Finite Group Databasebeta

← #41013762

Subgroups — #41013762

Order 27008.

310 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 93 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#27213913C211yesmaximal, characteristic, complemented
2#27213163C211yesmaximal, characteristic, complemented
3#27213142C211yesmaximal, characteristic, complemented
4#27213113C211yesmaximal, characteristic, complemented
5#27212918C211yesmaximal, characteristic, complemented
6#27212910C211yesmaximal, characteristic, complemented
7#575×C211C211yesmaximal, characteristic
8#20269984—22no—
9#20269984—22no—
10C2×#16809946—22no—
11C2×#16809946C2211yescharacteristic
12C2×#16809946C2211yescharacteristic
13C22×C211⋊C8C2211yescharacteristic
14#20269936C2211yescharacteristic
15#20269904—22no—
16#20269901—21no—
17#20269898—21no—
18C22⋊(C211⋊C8)—22no—
19C2×#41×C211C2211yescharacteristic
20C2×#41×C211C2211yescharacteristic
21#137×C211—21no—
22#137×C211C412yescomplemented
23AGL(1,ℤ/8)×C211—21no—
24AGL(1,ℤ/8)×C211C412yescomplemented
25#135×C211—22no—
26C2×M16×C211C2211yescharacteristic
27#16809946—24no—
28#16809946—21no—
29#16809946—21no—
30#16809946—21no—
31#16809946D412yes—
32C2×C211⋊C8—24no—
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C2×C211⋊C8—21no—
36C2×C211⋊C8—21no—
37C2×C211⋊C8D412yes—
38C4⋊Dic211—21no—
39C22⋊Dic211—21no—
40C4×Dic211—21no—
41#41×C211—21no—
42#41×C211—21no—
43#41×C211—21no—
44#41×C211—21no—
45#41×C211—21no—
46#41×C211—21no—
47#41×C211C2×C412yes—
48#41×C211D412yes—
49C2×Q8×C211C2×C411yescharacteristic
50C2×Q8×C211D411yescharacteristic
51C2×D4×C211—21no—
52C2×D4×C211—21no—
53C2×D4×C211C2×C411yescharacteristic
54C2×D4×C211D411yescharacteristic
55C22×C4×C211—21no—
56C22×C4×C211—21no—
57C22×C4×C211C2311yescharacteristic
58Q16×C211—22no—
59SD16×C211—22no—
60SD16×C211—22no—
61D8×C211—22no—
62M16×C211—21no—
63M16×C211C2×C412yes—
64C2×C8×C211—21no—
65C2×Dic211—41no—
66C211⋊C8—22no—
67C211⋊C8—22no—
68C211⋊C8—41no—
69C23×C211—21no—
70C23×C211—21no—
71C23×C211#4111yescharacteristic
72Q8×C211—21no—
73Q8×C211—21no—
74Q8×C211—21no—
75Q8×C211C22⋊C412yes—
76D4×C211—22no—
77D4×C211—22no—
78D4×C211—21no—
79D4×C211—21no—
80D4×C211—21no—
81D4×C211—41no—
82D4×C211—41no—
83D4×C211C22⋊C412yes—
84C2×C4×C211—21no—
85C2×C4×C211—22no—
86C2×C4×C211—22no—
87C2×C4×C211—21no—
88C2×C4×C211—41no—
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—22no—
97C22×C211—22no—
98C22×C211—41no—
99C22×C211—42no—
100C22×C211—41no—
101C22×C211—81no—
102C22×C211C4×D411yescharacteristic
103C22×C211#12111yescharacteristic
104C22×C211#12311yescharacteristic
105C4×C211—22no—
106C4×C211—41no—
107C4×C211C2×C22⋊C411yescharacteristic
108C4×C211C4×D411yescharacteristic
109C4×C211#12011yescharacteristic
110C4×C211#12111yescharacteristic
111Dic211—81no—
112C2×C211—21no—
113C2×C211—21no—
114C2×C211—21no—
115C2×C211—42no—
116C2×C211—81no—
117C2×C211#38611yescharacteristic
