Finite Group Databasebeta

← #41014703

Subgroups — #41014703

Order 27008.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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#27213173C211yesmaximal, characteristic, complemented
3#27213142C211yesmaximal, characteristic, complemented
4#27213140C211yesmaximal, characteristic, complemented
5#27213119C211yesmaximal, characteristic, complemented
6#27212902C211yesmaximal, characteristic, complemented
7#410×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20270008—21no—
10#20270007—21no—
11#20270003—21no—
12#20270002—21no—
13#20269997—22no—
14C2×#16809946C2211yescharacteristic
15C22×C211⋊C8C2211yescharacteristic
16#20269936C2211yescharacteristic
17C4⋊(C211⋊C8)—22no—
18#20269901—21no—
19#20269900—21no—
20#20269898—21no—
21#20269897—21no—
22#20269891—22no—
23C2×#41×C211C2211yescharacteristic
24#117×C211C2211yescharacteristic
25#117×C211C2211yescharacteristic
26#99×C211—22no—
27#16809962—21no—
28#16809962—21no—
29#16809962—21no—
30#16809962—21no—
31C2×Dic422D411yescharacteristic
32C2×D844D411yescharacteristic
33C2×C22⋊D211—21no—
34C2×C4×D211—21no—
35#16809952—22no—
36#16809950—22no—
37#16809949—22no—
38#16809947—22no—
39#16809946—21no—
40#16809946—21no—
41C2×C211⋊C8—21no—
42C2×C211⋊C8—21no—
43C2×C211⋊C8—21no—
44C2×C211⋊C8D412yes—
45C4⋊Dic211—21no—
46C22⋊Dic211—21no—
47C4×Dic211—21no—
48#41×C211—21no—
49#41×C211—21no—
50#41×C211—21no—
51#41×C211—21no—
52C2×Q8×C211D411yescharacteristic
53C2×D4×C211—21no—
54C2×D4×C211D411yescharacteristic
55C22×C4×C211—21no—
56C22×C4×C211C2311yescharacteristic
57C4⋊C4×C211—21no—
58C4⋊C4×C211—21no—
59C22⋊C4×C211—21no—
60C22⋊C4×C211D412yes—
61C42×C211—21no—
62C42×C211—21no—
63C22×D211—21no—
64Dic422—21no—
65Dic422—21no—
66D844—21no—
67D844—21no—
68C22⋊D211—41no—
69C22⋊D211—41no—
70C2×Dic211—21no—
71C2×Dic211—41no—
72C4×D211—41no—
73C4×D211—41no—
74C211⋊C8—22no—
75C211⋊C8—41no—
76C23×C211—21no—
77C23×C211C2×D411yescharacteristic
78Q8×C211—21no—
79Q8×C211—21no—
80D4×C211—21no—
81D4×C211—21no—
82D4×C211—41no—
83D4×C211—41no—
84C2×C4×C211—22no—
85C2×C4×C211—21no—
86C2×C4×C211—41no—
87C2×C4×C211—41no—
88C2×C4×C211—41no—
89C2×C4×C211C2×D411yescharacteristic
90C2×C4×C211C2×D411yescharacteristic
91C2×C4×C211C2×D411yescharacteristic
92C2×C4×C211#4111yescharacteristic
93C2×C4×C211#4111yescharacteristic
94C2×C4×C211#4111yescharacteristic
95C22×C211—41no—
96C22×C211—41no—
97C22×C211—81no—
98C22×C211#12111yescharacteristic
99C22×C211#12111yescharacteristic
100C22×C211#12111yescharacteristic
101C2×D211—41no—
102C2×D211—81no—
103C4×C211—42no—
104C4×C211—41no—
105C4×C211—81no—
106C4×C211#12011yescharacteristic
107C4×C211#12311yescharacteristic
108C4×C211#12311yescharacteristic
109C4×C211#12411yescharacteristic
110Dic211—41no—
111Dic211—81no—
112C2×C211—21no—
113C2×C211—21no—
114C2×C211—21no—
115C2×C211—81no—
116C2×C211#39411yescharacteristic
117D211—81no—
