Finite Group Databasebeta

← #13154220

Subgroups — #13154220

Order 1728.

1056 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#125C211yesmaximal, characteristic, complemented
2#11721766C211yesmaximal, characteristic
3#11721766C211yesmaximal, characteristic
4#11721766C211yesmaximal, characteristic
5#11721764C211yesmaximal, characteristic
6#11721764C211yesmaximal, characteristic
7#11721760C211yesmaximal, characteristic
8#11721758C211yesmaximal, characteristic, complemented
9#11721757C211yesmaximal, characteristic, complemented
10#11721757C211yesmaximal, characteristic, complemented
11#11721757C211yesmaximal, characteristic, complemented
12#11721756C211yesmaximal, characteristic, complemented
13#11721755C211yesmaximal, characteristic, complemented
14#11721722C211yesmaximal, characteristic
15C4×#6229C211yesmaximal, characteristic, complemented
16C3×#5696—34nomaximal
17C2×C4×#272C2211yescharacteristic
18C2×#6229C2211yescharacteristic
19C2×#6229C2211yescharacteristic
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C4⋊C4×H(3,𝔽3)C2211yescharacteristic
22C4⋊C4×H(3,𝔽3)C2211yescharacteristic
23C4⋊C4×H(3,𝔽3)C2211yescharacteristic
24C4⋊C4×H(3,𝔽3)C2211yescharacteristic
25C4⋊C4×H(3,𝔽3)C2211yescharacteristic
26C42×H(3,𝔽3)C2211yescharacteristic
27#88542C2211yescharacteristic
28#88542C2211yescharacteristic
29#88542C2211yescharacteristic
30#88542C2211yescharacteristic
31#88542C2211yescharacteristic
32#88542C2211yescharacteristic
33C4⋊#1084C2211yescharacteristic
34C4⋊#1084C2211yescharacteristic
35C4⋊#1084C2211yescharacteristic
36C4⋊#1084C2211yescharacteristic
37#88540C2211yescharacteristic
38#88540C2211yescharacteristic
39#88540C2211yescharacteristic
40#88540C2211yescharacteristic
41#88540C2211yescharacteristic
42#88540C2211yescharacteristic
43#88540C2211yescharacteristic
44#88540C2211yescharacteristic
45#88540C2211yescharacteristic
46#88540C2211yescharacteristic
47C4×#1084C2211yescharacteristic
48C4×#1084C2211yescharacteristic
49C4×#1084C2211yescharacteristic
50C4×#1084C2211yescharacteristic
51C4×#1084C2211yescharacteristic
52C32×#125D314yescomplemented
53C3×#898—34no—
54C3×#898—34no—
55C3×#898—34no—
56C3×#896—34no—
57C3×#896—34no—
58C3×#892—34no—
59C3×#890—34no—
60C3×#889—34no—
61C3×#889—34no—
62C3×#889—34no—
63C3×#888—34no—
64C3×#887—34no—
65C3×#876—34no—
66C3×C4×Dic6—34no—
67C22×#272C2311yescharacteristic
68C2×C4×H(3,𝔽3)C2311yescharacteristic
69C2×C4×H(3,𝔽3)C2311yescharacteristic
