Finite Group Databasebeta

← #41013760

Subgroups — #41013760

Order 27008.

310 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 89 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#27213162C211yesmaximal, characteristic, complemented
3#27213142C211yesmaximal, characteristic, complemented
4#27213114C211yesmaximal, characteristic, complemented
5#27212917C211yesmaximal, characteristic, complemented
6#27212910C211yesmaximal, characteristic, complemented
7#568×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20270008—21no—
10#20270007C412yescomplemented
11#20270003—21no—
12#20270002C412yescomplemented
13#20269997—22no—
14C2×#16809946C2211yescharacteristic
15C22×C211⋊C8C2211yescharacteristic
16#20269936C2211yescharacteristic
17#20269903—22no—
18#20269901—21no—
19#20269898—21no—
20(C2×D211)⋊C8—22no—
21C2×#41×C211C2211yescharacteristic
22#131×C211—22no—
23#131×C211—22no—
24C2×M16×C211—22no—
25C2×M16×C211C2211yescharacteristic
26C2×M16×C211C2211yescharacteristic
27#16809962—21no—
28#16809962—21no—
29#16809962—21no—
30#16809962—21no—
31C2×Dic422C2×C411yescharacteristic
32C2×D844C2×C411yescharacteristic
33C2×C22⋊D211—21no—
34C2×C4×D211—21no—
35#16809952—22no—
36#16809950—22no—
37#16809949—22no—
38#16809947—22no—
39#16809946—21no—
40#16809946C2×C412yes—
41C2×C211⋊C8—21no—
42C2×C211⋊C8—21no—
43C2×C211⋊C8—21no—
44C2×C211⋊C8—21no—
45C4⋊Dic211—21no—
46C22⋊Dic211—21no—
47C4×Dic211—21no—
48#41×C211—21no—
49#41×C211—21no—
50#41×C211C2×C412yes—
51#41×C211D412yes—
52C2×Q8×C211D411yescharacteristic
53C2×D4×C211—21no—
54C2×D4×C211D411yescharacteristic
55C22×C4×C211—21no—
56C22×C4×C211C2311yescharacteristic
57M16×C211—24no—
58M16×C211—21no—
59M16×C211—21no—
60M16×C211—21no—
61M16×C211D412yes—
62C2×C8×C211—24no—
63C2×C8×C211—21no—
64C2×C8×C211D412yes—
65C22×D211—21no—
66Dic422—21no—
67Dic422—21no—
68D844—21no—
69D844—21no—
70C22⋊D211—41no—
71C22⋊D211—41no—
72C2×Dic211—21no—
73C2×Dic211—41no—
74C4×D211—41no—
75C4×D211—41no—
76C211⋊C8—22no—
77C211⋊C8—41no—
78C23×C211—21no—
79C23×C211#4111yescharacteristic
80Q8×C211—21no—
81Q8×C211C22⋊C412yes—
82D4×C211—22no—
83D4×C211—22no—
84D4×C211—21no—
85D4×C211C22⋊C412yes—
86C2×C4×C211—21no—
87C2×C4×C211—22no—
88C2×C4×C211—22no—
89C2×C4×C211C22×C411yescharacteristic
90C2×C4×C211C2×D411yescharacteristic
91C2×C4×C211C2×D411yescharacteristic
92C2×C4×C211C2×D411yescharacteristic
93C2×C4×C211C2×D411yescharacteristic
94C2×C4×C211#4111yescharacteristic
95C8×C211—22no—
96C8×C211—22no—
97C22×C211—22no—
98C22×C211—41no—
99C22×C211—81no—
100C22×C211C4×D411yescharacteristic
101C22×C211#12111yescharacteristic
102C22×C211#12311yescharacteristic
103C2×D211—42no—
104C2×D211—41no—
105C4×C211—22no—
106C4×C211C2×C22⋊C411yescharacteristic
107C4×C211C4×D411yescharacteristic
108C4×C211#12011yescharacteristic
109C4×C211#12111yescharacteristic
110Dic211—41no—
111Dic211—81no—
112C2×C211—21no—
113C2×C211—21no—
114C2×C211—21no—
