Finite Group Databasebeta

← #13153902

Subgroups — #13153902

Order 1728.

1309 conjugacy classes of subgroups.

The subgroup lattice has 395 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, complemented
3C4×#6235C211yesmaximal, characteristic, complemented
4#11721763C211yesmaximal, characteristic, complemented
5#11721761C211yesmaximal, characteristic, complemented
6C4⋊C4×#272C211yesmaximal, characteristic, complemented
7#11721758C211yesmaximal, characteristic, complemented
8#11721740C212yesmaximal, complemented
9#11721739C212yesmaximal, complemented
10#11721722C212yesmaximal, complemented
11C4×C4⋊#272C212yesmaximal, complemented
12C3×#5530—34nomaximal
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C2×#6235C2211yescharacteristic
15C2×#6235C2212yes—
16C2×C4⋊#272C2212yes—
17C2×C4⋊#272C2211yescharacteristic
18C2×C4×#272C2211yescharacteristic
19C2×C4×#272C2212yes—
20C2×C4×#272C2212yes—
21C4⋊C4×H(3,𝔽3)C2211yescharacteristic
22C4⋊C4×H(3,𝔽3)C2211yescharacteristic
23C22⋊C4×H(3,𝔽3)C2212yes—
24C42×H(3,𝔽3)C2212yes—
25#88552C2211yescharacteristic
26#88542C2212yes—
27#88542C2212yes—
28#88542C2211yescharacteristic
29#88542C2212yes—
30C4⋊#1084C2212yes—
31C4⋊#1084C2211yescharacteristic
32#88540C2212yes—
33#88540C2212yes—
34#88540C2211yescharacteristic
35C4×#1084C2211yescharacteristic
36C32×#117D314yescomplemented
37C3×#930—34no—
38C3×C4×#67—34no—
39C3×#895—34no—
40C3×#893—34no—
41C3×D3×C4⋊C4—34no—
42C3×#890—34no—
43C3×#884—38no—
44C3×#883—38no—
45C3×#876—38no—
46C3×C4×D12—38no—
47C23×H(3,𝔽3)C2311yescharacteristic
48C22×#272C2312yes—
49C22×#272C2311yescharacteristic
50C2×C4×H(3,𝔽3)—22no—
51C2×C4×H(3,𝔽3)C2312yes—
52C2×C4×H(3,𝔽3)C2312yes—
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55#6235—22no—
56#6235—42no—
57C2×#1084C2311yescharacteristic
58C2×#1084C2312yes—
59C2×#1084C2311yescharacteristic
60C2×#1084C2311yescharacteristic
61C4⋊#272—24no—
62C4⋊#272—22no—
63C4×#272—22no—
64C4×#272—42no—
65C4×#272—42no—
66C3×#553—91no—
67#5530—34no—
68C22×C32×C4C2×D314yes—
69C2×C3×#67—34no—
70C2×C3×#67—38no—
71C2×C3×D12—38no—
72C2×C3×D12—34no—
73C2×C3×C4×D3—34no—
74C2×C3×C4×D3—38no—
75C2×C3×C4×D3—38no—
76C32×C4⋊C4C2×D314yes—
77C32×C4⋊C4C2×D314yes—
78C32×C22⋊C4C2×D318yes—
79C32×C42C2×D318yes—
80C3×#217—34no—
81C3×(C2×D3)⋊C4—38no—
82C3×(C2×D3)⋊C4—38no—
83C3×(C2×D3)⋊C4—34no—
84C3×(C2×D3)⋊C4—38no—
85C3×C4⋊Dic3—38no—
86C3×C4⋊Dic3—34no—
87C3×Dic3⋊C4—38no—
88C3×Dic3⋊C4—38no—
89C3×Dic3⋊C4—34no—
90C3×C4×Dic3—34no—
91C22×H(3,𝔽3)—21no—
92C22×H(3,𝔽3)—21no—
93C22×H(3,𝔽3)—21no—
94C22×H(3,𝔽3)C2411yescharacteristic
95C2×#272—21no—
96C2×#272—21no—
97C2×#272—22no—
98C2×#272—22no—
99C2×#272—22no—
100C2×#272#4112yes—
101C4×H(3,𝔽3)—22no—
102C4×H(3,𝔽3)—21no—
103C4×H(3,𝔽3)—22no—
104C4×H(3,𝔽3)#4112yes—
105#1084—21no—
106#1084—21no—
107#1084—21no—
108#1084—22no—
109C3×#126—92no—
110C3×#125—91no—
111C3×#123—92no—
112C3×#121—91no—
113C3×#121—91no—
114C3×#121—92no—
115C3×C4×D4—91no—
116C3×C4×D4—91no—
117C3×C4×D4—92no—
