Finite Group Databasebeta

← #10589413⋊C3 #13135638

Subgroups — #10589413⋊C3 #13135638

Order 1728.

292 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 86 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
1H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic
2C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic
3C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic
4#10589413C311yesmaximal, characteristic, complemented
5C3×#4547—31nomaximal
6C2×Q8×H(3,𝔽3)—21no—
7C4⋊C4×H(3,𝔽3)—21no—
8C4⋊C4×H(3,𝔽3)C412yes—
9C42×H(3,𝔽3)C2211yescharacteristic
10C2×(C3⋊(C3⋊C8))⋊C3—21no—
11C2×(C3⋊(C3⋊C8))⋊C3—21no—
12C32×C4⋊Q8—21no—
13C32×C4⋊Q8D311yescharacteristic
14C32×C4⋊Q8C2×C311yescharacteristic
15C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic
16C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic
17C3×C4⋊(C3⋊C8)—31no—
18C3×C4⋊(C3⋊C8)—31no—
19Q8×H(3,𝔽3)—41no—
20Q8×H(3,𝔽3)—41no—
21C2×C4×H(3,𝔽3)—22no—
22C2×C4×H(3,𝔽3)C2×C411yescharacteristic
23C2×C4×H(3,𝔽3)D411yescharacteristic
24C2×C4×H(3,𝔽3)D411yescharacteristic
25(C3⋊(C3⋊C8))⋊C3—41no—
26(C3⋊(C3⋊C8))⋊C3—41no—
27C3×#332—91no—
28#4547—31no—
29#4547—91no—
30C2×C32×Q8—22no—
31C2×C32×Q8—21no—
32C2×C32×Q8—21no—
33C32×C4⋊C4—22no—
34C32×C4⋊C4—22no—
35C32×C4⋊C4—21no—
36C32×C4⋊C4—21no—
37C32×C4⋊C4Dic312yes—
38C32×C4⋊C4C3×C412yes—
39C32×C42—21no—
40C32×C42C2×D311yescharacteristic
41C32×C42C22×C311yescharacteristic
42C2×C3⋊(C3⋊C8)—21no—
43C2×C3⋊(C3⋊C8)—21no—
44C2×C3×C3⋊C8—61no—
45C2×C3×C3⋊C8—61no—
46C22×H(3,𝔽3)C22⋊C411yescharacteristic
47C4×H(3,𝔽3)—21no—
48C4×H(3,𝔽3)—42no—
49C4×H(3,𝔽3)D811yescharacteristic
50C4×H(3,𝔽3)SD1611yescharacteristic
51C4×H(3,𝔽3)SD1611yescharacteristic
52C4×H(3,𝔽3)Q1611yescharacteristic
53C3×C4⋊Q8—31no—
54C3×C4⋊Q8—31no—
55C3×C4⋊Q8—61no—
56C3×C4⋊Q8C3×D311yescharacteristic
57C3×C4⋊C8—91no—
58C3×C4⋊C8—91no—
59C4⋊(C3⋊C8)—31no—
60C4⋊(C3⋊C8)—31no—
61C4⋊(C3⋊C8)—91no—
62C4⋊(C3⋊C8)—91no—
63C32×Q8—42no—
64C32×Q8—42no—
65C32×Q8—41no—
66C32×Q8—41no—
67C32×Q8—41no—
68C32×Q8—41no—
69C2×C32×C4—24no—
70C2×C32×C4—22no—
71C2×C32×C4—22no—
72C2×C32×C4—21no—
73C2×C32×C4—21no—
74C2×C32×C4—21no—
75C2×C32×C4C2×Dic311yescharacteristic
76C2×C32×C4#6711yescharacteristic
77C2×C32×C4#6711yescharacteristic
78C2×C32×C4C2×C3×C411yescharacteristic
79C2×C32×C4C3×D411yescharacteristic
80C2×C32×C4C3×D411yescharacteristic
81C3⋊(C3⋊C8)—41no—
82C3⋊(C3⋊C8)—41no—
83C3×C3⋊C8—121no—
84C3×C3⋊C8—121no—
85#332—91noSylow
86C2×H(3,𝔽3)#10111yescharacteristic
87C2×H(3,𝔽3)#10211yescharacteristic
88C2×H(3,𝔽3)#10311yescharacteristic
89C2×C3×Q8—21no—
90C2×C3×Q8—61no—
91C2×C3×Q8—61no—
92C2×C3×Q8—62no—
93C2×C3×C8—181no—
94C2×C3×C8—181no—
95C3×C4⋊C4—21no—
96C3×C4⋊C4—32no—
97C3×C4⋊C4—32no—
98C3×C4⋊C4—61no—
99C3×C4⋊C4—61no—
100C3×C4⋊C4—62no—
101C3×C4⋊C4—62no—
102C3×C4⋊C4C3×Dic312yes—
103C3×C42—31no—
104C3×C42—31no—
105C3×C42—61no—
106C3×C42C2×C3×D311yescharacteristic
107C2×C3⋊C8—61no—
108C2×C3⋊C8—61no—
109C2×C3⋊C8—181no—
110C2×C3⋊C8—181no—
111C22×C32—21no—
112C22×C32#21711yescharacteristic
113C22×C32C3×C22⋊C411yescharacteristic
114C32×C4—22no—
115C32×C4—21no—
116C32×C4—21no—
117C32×C4—21no—
118C32×C4—21no—
119C32×C4—21no—
120C32×C4—21no—
121C32×C4—44no—
122C32×C4—42no—
123C32×C4—42no—
124C32×C4#21311yescharacteristic
125C32×C4#21411yescharacteristic
