Finite Group Databasebeta

← #13153903

Subgroups — #13153903

Order 1728.

1188 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
2#11721830C211yesmaximal, characteristic
3#11721824C211yesmaximal, characteristic, complemented
4#11721766C212yesmaximal, complemented
5#11721764C212yesmaximal, complemented
6#11721741C212yesmaximal, complemented
7#11721739C212yesmaximal, complemented
8#11721721C211yesmaximal, characteristic, complemented
9C4×C4⋊#272C211yesmaximal, characteristic, complemented
10#11721715C211yesmaximal, characteristic, complemented
11C4×#6229C211yesmaximal, characteristic, complemented
12C3×#5534—34nomaximal
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C2×#6235C2212yes—
15C2×C4⋊#272C2211yescharacteristic
16C2×C4×#272C2212yes—
17C2×#6229C2211yescharacteristic
18C2×#6229C2212yes—
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C22⋊C4×H(3,𝔽3)C2211yescharacteristic
22C22⋊C4×H(3,𝔽3)C2211yescharacteristic
23C42×H(3,𝔽3)C2211yescharacteristic
24C42×H(3,𝔽3)C2211yescharacteristic
25#88552C2212yes—
26#88542C2212yes—
27#88542C2212yes—
28#88542C2212yes—
29#88542C2212yes—
30C4⋊#1084C2212yes—
31C4⋊#1084C2211yescharacteristic
32C4⋊#1084C2211yescharacteristic
33#88540C2212yes—
34#88540C2212yes—
35C4×#1084C2212yes—
36C32×#117D314yescomplemented
37C3×#930—34no—
38C3×#924—34no—
39C3×#898—38no—
40C3×#896—38no—
41C3×#885—38no—
42C3×#883—38no—
43C3×#875—34no—
44C3×C4×D12—34no—
45C3×#869—34no—
46C3×C4×Dic6—34no—
47C23×H(3,𝔽3)C2311yescharacteristic
48C22×#272C2312yes—
49C2×C4×H(3,𝔽3)—22no—
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51C2×C4×H(3,𝔽3)C2311yescharacteristic
52C2×C4×H(3,𝔽3)C2311yescharacteristic
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56#6235—42no—
57C2×#1084C2312yes—
58C2×#1084C2312yes—
59C2×#1084C2312yes—
60C4⋊#272—22no—
61C4×#272—42no—
62#6229—22no—
63#6229—42no—
64C3×#542—91no—
65#5534—34no—
66C22×C32×C4C2×D314yes—
67C2×C3×#67—38no—
68C2×C3×D12—34no—
69C2×C3×C4×D3—38no—
70C2×C3×Dic6—34no—
71C2×C3×Dic6—38no—
72C32×C4⋊C4C2×D314yes—
73C32×C4⋊C4C2×D314yes—
74C32×C22⋊C4C2×D314yes—
75C32×C22⋊C4C2×D314yes—
76C32×C42C2×D314yes—
77C32×C42C2×D314yes—
78C3×#217—38no—
79C3×(C2×D3)⋊C4—38no—
80C3×(C2×D3)⋊C4—38no—
81C3×(C2×D3)⋊C4—38no—
82C3×(C2×D3)⋊C4—38no—
83C3×C4⋊Dic3—38no—
84C3×C4⋊Dic3—34no—
85C3×C4⋊Dic3—34no—
86C3×Dic3⋊C4—38no—
87C3×Dic3⋊C4—38no—
88C3×C4×Dic3—38no—
89C22×H(3,𝔽3)—21no—
90C22×H(3,𝔽3)—21no—
91C22×H(3,𝔽3)—21no—
92C22×H(3,𝔽3)C2411yescharacteristic
93C2×#272—22no—
94C2×#272—22no—
95C2×#272—22no—
96C4×H(3,𝔽3)—21no—
97C4×H(3,𝔽3)—21no—
98C4×H(3,𝔽3)—21no—
99C4×H(3,𝔽3)—21no—
100C4×H(3,𝔽3)—21no—
101C4×H(3,𝔽3)#4112yes—
102#1084—22no—
103#1084—22no—
104#1084—22no—
105C3×C4⋊Q8—91no—
106C3×#126—92no—
107C3×#124—92no—
108C3×#124—91no—
109C3×#123—92no—
110C3×C22⋊Q8—91no—
111C3×C22⋊Q8—92no—
112C3×#121—91no—
113C3×C4×Q8—91no—
114C3×C4×D4—91no—
115C3×#117—34no—
