Finite Group Databasebeta

← #13154226

Subgroups — #13154226

Order 1728.

1287 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#125C211yesmaximal, characteristic, complemented
2#11721766C212yesmaximal, complemented
3#11721765C211yesmaximal, characteristic, complemented
4#11721763C211yesmaximal, characteristic
5#11721762C212yesmaximal, complemented
6#11721761C211yesmaximal, characteristic, complemented
7#11721761C212yesmaximal, complemented
8#11721760C212yesmaximal, complemented
9#11721755C211yesmaximal, characteristic, complemented
10#11721721C211yesmaximal, characteristic, complemented
11C42×#272C211yesmaximal, characteristic, complemented
12C3×#5691—34nomaximal
13C2×C4⋊#272C2212yes—
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272C2212yes—
16C2×C4×#272C2211yescharacteristic
17C2×C4×#272C2211yescharacteristic
18C2×C4×#272C2211yescharacteristic
19C2×#6229C2211yescharacteristic
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C4⋊C4×H(3,𝔽3)C2212yes—
22C4⋊C4×H(3,𝔽3)C2211yescharacteristic
23C4⋊C4×H(3,𝔽3)C2212yes—
24C42×H(3,𝔽3)C2211yescharacteristic
25#88542C2212yes—
26#88542C2212yes—
27#88542C2212yes—
28#88542C2212yes—
29#88542C2212yes—
30C4⋊#1084C2212yes—
31#88540C2212yes—
32C4×#1084C2212yes—
33C4×#1084C2211yescharacteristic
34C4×#1084C2211yescharacteristic
35C4×#1084C2211yescharacteristic
36C32×#125D314yescomplemented
37C3×#898—38no—
38C3×#897—34no—
39C3×#895—34no—
40C3×#894—38no—
41C3×#893—38no—
42C3×#893—34no—
43C3×#892—38no—
44C3×#887—34no—
45C3×#875—34no—
46C3×C42×D3—34no—
47C22×#272C2312yes—
48C22×#272C2311yescharacteristic
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51C2×C4×H(3,𝔽3)C2312yes—
52C2×C4×H(3,𝔽3)C2312yes—
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×#1084C2312yes—
55C2×#1084C2311yescharacteristic
56C2×#1084C2311yescharacteristic
57C2×#1084C2311yescharacteristic
58C4⋊#272—24no—
59C4⋊#272—22no—
60C4×#272—22no—
61C4×#272—22no—
62C4×#272—22no—
63C4×#272—24no—
64#6229—22no—
65C3×#529—91no—
66#5691—34no—
67C2×C3×D12—38no—
68C2×C3×D12—34no—
69C2×C3×C4×D3—38no—
70C2×C3×C4×D3—34no—
71C2×C3×C4×D3—34no—
72C2×C3×C4×D3—34no—
73C2×C3×Dic6—34no—
74C32×C4⋊C4C2×D314yes—
75C32×C4⋊C4C2×D318yes—
76C32×C4⋊C4C2×D314yes—
77C32×C4⋊C4C2×D318yes—
78C32×C42C2×D314yes—
79C3×(C2×D3)⋊C4—38no—
80C3×(C2×D3)⋊C4—38no—
81C3×(C2×D3)⋊C4—38no—
82C3×(C2×D3)⋊C4—38no—
83C3×(C2×D3)⋊C4—38no—
84C3×C4⋊Dic3—38no—
85C3×Dic3⋊C4—38no—
86C3×C4×Dic3—38no—
87C3×C4×Dic3—34no—
88C3×C4×Dic3—34no—
89C3×C4×Dic3—34no—
90C22×H(3,𝔽3)C2411yescharacteristic
91C2×#272—21no—
92C2×#272—21no—
93C2×#272—22no—
94C2×#272—22no—
95C2×#272—22no—
96C2×#272#4112yes—
97C4×H(3,𝔽3)—21no—
98C4×H(3,𝔽3)—21no—
99C4×H(3,𝔽3)—22no—
100C4×H(3,𝔽3)—22no—
101C4×H(3,𝔽3)#4112yes—
102#1084—22no—
103#1084—21no—
104#1084—21no—
105#1084#4112yes—
106C3×#126—92no—
107C3×#125—34no—
108C3×#125C3⋊D311yescharacteristic
109C3×#124—91no—
