Finite Group Databasebeta

← #41014605

Subgroups — #41014605

Order 27008.

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

normal subgroups only

This group has 326 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#27213189C211yesmaximal, characteristic, complemented
2C2×#20269904C211yesmaximal, characteristic, complemented
3#27213147C211yesmaximal, characteristic, complemented
4C2×#20269894C211yesmaximal, characteristic, complemented
5#27213049C211yesmaximal, characteristic, complemented
6#27212905C211yesmaximal, characteristic, complemented
7#535×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C2×C4⋊Dic211C2211yescharacteristic
11C2×C22⋊Dic211—22no—
12C2×C4×Dic211C2211yescharacteristic
13#20269910—21no—
14#20269908—21no—
15#20269904—22no—
16#20269901—21no—
17#20269898—21no—
18#20269894—22no—
19C2×#41×C211C2211yescharacteristic
20C22×D4×C211C2211yescharacteristic
21#127×C211—21no—
22#124×C211—21no—
23#121×C211—21no—
24#121×C211—21no—
25#120×C211—22no—
26#117×C211C2211yescharacteristic
27C22×Dic211—21no—
28C22×Dic211—21no—
29#16809946—21no—
30#16809946—21no—
31#16809946—21no—
32#16809946D412yescomplemented
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211—21no—
37C22⋊Dic211—44no—
38C4×Dic211—21no—
39C4×Dic211—21no—
40C24×C211D412yes—
41#41×C211—21no—
42#41×C211—21no—
43#41×C211—21no—
44#41×C211—21no—
45C2×Q8×C211D411yescharacteristic
46C2×D4×C211—21no—
47C2×D4×C211—22no—
48C2×D4×C211—22no—
49C2×D4×C211—21no—
50C2×D4×C211—21no—
51C2×D4×C211—21no—
52C2×D4×C211—21no—
53C2×D4×C211—21no—
54C2×D4×C211—21no—
55C2×D4×C211D411yescharacteristic
56C22×C4×C211—21no—
57C22×C4×C211C2311yescharacteristic
58C4⋊C4×C211D411yescharacteristic
59C4⋊C4×C211D411yescharacteristic
60C22⋊C4×C211—22no—
61C22⋊C4×C211—22no—
62C22⋊C4×C211—21no—
63C42×C211—21no—
64C2×Dic211—22no—
65C2×Dic211—41no—
66C2×Dic211—41no—
67C2×Dic211—42no—
68C2×Dic211—41no—
69C211⋊C8—22no—
70C211⋊C8—41no—
71C23×C211—22no—
72C23×C211—21no—
73C23×C211—22no—
74C23×C211—22no—
75C23×C211—42no—
76C23×C211—42no—
77C23×C211C2×D411yescharacteristic
78Q8×C211—21no—
79Q8×C211—21no—
80D4×C211—21no—
81D4×C211—21no—
82D4×C211—41no—
83D4×C211—41no—
84D4×C211—41no—
85D4×C211—41no—
86D4×C211—41no—
87D4×C211—41no—
88D4×C211—82no—
89D4×C211—82no—
90C2×C4×C211—21no—
91C2×C4×C211—21no—
92C2×C4×C211—21no—
93C2×C4×C211—41no—
94C2×C4×C211—41no—
95C2×C4×C211C2×D411yescharacteristic
96C2×C4×C211C2×D411yescharacteristic
97C2×C4×C211C2×D411yescharacteristic
98C2×C4×C211#4111yescharacteristic
99C2×C4×C211#4111yescharacteristic
100C2×C4×C211#4111yescharacteristic
101C22×C211—21no—
102C22×C211—21no—
103C22×C211—44no—
104C22×C211—41no—
105C22×C211—42no—
106C22×C211—41no—
107C22×C211—42no—
108C22×C211—41no—
109C22×C211—42no—
110C22×C211—82no—
111C22×C211#12011yescharacteristic
112C22×C211#12111yescharacteristic
113C22×C211#12311yescharacteristic
114C4×C211—41no—
115C4×C211—41no—
116C4×C211—41no—
117C4×C211#12111yescharacteristic
118C4×C211#12111yescharacteristic
119C4×C211#12311yescharacteristic
120C4×C211#12411yescharacteristic
121Dic211—42no—
122Dic211—81no—
123C2×C211—21no—
124C2×C211—21no—
125C2×C211—81no—