118C3×#117—34no—
119C3×#117C3⋊D311yescharacteristic
120C2×C3×C4⋊C4—91no—
121#930—34no—
122C4×#67—34no—
123#895—34no—
124#893—34no—
125D3×C4⋊C4—34no—
126#890—34no—
127#884—38no—
128#883—38no—
129#876—38no—
130C4×D12—38no—
131C23×C32C22×D314yes—
132C22×C3×D3—34no—
133C22×C3×D3—38no—
134C2×C32×C4—28no—
135C2×C32×C4C22×D318yes—
136C2×C32×C4C22×D314yes—
137C2×C32×C4C22×D318yes—
138C2×C32×C4C22×D314yes—
139C3×#67—68no—
140C3×#67—128no—
141C2×C3×Dic3—34no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—34no—
144C2×C3×Dic3—38no—
145C3×D12—616no—
146C3×D12—68no—
147C3×C4×D3—68no—
148C3×C4×D3—128no—
149C3×C4×D3—128no—
150#553—91noSylow
151C2×H(3,𝔽3)—41no—
152C2×H(3,𝔽3)C2×#4111yescharacteristic
153C2×H(3,𝔽3)C2×#4111yescharacteristic
154C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
155#272—22no—
156#272—42no—
157C2×C3×D4—91no—
158C2×C3×D4—92no—
159C2×C3×D4—92no—
160C2×C3×D4—91no—
161C22×C3×C4—34no—
162C22×C3×C4—91no—
163C22×C3×C4—92no—
164C22×C3×C4—92no—
165C22×C3×C4C2×C3⋊D311yescharacteristic
166C2×#67—34no—
167C2×#67—38no—
168C2×D12—38no—
169C2×D12—34no—
170C2×C4×D3—34no—
171C2×C4×D3—38no—
172C2×C4×D3—38no—
173C3×C4⋊C4—34no—
174C3×C4⋊C4—34no—
175C3×C4⋊C4—92no—
176C3×C4⋊C4—91no—
177C3×C4⋊C4—92no—
178C3×C4⋊C4—91no—
179C3×C4⋊C4—92no—
180C3×C4⋊C4C2×C3⋊D311yescharacteristic
181C3×C4⋊C4C2×C3⋊D311yescharacteristic
182C3×C22⋊C4—38no—
183C3×C22⋊C4—91no—
184C3×C22⋊C4—91no—
185C3×C22⋊C4—92no—
186C3×C22⋊C4—92no—
187C3×C22⋊C4—92no—
188C3×C22⋊C4C2×C3⋊D312yes—
189C3×C42—38no—
190C3×C42—91no—
191C3×C42C2×C3⋊D312yes—
192#217—34no—
193(C2×D3)⋊C4—34no—
194(C2×D3)⋊C4—38no—
195(C2×D3)⋊C4—38no—
196(C2×D3)⋊C4—38no—
197C4⋊Dic3—38no—
198C4⋊Dic3—34no—
199Dic3⋊C4—34no—
200Dic3⋊C4—38no—
201Dic3⋊C4—38no—
202C4×Dic3—34no—
203C22×C32—24no—
204C22×C32—24no—
205C22×C32—24no—
206C22×C32C23×D314yes—
207C2×C3×D3—38no—
208C2×C3×D3—64no—
209C2×C3×D3—64no—
210C2×C3×D3—68no—
211C2×C3×D3—68no—
212C2×C3×D3—68no—
213C32×C4—24no—
214C32×C4—28no—
215C32×C4—28no—
216C32×C4#23518yes—
217C3×Dic3—64no—
218C3×Dic3—64no—
219C3×Dic3—64no—
220C3×Dic3—68no—
221#126—92no—
222#125—91no—
223#123—92no—
224#121—91no—
225#121—91no—
226#121—92no—
227C4×D4—91no—
228C4×D4—91no—
229C4×D4—92no—
230#117#27211yescharacteristic, complemented
231C2×C4⋊C4—91no—
232H(3,𝔽3)#55311yescharacteristic, complemented, Sylow
233C23×C3—34no—
234C23×C3—91no—
235C23×C3—92no—
236C23×C3C22×C3⋊D311yescharacteristic
237C22×D3—34no—
238C22×D3—38no—
239C3×D4—184no—
240C3×D4—182no—
241C3×D4—182no—
242C3×D4—362no—
243C2×C3×C4—22no—
244C2×C3×C4—34no—
245C2×C3×C4—38no—
246C2×C3×C4—34no—
247C2×C3×C4—38no—
248C2×C3×C4—68no—
249C2×C3×C4—91no—
250C2×C3×C4—91no—
251C2×C3×C4—91no—
252C2×C3×C4—92no—
253C2×C3×C4—182no—
254C2×C3×C4—362no—
255C2×C3×C4—362no—
256C2×C3×C4C22×C3⋊D311yescharacteristic
257C2×C3×C4C22×C3⋊D312yes—
258C2×C3×C4C22×C3⋊D312yes—
259C2×C3×C4C22×C3⋊D311yescharacteristic
260#67—68no—
261#67—128no—
262C2×Dic3—34no—
263C2×Dic3—34no—
264C2×Dic3—34no—
265C2×Dic3—38no—