110C3×#123—92no—
111C3×C22⋊Q8—91no—
112C3×#121—91no—
113C3×C4×Q8—91no—
114C3×C4×D4—91no—
115C3×C4×D4—92no—
116C3×#117—92no—
117C2×C3×C42—91no—
118#898—38no—
119#897—34no—
120#895—34no—
121#894—38no—
122#893—34no—
123#893—38no—
124#892—38no—
125#887—34no—
126#875—34no—
127C42×D3—34no—
128C22×C3×D3—34no—
129C22×C3×D3—38no—
130C2×C32×C4C22×D318yes—
131C2×C32×C4C22×D314yes—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D318yes—
134C2×C32×C4C22×D314yes—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—34no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—38no—
139C3×D12—616no—
140C3×D12—68no—
141C3×C4×D3—68no—
142C3×C4×D3—616no—
143C3×C4×D3—68no—
144C3×C4×D3—68no—
145C3×Dic6—68no—
146#529—91noSylow
147C2×H(3,𝔽3)C2×#4111yescharacteristic
148C2×H(3,𝔽3)C2×#4111yescharacteristic
149C2×H(3,𝔽3)C2×#4111yescharacteristic
150#272—22no—
151#272—42no—
152C2×C3×Q8—91no—
153C2×C3×D4—92no—
154C2×C3×D4—91no—
155C22×C3×C4—91no—
156C22×C3×C4—91no—
157C22×C3×C4—91no—
158C22×C3×C4—92no—
159C2×D12—34no—
160C2×D12—38no—
161C2×C4×D3—34no—
162C2×C4×D3—34no—
163C2×C4×D3—34no—
164C2×C4×D3—38no—
165C2×Dic6—34no—
166C3×C4⋊C4—34no—
167C3×C4⋊C4—38no—
168C3×C4⋊C4—34no—
169C3×C4⋊C4—38no—
170C3×C4⋊C4—92no—
171C3×C4⋊C4—92no—
172C3×C4⋊C4C2×C3⋊D312yes—
173C3×C4⋊C4C2×C3⋊D311yescharacteristic
174C3×C4⋊C4C2×C3⋊D312yes—
175C3×C4⋊C4C2×C3⋊D311yescharacteristic
176C3×C22⋊C4—92no—
177C3×C22⋊C4—92no—
178C3×C22⋊C4—92no—
179C3×C22⋊C4—92no—
180C3×C22⋊C4—92no—
181C3×C42—34no—
182C3×C42—91no—
183C3×C42—91no—
184C3×C42—91no—
185C3×C42—92no—
186C3×C42C2×C3⋊D311yescharacteristic
187(C2×D3)⋊C4—38no—
188(C2×D3)⋊C4—38no—
189(C2×D3)⋊C4—38no—
190(C2×D3)⋊C4—38no—
191(C2×D3)⋊C4—38no—
192C4⋊Dic3—38no—
193Dic3⋊C4—38no—
194C4×Dic3—34no—
195C4×Dic3—34no—
196C4×Dic3—34no—
197C4×Dic3—38no—
198C22×C32C23×D314yes—
199C2×C3×D3—38no—
200C2×C3×D3—64no—
201C2×C3×D3—64no—
202C2×C3×D3—68no—
203C2×C3×D3—68no—
204C2×C3×D3—68no—
205C32×C4—24no—
206C32×C4—28no—
207C32×C4—24no—
208C32×C4—28no—
209C32×C4#23918yes—
210C3×Dic3—38no—
211C3×Dic3—68no—
212C3×Dic3—64no—
213C3×Dic3—64no—
214#126—92no—
215#125#27211yescharacteristic, complemented
216#124—91no—
217#123—92no—
218C22⋊Q8—91no—
219#121—91no—
220C4×Q8—91no—
221C4×D4—91no—
222C4×D4—92no—
223#117—92no—
224C2×C42—91no—
225H(3,𝔽3)#52911yescharacteristic, complemented, Sylow
226C23×C3—91no—
227C23×C3—92no—
228C22×D3—34no—
229C22×D3—38no—
230C3×Q8—182no—
231C3×D4—184no—
232C3×D4—182no—
233C2×C3×C4—34no—
234C2×C3×C4—38no—
235C2×C3×C4—34no—
236C2×C3×C4—34no—
237C2×C3×C4—38no—
238C2×C3×C4—91no—
239C2×C3×C4—91no—
240C2×C3×C4—91no—
241C2×C3×C4—92no—
242C2×C3×C4—182no—
243C2×C3×C4—184no—
244C2×C3×C4—182no—
245C2×C3×C4—182no—
246C2×C3×C4C22×C3⋊D311yescharacteristic
247C2×C3×C4C22×C3⋊D311yescharacteristic
248C2×C3×C4C22×C3⋊D312yes—
249C2×C3×C4C22×C3⋊D312yes—
250C2×C3×C4C22×C3⋊D311yescharacteristic
251C2×Dic3—34no—
