Finite Group Databasebeta

← #41014089

Subgroups — #41014089

Order 27008.

248 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 91 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×#20269902C211yesmaximal, characteristic
2C2×#20269900C211yesmaximal, characteristic
3C2×#20269887C211yesmaximal, characteristic
4#27213051C211yesmaximal, characteristic
5#27213045C211yesmaximal, characteristic
6#27212908C211yesmaximal, characteristic
7#340×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10C2×C4⋊Dic211—21no—
11C2×C4⋊Dic211C2211yescharacteristic
12C2×C4×Dic211C2211yescharacteristic
13#20269902—22no—
14#20269900C414yescomplemented
15#20269887—22no—
16C211⋊#95—21no—
17C211⋊#95—21no—
18C211⋊#95—21no—
19C22×C8×C211C2211yescharacteristic
20C2×C4⋊C4×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22C2×Dic422—21no—
23C2×Dic422—21no—
24C2×Dic422—21no—
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422C2×C412yes—
28C22×Dic211—21no—
29C22×Dic211—21no—
30C22×Dic211D411yescharacteristic
31C22×Dic211D411yescharacteristic
32C2×C211⋊C8—21no—
33C2×C211⋊C8—21no—
34C2×C211⋊C8C2×C412yes—
35C4⋊Dic211—21no—
36C4⋊Dic211—21no—
37C4⋊Dic211—42no—
38C4×Dic211—21no—
39C4×Dic211—21no—
40C22×C4×C211—21no—
41C22×C4×C211—21no—
42C22×C4×C211C2311yescharacteristic
43C2×C8×C211—21no—
44C2×C8×C211—21no—
45C2×C8×C211D412yes—
46C4⋊C4×C211—21no—
47C4⋊C4×C211—21no—
48C4⋊C4×C211—21no—
49C4⋊C4×C211C2×C412yes—
50Dic422—22no—
51Dic422—22no—
52Dic422—41no—
53Dic422—41no—
54C2×Dic211—22no—
55C2×Dic211—21no—
56C2×Dic211—21no—
57C2×Dic211—21no—
58C2×Dic211—21no—
59C2×Dic211—21no—
60C2×Dic211—21no—
61C2×Dic211—41no—
62C2×Dic211—41no—
63C2×Dic211—41no—
64C2×Dic211—41no—
65C2×Dic211—41no—
66C211⋊C8—22no—
67C23×C211C2×D411yescharacteristic
68C2×C4×C211—22no—
69C2×C4×C211—41no—
70C2×C4×C211—41no—
71C2×C4×C211—41no—
72C2×C4×C211—41no—
73C2×C4×C211—41no—
74C2×C4×C211C22×C411yescharacteristic
75C2×C4×C211C2×D411yescharacteristic
76C2×C4×C211#4111yescharacteristic
77C2×C4×C211#4111yescharacteristic
78C2×C4×C211#4111yescharacteristic
79C2×C4×C211#4111yescharacteristic
80C8×C211—22no—
81C22×C211C4×D411yescharacteristic
82C22×C211#12111yescharacteristic
83C22×C211#12411yescharacteristic
84C22×C211AGL(1,ℤ/8)11yescharacteristic
85C22×C211#13711yescharacteristic
86C22×C211#13711yescharacteristic
87C22×C211#13711yescharacteristic
88C4×C211—42no—
89C4×C211—81no—
90C4×C211#11711yescharacteristic
91C4×C211C4×D411yescharacteristic
92C4×C211#12311yescharacteristic
93C4×C211#12611yescharacteristic
94Dic211—42no—
95Dic211—41no—
96Dic211—41no—
97Dic211—81no—
98C2×C211#38811yescharacteristic
99C2×C211#44011yescharacteristic
100C2×C211#44111yescharacteristic
101C2×C211#46811yescharacteristic
102C2×C211#47011yescharacteristic
103C2×C211#48911yescharacteristic
104C2×C211#49111yescharacteristic
105C211#193511yescharacteristic, complemented, Sylow
106#1935—2111nomaximal, Sylow
107C2×#103—2111no—
108C2×#103—2111no—
109C2×#97—2111no—
110#391—2111no—
111#389—2111no—
112#340—2111no—
113#340D21111yescharacteristic
114C22×Q8—2111no—
115C22×C8—2111no—
116C22×C8C2×D21111yescharacteristic
117C2×C4⋊C4—4221no—
118C2×C4⋊C4—2111no—
119C2×C4⋊C4C2×D21111yescharacteristic
120C2×C4⋊C4C2×D21111yescharacteristic
121C2×C42—2111no—
122#103—4222no—
123#103—2114no—
124#97—4222no—
125#95—4221no—
126#95—4221no—
127#95—4221no—
128C2×Q8—4221no—
129C2×Q8—4221no—
130C2×Q8—4221no—
131C2×Q8—4221no—
132C2×Q8—4221no—
133C2×Q8—2112no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—4221no—
137C22×C4—4221no—
138C22×C4—2111no—
139C22×C4—2111no—
140C22×C4C22×D21111yescharacteristic
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—4221no—
144C2×C8—4221no—
145C2×C8—2112no—
146C2×C8D84412yes—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—8442no—
151C4⋊C4—4221no—
152C4⋊C4—4221no—
153C4⋊C4C4×D21112yes—
154C42—4221no—
155C42—4221no—
156C23D4×D21111yescharacteristic
157Q8—8441no—
158Q8—8441no—
159Q8—4222no—
160Q8—4222no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—8441no—
168C2×C4—8441no—
169C2×C4—8441no—
170C2×C4—8441no—
171C2×C4—8441no—
172C2×C4—4222no—
173C2×C4—4221no—
174C2×C4—4221no—
175C2×C4—4221no—
176C2×C4—4221no—
177C2×C4—4221no—
178C2×C4—4221no—
179C2×C4C2×C4×D21111yescharacteristic
180C2×C4C2×D84411yescharacteristic
181C2×C4#1680996111yescharacteristic
182C2×C4#1680996111yescharacteristic
183C2×C4#1680996211yescharacteristic
184C2×C4#1680996311yescharacteristic
185C8—22no—
186C8—4222no—
187C22(C22⋊D211)⋊C411yescharacteristic
188C22#2026995211yescharacteristic
189C22#2026996311yescharacteristic
190C22#2027000111yescharacteristic
191C22#2027000611yescharacteristic
192C22#2027000611yescharacteristic
193C22#2027000911yescharacteristic
194C4—42no—
195C4—81no—
196C4—8441no—
197C4—8441no—
198C4—8442no—
199C4—16881no—
200C4#2026993511yescharacteristic
201C4C4×D84411yescharacteristic
202C4#2026996111yescharacteristic
203C4#2026997211yescharacteristic
204C2#2721303811yescharacteristic
205C2#2721324311yescharacteristic
206C2#2721325011yescharacteristic
207C2#2721339311yescharacteristic
208C2#2721340511yescharacteristic
209C2#2721350411yescharacteristic
210C2#2721351311yescharacteristic
subgroupG/H
centerC23D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#340×C211C2
socle, prime radicalC23×C211C2×D4
cosocleC22×C4×C211C23
hypercenter#340D211
hypocenterC211#1935
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
14. #20269900C4semidirect
105. C211#1935semidirect

Maximal subgroups

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

subgroupindex
C2×#202699022
C2×#202699002
C2×#202698872
#272130512
#272130452
#272129082
#340×C2112
#1935211