Finite Group Databasebeta

← #41014687

Subgroups — #41014687

Order 27008.

282 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 71 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×#20269912C211yesmaximal, characteristic, complemented
2#27213147C211yesmaximal, characteristic, complemented
3C2×#20269898C211yesmaximal, characteristic, complemented
4C2×#20269894C211yesmaximal, characteristic, complemented
5#27213113C211yesmaximal, characteristic, complemented
6#27212914C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C22×C211⋊C8C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12#20269912—22no—
13#20269901—21no—
14#20269898—22no—
15#20269898—21no—
16#20269894—22no—
17C22⋊(C211⋊C8)—22no—
18C2×#41×C211C2211yescharacteristic
19C22×D4×C211C2211yescharacteristic
20AGL(1,ℤ/8)×C211—22no—
21AGL(1,ℤ/8)×C211—22no—
22C2×SD16×C211—21no—
23C2×D8×C211—21no—
24C2×M16×C211C2211yescharacteristic
25C22×Dic211—21no—
26#16809946—21no—
27#16809946—21no—
28#16809946—21no—
29#16809946—21no—
30C2×C211⋊C8—21no—
31C2×C211⋊C8—21no—
32C2×C211⋊C8—21no—
33C2×C211⋊C8—21no—
34C2×C211⋊C8D412yes—
35C4⋊Dic211—21no—
36C4⋊Dic211—21no—
37C24×C211—21no—
38#41×C211—21no—
39#41×C211—21no—
40#41×C211—21no—
41#41×C211—21no—
42C2×Q8×C211D411yescharacteristic
43C2×D4×C211—21no—
44C2×D4×C211—21no—
45C2×D4×C211—21no—
46C2×D4×C211—21no—
47C2×D4×C211—21no—
48C2×D4×C211—21no—
49C2×D4×C211D411yescharacteristic
50C2×D4×C211D412yes—
51C22×C4×C211—21no—
52C22×C4×C211C2311yescharacteristic
53SD16×C211—24no—
54D8×C211—24no—
55M16×C211—21no—
56M16×C211D412yes—
57C2×C8×C211—21no—
58C2×Dic211—41no—
59C2×Dic211—81no—
60C211⋊C8—22no—
61C211⋊C8—41no—
62C211⋊C8—41no—
63C23×C211—21no—
64C23×C211—22no—
65C23×C211—41no—
66C23×C211—41no—
67C23×C211—81no—
68C23×C211—81no—
69C23×C211#4111yescharacteristic
70Q8×C211—21no—
71Q8×C211—21no—
72D4×C211—21no—
73D4×C211—22no—
74D4×C211—22no—
75D4×C211—21no—
76D4×C211—41no—
77D4×C211—41no—
78D4×C211—41no—
79D4×C211—41no—
80C2×C4×C211—21no—
81C2×C4×C211—41no—
82C2×C4×C211—41no—
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×D411yescharacteristic
85C2×C4×C211C2×D411yescharacteristic
86C2×C4×C211C2×D411yescharacteristic
87C2×C4×C211#4111yescharacteristic
88C2×C4×C211#4111yescharacteristic
89C8×C211—22no—
90C22×C211—21no—
91C22×C211—21no—
92C22×C211—41no—
93C22×C211—42no—
94C22×C211—41no—
95C22×C211—41no—
96C22×C211—82no—
97C22×C211—82no—
98C22×C211—81no—
99C22×C211—81no—
100C22×C211#12111yescharacteristic
101C22×C211#12311yescharacteristic
102C22×C211#12411yescharacteristic
103C4×C211—41no—
104C4×C211#12011yescharacteristic
105C4×C211#12111yescharacteristic
106C4×C211#12111yescharacteristic
107C4×C211#12311yescharacteristic
108Dic211—81no—
109C2×C211—21no—
110C2×C211—21no—
111C2×C211—81no—
112C2×C211—82no—
113C2×C211#39411yescharacteristic
114C2×C211#47111yescharacteristic
115C2×C211#47211yescharacteristic