252C2×Dic3—34no—
253C2×Dic3—34no—
254C2×Dic3—38no—
255D12—68no—
256D12—616no—
257C4×D3—68no—
258C4×D3—616no—
259C4×D3—68no—
260C4×D3—68no—
261Dic6—68no—
262C2×C32C2×#23914yes—
263C2×C32D3×#4114yes—
264C2×C32D3×#4114yes—
265C3×D3—68no—
266C3×D3—128no—
267C2×Q8—91no—
268C2×D4—92no—
269C2×D4—91no—
270C22×C4—91no—
271C22×C4—91no—
272C22×C4—91no—
273C22×C4—92no—
274C4⋊C4—92no—
275C4⋊C4—92no—
276C4⋊C4C2×#27212yes—
277C4⋊C4C2×#27211yescharacteristic
278C4⋊C4C2×#27212yes—
279C4⋊C4C2×#27211yescharacteristic
280C22⋊C4—92no—
281C22⋊C4—92no—
282C22⋊C4—92no—
283C22⋊C4—92no—
284C22⋊C4—92no—
285C42—91no—
286C42—91no—
287C42—91no—
288C42—92no—
289C42C2×#27211yescharacteristic
290C22×C3—34no—
291C22×C3—92no—
292C22×C3—181no—
293C22×C3—181no—
294C22×C3—182no—
295C22×C3—182no—
296C22×C3—182no—
297C22×C3C23×C3⋊D311yescharacteristic
298C2×D3—38no—
299C2×D3—64no—
300C2×D3—64no—
301C2×D3—68no—
302C2×D3—68no—
303C2×D3—68no—
304C3×C4—21no—
305C3×C4—22no—
306C3×C4—21no—
307C3×C4—22no—
308C3×C4—38no—
309C3×C4—64no—
310C3×C4—68no—
311C3×C4—64no—
312C3×C4—68no—
313C3×C4—92no—
314C3×C4—181no—
315C3×C4—182no—
316C3×C4—181no—
317C3×C4#383212yes—
318Dic3—38no—
319Dic3—68no—
320Dic3—64no—
321Dic3—64no—
322C32#569114yescomplemented
323C23—91no—
324C23—92no—
325Q8—182no—
326D4—184no—
327D4—182no—
328C2×C4—91no—
329C2×C4—91no—
330C2×C4—91no—
331C2×C4—92no—
332C2×C4—182no—
333C2×C4—184no—
334C2×C4—182no—
335C2×C4—182no—
336C2×C4C22×#27211yescharacteristic
337C2×C4C22×#27211yescharacteristic
338C2×C4C22×#27212yes—
339C2×C4C22×#27212yes—
340C2×C4C22×#27211yescharacteristic
341C2×C3—34no—
342C2×C3—34no—
343C2×C3—34no—
344C2×C3—182no—
345C2×C3—362no—
346C2×C3C2×#383211yescharacteristic
347C2×C3#41×C3⋊D311yescharacteristic
348C2×C3#41×C3⋊D311yescharacteristic
349D3—68no—
350D3—128no—
351C22—92no—
352C22—181no—
353C22—181no—
354C22—182no—
355C22—182no—
356C22—182no—
357C22C23×#27211yescharacteristic
358C4—21no—
359C4—22no—
360C4—21no—
361C4—22no—
362C4—92no—
363C4—181no—
364C4—182no—
365C4—181no—
366C4#8874812yes—
367C3—34no—
368C3#1059616211yescharacteristic
369C2—182no—
370C2—362no—
371C2C2×#8874811yescharacteristic
372C2#41×#27211yescharacteristic
373C2#41×#27211yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#125C2
prime radical#11721763C2
hypercenterC3×#125C3⋊D3
hypocenterH(3,𝔽3)#529
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#125C2semidirect
2. #11721766C2semidirect
3. #11721765C2semidirect
5. #11721762C2semidirect
6. #11721761C2semidirect
7. #11721761C2semidirect
8. #11721760C2semidirect
9. #11721755C2semidirect
10. #11721721C2semidirect
11. C42×#272C2semidirect
36. C32×#125D3semidirect
215. #125#272semidirect
225. H(3,𝔽3)#529semidirect
322. C32#5691semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#117217662
#117217652
#117217632
#117217622
#117217612
#117217602
#117217552
#117217212
C42×#2722
C3×#56913