Finite Group Databasebeta

← #13153889

Subgroups — #13153889

Order 1728.

1166 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
2C4×#6235C211yesmaximal, characteristic
3#11721824C211yesmaximal, characteristic, complemented
4#11721765C211yesmaximal, characteristic, complemented
5#11721760C211yesmaximal, characteristic, complemented
6#11721756C211yesmaximal, characteristic, complemented
7#11721755C211yesmaximal, characteristic, complemented
8#11721741C212yesmaximal
9#11721735C212yesmaximal, complemented
10#11721722C212yesmaximal, complemented
11C4×#6229C212yesmaximal, complemented
12C3×#5533—34nomaximal
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C2×#6235C2211yescharacteristic
15C2×C4×#272C2211yescharacteristic
16C2×#6229C2212yes—
17C2×#6229C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20C22⋊C4×H(3,𝔽3)C2212yes—
21C42×H(3,𝔽3)C2212yes—
22#88552C2211yescharacteristic
23#88552C2212yes—
24#88542C2212yes—
25#88542C2212yes—
26#88542C2211yescharacteristic
27C4⋊#1084C2212yes—
28C4⋊#1084C2211yescharacteristic
29#88540C2212yes—
30#88540C2212yes—
31#88540C2212yes—
32#88540C2211yescharacteristic
33C4×#1084C2212yes—
34C4×#1084C2212yes—
35C4×#1084C2211yescharacteristic
36C32×#117D314yescomplemented
37C3×C4×#67—34no—
38C3×#924—34no—
39C3×#897—34no—
40C3×#892—34no—
41C3×#888—34no—
42C3×#887—34no—
43C3×#885—38no—
44C3×#879—38no—
45C3×#876—38no—
46C3×C4×Dic6—38no—
47C23×H(3,𝔽3)C2311yescharacteristic
48C22×#272C2311yescharacteristic
49C2×C4×H(3,𝔽3)—22no—
50C2×C4×H(3,𝔽3)C2312yes—
51C2×C4×H(3,𝔽3)C2311yescharacteristic
52C2×C4×H(3,𝔽3)C2312yes—
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54#6235—22no—
55C2×#1084C2312yes—
56C2×#1084C2312yes—
57C2×#1084C2311yescharacteristic
58C2×#1084C2311yescharacteristic
59C2×#1084C2311yescharacteristic
60C4×#272—22no—
61#6229—22no—
62#6229—24no—
63C3×#556—91no—
64#5533—34no—
65C22×C32×C4C2×D314yes—
66C2×C3×#67—34no—
67C2×C3×C4×D3—34no—
68C2×C3×Dic6—38no—
69C2×C3×Dic6—34no—
70C32×C4⋊C4C2×D314yes—
71C32×C4⋊C4C2×D314yes—
72C32×C22⋊C4C2×D318yes—
73C32×C42C2×D318yes—
74C3×#217—34no—
75C3×#217—38no—
76C3×(C2×D3)⋊C4—38no—
77C3×(C2×D3)⋊C4—38no—
78C3×(C2×D3)⋊C4—34no—
79C3×C4⋊Dic3—38no—
80C3×C4⋊Dic3—34no—
81C3×Dic3⋊C4—38no—
82C3×Dic3⋊C4—38no—
83C3×Dic3⋊C4—38no—
84C3×Dic3⋊C4—34no—
85C3×C4×Dic3—38no—
86C3×C4×Dic3—38no—
87C3×C4×Dic3—34no—
88C22×H(3,𝔽3)—21no—
89C22×H(3,𝔽3)—21no—
90C22×H(3,𝔽3)—21no—
91C22×H(3,𝔽3)C2411yescharacteristic
92C2×#272—21no—
93C2×#272—21no—
94C2×#272—21no—
95C4×H(3,𝔽3)—22no—
96C4×H(3,𝔽3)—21no—
97C4×H(3,𝔽3)—22no—
98C4×H(3,𝔽3)#4112yes—
99#1084—22no—
100#1084—22no—
101#1084—21no—
102#1084—21no—
103#1084#4112yes—
104C3×C4⋊Q8—91no—
105C3×#126—92no—
106C3×#126—92no—
107C3×#124—92no—
108C3×C22⋊Q8—91no—
109C3×C22⋊Q8—91no—
110C3×C4×Q8—91no—
