Finite Group Databasebeta

← #41014668

Subgroups — #41014668

Order 27008.

274 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 67 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#27213194C211yesmaximal, characteristic, complemented
2#27213163C211yesmaximal, characteristic, complemented
3#27213151C211yesmaximal, characteristic, complemented
4#27213130C211yesmaximal, characteristic, complemented
5C2×#20269894C211yesmaximal, characteristic, complemented
6#27212918C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11#20269936C2211yescharacteristic
12#20269912—22no—
13#20269904—22no—
14#20269898—22no—
15#20269896—21no—
16#20269894—22no—
17#20269894—21no—
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
25#16809946—21no—
26#16809946—21no—
27#16809946—21no—
28#16809946—21no—
29#16809946—21no—
30#16809946D412yes—
31C2×C211⋊C8—21no—
32C2×C211⋊C8—21no—
33C2×C211⋊C8—21no—
34C4⋊Dic211—21no—
35C22⋊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—
59C211⋊C8—22no—
60C211⋊C8—41no—
61C211⋊C8—41no—
62C23×C211—22no—
63C23×C211—21no—
64C23×C211—41no—
65C23×C211—41no—
66C23×C211—82no—
67C23×C211#4111yescharacteristic
68Q8×C211—21no—
69Q8×C211—21no—
70D4×C211—21no—
71D4×C211—22no—
72D4×C211—21no—
73D4×C211—22no—
74D4×C211—41no—
75D4×C211—41no—
76D4×C211—41no—
77D4×C211—41no—
78C2×C4×C211—21no—
79C2×C4×C211—41no—
80C2×C4×C211—41no—
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×D411yescharacteristic
85C2×C4×C211#4111yescharacteristic
86C2×C4×C211#4111yescharacteristic
87C8×C211—22no—
88C22×C211—41no—
89C22×C211—41no—
90C22×C211—42no—
91C22×C211—81no—
92C22×C211—82no—
93C22×C211—82no—
94C22×C211—82no—
95C22×C211#12111yescharacteristic
96C22×C211#12311yescharacteristic
97C22×C211#12411yescharacteristic
98C4×C211—41no—
99C4×C211#12011yescharacteristic
100C4×C211#12111yescharacteristic
101C4×C211#12111yescharacteristic
102C4×C211#12311yescharacteristic
103Dic211—81no—
104C2×C211—21no—
105C2×C211—21no—
106C2×C211—21no—
107C2×C211—81no—
108C2×C211—82no—
109C2×C211#39411yescharacteristic
110C211#204811yescharacteristic, complemented, Sylow
111#2048—2111nomaximal, Sylow
112C2×AGL(1,ℤ/8)D21111yescharacteristic
113#430—2111no—
114#421—2111no—
115#418—2111no—
116#413—2111no—
117C2×#100—2111no—
118#344—2111no—
119C2×#41C2×D21111yescharacteristic
120C22×D4C2×D21111yescharacteristic
121AGL(1,ℤ/8)—22no—
122AGL(1,ℤ/8)—22no—
123C2×SD16—21no—
124C2×D8—21no—
125C2×M16—2111no—
126C2×M16—2111no—
127C2×M16—2111no—
128C2×M16C2×D21111yescharacteristic
129#117—2111no—
130#108—4222no—
131#104—4222no—
132#102—4222no—
133#101—4221no—
134#100—4221no—
135#100—4222no—
136C24—21no—
137#41—21no—
138#41—21no—
139#41—21no—
140#41—21no—
141C2×Q8C22⋊D21111yescharacteristic
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4C22⋊D21111yescharacteristic
149C2×D4C22⋊D21112yes—
150C22×C4—21no—
151C22×C4C22×D21111yescharacteristic
152SD16—24no—
153D8—24no—
154M16—21no—
155M16—4221no—
156M16—4221no—
157M16—4221no—
158M16—4221no—
159M16—4221no—
160M16—2112no—
161M16C22⋊D21112yes—
162C2×C8—21no—
163C2×C8—4221no—
164C2×C8—4221no—
165C2×C8—4221no—
166C4⋊C4—4221no—
167C22⋊C4—4221no—
168C42—4221no—
169C23—22no—
170C23—21no—
171C23—41no—
172C23—41no—
173C23—82no—
174C23#1680996111yescharacteristic
175Q8—21no—
176Q8—21no—
177D4—21no—
178D4—22no—
179D4—21no—
180D4—22no—
181D4—41no—
182D4—41no—
183D4—41no—
184D4—41no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—8441no—
189C2×C4D4×D21111yescharacteristic
190C2×C4C2×C22⋊D21111yescharacteristic
191C2×C4C2×C22⋊D21111yescharacteristic
192C2×C4C2×C22⋊D21111yescharacteristic
193C2×C4#1680996111yescharacteristic
194C2×C4#1680996111yescharacteristic
195C8—22no—
196C8—8441no—
197C8—8441no—
198C8—4222no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—82no—
203C22—81no—
204C22—82no—
205C22—82no—
206C22#2026994911yescharacteristic
207C22#2026995811yescharacteristic
208C22#2026996411yescharacteristic
209C4—41no—
210C4—16881no—
211C4#2026994611yescharacteristic
212C4#2026994811yescharacteristic
213C4#2026994911yescharacteristic
214C4#2026995811yescharacteristic
215C2—21no—
216C2—21no—
217C2—21no—
218C2—82no—
219C2—81no—
220C2#2721306011yescharacteristic
subgroupG/H
centerC2#27213060
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radicalC2×AGL(1,ℤ/8)×C211C2
socleC2×C211#394
hypercenterC2×AGL(1,ℤ/8)D211
hypocenterC211#2048
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #27213194C2semidirect
2. #27213163C2semidirect
3. #27213151C2semidirect
4. #27213130C2semidirect
5. C2×#20269894C2semidirect
6. #27212918C2semidirect
110. C211#2048semidirect

Maximal subgroups

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

subgroupindex
#272131942
#272131632
#272131512
#272131302
C2×#202698942
#272129182
C2×AGL(1,ℤ/8)×C2112
#2048211