266D12—68no—
267D12—616no—
268C4×D3—68no—
269C4×D3—128no—
270C4×D3—128no—
271C2×C32—44no—
272C2×C32C2×#23514yes—
273C2×C32D3×#4114yes—
274C2×C32#100914yes—
275C3×D3—68no—
276C3×D3—128no—
277C2×D4—91no—
278C2×D4—92no—
279C2×D4—92no—
280C2×D4—91no—
281C22×C4—91no—
282C22×C4—92no—
283C22×C4—92no—
284C22×C4C2×#27211yescharacteristic
285C4⋊C4—91no—
286C4⋊C4—92no—
287C4⋊C4—92no—
288C4⋊C4—91no—
289C4⋊C4—92no—
290C4⋊C4C2×#27211yescharacteristic
291C4⋊C4C2×#27211yescharacteristic
292C22⋊C4—91no—
293C22⋊C4—91no—
294C22⋊C4—92no—
295C22⋊C4—92no—
296C22⋊C4—92no—
297C22⋊C4C2×#27212yes—
298C42—91no—
299C42C2×#27212yes—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—34no—
304C22×C3—64no—
305C22×C3—64no—
306C22×C3—64no—
307C22×C3—92no—
308C22×C3—181no—
309C22×C3—181no—
310C22×C3—182no—
311C22×C3—182no—
312C22×C3—182no—
313C22×C3C23×C3⋊D311yescharacteristic
314C2×D3—38no—
315C2×D3—64no—
316C2×D3—64no—
317C2×D3—68no—
318C2×D3—68no—
319C2×D3—68no—
320C3×C4—21no—
321C3×C4—22no—
322C3×C4—22no—
323C3×C4—38no—
324C3×C4—68no—
325C3×C4—68no—
326C3×C4—64no—
327C3×C4—181no—
328C3×C4—181no—
329C3×C4—181no—
330C3×C4—182no—
331C3×C4#382812yes—
332Dic3—64no—
333Dic3—64no—
334Dic3—68no—
335Dic3—64no—
336C32#553014yescomplemented
337C23—91no—
338C23—92no—
339C23C22×#27211yescharacteristic
340D4—184no—
341D4—182no—
342D4—182no—
343D4—362no—
344C2×C4—22no—
345C2×C4—91no—
346C2×C4—91no—
347C2×C4—91no—
348C2×C4—92no—
349C2×C4—182no—
350C2×C4—362no—
351C2×C4—362no—
352C2×C4C22×#27211yescharacteristic
353C2×C4C22×#27212yes—
354C2×C4C22×#27212yes—
355C2×C4C22×#27211yescharacteristic
356C2×C3—34no—
357C2×C3—34no—
358C2×C3—34no—
359C2×C3—41no—
360C2×C3—124no—
361C2×C3—182no—
362C2×C3—362no—
363C2×C3C2×#382811yescharacteristic
364C2×C3#41×C3⋊D311yescharacteristic
365C2×C3#6436611yescharacteristic
366D3—68no—
367D3—128no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—92no—
372C22—181no—
373C22—181no—
374C22—182no—
375C22—182no—
376C22—182no—
377C22C23×#27211yescharacteristic
378C4—21no—
379C4—22no—
380C4—22no—
381C4—181no—
382C4—181no—
383C4—181no—
384C4—182no—
385C4#8874012yes—
386C3—34no—
387C3#1059600111yescharacteristic
388C2—41no—
389C2—182no—
390C2—362no—
391C2C2×#8874011yescharacteristic
392C2#41×#27211yescharacteristic
393C2#1172267511yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#117C2
hypercenterC3×#117C3⋊D3
hypocenterH(3,𝔽3)#553
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
2. #11721830C2semidirect
3. C4×#6235C2semidirect
4. #11721763C2semidirect
5. #11721761C2semidirect
6. C4⋊C4×#272C2semidirect
7. #11721758C2semidirect
8. #11721740C2semidirect
9. #11721739C2semidirect
10. #11721722C2semidirect
11. C4×C4⋊#272C2semidirect
36. C32×#117D3semidirect
230. #117#272semidirect
232. H(3,𝔽3)#553semidirect
336. C32#5530semidirect

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
C4×#62352
#117217632
#117217612
C4⋊C4×#2722
#117217582
#117217402
#117217392
#117217222
C4×C4⋊#2722
C3×#55303