116C211#205111yescharacteristic, complemented, Sylow
117#2051—2111nomaximal, Sylow
118C2×AGL(1,ℤ/8)D21111yescharacteristic
119C2×#108—2111no—
120#417—2111no—
121C2×#102—2111no—
122C2×#100—2111no—
123#408—2111no—
124#343—2111no—
125C2×#41C2×D21111yescharacteristic
126C22×D4C2×D21111yescharacteristic
127AGL(1,ℤ/8)—22no—
128AGL(1,ℤ/8)—22no—
129C2×SD16—21no—
130C2×D8—21no—
131C2×M16—2111no—
132C2×M16—2111no—
133C2×M16C2×D21111yescharacteristic
134C22×C8—2111no—
135C2×C4⋊C4—2111no—
136#108—4222no—
137#103—4221no—
138#102—4221no—
139#102—4222no—
140#100—4222no—
141C22⋊C8—4222no—
142C24—21no—
143#41—21no—
144#41—21no—
145#41—21no—
146#41—21no—
147C2×Q8C22⋊D21111yescharacteristic
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4C22⋊D21112yes—
155C2×D4C22⋊D21111yescharacteristic
156C22×C4—21no—
157C22×C4—4221no—
158C22×C4C22×D21111yescharacteristic
159SD16—24no—
160D8—24no—
161M16—21no—
162M16—4221no—
163M16—4221no—
164M16—4221no—
165M16—4221no—
166M16C22⋊D21112yes—
167C2×C8—21no—
168C2×C8—4221no—
169C2×C8—4221no—
170C2×C8—4221no—
171C2×C8—4221no—
172C2×C8—2112no—
173C4⋊C4—4221no—
174C4⋊C4—4221no—
175C23—21no—
176C23—22no—
177C23—41no—
178C23—41no—
179C23—81no—
180C23—81no—
181C23#1680996111yescharacteristic
182Q8—21no—
183Q8—21no—
184D4—21no—
185D4—22no—
186D4—22no—
187D4—21no—
188D4—41no—
189D4—41no—
190D4—41no—
191D4—41no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—8441no—
196C2×C4—16881no—
197C2×C4D4×D21111yescharacteristic
198C2×C4C2×C22⋊D21111yescharacteristic
199C2×C4C2×C22⋊D21111yescharacteristic
200C2×C4C2×C22⋊D21111yescharacteristic
201C2×C4#1680996111yescharacteristic
202C2×C4#1680996111yescharacteristic
203C8—22no—
204C8—8441no—
205C8—8441no—
206C8—4222no—
207C22—21no—
208C22—21no—
209C22—41no—
210C22—42no—
211C22—41no—
212C22—41no—
213C22—82no—
214C22—82no—
215C22—81no—
216C22—81no—
217C22#2026994911yescharacteristic
218C22#2026995811yescharacteristic
219C22#2026996411yescharacteristic
220C4—41no—
221C4—16881no—
222C4#2026994611yescharacteristic
223C4#2026994811yescharacteristic
224C4#2026994911yescharacteristic
225C4#2026995811yescharacteristic
226C2—21no—
227C2—21no—
228C2—81no—
229C2—82no—
230C2#2721306011yescharacteristic
231C2#2721340711yescharacteristic
232C2#2721341411yescharacteristic
subgroupG/H
centerC22#20269949
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radicalC2×AGL(1,ℤ/8)×C211C2
socleC22×C211#121
hypercenterC2×AGL(1,ℤ/8)D211
hypocenterC211#2051
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#20269912C2semidirect
2. #27213147C2semidirect
3. C2×#20269898C2semidirect
4. C2×#20269894C2semidirect
5. #27213113C2semidirect
6. #27212914C2semidirect
116. C211#2051semidirect

Maximal subgroups

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

subgroupindex
C2×#202699122
#272131472
C2×#202698982
C2×#202698942
#272131132
#272129142
C2×AGL(1,ℤ/8)×C2112
#2051211