Finite Group Databasebeta

← #41014457

Subgroups — #41014457

Order 27008.

348 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 69 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
2C2×#20270007C211yesmaximal, characteristic, complemented
3#27213722C211yesmaximal, characteristic, complemented
4#27213164C211yesmaximal, characteristic, complemented
5#27213154C211yesmaximal, characteristic, complemented
6#27212916C211yesmaximal, characteristic, complemented
7#410×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C22×Dic422C2211yescharacteristic
10#20270008—21no—
11#20270007—22no—
12#20270007—21no—
13#20270007—22no—
14#20270003—21no—
15#20270002—21no—
16#20269997—22no—
17C2×#16809952—21no—
18C2×#16809950—21no—
19C2×#16809946C2211yescharacteristic
20C2×#16809946C2211yescharacteristic
21#20269977—21no—
22#20269965—21no—
23#20269961—22no—
24#20269957—22no—
25#20269905—22no—
26#20269900—22no—
27C2×#41×C211C2211yescharacteristic
28#117×C211C2211yescharacteristic
29#117×C211C2211yescharacteristic
30#99×C211—22no—
31#16809962—21no—
32#16809962—21no—
33#16809962—21no—
34#16809962—21no—
35C2×Dic422—21no—
36C2×Dic422—21no—
37C2×Dic422—21no—
38C2×Dic422—21no—
39C2×Dic422—21no—
40C2×Dic422D412yes—
41C2×Dic422D411yescharacteristic
42C2×D844D411yescharacteristic
43C2×C22⋊D211—21no—
44C22×Dic211—21no—
45C2×C4×D211—21no—
46#16809952—24no—
47#16809952—22no—
48#16809950—24no—
49#16809950—22no—
50#16809949—22no—
51#16809947—22no—
52#16809946—21no—
53#16809946—21no—
54#16809946—21no—
55#16809946D412yes—
56C2×C211⋊C8—21no—
57C2×C211⋊C8D412yes—
58C4⋊Dic211—24no—
59(C2×D211)⋊C4—24no—
60#41×C211—21no—
61#41×C211—21no—
62#41×C211—21no—
63#41×C211—21no—
64C2×Q8×C211D411yescharacteristic
65C2×D4×C211—21no—
66C2×D4×C211D411yescharacteristic
67C22×C4×C211—21no—
68C22×C4×C211C2311yescharacteristic
69C4⋊C4×C211—21no—
70C4⋊C4×C211—21no—
71C22⋊C4×C211—21no—
72C22⋊C4×C211D412yes—
73C42×C211—21no—
74C42×C211—21no—
75C22×D211—21no—
76Dic422—22no—
77Dic422—21no—
78Dic422—22no—
79Dic422—21no—
80Dic422—41no—
81Dic422—41no—
82D844—21no—
83D844—21no—
84C22⋊D211—41no—
85C22⋊D211—41no—
86C2×Dic211—22no—
87C2×Dic211—21no—
88C2×Dic211—41no—
89C2×Dic211—41no—
90C4×D211—41no—
91C4×D211—41no—
92C211⋊C8—22no—
93C211⋊C8—22no—
94C23×C211—21no—
95C23×C211C2×D411yescharacteristic
96Q8×C211—21no—
97Q8×C211—21no—
98D4×C211—21no—
99D4×C211—21no—
100D4×C211—41no—
101D4×C211—41no—
102C2×C4×C211—22no—
103C2×C4×C211—21no—
104C2×C4×C211—41no—
105C2×C4×C211—41no—
106C2×C4×C211—41no—
107C2×C4×C211C2×D411yescharacteristic
108C2×C4×C211C2×D411yescharacteristic
109C2×C4×C211C2×D411yescharacteristic
110C2×C4×C211C2×D411yescharacteristic
111C2×C4×C211C2×D411yescharacteristic
112C2×C4×C211#4111yescharacteristic
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—42no—
122C4×C211—41no—
123C4×C211—81no—
124C4×C211#12011yescharacteristic
125C4×C211#12011yescharacteristic
126C4×C211#12111yescharacteristic
127C4×C211#12111yescharacteristic
128Dic211—41no—
129Dic211—42no—
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×#137—2111no—
140#536—2111no—
141#421—2111no—
142#419—2111no—
143#410D21111yescharacteristic, complemented
144#344—2111no—
145C2×#41—2111no—
146C2×#41C2×D21111yescharacteristic
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×M16—2111no—
158C2×M16—2111no—
159C4⋊Q8—4221no—
160#124—4221no—
161#123—4222no—
162C22⋊Q8—4222no—
163#117C2×D21111yescharacteristic
164#117C2×D21111yescharacteristic
165#104—4222no—
166#103—4222no—
167#99—22no—
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—4221no—
177C2×Q8—4221no—
178C2×Q8—4221no—
179C2×Q8—4221no—
180C2×Q8—4221no—
181C2×Q8—2112no—
182C2×Q8—2111no—
183C2×Q8D84411yescharacteristic
184C2×D4—21no—
185C2×D4—4221no—
186C2×D4—2111no—
187C2×D4D84411yescharacteristic
188C22×C4—21no—
189C22×C4—4221no—
190C22×C4—4221no—
191C22×C4C22×D21111yescharacteristic
192Q16—4224no—
193Q16—4222no—
194SD16—4224no—
195SD16—4222no—
196SD16—4222no—
197D8—4222no—
198M16—4221no—
199M16—4221no—
200M16—4221no—
201M16—2112no—
202C2×C8—4221no—
203C2×C8—2112no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—4224no—
207C22⋊C4—21no—
208C22⋊C4—4224no—
209C22⋊C4C22⋊D21112yes—
210C42—21no—
211C42—21no—
212C23—21no—
213C23—4221no—
214C23C2×C22⋊D21111yescharacteristic
215Q8—21no—
216Q8—21no—
217Q8—8441no—
218Q8—8441no—
219Q8—4222no—
220Q8—4221no—
221Q8—4222no—
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—8441no—
237C2×C4—8441no—
238C2×C4—8441no—
239C2×C4—8441no—
240C2×C4—4222no—
241C2×C4—4221no—
242C2×C4D4×D21111yescharacteristic
243C2×C4D4×D21111yescharacteristic
244C2×C4D4×D21111yescharacteristic
245C2×C4D4×D21111yescharacteristic
246C2×C4C2×D84411yescharacteristic
247C2×C4#1680996211yescharacteristic
248C8—4222no—
249C8—4222no—
250C22—41no—
251C22—41no—
252C22—81no—
253C22—8442no—
254C22—8441no—
255C22#2026994511yescharacteristic
256C22#2026995111yescharacteristic
257C22#2026997311yescharacteristic
258C4—42no—
259C4—41no—
260C4—81no—
261C4—8441no—
262C4—8442no—
263C4#2026994711yescharacteristic
264C4#2026994711yescharacteristic
265C4#2026995011yescharacteristic
266C4#2026995011yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—81no—
271C2—16881no—
272C2#2721305511yescharacteristic
subgroupG/H
centerC2#27213055
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#410×C211C2
socleC2×C211#392
prime radical#27213944C2
hypercenter#410D211
hypocenterC211#2009
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. C2×#20270007C2semidirect
3. #27213722C2semidirect
4. #27213164C2semidirect
5. #27213154C2semidirect
6. #27212916C2semidirect
7. #410×C211C2semidirect
136. C211#2009semidirect
143. #410D211semidirect

Maximal subgroups

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

subgroupindex
#272139442
C2×#202700072
#272137222
#272131642
#272131542
#272129162
#410×C2112
#2009211