Finite Group Databasebeta

← #13144020

Subgroups — #13144020

Order 1728.

836 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 113 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×#88748C211yesmaximal, characteristic
2C2×#88550C211yesmaximal, characteristic, complemented
3C2×#88528C211yesmaximal, characteristic, complemented
4#11721765C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
6#11720553C211yesmaximal, characteristic, complemented
7#11720531C211yesmaximal, characteristic, complemented
8C3×#4808—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10#88748—22no—
11#88748—22no—
12C2×C4⋊#272—21no—
13C2×C4⋊#272C2211yescharacteristic
14C2×C4×#272—21no—
15C2×C4×#272C2211yescharacteristic
16C2×#6229C2211yescharacteristic
17C2×C8×H(3,𝔽3)C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19#88550—22no—
20#88542—21no—
21C4⋊#1084—21no—
22C2×#6180C2211yescharacteristic
23#88528—22no—
24C2×C3×#239—34no—
25C2×C3×#216—34no—
26C2×C3×#204—34no—
27C3×#897—34no—
28C32×#103D314yescomplemented
29C3×(C2×D3)⋊C8—34no—
30C3×#809—34no—
31C22×#272—21no—
32C22×#272D411yescharacteristic
33Q8×H(3,𝔽3)—21no—
34Q8×H(3,𝔽3)D412yes—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C2×#1084D411yescharacteristic
40C4⋊#272—24no—
41C4⋊#272—21no—
42C4⋊#272D412yes—
43C4×#272—24no—
44C4×#272—21no—
45C4×#272—21no—
46#6229—21no—
47#6229—21no—
48C8×H(3,𝔽3)—21no—
49#6180—21no—
50C3×#452—91no—
51#4808—34no—
52C2×C32×Q8C2×D314yes—
53C3×#239—68no—
54C3×#239—68no—
55C2×C3×D12—34no—
56C2×C3×D12—64no—
57C2×C3×C4×D3—34no—
58C2×C3×C4×D3—64no—
59C2×C3×Dic6—34no—
60C2×C32×C8C2×D314yes—
61C32×C4⋊C4C2×D314yes—
62C3×#216—68no—
63C3×(C2×D3)⋊C4—64no—
64C3×C4⋊Dic3—64no—
65C2×C3×C3⋊C8—34no—
66C3×#204—68no—
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C2×#272—21no—
69C2×#272—22no—
70C2×#272—21no—
71C2×#272—21no—
72C2×#272—41no—
73C2×#272—41no—
74C4×H(3,𝔽3)—22no—
75C4×H(3,𝔽3)—41no—
76C4×H(3,𝔽3)C2×D411yescharacteristic
77C4×H(3,𝔽3)C2×D411yescharacteristic
78#1084—21no—
79#1084—41no—
80C2×C3×#41—91no—
81C2×#239—34no—
82C2×C3×Q16—91no—
83C2×C3×SD16—91no—
84C3×C22⋊Q8—91no—
85C2×#216—34no—
86C2×#204—34no—
87#897—34no—
88C3×#103—34no—
89C3×#103C3⋊D311yescharacteristic
90C3×#102—91no—
91C3×C22⋊C8—91no—
92(C2×D3)⋊C8—34no—
93#809—34no—
94C22×C3×D3—34no—
95C22×C3×D3—64no—
96C32×Q8—24no—
97C32×Q8D1218yes—
98C2×C32×C4—24no—
99C2×C32×C4—24no—
100C2×C32×C4C22×D314yes—
101C2×C3×Dic3—34no—
102C2×C3×Dic3—64no—
103C3×D12—38no—
104C3×D12—616no—
105C3×D12—64no—
106C3×C4×D3—616no—
107C3×C4×D3—64no—
108C3×C4×D3—64no—
109C3×Dic6—64no—
110C3×Dic6—64no—
111C32×C8—24no—
112C3×C3⋊C8—64no—
113#452—91noSylow
114C2×H(3,𝔽3)#12011yescharacteristic
115C2×H(3,𝔽3)#13511yescharacteristic
116C2×H(3,𝔽3)#13711yescharacteristic
117#272—41no—
118#272—42no—
119C3×#41—182no—
120C3×#41—182no—
121C2×C3×Q8—34no—
122C2×C3×Q8—91no—
123C2×C3×Q8C2×C3⋊D311yescharacteristic
124C2×C3×D4—91no—
125C2×C3×D4—181no—
126C22×C3×C4—91no—
127C22×C3×C4—181no—
128#239—68no—
129#239—68no—
130C2×D12—34no—
131C2×D12—64no—
132C2×C4×D3—34no—
133C2×C4×D3—64no—
134C2×Dic6—34no—
135C3×Q16—182no—
136C3×SD16—182no—
137C2×C3×C8—34no—
138C2×C3×C8—91no—
139C2×C3×C8C2×C3⋊D311yescharacteristic
140C3×C4⋊C4—34no—
141C3×C4⋊C4—181no—
142C3×C4⋊C4C2×C3⋊D311yescharacteristic
143C3×C22⋊C4—181no—
144#216—68no—
145(C2×D3)⋊C4—64no—
146C4⋊Dic3—64no—
147C2×C3⋊C8—34no—
148#204—68no—
149C22×C32D3×D414yes—
150C2×C3×D3—68no—
151C2×C3×D3—64no—
152C2×C3×D3—64no—
153C2×C3×D3—64no—