70C2×C4×H(3,𝔽3)C2311yescharacteristic
71C2×C4×H(3,𝔽3)C2311yescharacteristic
72C2×C4×H(3,𝔽3)C2311yescharacteristic
73C2×C4×H(3,𝔽3)C2311yescharacteristic
74C2×C4×H(3,𝔽3)C2311yescharacteristic
75C2×#1084C2311yescharacteristic
76C2×#1084C2311yescharacteristic
77C2×#1084C2311yescharacteristic
78C2×#1084C2311yescharacteristic
79C2×#1084C2311yescharacteristic
80C2×#1084C2311yescharacteristic
81C2×#1084C2311yescharacteristic
82C4×#272—22no—
83#6229—22no—
84#6229—22no—
85C3×#541—91no—
86#5696—34no—
87C2×C3×C4×D3—34no—
88C2×C3×Dic6—34no—
89C2×C3×Dic6—34no—
90C32×C4⋊C4C2×D314yes—
91C32×C4⋊C4C2×D314yes—
92C32×C4⋊C4C2×D314yes—
93C32×C4⋊C4C2×D314yes—
94C32×C4⋊C4C2×D314yes—
95C32×C4⋊C4C2×D314yes—
96C32×C42C2×D314yes—
97C3×(C2×D3)⋊C4—34no—
98C3×(C2×D3)⋊C4—34no—
99C3×(C2×D3)⋊C4—34no—
100C3×(C2×D3)⋊C4—34no—
101C3×(C2×D3)⋊C4—34no—
102C3×(C2×D3)⋊C4—34no—
103C3×C4⋊Dic3—34no—
104C3×C4⋊Dic3—34no—
105C3×C4⋊Dic3—34no—
106C3×C4⋊Dic3—34no—
107C3×Dic3⋊C4—34no—
108C3×Dic3⋊C4—34no—
109C3×Dic3⋊C4—34no—
110C3×Dic3⋊C4—34no—
111C3×Dic3⋊C4—34no—
112C3×Dic3⋊C4—34no—
113C3×Dic3⋊C4—34no—
114C3×Dic3⋊C4—34no—
115C3×Dic3⋊C4—34no—
116C3×Dic3⋊C4—34no—
117C3×C4×Dic3—34no—
118C3×C4×Dic3—34no—
119C3×C4×Dic3—34no—
120C3×C4×Dic3—34no—
121C3×C4×Dic3—34no—
122C22×H(3,𝔽3)C2411yescharacteristic
123C2×#272—21no—
124C2×#272—21no—
125C2×#272—21no—
126C4×H(3,𝔽3)—21no—
127C4×H(3,𝔽3)—21no—
128C4×H(3,𝔽3)—21no—
129C4×H(3,𝔽3)—21no—
130C4×H(3,𝔽3)—21no—
131C4×H(3,𝔽3)—21no—
132C4×H(3,𝔽3)—21no—
133#1084—21no—
134#1084—21no—
135#1084—21no—
136#1084—21no—
137#1084—21no—
138#1084—21no—
139#1084#4112yes—
140C3×C4⋊Q8—91no—
141C3×#126—91no—
142C3×#126—91no—
143C3×#126—91no—
144C3×#126—91no—
145C3×#125—34no—
146C3×#125—91no—
147C3×#125—91no—
148C3×#125—91no—
149C3×#125—91no—
150C3×#125C3⋊D311yescharacteristic
151C3×C22⋊Q8—91no—
152C3×C22⋊Q8—91no—
153C3×C4×Q8—91no—
154C3×C4×Q8—91no—
155C3×#117—91no—
156#898—34no—
157#898—34no—
158#898—34no—
159#896—34no—
160#896—34no—
161#892—34no—
162#890—34no—
163#889—34no—
164#889—34no—
165#889—34no—
166#888—34no—
167#887—34no—
168#876—34no—
169C4×Dic6—34no—
170C22×C3×D3—34no—
171C2×C32×C4C22×D314yes—
172C2×C32×C4C22×D314yes—
173C2×C32×C4C22×D314yes—
174C2×C32×C4C22×D314yes—
175C2×C32×C4C22×D314yes—
176C2×C32×C4C22×D314yes—
177C2×C32×C4C22×D314yes—
178C2×C3×Dic3—34no—
179C2×C3×Dic3—34no—
180C2×C3×Dic3—34no—
181C2×C3×Dic3—34no—
182C2×C3×Dic3—34no—
183C2×C3×Dic3—34no—
184C2×C3×Dic3—34no—
