no—
90Q8×C211—21no—
91Q8×C211—21no—
92D4×C211—21no—
93D4×C211—21no—
94D4×C211—21no—
95D4×C211—21no—
96D4×C211—41no—
97D4×C211—41no—
98D4×C211—41no—
99D4×C211—41no—
100C2×C4×C211—21no—
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—42no—
115C22×C211—41no—
116C22×C211—41no—
117C22×C211—81no—
118C22×C211#12011yescharacteristic
119C22×C211#12111yescharacteristic
120C22×C211#12711yescharacteristic
121C4×C211—41no—
122C4×C211—41no—
123C4×C211#12011yescharacteristic
124C4×C211#12011yescharacteristic
125C4×C211#12111yescharacteristic
126C4×C211#12111yescharacteristic
127Dic211—42no—
128Dic211—42no—
129Dic211—81no—
130C2×C211—21no—
131C2×C211—21no—
132C2×C211—21no—
133C2×C211—81no—
134C2×C211—81no—
135C2×C211#39211yescharacteristic
136C211#200911yescharacteristic, complemented, Sylow
137#2009—2111nomaximal, Sylow
138#575D21111yescharacteristic
139C2×#137—2111no—
140#536—2111no—
141#421—2111no—
142#419—2111no—
143#410—2111no—
144#344—2111no—
145C2×#41C2×D21111yescharacteristic
146C2×#41C2×D21111yescharacteristic
147C22×Q8—2111no—
148#137—21no—
149#137—21no—
150#137—4222no—
151#137—4222no—
152AGL(1,ℤ/8)—21no—
153AGL(1,ℤ/8)—21no—
154#135—22no—
155C2×Q16—4221no—
156C2×SD16—4221no—
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—21no—
173#41—21no—
174#41—21no—
175#41—21no—
176C2×Q8—4221no—
177C2×Q8—4221no—
178C2×Q8—4221no—
179C2×Q8—4221no—
180C2×Q8—4221no—
181C2×Q8—2112no—
182C2×Q8C22⋊D21111yescharacteristic
183C2×Q8C22⋊D21111yescharacteristic
184C2×D4—21no—
185C2×D4—21no—
186C2×D4C22⋊D21111yescharacteristic
187C2×D4C22⋊D21111yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—4221no—
191C22×C4C22×D21111yescharacteristic
192Q16—22no—
193Q16—4224no—
194SD16—22no—
195SD16—22no—
196SD16—4224no—
197D8—22no—
198M16—21no—
199M16—21no—
200M16—4221no—
201M16—2112no—
202C2×C8—4221no—
203C2×C8C22⋊D21112yes—
204C4⋊C4—4224no—
205C4⋊C4—4221no—
206C4⋊C4—4221no—
207C22⋊C4—4221no—
208C22⋊C4—4224no—
209C22⋊C4—2112no—
210C42—4221no—
211C42—4221no—
212C23—21no—
213C23—21no—
214C23D4×D21111yescharacteristic
215Q8—21no—
216Q8—21no—
217Q8—21no—
218Q8—21no—
219Q8—8441no—
220Q8—8441no—
221Q8—4222no—
222Q8—4222no—
223D4—21no—
224D4—21no—
225D4—21no—
226D4—21no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—41no—
231C2×C4—21no—
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—4222no—
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—42no—
252C22—41no—
253C22—41no—
254C22—81no—
255C22#2026994511yescharacteristic
256C22#2026994911yescharacteristic
257C22#2026997311yescharacteristic
258C4—41no—
259C4—41no—
260C4—8442no—
261C4—8442no—
262C4—16881no—
263C4#2026994511yescharacteristic
264C4#2026994611yescharacteristic
265C4#2026994811yescharacteristic
266C4#2026994911yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—81no—
271C2—81no—
272C2#2721305411yescharacteristic
subgroupG/H
centerC2#27213054
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radical#575×C211C2
socleC2×C211#392
hypercenter#575D211
hypocenterC211#2009
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#20270007C2semidirect
2. #27213717C2semidirect
3. #27213163C2semidirect
4. #27213156C2semidirect
5. #27213120C2semidirect
6. #27212917C2semidirect
136. C211#2009semidirect

Maximal subgroups

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

subgroupindex
C2×#202700072
#272137172
#272131632
#272131562
#272131202
#272129172
#575×C2112
#2009211