154C2×C3×D3—124no—
155C2×C3×D3—124no—
156C32×C4—28no—
157C32×C4—44no—
158C32×C4C2×D1214yes—
159C32×C4D3×D414yes—
160C3×Dic3—64no—
161C3×Dic3—124no—
162C2×#41—91no—
163C2×Q16—91no—
164C2×SD16—91no—
165C22⋊Q8—91no—
166#103#27211yescharacteristic, complemented
167#102—91no—
168C22⋊C8—91no—
169H(3,𝔽3)#45211yescharacteristic, complemented, Sylow
170C23×C3—91no—
171C23×C3—181no—
172C22×D3—34no—
173C22×D3—64no—
174C3×Q8—21no—
175C3×Q8—38no—
176C3×Q8—64no—
177C3×Q8—181no—
178C3×Q8—181no—
179C3×Q8C4⋊(C3⋊D3)12yes—
180C3×D4—92no—
181C3×D4—184no—
182C3×D4—181no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—34no—
186C2×C3×C4—64no—
187C2×C3×C4—64no—
188C2×C3×C4—91no—
189C2×C3×C4—184no—
190C2×C3×C4—181no—
191C2×C3×C4—181no—
192C2×C3×C4—181no—
193C2×C3×C4C22×C3⋊D311yescharacteristic
194C2×Dic3—34no—
195C2×Dic3—64no—
196D12—38no—
197D12—64no—
198D12—616no—
199C4×D3—64no—
200C4×D3—64no—
201C4×D3—616no—
202Dic6—64no—
203Dic6—64no—
204C3×C8—21no—
205C3×C8—64no—
206C3×C8—181no—
207C3⋊C8—64no—
208C2×C32#88214yes—
209C2×C32#91614yes—
210C2×C32#91814yes—
211C3×D3—124no—
212C3×D3—128no—
213#41—182no—
214#41—182no—
215C2×Q8—91no—
216C2×Q8C2×#27211yescharacteristic
217C2×D4—91no—
218C2×D4—181no—
219C22×C4—91no—
220C22×C4—181no—
221Q16—182no—
222SD16—182no—
223C2×C8—91no—
224C2×C8C2×#27211yescharacteristic
225C4⋊C4—181no—
226C4⋊C4C2×#27211yescharacteristic
227C22⋊C4—181no—
228C22×C3—34no—
229C22×C3—182no—
230C22×C3—181no—
231C22×C3—181no—
232C22×C3—181no—
233C22×C3—361no—
234C22×C3—361no—
235C22×C3D4×C3⋊D311yescharacteristic
236C2×D3—68no—
237C2×D3—64no—
238C2×D3—64no—
239C2×D3—64no—
240C2×D3—124no—
241C2×D3—124no—
242C3×C4—22no—
243C3×C4—34no—
244C3×C4—34no—
245C3×C4—41no—
246C3×C4—68no—
247C3×C4—124no—
248C3×C4—181no—
249C3×C4—361no—
250C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
251C3×C4D4×C3⋊D311yescharacteristic
252Dic3—64no—
253Dic3—124no—
254C32#480814yescomplemented
255C23—91no—
256C23—181no—
257Q8—21no—
258Q8—181no—
259Q8—181no—
260Q8C4⋊#27212yes—
261D4—92no—
262D4—184no—
263D4—181no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—91no—
267C2×C4—184no—
268C2×C4—181no—
269C2×C4—181no—
270C2×C4—181no—
271C2×C4C22×#27211yescharacteristic
272C8—21no—
273C8—181no—
274C2×C3—34no—
275C2×C3—34no—
276C2×C3—34no—
277C2×C3—361no—
278C2×C3—362no—
279C2×C3#6409111yescharacteristic
280C2×C3#6412511yescharacteristic
281C2×C3#6412711yescharacteristic
282D3—124no—
283D3—128no—
284C22—182no—
285C22—181no—
286C22—181no—
287C22—181no—
288C22—361no—
289C22—361no—
290C22D4×#27211yescharacteristic
291C4—22no—
292C4—41no—
293C4—181no—
294C4—361no—
295C4C2×C4⋊#27211yescharacteristic
296C4D4×#27211yescharacteristic
297C3—34no—
298C3#1059250711yescharacteristic
299C2—361no—
300C2—362no—
301C2#1172173811yescharacteristic
302C2#1172180211yescharacteristic
303C2#1172180711yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#103C2
prime radicalC2×#88748C2
hypercenterC3×#103C3⋊D3
hypocenterH(3,𝔽3)#452
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
2. C2×#88550C2semidirect
3. C2×#88528C2semidirect
4. #11721765C2semidirect
5. H(3,𝔽3)×#103C2semidirect
6. #11720553C2semidirect
7. #11720531C2semidirect
28. C32×#103D3semidirect
166. #103#272semidirect
169. H(3,𝔽3)#452semidirect
254. C32#4808semidirect

Maximal subgroups

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

subgroupindex
C2×#887482
C2×#885502
C2×#885282
#117217652
H(3,𝔽3)×#1032
#117205532
#117205312
C3×#48083