116C3×#117C3⋊D311yescharacteristic
117#930—34no—
118#924—34no—
119#898—38no—
120#896—38no—
121#885—38no—
122#883—38no—
123#875—34no—
124C4×D12—34no—
125#869—34no—
126C4×Dic6—34no—
127C23×C32C22×D314yes—
128C22×C3×D3—38no—
129C2×C32×C4—28no—
130C2×C32×C4C22×D314yes—
131C2×C32×C4C22×D314yes—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D314yes—
134C2×C32×C4C22×D314yes—
135C2×C32×C4C22×D314yes—
136C3×#67—128no—
137C2×C3×Dic3—38no—
138C2×C3×Dic3—38no—
139C2×C3×Dic3—38no—
140C3×D12—68no—
141C3×C4×D3—128no—
142C3×Dic6—68no—
143C3×Dic6—128no—
144#542—91noSylow
145C2×H(3,𝔽3)—41no—
146C2×H(3,𝔽3)C2×#4111yescharacteristic
147C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
148C2×H(3,𝔽3)2−1+411yescharacteristic
149#272—42no—
150C2×C3×Q8—91no—
151C2×C3×Q8—92no—
152C2×C3×D4—92no—
153C2×C3×D4—91no—
154C22×C3×C4—34no—
155C22×C3×C4—92no—
156C22×C3×C4C2×C3⋊D311yescharacteristic
157C2×#67—38no—
158C2×D12—34no—
159C2×C4×D3—38no—
160C2×Dic6—34no—
161C2×Dic6—38no—
162C3×C4⋊C4—34no—
163C3×C4⋊C4—34no—
164C3×C4⋊C4—92no—
165C3×C4⋊C4—92no—
166C3×C4⋊C4—91no—
167C3×C4⋊C4—91no—
168C3×C4⋊C4—92no—
169C3×C4⋊C4C2×C3⋊D311yescharacteristic
170C3×C4⋊C4C2×C3⋊D311yescharacteristic
171C3×C22⋊C4—34no—
172C3×C22⋊C4—34no—
173C3×C22⋊C4—92no—
174C3×C22⋊C4—92no—
175C3×C22⋊C4—92no—
176C3×C22⋊C4—92no—
177C3×C22⋊C4—92no—
178C3×C22⋊C4C2×C3⋊D311yescharacteristic
179C3×C22⋊C4C2×C3⋊D311yescharacteristic
180C3×C42—34no—
181C3×C42—34no—
182C3×C42—92no—
183C3×C42C2×C3⋊D311yescharacteristic
184C3×C42C2×C3⋊D311yescharacteristic
185#217—38no—
186(C2×D3)⋊C4—38no—
187(C2×D3)⋊C4—38no—
188(C2×D3)⋊C4—38no—
189(C2×D3)⋊C4—38no—
190C4⋊Dic3—34no—
191C4⋊Dic3—34no—
192C4⋊Dic3—38no—
193Dic3⋊C4—38no—
194Dic3⋊C4—38no—
195C4×Dic3—38no—
196C22×C32—24no—
197C22×C32—24no—
198C22×C32—24no—
199C22×C32C23×D314yes—
200C2×C3×D3—68no—
201C2×C3×D3—68no—
202C2×C3×D3—68no—
203C32×C4—24no—
204C32×C4—24no—
205C32×C4—24no—
206C32×C4—24no—
207C32×C4—24no—
208C32×C4#23518yes—
209C3×Dic3—68no—
210C3×Dic3—68no—
211C3×Dic3—68no—
212C4⋊Q8—91no—
213#126—92no—
214#124—92no—
215#124—91no—
216#123—92no—
217C22⋊Q8—91no—
218C22⋊Q8—92no—
219#121—91no—
220C4×Q8—91no—
221C4×D4—91no—
222#117#27211yescharacteristic, complemented
223H(3,𝔽3)#54211yescharacteristic, complemented, Sylow
224C23×C3—34no—
225C23×C3—92no—
226C23×C3C22×C3⋊D311yescharacteristic
227C22×D3—38no—
228C3×Q8—182no—
229C3×Q8—362no—
230C3×D4—182no—
231C3×D4—362no—
232C2×C3×C4—22no—
233C2×C3×C4—34no—
234C2×C3×C4—34no—
235C2×C3×C4—34no—
236C2×C3×C4—34no—
237C2×C3×C4—34no—
238C2×C3×C4—34no—
239C2×C3×C4—68no—
240C2×C3×C4—92no—
241C2×C3×C4—92no—
242C2×C3×C4—92no—
243C2×C3×C4—362no—
244C2×C3×C4C22×C3⋊D311yescharacteristic
245C2×C3×C4C22×C3⋊D311yescharacteristic
246C2×C3×C4C22×C3⋊D311yescharacteristic
247C2×C3×C4C22×C3⋊D311yescharacteristic
248C2×C3×C4C22×C3⋊D311yescharacteristic
249C2×C3×C4C22×C3⋊D311yescharacteristic