126C32×C4#21511yescharacteristic
127C32×C4#21611yescharacteristic
128C32×C4C3×D811yescharacteristic
129C32×C4C3×SD1611yescharacteristic
130C32×C4C3×SD1611yescharacteristic
131C32×C4C3×Q1611yescharacteristic
132C4⋊Q8(C3⋊D3)⋊C311yescharacteristic
133C4⋊C8—91no—
134C4⋊C8—91no—
135H(3,𝔽3)#33211yescharacteristic, complemented, Sylow
136C3×Q8—41no—
137C3×Q8—41no—
138C3×Q8—121no—
139C3×Q8—121no—
140C3×Q8—121no—
141C3×Q8—121no—
142C3×Q8—122no—
143C3×Q8—122no—
144C2×C3×C4—22no—
145C2×C3×C4—31no—
146C2×C3×C4—31no—
147C2×C3×C4—31no—
148C2×C3×C4—31no—
149C2×C3×C4—31no—
150C2×C3×C4—31no—
151C2×C3×C4—62no—
152C2×C3×C4—62no—
153C2×C3×C4—64no—
154C2×C3×C4—61no—
155C2×C3×C4—61no—
156C2×C3×C4—61no—
157C2×C3×C4C2×C3×Dic311yescharacteristic
158C2×C3×C4C3×#6711yescharacteristic
159C2×C3×C4C3×#6711yescharacteristic
160C3×C8—361no—
161C3×C8—361no—
162C3⋊C8—121no—
163C3⋊C8—121no—
164C3⋊C8—361no—
165C3⋊C8—361no—
166C2×C32—21no—
167C2×C32—21no—
168C2×C32—21no—
169C2×C32#83211yescharacteristic
170C2×C32#83511yescharacteristic
171C2×C32#83611yescharacteristic
172C2×C32C3×#10111yescharacteristic
173C2×C32C3×#10211yescharacteristic
174C2×C32C3×#10311yescharacteristic
175C2×Q8—21no—
176C2×C8—181no—
177C2×C8—181no—
178C4⋊C4—21no—
179C4⋊C4(C3⋊Dic3)⋊C312yes—
180C42C2×(C3⋊D3)⋊C311yescharacteristic
181C22×C3—31no—
182C22×C3—31no—
183C22×C3—61no—
184C22×C3C3×#21711yescharacteristic
185C3×C4—21no—
186C3×C4—31no—
187C3×C4—31no—
188C3×C4—31no—
189C3×C4—31no—
190C3×C4—31no—
191C3×C4—31no—
192C3×C4—31no—
193C3×C4—31no—
194C3×C4—42no—
195C3×C4—61no—
196C3×C4—61no—
197C3×C4—62no—
198C3×C4—61no—
199C3×C4—61no—
200C3×C4—61no—
201C3×C4—61no—
202C3×C4—122no—
203C3×C4—122no—
204C3×C4—124no—
205C3×C4C3×#21311yescharacteristic
206C3×C4C3×#21411yescharacteristic
207C3×C4C3×#21511yescharacteristic
208C3×C4C3×#21611yescharacteristic
209C32—21no—
210C32#454711yescharacteristic, complemented
211C32C3×#33211yescharacteristic, complemented
212Q8—41no—
213Q8—41no—
214C2×C4—22no—
215C2×C4C2×(C3⋊Dic3)⋊C311yescharacteristic
216C2×C4#638⋊C311yescharacteristic
217C2×C4#638⋊C311yescharacteristic
218C8—361no—
219C8—361no—
220C2×C3—31no—
221C2×C3—31no—
222C2×C3—31no—
223C2×C3—31no—
224C2×C3—31no—
225C2×C3—31no—
226C2×C3—61no—
227C2×C3—61no—
228C2×C3—61no—
229C2×C3C3×#83211yescharacteristic
230C2×C3C3×#83511yescharacteristic
231C2×C3C3×#83611yescharacteristic
232C22#3757⋊C311yescharacteristic
233C4—21no—
234C4—42no—
235C4#3754⋊C311yescharacteristic
236C4#3753⋊C311yescharacteristic
237C4#3756⋊C311yescharacteristic
238C4#3755⋊C311yescharacteristic
239C3—31no—
240C3—31no—
241C3—61no—
242C3C3×#454711yescharacteristic
243C2#63659⋊C311yescharacteristic
244C2#63663⋊C311yescharacteristic
245C2#63662⋊C311yescharacteristic
subgroupG/H
centerC22#3757⋊C3
commutatorC2×C32×C4C2×C3×C4
FrattiniC3×C42C2×C3×D3
FittingH(3,𝔽3)×C4⋊Q8C2
socleC22×C3C3×#217
prime radicalC22×H(3,𝔽3)C22⋊C4
cosocleC32×C42C22×C3
hypercenterC4⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#332
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #10589413C3semidirect
135. H(3,𝔽3)#332semidirect
210. C32#4547semidirect
211. C32C3×#332semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C4⋊Q82
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105894133
C3×#45473