115C2×C211—42no—
116C2×C211#38611yescharacteristic
117D211—81no—
118C211#188111yescharacteristic, complemented, Sylow
119#1881—2111nomaximal, Sylow
120#575—2111no—
121#568D21111yescharacteristic, complemented
122#421—2111no—
123#416—2111no—
124#408—2111no—
125#344—2111no—
126#341—2111no—
127C2×#41—2111no—
128C2×#41C2×D21111yescharacteristic
129#137—4221no—
130#137—2112no—
131AGL(1,ℤ/8)—4221no—
132AGL(1,ℤ/8)—2112no—
133#135—4222no—
134#131—22no—
135#131—22no—
136C2×M16—22no—
137C2×M16—2111no—
138C2×M16C2×D21111yescharacteristic
139C2×M16C2×D21111yescharacteristic
140C22×C8—2111no—
141#117—2111no—
142#104—4222no—
143#103—4221no—
144#102—4221no—
145C22⋊C8—4222no—
146#41—21no—
147#41—21no—
148#41—4221no—
149#41—4221no—
150#41—4221no—
151#41—4221no—
152#41C4×D21112yes—
153#41C22⋊D21112yes—
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—24no—
168M16—21no—
169M16—21no—
170M16—21no—
171M16—4221no—
172M16—2112no—
173M16C22⋊D21112yes—
174C2×C8—24no—
175C2×C8—21no—
176C2×C8—4221no—
177C2×C8—4221no—
178C2×C8—4221no—
179C2×C8—4221no—
180C2×C8C22⋊D21112yes—
181C4⋊C4—4221no—
182C22⋊C4—4221no—
183C42—4221no—
184C23—21no—
185C23—4221no—
186C23#1680996211yescharacteristic
187Q8—21no—
188Q8—4221no—
189Q8—4221no—
190Q8(C2×D211)⋊C412yes—
191D4—22no—
192D4—22no—
193D4—21no—
194D4—8441no—
195D4—8441no—
196D4—4221no—
197D4—4221no—
198D4(C2×D211)⋊C412yes—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—8441no—
203C2×C4—8441no—
204C2×C4—8441no—
205C2×C4—4221no—
206C2×C4C2×C4×D21111yescharacteristic
207C2×C4C2×C22⋊D21111yescharacteristic
208C2×C4C2×C22⋊D21111yescharacteristic
209C2×C4C2×C22⋊D21111yescharacteristic
210C2×C4C2×D84411yescharacteristic
211C2×C4#1680996211yescharacteristic
212C8—22no—
213C8—22no—
214C8—8441no—
215C8—4222no—
216C22—22no—
217C22—41no—
218C22—81no—
219C22—8442no—
220C22—8441no—
221C22C4×C22⋊D21111yescharacteristic
222C22#2026995111yescharacteristic
223C22#2026996011yescharacteristic
224C4—22no—
225C4—8441no—
226C4—16881no—
227C4C2×(C2×D211)⋊C411yescharacteristic
228C4C4×C22⋊D21111yescharacteristic
229C4#2026994611yescharacteristic
230C4#2026995111yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—42no—
235C2—16881no—
236C2#2721302511yescharacteristic
subgroupG/H
centerC4#20269946
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#568×C211C2
socleC2×C211#386
prime radical#27213944C2
cosocleC22×C4×C211C23
hypercenter#568D211
hypocenterC211#1881
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #27213162C2semidirect
3. #27213142C2semidirect
4. #27213114C2semidirect
5. #27212917C2semidirect
6. #27212910C2semidirect
7. #568×C211C2semidirect
10. #20270007C4semidirect
12. #20270002C4semidirect
118. C211#1881semidirect
121. #568D211semidirect

Maximal subgroups

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

subgroupindex
#272139442
#272131622
#272131422
#272131142
#272129172
#272129102
#568×C2112
#1881211