111C3×C4×Q8—92no—
112C3×C4×D4—91no—
113C3×#117—34no—
114C3×#117—91no—
115C3×#117C3⋊D311yescharacteristic
116C4×#67—34no—
117#924—34no—
118#897—34no—
119#892—34no—
120#888—34no—
121#887—34no—
122#885—38no—
123#879—38no—
124#876—38no—
125C4×Dic6—38no—
126C23×C32C22×D314yes—
127C22×C3×D3—34no—
128C2×C32×C4—28no—
129C2×C32×C4C22×D318yes—
130C2×C32×C4C22×D314yes—
131C2×C32×C4C22×D318yes—
132C2×C32×C4C22×D314yes—
133C3×#67—68no—
134C2×C3×Dic3—34no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—34no—
137C2×C3×Dic3—38no—
138C2×C3×Dic3—38no—
139C3×C4×D3—68no—
140C3×Dic6—616no—
141C3×Dic6—68no—
142#556—91noSylow
143C2×H(3,𝔽3)—41no—
144C2×H(3,𝔽3)C2×#4111yescharacteristic
145C2×H(3,𝔽3)C2×#4111yescharacteristic
146C2×H(3,𝔽3)2−1+411yescharacteristic
147#272—41no—
148C2×C3×Q8—91no—
149C2×C3×Q8—92no—
150C2×C3×D4—91no—
151C22×C3×C4—34no—
152C22×C3×C4—91no—
153C22×C3×C4C2×C3⋊D311yescharacteristic
154C2×#67—34no—
155C2×C4×D3—34no—
156C2×Dic6—38no—
157C2×Dic6—34no—
158C3×C4⋊C4—34no—
159C3×C4⋊C4—34no—
160C3×C4⋊C4—91no—
161C3×C4⋊C4—92no—
162C3×C4⋊C4—92no—
163C3×C4⋊C4—92no—
164C3×C4⋊C4—91no—
165C3×C4⋊C4—92no—
166C3×C4⋊C4C2×C3⋊D311yescharacteristic
167C3×C4⋊C4C2×C3⋊D311yescharacteristic
168C3×C22⋊C4—38no—
169C3×C22⋊C4—91no—
170C3×C22⋊C4—92no—
171C3×C22⋊C4—92no—
172C3×C22⋊C4—91no—
173C3×C22⋊C4—92no—
174C3×C22⋊C4C2×C3⋊D312yes—
175C3×C42—38no—
176C3×C42—92no—
177C3×C42—91no—
178C3×C42—92no—
179C3×C42C2×C3⋊D312yes—
180#217—34no—
181#217—38no—
182(C2×D3)⋊C4—34no—
183(C2×D3)⋊C4—38no—
184(C2×D3)⋊C4—38no—
185C4⋊Dic3—38no—
186C4⋊Dic3—34no—
187Dic3⋊C4—34no—
188Dic3⋊C4—38no—
189Dic3⋊C4—38no—
190Dic3⋊C4—38no—
191C4×Dic3—38no—
192C4×Dic3—38no—
193C4×Dic3—34no—
194C22×C32—24no—
195C22×C32—24no—
196C22×C32—24no—
197C22×C32C23×D314yes—
198C2×C3×D3—64no—
199C2×C3×D3—64no—
200C2×C3×D3—64no—
201C32×C4—24no—
202C32×C4—28no—
203C32×C4—28no—
204C32×C4#23518yes—
205C3×Dic3—38no—
206C3×Dic3—68no—
207C3×Dic3—64no—
208C3×Dic3—68no—
209C3×Dic3—64no—
210C4⋊Q8—91no—
211#126—92no—
212#126—92no—
213#124—92no—
214C22⋊Q8—91no—
215C22⋊Q8—91no—
216C4×Q8—91no—
217C4×Q8—92no—
218C4×D4—91no—
219#117—91no—
220#117#27211yescharacteristic, complemented
221H(3,𝔽3)#55611yescharacteristic, complemented, Sylow
222C23×C3—34no—
223C23×C3—91no—
224C23×C3C22×C3⋊D311yescharacteristic
225C22×D3—34no—
226C3×Q8—184no—
227C3×Q8—182no—
228C3×D4—182no—
229C2×C3×C4—22no—
230C2×C3×C4—34no—
231C2×C3×C4—38no—
232C2×C3×C4—34no—
233C2×C3×C4—38no—
234C2×C3×C4—68no—
235C2×C3×C4—91no—
236C2×C3×C4—91no—
237C2×C3×C4—91no—
238C2×C3×C4—92no—
239C2×C3×C4—92no—
240C2×C3×C4—182no—
241C2×C3×C4C22×C3⋊D312yes—
242C2×C3×C4C22×C3⋊D311yescharacteristic
243C2×C3×C4C22×C3⋊D312yes—
244C2×C3×C4C22×C3⋊D311yescharacteristic
245#67—68no—
246C2×Dic3—34no—
247C2×Dic3—34no—
248C2×Dic3—34no—
