Finite Group Databasebeta

← #41014789

Subgroups — #41014789

Order 27008.

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

normal subgroups only

This group has 248 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×#20269910C211yesmaximal, characteristic
2C2×Dic211⋊C8C211yesmaximal, characteristic
3C2×#20269901C211yesmaximal, characteristic
4#27213053C211yesmaximal, characteristic
5#27213045C211yesmaximal, characteristic
6#27212909C211yesmaximal, characteristic
7C2×#103×C211C211yesmaximal, characteristic
8C22×C211⋊C8C2211yescharacteristic
9C2×C4⋊Dic211—21no—
10C2×C4⋊Dic211C2211yescharacteristic
11C2×C4⋊Dic211C2211yescharacteristic
12C2×C4×Dic211C2211yescharacteristic
13#20269910—22no—
14Dic211⋊C8—22no—
15#20269901—22no—
16C211⋊#95—21no—
17C211⋊#95—21no—
18C211⋊#95—21no—
19C22×Q8×C211C2211yescharacteristic
20C22×C8×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22#103×C211—22no—
23C22×Dic211—21no—
24C22×Dic211—21no—
25C22×Dic211D411yescharacteristic
26C22×Dic211D411yescharacteristic
27C2×C211⋊C8—21no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8D412yes—
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32C4⋊Dic211—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211—42no—
35C4×Dic211—21no—
36C4×Dic211—21no—
37C2×Q8×C211—21no—
38C2×Q8×C211—21no—
39C2×Q8×C211—21no—
40C2×Q8×C211—21no—
41C2×Q8×C211—21no—
42C2×Q8×C211Q812yes—
43C22×C4×C211—21no—
44C22×C4×C211—21no—
45C22×C4×C211C2311yescharacteristic
46C2×C8×C211—21no—
47C2×C8×C211—21no—
48C2×C8×C211—21no—
49C4⋊C4×C211—21no—
50C4⋊C4×C211D412yes—
51C2×Dic211—21no—
52C2×Dic211—21no—
53C2×Dic211—21no—
54C2×Dic211—21no—
55C2×Dic211—21no—
56C2×Dic211—41no—
57C2×Dic211—41no—
58C2×Dic211—41no—
59C2×Dic211—41no—
60C2×Dic211—41no—
61C2×Dic211—41no—
62C2×Dic211Q1612yes—
63C211⋊C8—22no—
64C23×C211C2×D411yescharacteristic
65Q8×C211—22no—
66Q8×C211—22no—
67Q8×C211—41no—
68Q8×C211—41no—
69C2×C4×C211—22no—
70C2×C4×C211—22no—
71C2×C4×C211—41no—
72C2×C4×C211—41no—
73C2×C4×C211—41no—
74C2×C4×C211—41no—
75C2×C4×C211C2×D411yescharacteristic
76C2×C4×C211C2×D411yescharacteristic
77C2×C4×C211C2×Q811yescharacteristic
78C2×C4×C211#4111yescharacteristic
79C2×C4×C211#4111yescharacteristic
80C2×C4×C211#4111yescharacteristic
81C8×C211—41no—
82C22×C211#12111yescharacteristic
83C22×C211#12111yescharacteristic
84C22×C211C22⋊Q811yescharacteristic
85C22×C211C2×Q1611yescharacteristic
86C22×C211#13511yescharacteristic
87C22×C211AGL(1,ℤ/8)11yescharacteristic
88C22×C211#13711yescharacteristic
89C4×C211—42no—
90C4×C211—42no—
91C4×C211#12111yescharacteristic
92C4×C211C22⋊Q811yescharacteristic
93C4×C211C22⋊Q811yescharacteristic
94C4×C211#12611yescharacteristic
95Dic211—22no—
96Dic211—41no—
97Dic211—81no—
98Dic211—81no—
99C2×C211#39611yescharacteristic
100C2×C211#46211yescharacteristic
101C2×C211#46311yescharacteristic
102C2×C211#46711yescharacteristic
103C2×C211#46811yescharacteristic
104C2×C211#47711yescharacteristic
105C2×C211#47911yescharacteristic
106C211#206811yescharacteristic, complemented, Sylow
107#2068—2111nomaximal, Sylow
108C2×#107—2111no—
109C2×C4⋊C8—2111no—
110C2×#103—2111no—
111C2×#103D21111yescharacteristic
112#391—2111no—
113#389—2111no—
114#340—2111no—
115C22×Q8C2×D21111yescharacteristic
116C22×C8—2111no—
117C22×C8C2×D21111yescharacteristic
118C2×C4⋊C4—4221no—
119C2×C4⋊C4—2111no—
120C2×C4⋊C4—2111no—
121C2×C4⋊C4C2×D21111yescharacteristic
122C2×C42—2111no—
123#107—4222no—
124C4⋊C8—4222no—
125#103—22no—
126#103—4222no—
127#95—4221no—
128#95—4221no—
129#95—4221no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8Dic42212yes—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—4221no—
139C22×C4—4221no—
140C22×C4—2111no—
141C22×C4—2111no—
142C22×C4C22×D21111yescharacteristic
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—4221no—
147C2×C8—4221no—
148C2×C8—2112no—
149C4⋊C4—21no—
150C4⋊C4—8442no—
151C4⋊C4—4221no—
152C4⋊C4—4221no—
153C4⋊C4—4221no—
154C4⋊C4—4221no—
155C4⋊C4C22⋊D21112yes—
156C42—4221no—
157C42—4221no—
158C23D4×D21111yescharacteristic
159Q8—22no—
160Q8—22no—
161Q8—41no—
162Q8—41no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—8441no—
170C2×C4—8441no—
171C2×C4—8441no—
172C2×C4—8441no—
173C2×C4—8441no—
174C2×C4—8441no—
175C2×C4—4221no—
176C2×C4—4221no—
177C2×C4—4221no—
178C2×C4—4221no—
179C2×C4—4221no—
180C2×C4—2112no—
181C2×C4D4×D21111yescharacteristic
182C2×C4C2×C22⋊D21111yescharacteristic
183C2×C4C2×Dic42211yescharacteristic
184C2×C4#1680996111yescharacteristic
185C2×C4#1680996111yescharacteristic
186C2×C4#1680996211yescharacteristic
187C8—41no—
188C8—4222no—
189C22#2026994911yescharacteristic
190C22#2026995011yescharacteristic
191C22#2026995411yescharacteristic
192C22Q16×D21111yescharacteristic
193C22#2026999811yescharacteristic
194C22#2027000411yescharacteristic
195C22#2027000911yescharacteristic
196C4—42no—
197C4—42no—
198C4—8441no—
199C4—16881no—
200C4—16881no—
201C4—4222no—
202C4#2026994811yescharacteristic
203C4#2026995411yescharacteristic
204C4#2026995611yescharacteristic
205C4#2026997011yescharacteristic
206C2#2721306911yescharacteristic
207C2#2721335411yescharacteristic
208C2#2721336311yescharacteristic
209C2#2721338611yescharacteristic
210C2#2721339511yescharacteristic
211C2#2721343911yescharacteristic
212C2#2721345111yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#103×C211C2
socle, prime radicalC23×C211C2×D4
hypercenterC2×#103D211
hypocenterC211#2068
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
106. C211#2068semidirect

Maximal subgroups

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

subgroupindex
C2×#202699102
C2×Dic211⋊C82
C2×#202699012
#272130532
#272130452
#272129092
C2×#103×C2112
#2068211