250#67—128no—
251C2×Dic3—38no—
252C2×Dic3—38no—
253C2×Dic3—38no—
254D12—68no—
255C4×D3—128no—
256Dic6—68no—
257Dic6—128no—
258C2×C32—44no—
259C2×C32C2×#23514yes—
260C2×C32#100914yes—
261C2×C32#101014yes—
262C3×D3—128no—
263C2×Q8—91no—
264C2×Q8—92no—
265C2×D4—92no—
266C2×D4—91no—
267C22×C4—92no—
268C22×C4C2×#27211yescharacteristic
269C4⋊C4—91no—
270C4⋊C4—91no—
271C4⋊C4—92no—
272C4⋊C4—92no—
273C4⋊C4—92no—
274C4⋊C4C2×#27211yescharacteristic
275C4⋊C4C2×#27211yescharacteristic
276C22⋊C4—92no—
277C22⋊C4—92no—
278C22⋊C4—92no—
279C22⋊C4—92no—
280C22⋊C4—92no—
281C22⋊C4C2×#27211yescharacteristic
282C22⋊C4C2×#27211yescharacteristic
283C42—92no—
284C42C2×#27211yescharacteristic
285C42C2×#27211yescharacteristic
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—34no—
290C22×C3—64no—
291C22×C3—64no—
292C22×C3—64no—
293C22×C3—182no—
294C22×C3—182no—
295C22×C3—182no—
296C22×C3C23×C3⋊D311yescharacteristic
297C2×D3—68no—
298C2×D3—68no—
299C2×D3—68no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—38no—
306C3×C4—64no—
307C3×C4—64no—
308C3×C4—64no—
309C3×C4—64no—
310C3×C4—64no—
311C3×C4—182no—
312C3×C4—182no—
313C3×C4—182no—
314C3×C4#382812yes—
315Dic3—68no—
316Dic3—68no—
317Dic3—68no—
318C32#553414yescomplemented
319C23—92no—
320C23C22×#27211yescharacteristic
321Q8—182no—
322Q8—362no—
323D4—182no—
324D4—362no—
325C2×C4—22no—
326C2×C4—92no—
327C2×C4—92no—
328C2×C4—92no—
329C2×C4—362no—
330C2×C4C22×#27211yescharacteristic
331C2×C4C22×#27211yescharacteristic
332C2×C4C22×#27211yescharacteristic
333C2×C4C22×#27211yescharacteristic
334C2×C4C22×#27211yescharacteristic
335C2×C4C22×#27211yescharacteristic
336C2×C3—34no—
337C2×C3—34no—
338C2×C3—34no—
339C2×C3—41no—
340C2×C3—124no—
341C2×C3—362no—
342C2×C3C2×#382811yescharacteristic
343C2×C3#6436611yescharacteristic
344C2×C3#6436711yescharacteristic
345D3—128no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—182no—
350C22—182no—
351C22—182no—
352C22C23×#27211yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—182no—
359C4—182no—
360C4—182no—
361C4#8874012yes—
362C3—34no—
363C3#1059600511yescharacteristic
364C2—41no—
365C2—362no—
366C2C2×#8874011yescharacteristic
367C2#1172267511yescharacteristic
368C2#1172267611yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#117C2
prime radical#11721830C2
hypercenterC3×#117C3⋊D3
hypocenterH(3,𝔽3)#542
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. #11721766C2semidirect
5. #11721764C2semidirect
6. #11721741C2semidirect
7. #11721739C2semidirect
8. #11721721C2semidirect
9. C4×C4⋊#272C2semidirect
10. #11721715C2semidirect
11. C4×#6229C2semidirect
36. C32×#117D3semidirect
222. #117#272semidirect
223. H(3,𝔽3)#542semidirect
318. C32#5534semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1172
#117218302
#117218242
#117217662
#117217642
#117217412
#117217392
#117217212
C4×C4⋊#2722
#117217152
C4×#62292
C3×#55343