249C2×Dic3—38no—
250C2×Dic3—38no—
251C4×D3—68no—
252Dic6—616no—
253Dic6—68no—
254C2×C32—44no—
255C2×C32C2×#23514yes—
256C2×C32D3×#4114yes—
257C2×C32#101014yes—
258C3×D3—124no—
259C2×Q8—91no—
260C2×Q8—92no—
261C2×D4—91no—
262C22×C4—91no—
263C22×C4C2×#27211yescharacteristic
264C4⋊C4—91no—
265C4⋊C4—91no—
266C4⋊C4—92no—
267C4⋊C4—92no—
268C4⋊C4—92no—
269C4⋊C4—92no—
270C4⋊C4C2×#27211yescharacteristic
271C4⋊C4C2×#27211yescharacteristic
272C22⋊C4—91no—
273C22⋊C4—92no—
274C22⋊C4—92no—
275C22⋊C4—91no—
276C22⋊C4—92no—
277C22⋊C4C2×#27212yes—
278C42—92no—
279C42—91no—
280C42—92no—
281C42C2×#27212yes—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—34no—
286C22×C3—64no—
287C22×C3—64no—
288C22×C3—64no—
289C22×C3—181no—
290C22×C3—181no—
291C22×C3—181no—
292C22×C3C23×C3⋊D311yescharacteristic
293C2×D3—64no—
294C2×D3—64no—
295C2×D3—64no—
296C3×C4—21no—
297C3×C4—22no—
298C3×C4—22no—
299C3×C4—38no—
300C3×C4—68no—
301C3×C4—64no—
302C3×C4—68no—
303C3×C4—92no—
304C3×C4—181no—
305C3×C4—182no—
306C3×C4—182no—
307C3×C4—181no—
308C3×C4#382812yes—
309Dic3—38no—
310Dic3—68no—
311Dic3—68no—
312Dic3—64no—
313Dic3—64no—
314C32#553314yescomplemented
315C23—91no—
316C23C22×#27211yescharacteristic
317Q8—184no—
318Q8—182no—
319D4—182no—
320C2×C4—22no—
321C2×C4—91no—
322C2×C4—91no—
323C2×C4—91no—
324C2×C4—92no—
325C2×C4—92no—
326C2×C4—182no—
327C2×C4C22×#27212yes—
328C2×C4C22×#27211yescharacteristic
329C2×C4C22×#27212yes—
330C2×C4C22×#27211yescharacteristic
331C2×C3—34no—
332C2×C3—34no—
333C2×C3—34no—
334C2×C3—41no—
335C2×C3—124no—
336C2×C3—361no—
337C2×C3C2×#382811yescharacteristic
338C2×C3#41×C3⋊D311yescharacteristic
339C2×C3#6436711yescharacteristic
340D3—124no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—181no—
345C22—181no—
346C22—181no—
347C22C23×#27211yescharacteristic
348C4—21no—
349C4—22no—
350C4—22no—
351C4—92no—
352C4—181no—
353C4—182no—
354C4—182no—
355C4—181no—
356C4#8874012yes—
357C3—34no—
358C3#1059600411yescharacteristic
359C2—41no—
360C2—361no—
361C2C2×#8874011yescharacteristic
362C2#41×#27211yescharacteristic
363C2#1172267611yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#117C2
prime radicalC2×#6235C22
hypercenterC3×#117C3⋊D3
hypocenterH(3,𝔽3)#556
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#117C2semidirect
3. #11721824C2semidirect
4. #11721765C2semidirect
5. #11721760C2semidirect
6. #11721756C2semidirect
7. #11721755C2semidirect
9. #11721735C2semidirect
10. #11721722C2semidirect
11. C4×#6229C2semidirect
36. C32×#117D3semidirect
220. #117#272semidirect
221. H(3,𝔽3)#556semidirect
314. C32#5533semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1172
C4×#62352
#117218242
#117217652
#117217602
#117217562
#117217552
#117217412
#117217352
#117217222
C4×#62292
C3×#55333