Finite Group Databasebeta

← #41014538

Subgroups — #41014538

Order 27008.

290 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 87 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×#20269976C211yesmaximal, characteristic
2C2×#20269902C211yesmaximal, characteristic
3C2×#20269900C211yesmaximal, characteristic
4#27213051C211yesmaximal, characteristic
5#27212908C211yesmaximal, characteristic
6#27212900C211yesmaximal, characteristic
7#340×C211C211yesmaximal, characteristic
8C22×Dic422—21no—
9C22×Dic422C2211yescharacteristic
10C22×C211⋊C8C2211yescharacteristic
11#20269976—22no—
12#20269976—22no—
13C2×C4⋊Dic211C2211yescharacteristic
14C2×Dic211⋊C4—21no—
15C2×C4×Dic211C2211yescharacteristic
16#20269902—22no—
17#20269900—22no—
18C211⋊#95—21no—
19C211⋊#95—21no—
20C22×C8×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23C2×Dic422—24no—
24C2×Dic422—21no—
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—42no—
30C2×Dic422—42no—
31C2×Dic422Q812yescomplemented
32C22×Dic211—21no—
33C22×Dic211—21no—
34C22×Dic211D411yescharacteristic
35C22×Dic211D411yescharacteristic
36C2×C211⋊C8—21no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39C4⋊Dic211—21no—
40C4⋊Dic211D412yes—
41Dic211⋊C4—24no—
42C4×Dic211—21no—
43C4×Dic211—21no—
44C22×C4×C211—21no—
45C22×C4×C211—21no—
46C22×C4×C211C2311yescharacteristic
47C2×C8×C211—21no—
48C2×C8×C211—21no—
49C2×C8×C211—21no—
50C4⋊C4×C211—21no—
51C4⋊C4×C211—21no—
52C4⋊C4×C211—21no—
53C4⋊C4×C211D412yes—
54Dic422—22no—
55Dic422—22no—
56Dic422—48no—
57Dic422—41no—
58Dic422—41no—
59C2×Dic211—21no—
60C2×Dic211—21no—
61C2×Dic211—21no—
62C2×Dic211—21no—
63C2×Dic211—22no—
64C2×Dic211—22no—
65C2×Dic211—41no—
66C2×Dic211—41no—
67C2×Dic211—41no—
68C2×Dic211—41no—
69C2×Dic211SD1612yes—
70C2×Dic211Q1612yes—
71C211⋊C8—41no—
72C23×C211C2×D411yescharacteristic
73C2×C4×C211—22no—
74C2×C4×C211—41no—
75C2×C4×C211—41no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211C2×D411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×Q811yescharacteristic
82C2×C4×C211#4111yescharacteristic
83C2×C4×C211#4111yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C8×C211—41no—
86C22×C211#12011yescharacteristic
87C22×C211C22⋊Q811yescharacteristic
88C22×C211#12411yescharacteristic
89C22×C211C2×SD1611yescharacteristic
90C22×C211C2×Q1611yescharacteristic
91C22×C211AGL(1,ℤ/8)11yescharacteristic
92C22×C211#13711yescharacteristic
93C4×C211—42no—
94C4×C211—81no—
95C4×C211C22⋊Q811yescharacteristic
96C4×C211C22⋊Q811yescharacteristic
97C4×C211#12411yescharacteristic
98C4×C211#12411yescharacteristic
99Dic211—22no—
100Dic211—22no—
101Dic211—42no—
102Dic211—42no—
103C2×C211#39311yescharacteristic
104C2×C211#45011yescharacteristic
105C2×C211#45111yescharacteristic
106C2×C211#47511yescharacteristic
107C2×C211#47711yescharacteristic
108C2×C211#48711yescharacteristic
109C2×C211#48911yescharacteristic
110C211#202411yescharacteristic, complemented, Sylow
111#2024—2111nomaximal, Sylow
112C2×C4⋊Q8—2111no—
113C2×#103—2111no—
114C2×#103—2111no—
115#391—2111no—
116#340—2111no—
117#340D21111yescharacteristic
118#336—2111no—
119C22×Q8—4221no—
120C22×Q8—2111no—
121C22×C8—2111no—
122C22×C8C2×D21111yescharacteristic
123C4⋊Q8—4222no—
124C4⋊Q8—4222no—
125C2×C4⋊C4—4221no—
126C2×C4⋊C4—2111no—
127C2×C4⋊C4C2×D21111yescharacteristic
128C2×C4⋊C4C2×D21111yescharacteristic
129C2×C42—2111no—
130#103—4222no—
131#103—4222no—
132#95—4221no—
133#95—4221no—
134C2×Q8—8442no—
135C2×Q8—8442no—
136C2×Q8—4224no—
137C2×Q8—4221no—
138C2×Q8—4221no—
139C2×Q8—4221no—
140C2×Q8—4221no—
141C2×Q8—4221no—
142C2×Q8—2112no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—4221no—
146C22×C4—4221no—
147C22×C4—2111no—
148C22×C4—2111no—
149C22×C4C22×D21111yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—4221no—
154C2×C8—4221no—
155C2×C8—4221no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—4224no—
160C4⋊C4—4221no—
161C4⋊C4—2112no—
162C4⋊C4D84412yes—
163C42—4221no—
164C42—4221no—
165C23D4×D21111yescharacteristic
166Q8—8448no—
167Q8—8441no—
168Q8—8441no—
169Q8—4222no—
170Q8—4222no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—8441no—
178C2×C4—8441no—
179C2×C4—8441no—
180C2×C4—8441no—
181C2×C4—4222no—
182C2×C4—4222no—
183C2×C4—4221no—
184C2×C4—4221no—
185C2×C4—4221no—
186C2×C4—4221no—
187C2×C4—2112no—
188C2×C4—2112no—
189C2×C4D4×D21111yescharacteristic
190C2×C4C2×D84411yescharacteristic
191C2×C4Q8×D21111yescharacteristic
192C2×C4#1680996111yescharacteristic
193C2×C4#1680996211yescharacteristic
194C2×C4#1680996211yescharacteristic
195C8—41no—
196C8—8441no—
197C22#2026994711yescharacteristic
198C22#2026995511yescharacteristic
199C22#2026996311yescharacteristic
200C22SD16×D21111yescharacteristic
201C22Q16×D21111yescharacteristic
202C22#2027000111yescharacteristic
203C22#2027000611yescharacteristic
204C4—42no—
205C4—81no—
206C4—8442no—
207C4—8442no—
208C4—4222no—
209C4—4222no—
210C4#2026995511yescharacteristic
211C4#2026995711yescharacteristic
212C4#2026996311yescharacteristic
213C4#2026996511yescharacteristic
214C2#2721305911yescharacteristic
215C2#2721329811yescharacteristic
216C2#2721330311yescharacteristic
217C2#2721343011yescharacteristic
218C2#2721344011yescharacteristic
219C2#2721349511yescharacteristic
220C2#2721350411yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#340×C211C2
socle, prime radicalC23×C211C2×D4
hypercenter#340D211
hypocenterC211#2024
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
31. C2×Dic422Q8semidirect
110. C211#2024semidirect

Maximal subgroups

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

subgroupindex
C2×#202699762
C2×#202699022
C2×#202699002
#272130512
#272129082
#272129002
#340×C2112
#2024211