Finite Group Databasebeta

← #13140011

Subgroups — #13140011

Order 1728.

545 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 119 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×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic
2C2×#3750⋊C3C211yesmaximal, characteristic
3#11720574C211yesmaximal, characteristic
4#11720334C212yesmaximal
5#11720334C212yesmaximal
6#10591227—31nomaximal
7#10591226—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2212yes—
10C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)—22no—
14#3750⋊C3—22no—
15C2×C4⋊(C3⋊Dic3)D311yescharacteristic
16C2×#3750D311yescharacteristic
17C2×C3×C4⋊Dic3—31no—
18C2×C3×Dic3⋊C4—31no—
19C3×C3⋊#95—31no—
20C3×C3⋊#95—31no—
21C3×C3⋊#95—61no—
22#63579—32no—
23#63579—32no—
24#63579—32no—
25#63579—32no—
26C23×H(3,𝔽3)C2311yescharacteristic
27C2×C4×H(3,𝔽3)—22no—
28C2×C4×H(3,𝔽3)D412yes—
29C2×#1084—22no—
30C2×#1084—22no—
31C2×#1084—22no—
32C2×(C3⋊Dic3)⋊C3—21no—
33C2×(C3⋊Dic3)⋊C3—21no—
34C2×(C3⋊Dic3)⋊C3—21no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—24no—
38C2×(C3⋊Dic3)⋊C3Q812yes—
39#4986—91no—
40#4985—91no—
41#4659—91no—
42C22×C32×C4—21no—
43C22×C32×C4C2×D311yescharacteristic
44C22×C32×C4C2×D311yescharacteristic
45C22×C3⋊Dic3C2×D311yescharacteristic
46C22×C3⋊Dic3C2×D312yes—
47C22×C3⋊Dic3C2×D311yescharacteristic
48C22×C3×Dic3—31no—
49C22×C3×Dic3—31no—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—32no—
53C22×C3×Dic3—62no—
54C4⋊(C3⋊Dic3)—22no—
55#3750—22no—
56C3×C4⋊Dic3—62no—
57C3×Dic3⋊C4—62no—
58C22×H(3,𝔽3)C2×D411yescharacteristic
59C22×H(3,𝔽3)C2×Q811yescharacteristic
60C22×H(3,𝔽3)#4111yescharacteristic
61C22×H(3,𝔽3)#4112yes—
62C22×H(3,𝔽3)#4112yes—
63C4×H(3,𝔽3)—22no—
64#1084—42no—
65(C3⋊Dic3)⋊C3—22no—
66(C3⋊Dic3)⋊C3—41no—
67(C3⋊Dic3)⋊C3—42no—
68C2×C3×C4⋊C4—91no—
69C2×C3×C4⋊C4—91no—
70C2×C4⋊Dic3—31no—
71C2×C4⋊Dic3—91no—
72C2×Dic3⋊C4—31no—
73C2×Dic3⋊C4—91no—
74C3×#95—91no—
75C3⋊#95—31no—
76C3⋊#95—31no—
77C3⋊#95—61no—
78C3⋊#95—92no—
79C3⋊#95—92no—
80C3⋊#95—92no—
81C3⋊#95—92no—
82C3⋊#95—92no—
83C3⋊#95—92no—
84C23×C32—21no—
85C23×C32C22×D311yescharacteristic
86C23×C32C22×D311yescharacteristic
87C2×C32×C4—22no—
88C2×C32×C4—22no—
89C2×C32×C4—22no—
90C2×C32×C4—42no—
91C2×C32×C4#6712yes—
92C2×C32×C4#6712yes—
93C2×C3⋊Dic3—21no—
94C2×C3⋊Dic3—21no—
95C2×C3⋊Dic3—24no—
96C2×C3⋊Dic3—22no—
97C2×C3⋊Dic3—21no—
98C2×C3⋊Dic3—22no—
99C2×C3⋊Dic3Dic612yes—
100C2×C3×Dic3—32no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—64no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—64no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—122no—
115C2×C3×Dic3—122no—
116#400—271noSylow
117C2×H(3,𝔽3)C22⋊Q811yescharacteristic
118C2×H(3,𝔽3)#12311yescharacteristic
119C2×H(3,𝔽3)#12411yescharacteristic
120C2×H(3,𝔽3)#12512yes—
121C2×H(3,𝔽3)#12612yes—
122C22×C3×C4—31no—
123C22×C3×C4—31no—
124C22×C3×C4—61no—
125C22×C3×C4—91no—
126C22×C3×C4—92no—
127C22×C3×C4—91no—
128C22×C3×C4—92no—
129C22×C3×C4D3211yescharacteristic
130C22×Dic3—31no—
131C22×Dic3—32no—
132C22×Dic3—32no—
133C22×Dic3—31no—
134C22×Dic3—32no—
135C22×Dic3—62no—
136C22×Dic3—91no—
137C22×Dic3—91no—
138C22×Dic3—92no—
139C3×C4⋊C4—182no—
140C3×C4⋊C4—182no—
141C4⋊Dic3—62no—
142C4⋊Dic3—182no—
143Dic3⋊C4—62no—
144Dic3⋊C4—182no—
145C22×C32—21no—
146C22×C32—22no—
147C22×C32—21no—
148C22×C32—22no—
149C22×C32—21no—
150C22×C32C2×Dic611yescharacteristic
151C22×C32#23511yescharacteristic
152C22×C32#23512yes—
153C22×C32#23712yes—
154C22×C32#23712yes—
155C22×C32D3×Q811yescharacteristic
156C22×C32#23912yes—
157C22×C32#23911yescharacteristic
158C22×C32C2×#6711yescharacteristic
159C22×C32C2×#6711yescharacteristic
160C32×C4—22no—
161C32×C4—22no—
162C32×C4—42no—
163C3⋊Dic3—22no—
164C3⋊Dic3—41no—