118C211#188111yescharacteristic, complemented, Sylow
119#1881—2111nomaximal, Sylow
120#575D21111yescharacteristic
121#568—2111no—
122#421—2111no—
123#416—2111no—
124#408—2111no—
125#344—2111no—
126#341—2111no—
127C2×#41C2×D21111yescharacteristic
128C2×#41C2×D21111yescharacteristic
129#137—21no—
130#137Dic21112yescomplemented
131AGL(1,ℤ/8)—21no—
132AGL(1,ℤ/8)Dic21112yescomplemented
133#135—22no—
134#131—4222no—
135#131—4222no—
136C2×M16—4222no—
137C2×M16—2111no—
138C2×M16—2111no—
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—21no—
149#41—21no—
150#41—21no—
151#41—21no—
152#41C2×Dic21112yes—
153#41C22⋊D21112yes—
154C2×Q8C2×Dic21111yescharacteristic
155C2×Q8C22⋊D21111yescharacteristic
156C2×D4—21no—
157C2×D4—21no—
158C2×D4C2×Dic21111yescharacteristic
159C2×D4C22⋊D21111yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4C22×D21111yescharacteristic
163Q16—22no—
164SD16—22no—
165SD16—22no—
166D8—22no—
167M16—21no—
168M16—4224no—
169M16—4221no—
170M16—4221no—
171M16—4221no—
172M16—2112no—
173M16C2×Dic21112yes—
174C2×C8—21no—
175C2×C8—4221no—
176C2×C8—4221no—
177C2×C8—4221no—
178C2×C8—4224no—
179C2×C8—4221no—
180C2×C8—2112no—
181C4⋊C4—4221no—
182C22⋊C4—4221no—
183C42—4221no—
184C23—21no—
185C23—21no—
186C23#1680996111yescharacteristic
187Q8—21no—
188Q8—21no—
189Q8—21no—
190Q8C22⋊Dic21112yes—
191D4—22no—
192D4—22no—
193D4—21no—
194D4—21no—
195D4—21no—
196D4—41no—
197D4—41no—
198D4C22⋊Dic21112yes—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—8441no—
206C2×C4C22×Dic21111yescharacteristic
207C2×C4D4×D21111yescharacteristic
208C2×C4D4×D21111yescharacteristic
209C2×C4C2×C22⋊D21111yescharacteristic
210C2×C4C2×C22⋊D21111yescharacteristic
211C2×C4#1680996111yescharacteristic
212C8—22no—
213C8—8441no—
214C8—4222no—
215C8—4222no—
216C22—22no—
217C22—41no—
218C22—42no—
219C22—41no—
220C22—81no—
221C22D4×Dic21111yescharacteristic
222C22#2026994911yescharacteristic
223C22#2026995811yescharacteristic
224C4—22no—
225C4—41no—
226C4—16881no—
227C4C2×C22⋊Dic21111yescharacteristic
228C4D4×Dic21111yescharacteristic
229C4#2026994511yescharacteristic
230C4#2026994911yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—42no—
235C2—81no—
236C2#2721302611yescharacteristic
subgroupG/H
centerC4#20269945
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting, prime radical#575×C211C2
socleC2×C211#386
cosocleC22×C4×C211C23
hypercenter#575D211
hypocenterC211#1881
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #27213913C2semidirect
2. #27213163C2semidirect
3. #27213142C2semidirect
4. #27213113C2semidirect
5. #27212918C2semidirect
6. #27212910C2semidirect
22. #137×C211C4semidirect
24. AGL(1,ℤ/8)×C211C4semidirect
118. C211#1881semidirect
130. #137Dic211semidirect
132. AGL(1,ℤ/8)Dic211semidirect

Maximal subgroups

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

subgroupindex
#272139132
#272131632
#272131422
#272131132
#272129182
#272129102
#575×C2112
#1881211