Finite Group Databasebeta

← #41013915

Subgroups — #41013915

Order 27008.

312 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#20269984C211yesmaximal, characteristic
2#27213147C211yesmaximal, characteristic
3C2×#20269901C211yesmaximal, characteristic, complemented
4C2×#20269896C211yesmaximal, characteristic, complemented
5#27212914C211yesmaximal, characteristic, complemented
6#27212910C211yesmaximal, characteristic, complemented
7C2×#137×C211C211yesmaximal, characteristic
8#20269984—22no—
9#20269984—22no—
10C2×#16809946—21no—
11C2×#16809946C2211yescharacteristic
12C2×#16809946C2211yescharacteristic
13C22×C211⋊C8—21no—
14C22×C211⋊C8C2211yescharacteristic
15C2×C4⋊Dic211C2211yescharacteristic
16#20269901—21no—
17#20269901—21no—
18#20269901—21no—
19#20269898—21no—
20#20269896—22no—
21C2×#41×C211C2211yescharacteristic
22C22×Q8×C211C2211yescharacteristic
23#137×C211—22no—
24#137×C211C414yescomplemented
25C2×Q16×C211—21no—
26C2×SD16×C211—21no—
27C2×M16×C211C2211yescharacteristic
28C22×Dic211—21no—
29#16809946—24no—
30#16809946—21no—
31#16809946—21no—
32#16809946—21no—
33#16809946D412yes—
34C2×C211⋊C8—24no—
35C2×C211⋊C8—21no—
36C2×C211⋊C8—21no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39C2×C211⋊C8D411yescharacteristic
40C2×C211⋊C8D411yescharacteristic
41C4⋊Dic211—21no—
42C4⋊Dic211—21no—
43#41×C211—21no—
44#41×C211—21no—
45#41×C211C2×C412yes—
46#41×C211D412yes—
47C2×Q8×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211—21no—
50C2×Q8×C211—21no—
51C2×Q8×C211—21no—
52C2×Q8×C211C2×C412yes—
53C2×Q8×C211D411yescharacteristic
54C2×D4×C211—21no—
55C2×D4×C211D411yescharacteristic
56C22×C4×C211—21no—
57C22×C4×C211—21no—
58C22×C4×C211C2311yescharacteristic
59Q16×C211—24no—
60SD16×C211—24no—
61M16×C211—21no—
62M16×C211C2×C412yes—
63C2×C8×C211—21no—
64C2×Dic211—41no—
65C2×Dic211—81no—
66C211⋊C8—21no—
67C211⋊C8—22no—
68C211⋊C8—21no—
69C211⋊C8—41no—
70C23×C211—21no—
71C23×C211#4111yescharacteristic
72Q8×C211—21no—
73Q8×C211—22no—
74Q8×C211—22no—
75Q8×C211—41no—
76Q8×C211—41no—
77Q8×C211C22⋊C412yes—
78D4×C211—22no—
79D4×C211—22no—
80D4×C211—21no—
81D4×C211C22⋊C412yes—
82C2×C4×C211—21no—
83C2×C4×C211—22no—
84C2×C4×C211—22no—
85C2×C4×C211—22no—
86C2×C4×C211—41no—
87C2×C4×C211—41no—
88C2×C4×C211C22×C411yescharacteristic
89C2×C4×C211C2×D411yescharacteristic
90C2×C4×C211C2×D411yescharacteristic
91C2×C4×C211C2×D411yescharacteristic
92C2×C4×C211C2×D411yescharacteristic
93C2×C4×C211#4111yescharacteristic
94C8×C211—22no—
95C22×C211—21no—
96C22×C211—21no—
97C22×C211—22no—
98C22×C211—41no—
99C22×C211—41no—
100C22×C211C4×D411yescharacteristic
101C22×C211#12111yescharacteristic
102C22×C211#12311yescharacteristic
103C4×C211—22no—
104C4×C211—42no—
105C4×C211C2×C22⋊C411yescharacteristic
106C4×C211C4×D411yescharacteristic
107C4×C211#12011yescharacteristic
108C4×C211#12111yescharacteristic
109Dic211—81no—
110C2×C211—21no—
111C2×C211—21no—
112C2×C211—42no—
113C2×C211#38611yescharacteristic
