Finite Group Databasebeta

← #41013758

Subgroups — #41013758

Order 27008.

310 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 85 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#27213913C211yesmaximal, characteristic, complemented
3#27213164C211yesmaximal, characteristic, complemented
4#27213112C211yesmaximal, characteristic, complemented
5#27212916C211yesmaximal, characteristic, complemented
6#27212911C211yesmaximal, characteristic, complemented
7#416×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20270008—21no—
10#20270007C412yescomplemented
11#20270003—21no—
12#20270002C412yescomplemented
13#20269997—22no—
14#20269984—22no—
15#20269984—22no—
16C2×#16809946—22no—
17C2×#16809946C2211yescharacteristic
18C2×#16809946C2211yescharacteristic
19C2×#16809946C2211yescharacteristic
20#20269905—22no—
21(C2×D211)⋊C8—22no—
22C2×#41×C211C2211yescharacteristic
23C22×C8×C211C2211yescharacteristic
24#117×C211C2211yescharacteristic
25#103×C211—21no—
26#102×C211—21no—
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—24no—
40#16809946—21no—
41#16809946—21no—
42#16809946—21no—
43#16809946—21no—
44#16809946C2×C412yes—
45#16809946D412yes—
46C2×C211⋊C8—24no—
47C2×C211⋊C8—21no—
48C2×C211⋊C8—21no—
49C2×C211⋊C8D412yes—
50#41×C211—21no—
51#41×C211—21no—
52#41×C211C2×C412yes—
53#41×C211D412yes—
54C2×Q8×C211D411yescharacteristic
55C2×D4×C211—21no—
56C2×D4×C211D411yescharacteristic
57C22×C4×C211—21no—
58C22×C4×C211C2311yescharacteristic
59C2×C8×C211—21no—
60C2×C8×C211—21no—
61C2×C8×C211—21no—
62C4⋊C4×C211—21no—
63C22⋊C4×C211—21no—
64C42×C211—21no—
65C22×D211—21no—
66Dic422—21no—
67Dic422—21no—
68D844—21no—
69D844—21no—
70C22⋊D211—41no—
71C22⋊D211—41no—
72C2×Dic211—21no—
73C4×D211—41no—
74C4×D211—41no—
75C211⋊C8—22no—
76C211⋊C8—22no—
77C211⋊C8—22no—
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×C211—41no—
90C2×C4×C211C22×C411yescharacteristic
91C2×C4×C211C2×D411yescharacteristic
92C2×C4×C211C2×D411yescharacteristic
93C2×C4×C211C2×D411yescharacteristic
94C2×C4×C211C2×D411yescharacteristic
95C2×C4×C211#4111yescharacteristic
96C8×C211—41no—
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×C211—81no—
107C4×C211C2×C22⋊C411yescharacteristic
108C4×C211C4×D411yescharacteristic
109C4×C211#12011yescharacteristic
110C4×C211#12111yescharacteristic
111Dic211—41no—
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#568—2111no—
122#421—2111no—
123#416D21111yescharacteristic, complemented
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—4222no—
135#131—4222no—
136C2×M16—4222no—
137C2×M16—2111no—
138C2×M16—2111no—
139C2×M16—2111no—
140C22×C8C2×D21111yescharacteristic
141#117C2×D21111yescharacteristic
142#104—4222no—
143#103—21no—
144#102—21no—
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#41D84412yes—
154C2×Q8—2111no—
155C2×Q8C22⋊D21111yescharacteristic
156C2×D4—21no—
157C2×D4—4221no—
158C2×D4—2111no—
159C2×D4C22⋊D21111yescharacteristic
160C22×C4—21no—
161C22×C4—4221no—
162C22×C4C22×D21111yescharacteristic
163Q16—4222no—
164SD16—4222no—
165SD16—4222no—
166D8—4222no—
167M16—4224no—
168M16—4221no—
169M16—4221no—
170M16—4221no—
171M16—4221no—
172M16—2112no—
173M16—2112no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—4224no—
178C2×C8—4221no—
179C2×C8—4221no—
180C2×C8—2112no—
181C4⋊C4—21no—
182C22⋊C4—21no—
183C42—21no—
184C23—21no—
185C23—4221no—
186C23#1680996111yescharacteristic
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—41no—
203C2×C4—8441no—
204C2×C4—8441no—
205C2×C4—4221no—
206C2×C4C2×C4×D21111yescharacteristic
207C2×C4D4×D21111yescharacteristic
208C2×C4D4×D21111yescharacteristic
209C2×C4C2×C22⋊D21111yescharacteristic
210C2×C4C2×D84411yescharacteristic
211C2×C4#1680996111yescharacteristic
212C8—41no—
213C8—4222no—
214C8—4222no—
215C8—4222no—
216C22—22no—
217C22—41no—
218C22—81no—
219C22—8442no—
220C22—8441no—
221C22(C22⋊D211)⋊C411yescharacteristic
222C22#2026994911yescharacteristic
223C22#2026996111yescharacteristic
224C4—22no—
225C4—81no—
226C4—8441no—
227C4C2×(C2×D211)⋊C411yescharacteristic
228C4(C22⋊D211)⋊C411yescharacteristic
229C4#2026994711yescharacteristic
230C4#2026994911yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—42no—
235C2—16881no—
236C2#2721302311yescharacteristic
subgroupG/H
centerC4#20269947
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#416×C211C2
socleC2×C211#386
prime radical#27213944C2
cosocleC22×C4×C211C23
hypercenter#416D211
hypocenterC211#1881
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. #27213913C2semidirect
3. #27213164C2semidirect
4. #27213112C2semidirect
5. #27212916C2semidirect
6. #27212911C2semidirect
7. #416×C211C2semidirect
10. #20270007C4semidirect
12. #20270002C4semidirect
118. C211#1881semidirect
123. #416D211semidirect

Maximal subgroups

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

subgroupindex
#272139442
#272139132
#272131642
#272131122
#272129162
#272129112
#416×C2112
#1881211