126C2×C211—82no—
127C2×C211#39411yescharacteristic
128C2×C211#45311yescharacteristic
129C2×C211#45511yescharacteristic
130C211#203711yescharacteristic, complemented, Sylow
131#2037—2111nomaximal, Sylow
132#535D21111yescharacteristic
133#428—2111no—
134C2×#104—2111no—
135#417—2111no—
136C2×#100—2111no—
137#390—2111no—
138#339—2111no—
139C2×#41C2×D21111yescharacteristic
140C22×D4C2×D21111yescharacteristic
141C2×M16—2111no—
142C2×M16—2111no—
143#127—21no—
144#124—21no—
145#121—21no—
146#121—21no—
147#120—22no—
148#117C2×D21111yescharacteristic
149C2×C4⋊C4—2111no—
150C2×C22⋊C4—4222no—
151C2×C42—2111no—
152#107—4221no—
153#106—4221no—
154#104—4222no—
155#103—4221no—
156#102—4221no—
157#100—4222no—
158C24C22⋊D21112yes—
159#41—21no—
160#41—21no—
161#41—21no—
162#41—21no—
163C2×Q8C22⋊D21111yescharacteristic
164C2×D4—21no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4C22⋊D21111yescharacteristic
174C22×C4—21no—
175C22×C4—4221no—
176C22×C4—4221no—
177C22×C4C22×D21111yescharacteristic
178M16—4221no—
179M16—4221no—
180M16—4221no—
181M16—2112no—
182C2×C8—4221no—
183C2×C8—4221no—
184C4⋊C4—4221no—
185C4⋊C4—4221no—
186C4⋊C4C22⋊D21111yescharacteristic
187C4⋊C4C22⋊D21111yescharacteristic
188C22⋊C4—22no—
189C22⋊C4—22no—
190C22⋊C4—21no—
191C22⋊C4—8444no—
192C42—21no—
193C42—4221no—
194C42—4221no—
195C23—22no—
196C23—21no—
197C23—22no—
198C23—22no—
199C23—42no—
200C23—42no—
201C23C2×C22⋊D21111yescharacteristic
202Q8—21no—
203Q8—21no—
204D4—21no—
205D4—21no—
206D4—41no—
207D4—41no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—41no—
212D4—82no—
213D4—82no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—8441no—
220C2×C4—8441no—
221C2×C4—8442no—
222C2×C4—8441no—
223C2×C4—4222no—
224C2×C4D4×D21111yescharacteristic
225C2×C4C2×C22⋊D21111yescharacteristic
226C2×C4C2×C22⋊D21111yescharacteristic
227C2×C4#1680996111yescharacteristic
228C2×C4#1680996111yescharacteristic
229C2×C4#1680996111yescharacteristic
230C8—8441no—
231C8—4222no—
232C22—21no—
233C22—21no—
234C22—44no—
235C22—41no—
236C22—42no—
237C22—41no—
238C22—42no—
239C22—41no—
240C22—42no—
241C22—82no—
242C22#2026994611yescharacteristic
243C22#2026994911yescharacteristic
244C22#2026995811yescharacteristic
245C4—41no—
246C4—41no—
247C4—41no—
248C4—8442no—
249C4—16881no—
250C4#2026994811yescharacteristic
251C4#2026994911yescharacteristic
252C4#2026995811yescharacteristic
253C4#2026996411yescharacteristic
254C2—21no—
255C2—21no—
256C2—82no—
257C2—81no—
258C2#2721306011yescharacteristic
259C2#2721331411yescharacteristic
260C2#2721332411yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radical#535×C211C2
socleC22×C211#120
hypercenter#535D211
hypocenterC211#2037
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #27213189C2semidirect
2. C2×#20269904C2semidirect
3. #27213147C2semidirect
4. C2×#20269894C2semidirect
5. #27213049C2semidirect
6. #27212905C2semidirect
32. #16809946D4semidirect
130. C211#2037semidirect

Maximal subgroups

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

subgroupindex
#272131892
C2×#202699042
#272131472
C2×#202698942
#272130492
#272129052
#535×C2112
#2037211