114C2×C211#47111yescharacteristic
115C2×C211#47311yescharacteristic
116C211#190811yescharacteristic, complemented, Sylow
117#1908—2111nomaximal, Sylow
118C2×#137D21111yescharacteristic
119C2×#131—2111no—
120#417—2111no—
121C2×#103—2111no—
122C2×#101—2111no—
123#343—2111no—
124#341—2111no—
125C2×#41C2×D21111yescharacteristic
126C22×Q8C2×D21111yescharacteristic
127#137—22no—
128#137Dic21114yescomplemented
129C2×Q16—21no—
130C2×SD16—21no—
131#131—4222no—
132#131—4222no—
133C2×M16—4221no—
134C2×M16—2111no—
135C2×M16—2111no—
136C2×M16C2×D21111yescharacteristic
137C22×C8—4221no—
138C22×C8—2111no—
139C2×C4⋊C4—2111no—
140#103—4221no—
141#103—4221no—
142#103—4221no—
143#102—4221no—
144#101—4222no—
145#41—21no—
146#41—21no—
147#41C2×Dic21112yes—
148#41C22⋊D21112yes—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8C2×Dic21112yes—
155C2×Q8C22⋊D21111yescharacteristic
156C2×D4—21no—
157C2×D4C22⋊D21111yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—4221no—
161C22×C4C22×D21111yescharacteristic
162Q16—24no—
163SD16—24no—
164M16—21no—
165M16—4221no—
166M16—4221no—
167M16—4224no—
168M16—4221no—
169M16—2112no—
170M16C2×Dic21112yes—
171C2×C8—21no—
172C2×C8—4221no—
173C2×C8—4224no—
174C2×C8—4221no—
175C2×C8—4221no—
176C2×C8—4221no—
177C2×C8—2111no—
178C2×C8—2111no—
179C4⋊C4—4221no—
180C4⋊C4—4221no—
181C23—21no—
182C23#1680996111yescharacteristic
183Q8—21no—
184Q8—22no—
185Q8—22no—
186Q8—41no—
187Q8—41no—
188Q8C22⋊Dic21112yes—
189D4—22no—
190D4—22no—
191D4—21no—
192D4C22⋊Dic21112yes—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—8441no—
200C2×C4—16881no—
201C2×C4C22×Dic21111yescharacteristic
202C2×C4D4×D21111yescharacteristic
203C2×C4D4×D21111yescharacteristic
204C2×C4C2×C22⋊D21111yescharacteristic
205C2×C4C2×C22⋊D21111yescharacteristic
206C2×C4#1680996111yescharacteristic
207C8—22no—
208C8—8441no—
209C8—4221no—
210C8—4222no—
211C8—4221no—
212C22—21no—
213C22—21no—
214C22—22no—
215C22—41no—
216C22—41no—
217C22D4×Dic21111yescharacteristic
218C22#2026994911yescharacteristic
219C22#2026995811yescharacteristic
220C4—22no—
221C4—42no—
222C4—16881no—
223C4C2×C22⋊Dic21111yescharacteristic
224C4D4×Dic21111yescharacteristic
225C4#2026994511yescharacteristic
226C4#2026994911yescharacteristic
227C2—21no—
228C2—21no—
229C2—42no—
230C2#2721302611yescharacteristic
231C2#2721341111yescharacteristic
232C2#2721341911yescharacteristic
subgroupG/H
centerC22#20269949
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#137×C211C2
socleC22×C211#121
prime radicalC2×#41×C211C22
cosocleC22×C4×C211C23
hypercenterC2×#137D211
hypocenterC211#1908
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
3. C2×#20269901C2semidirect
4. C2×#20269896C2semidirect
5. #27212914C2semidirect
6. #27212910C2semidirect
24. #137×C211C4semidirect
116. C211#1908semidirect
128. #137Dic211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699842
#272131472
C2×#202699012
C2×#202698962
#272129142
#272129102
C2×#137×C2112
#1908211