118C211#205311yescharacteristic, complemented, Sylow
119#2053—2111nomaximal, Sylow
120#575—2111no—
121#424—2111no—
122#416—2111no—
123#416—2111no—
124#410—2111no—
125#410D21111yescharacteristic, complemented
126H(3,ℤ/4)—2111no—
127C2×#41—2111no—
128C2×#41C2×D21111yescharacteristic
129#137—4221no—
130#137—4221no—
131AGL(1,ℤ/8)—4221no—
132AGL(1,ℤ/8)—4221no—
133#135—4222no—
134C2×M16—2111no—
135C22×C8—2111no—
136#117—2111no—
137#117C2×D21111yescharacteristic
138#117C2×D21111yescharacteristic
139C4⋊C8—4222no—
140#103—4221no—
141#103—4221no—
142#102—4221no—
143#102—4221no—
144#99—22no—
145#99—4222no—
146#41—21no—
147#41—21no—
148#41—21no—
149#41—21no—
150#41—4221no—
151#41—4221no—
152#41—4221no—
153#41—4221no—
154C2×Q8—2111no—
155C2×Q8D84411yescharacteristic
156C2×D4—21no—
157C2×D4—4221no—
158C2×D4—2111no—
159C2×D4D84411yescharacteristic
160C22×C4—21no—
161C22×C4—4221no—
162C22×C4C22×D21111yescharacteristic
163Q16—4222no—
164SD16—4222no—
165SD16—4222no—
166D8—4222no—
167M16—4221no—
168M16—4221no—
169C2×C8—4221no—
170C2×C8—4221no—
171C2×C8—4221no—
172C2×C8—2112no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—4221no—
176C22⋊C4—21no—
177C22⋊C4—4221no—
178C22⋊C4C22⋊D21112yes—
179C42—21no—
180C42—21no—
181C42—4221no—
182C23—21no—
183C23—4221no—
184C23C2×C22⋊D21111yescharacteristic
185Q8—21no—
186Q8—21no—
187Q8—4221no—
188Q8—4221no—
189D4—21no—
190D4—21no—
191D4—41no—
192D4—41no—
193D4—8441no—
194D4—8441no—
195D4—4221no—
196D4—4221no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—8441no—
203C2×C4—8441no—
204C2×C4—8441no—
205C2×C4—4221no—
206C2×C4D4×D21111yescharacteristic
207C2×C4D4×D21111yescharacteristic
208C2×C4C2×D84411yescharacteristic
209C2×C4#1680996111yescharacteristic
210C2×C4#1680996111yescharacteristic
211C2×C4#1680996211yescharacteristic
212C8—8441no—
213C8—4222no—
214C22—41no—
215C22—41no—
216C22—81no—
217C22—8441no—
218C22—16881no—
219C22#2026994811yescharacteristic
220C22#2026994911yescharacteristic
221C22#2026995111yescharacteristic
222C4—42no—
223C4—41no—
224C4—81no—
225C4—8441no—
226C4—16881no—
227C4#2026994711yescharacteristic
228C4#2026995911yescharacteristic
229C4#2026996111yescharacteristic
230C4#2026996311yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—16881no—
236C2#2721306211yescharacteristic
subgroupG/H
centerC4#20269963
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#410×C211C2
socleC2×C211#394
prime radical#27213944C2
hypercenter#410D211
hypocenterC211#2053
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #27213173C2semidirect
3. #27213142C2semidirect
4. #27213140C2semidirect
5. #27213119C2semidirect
6. #27212902C2semidirect
7. #410×C211C2semidirect
118. C211#2053semidirect
125. #410D211semidirect

Maximal subgroups

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

subgroupindex
#272139442
#272131732
#272131422
#272131402
#272131192
#272129022
#410×C2112
#2053211