165C3⋊Dic3—42no—
166C3×Dic3—62no—
167C3×Dic3—124no—
168C3×Dic3—122no—
169C3×Dic3—122no—
170C3×Dic3—121no—
171C3×Dic3—122no—
172C2×C4⋊C4—91no—
173C2×C4⋊C4—91no—
174#95—91no—
175#95—272no—
176#95—272no—
177H(3,𝔽3)#40011yescharacteristic, complemented, Sylow
178C23×C3—31no—
179C23×C3—31no—
180C23×C3—61no—
181C23×C3C2×D3211yescharacteristic
182C2×C3×C4—22no—
183C2×C3×C4—32no—
184C2×C3×C4—32no—
185C2×C3×C4—62no—
186C2×C3×C4—62no—
187C2×C3×C4—62no—
188C2×C3×C4—92no—
189C2×C3×C4—122no—
190C2×C3×C4—181no—
191C2×C3×C4—181no—
192C2×C3×C4—181no—
193C2×C3×C4—182no—
194C2×C3×C4—184no—
195C2×C3×C4—182no—
196C2×C3×C4—182no—
197C2×C3×C4—182no—
198C2×C3×C4—182no—
199C2×C3×C4#62512yes—
200C2×Dic3—32no—
201C2×Dic3—61no—
202C2×Dic3—61no—
203C2×Dic3—62no—
204C2×Dic3—62no—
205C2×Dic3—62no—
206C2×Dic3—62no—
207C2×Dic3—62no—
208C2×Dic3—62no—
209C2×Dic3—64no—
210C2×Dic3—64no—
211C2×Dic3—62no—
212C2×Dic3—61no—
213C2×Dic3—62no—
214C2×Dic3—92no—
215C2×Dic3—122no—
216C2×Dic3—122no—
217C2×Dic3—181no—
218C2×Dic3—181no—
219C2×Dic3—181no—
220C2×Dic3—182no—
221C2×Dic3—184no—
222C2×Dic3—182no—
223C2×C32—22no—
224C2×C32—22no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32#88912yes—
229C2×C32#89012yes—
230C2×C32#89812yes—
231C2×C32#89812yes—
232C2×C32#92411yescharacteristic
233C2×C32#92911yescharacteristic
234C2×C32#93511yescharacteristic
235C2×C32#93611yescharacteristic
236C2×C32#94611yescharacteristic
237C2×C32#94711yescharacteristic
238C22×C4—91no—
239C22×C4—92no—
240C22×C4—92no—
241C22×C4—91no—
242C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
243C4⋊C4—182no—
244C4⋊C4—182no—
245C22×C3—31no—
246C22×C3—31no—
247C22×C3—32no—
248C22×C3—32no—
249C22×C3—31no—
250C22×C3—31no—
251C22×C3—32no—
252C22×C3—32no—
253C22×C3—31no—
254C22×C3—31no—
255C22×C3—61no—
256C22×C3—62no—
257C22×C3—61no—
258C22×C3—62no—
259C22×C3—61no—
260C22×C3D3×Dic611yescharacteristic
261C22×C3#379612yes—
262C22×C3#379911yescharacteristic
263C22×C3#380412yes—
264C22×C3C2×#62511yescharacteristic
265C3×C4—22no—
266C3×C4—62no—
267C3×C4—62no—
268C3×C4—122no—
269C3×C4—182no—
270C3×C4—361no—
271C3×C4—362no—
272C3×C4—362no—
273Dic3—62no—
274Dic3—122no—
275Dic3—122no—
276Dic3—124no—
277Dic3—122no—
278Dic3—121no—
279Dic3—182no—
280Dic3—361no—
281Dic3—362no—
282C32—21no—
283C32#498511yescharacteristic, complemented
284C32#498611yescharacteristic, complemented
285C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
286C2×C4—22no—
287C2×C4—92no—
288C2×C4—181no—
289C2×C4—181no—
290C2×C4—182no—
291C2×C4—184no—
292C2×C4—182no—
293C2×C4—182no—
294C2×C4—182no—
295C2×C4—181no—
296C2×C4—182no—
297C2×C4#619812yes—
298C2×C3—31no—
299C2×C3—31no—
300C2×C3—32no—
301C2×C3—32no—
302C2×C3—32no—
303C2×C3—32no—
304C2×C3—31no—
305C2×C3—31no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—61no—
309C2×C3—62no—
310C2×C3—62no—
311C2×C3—61no—
312C2×C3—61no—
313C2×C3#6384612yes—
314C2×C3#6384812yes—
315C2×C3#6385611yescharacteristic
316C2×C3#6386311yescharacteristic
317C2×C3#6396111yescharacteristic
318C22((C3⋊D3)⋊C3)⋊Q811yescharacteristic
319C22#8864912yes—
320C22#8865211yescharacteristic
321C22#8867112yes—
322C22C2×#619811yescharacteristic
323C4—22no—
324C4—182no—
325C4—361no—
326C4—362no—
327C4—362no—
328C3—31no—
329C3—31no—
330C3—61no—
331C3#1059122511yescharacteristic
332C2#1172108712yes—
333C2#1172108812yes—
334C2#1172110611yescharacteristic
335C2#1172111311yescharacteristic
336C2#1172130611yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#400
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
177. H(3,𝔽3)#400semidirect
283. C32#4985semidirect
284. C32#4986semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#3750⋊C32
#117205742
#117203342
#105912273
#105912263