Finite Group Databasebeta

← C3×#2059 #72774

Subgroups — C3×#2059 #72774

Order 384.

250 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 108 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×C3×#125C211yesmaximal, characteristic
2C2×C3×#107C211yesmaximal, characteristic
3C2×C3×#106C211yesmaximal, characteristic
4C3×#389C211yesmaximal, characteristic
5C3×#340C211yesmaximal, characteristic
6C3×#340C211yesmaximal, characteristic
7C3×#336C211yesmaximal, characteristic
8#2059C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×C8C2211yescharacteristic
10C22×C3×C8C2211yescharacteristic
11C3×#125—22no—
12C3×#125—22no—
13C2×C3×C4⋊C4—21no—
14C2×C3×C4⋊C4—21no—
15C2×C3×C4⋊C4—21no—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C42C2211yescharacteristic
21C3×#107—22no—
22C3×#106—22no—
23C3×#95—21no—
24C2×#125C2×C311yescharacteristic
25C2×#107C2×C311yescharacteristic
26C2×#106C2×C311yescharacteristic
27#389C2×C311yescharacteristic
28#340C2×C311yescharacteristic
29#340C2×C311yescharacteristic
30#336C2×C311yescharacteristic
31C22×C3×C4—21no—
32C22×C3×C4—21no—
33C22×C3×C4—21no—
34C22×C3×C4—21no—
35C22×C3×C4D411yescharacteristic
36C22×C3×C4D411yescharacteristic
37C22×C3×C4C2311yescharacteristic
38C2×C3×C8—21no—
39C2×C3×C8—21no—
40C2×C3×C8—21no—
41C2×C3×C8—21no—
42C2×C3×C8Q812yes—
43C2×C3×C8Q812yes—
44C3×C4⋊C4—24no—
45C3×C4⋊C4—24no—
46C3×C4⋊C4—21no—
47C3×C4⋊C4—21no—
48C3×C4⋊C4—21no—
49C3×C4⋊C4—21no—
50C3×C4⋊C4—21no—
51C3×C4⋊C4—42no—
52C3×C4⋊C4D412yes—
53C3×C4⋊C4D412yes—
54C3×C4⋊C4Q812yes—
55C3×C42—21no—
56C3×C42—21no—
57C22×C8C22×C311yescharacteristic
58C22×C8C22×C311yescharacteristic
59#125—22no—
60#125—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4C22×C311yescharacteristic
65C2×C4⋊C4C22×C311yescharacteristic
66C2×C4⋊C4C22×C311yescharacteristic
67C2×C4⋊C4C22×C311yescharacteristic
68C2×C42C22×C311yescharacteristic
69#107—22no—
70#106—22no—
71#95—21no—
72C23×C3C2×D411yescharacteristic
73C2×C3×C4—21no—
74C2×C3×C4—21no—
75C2×C3×C4—22no—
76C2×C3×C4—22no—
77C2×C3×C4—22no—
78C2×C3×C4—21no—
79C2×C3×C4—21no—
80C2×C3×C4—21no—
81C2×C3×C4—21no—
82C2×C3×C4—41no—
83C2×C3×C4—41no—
84C2×C3×C4—41no—
85C2×C3×C4—41no—
86C2×C3×C4—41no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89C2×C3×C4—41no—
90C2×C3×C4—41no—
91C2×C3×C4C2×D411yescharacteristic
92C2×C3×C4C2×D411yescharacteristic
93C2×C3×C4C2×Q811yescharacteristic
94C2×C3×C4C2×Q811yescharacteristic
95C2×C3×C4C2×Q811yescharacteristic
96C2×C3×C4#4111yescharacteristic
97C3×C8—22no—
98C3×C8—22no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4C3×D411yescharacteristic
104C22×C4C3×D411yescharacteristic
105C22×C4C23×C311yescharacteristic
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8C3×Q812yes—
111C2×C8C3×Q812yes—
112C4⋊C4—24no—
113C4⋊C4—24no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—42no—
120C4⋊C4C3×D412yes—
121C4⋊C4C3×D412yes—
122C4⋊C4C3×Q812yes—
123C42—21no—
124C42—21no—
125C22×C3#12011yescharacteristic
126C22×C3C22⋊Q811yescharacteristic
127C22×C3C4⋊Q811yescharacteristic
128C22×C3#13511yescharacteristic
129C22×C3#13511yescharacteristic
130C22×C3AGL(1,ℤ/8)11yescharacteristic
131C22×C3#13711yescharacteristic
132C3×C4—41no—
133C3×C4—42no—
134C3×C4—41no—
135C3×C4—42no—
136C3×C4—42no—
137C3×C4—81no—
138C3×C4C22⋊Q811yescharacteristic
139C3×C4C22⋊Q811yescharacteristic
140C3×C4C4⋊Q811yescharacteristic
141C3×C4C4⋊Q811yescharacteristic
142C23C2×C3×D411yescharacteristic
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4C2×C3×D411yescharacteristic
162C2×C4C2×C3×D411yescharacteristic
163C2×C4C2×C3×Q811yescharacteristic
164C2×C4C2×C3×Q811yescharacteristic
165C2×C4C2×C3×Q811yescharacteristic
166C2×C4C3×#4111yescharacteristic
167C8—22no—
168C8—22no—
169C2×C3#39511yescharacteristic
170C2×C3#44911yescharacteristic
171C2×C3#45211yescharacteristic
172C2×C3#47811yescharacteristic
173C2×C3#47911yescharacteristic
174C2×C3#49911yescharacteristic
175C2×C3#50111yescharacteristic
176C22C3×#12011yescharacteristic
177C22C3×C22⋊Q811yescharacteristic
178C22C3×C4⋊Q811yescharacteristic
179C22C3×#13511yescharacteristic
180C22C3×#13511yescharacteristic
181C22C3×AGL(1,ℤ/8)11yescharacteristic
182C22C3×#13711yescharacteristic
183C4—41no—
184C4—42no—
185C4—41no—
186C4—42no—
187C4—42no—
188C4—81no—
189C4C3×C22⋊Q811yescharacteristic
190C4C3×C22⋊Q811yescharacteristic
191C4C3×C4⋊Q811yescharacteristic
192C4C3×C4⋊Q811yescharacteristic
193C3#205911yescharacteristic, direct product factor, Sylow
194C2C3×#39511yescharacteristic
195C2C3×#44911yescharacteristic
196C2C3×#45211yescharacteristic
197C2C3×#47811yescharacteristic
198C2C3×#47911yescharacteristic
199C2C3×#49911yescharacteristic
200C2C3×#50111yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C3C2×D4
commutator, Frattini, cosocleC22×C4C23×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
8. #2059C3direct (N × H)
193. C3#2059direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1252
C2×C3×#1072
C2×C3×#1062
C3×#3892
C3×#3402
C3×#3362
#20593