185C3×C4×D3—68no—
186C3×Dic6—68no—
187C3×Dic6—68no—
188#541—91noSylow
189C2×H(3,𝔽3)C2×#4111yescharacteristic
190C2×H(3,𝔽3)2−1+411yescharacteristic
191C2×H(3,𝔽3)2−1+411yescharacteristic
192#272—41no—
193C2×C3×Q8—91no—
194C2×C3×Q8—91no—
195C22×C3×C4—91no—
196C2×C4×D3—34no—
197C2×Dic6—34no—
198C2×Dic6—34no—
199C3×C4⋊C4—34no—
200C3×C4⋊C4—34no—
201C3×C4⋊C4—34no—
202C3×C4⋊C4—34no—
203C3×C4⋊C4—34no—
204C3×C4⋊C4—34no—
205C3×C4⋊C4—91no—
206C3×C4⋊C4—91no—
207C3×C4⋊C4—91no—
208C3×C4⋊C4—91no—
209C3×C4⋊C4—91no—
210C3×C4⋊C4—91no—
211C3×C4⋊C4—91no—
212C3×C4⋊C4—91no—
213C3×C4⋊C4—91no—
214C3×C4⋊C4—91no—
215C3×C4⋊C4—91no—
216C3×C4⋊C4—91no—
217C3×C4⋊C4—91no—
218C3×C4⋊C4—91no—
219C3×C4⋊C4C2×C3⋊D311yescharacteristic
220C3×C4⋊C4C2×C3⋊D311yescharacteristic
221C3×C4⋊C4C2×C3⋊D311yescharacteristic
222C3×C4⋊C4C2×C3⋊D311yescharacteristic
223C3×C4⋊C4C2×C3⋊D311yescharacteristic
224C3×C4⋊C4C2×C3⋊D311yescharacteristic
225C3×C22⋊C4—91no—
226C3×C22⋊C4—91no—
227C3×C22⋊C4—91no—
228C3×C22⋊C4—91no—
229C3×C22⋊C4—91no—
230C3×C22⋊C4—91no—
231C3×C42—34no—
232C3×C42—91no—
233C3×C42—91no—
234C3×C42—91no—
235C3×C42—91no—
236C3×C42—91no—
237C3×C42C2×C3⋊D311yescharacteristic
238(C2×D3)⋊C4—34no—
239(C2×D3)⋊C4—34no—
240(C2×D3)⋊C4—34no—
241(C2×D3)⋊C4—34no—
242(C2×D3)⋊C4—34no—
243(C2×D3)⋊C4—34no—
244C4⋊Dic3—34no—
245C4⋊Dic3—34no—
246C4⋊Dic3—34no—
247C4⋊Dic3—34no—
248Dic3⋊C4—34no—
249Dic3⋊C4—34no—
250Dic3⋊C4—34no—
251Dic3⋊C4—34no—
252Dic3⋊C4—34no—
253Dic3⋊C4—34no—
254Dic3⋊C4—34no—
255Dic3⋊C4—34no—
256Dic3⋊C4—34no—
257Dic3⋊C4—34no—
258C4×Dic3—34no—
259C4×Dic3—34no—
260C4×Dic3—34no—
261C4×Dic3—34no—
262C4×Dic3—34no—
263C22×C32C23×D314yes—
264C2×C3×D3—64no—
265C2×C3×D3—64no—
266C2×C3×D3—64no—
267C32×C4—24no—
268C32×C4—24no—
269C32×C4—24no—
270C32×C4—24no—
271C32×C4—24no—
272C32×C4—24no—
273C32×C4—24no—
274C3×Dic3—38no—
275C3×Dic3—64no—
276C3×Dic3—64no—
277C3×Dic3—64no—
278C3×Dic3—64no—
279C3×Dic3—64no—
280C3×Dic3—64no—
281C4⋊Q8—91no—
282#126—91no—
283#126—91no—
284#126—91no—
285#126—91no—
286#125—91no—
287#125—91no—
288#125—91no—
289#125—91no—
290#125#27211yescharacteristic, complemented
291C22⋊Q8—91no—
292C22⋊Q8—91no—
293C4×Q8—91no—
294C4×Q8—91no—
295#117—91no—
296H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
297C23×C3—91no—
298C22×D3—34no—
299C3×Q8—182no—
300C3×Q8—182no—
301C2×C3×C4—34no—
302C2×C3×C4—34no—
303C2×C3×C4—34no—
304C2×C3×C4—34no—
305C2×C3×C4—34no—
306C2×C3×C4—34no—
307C2×C3×C4—34no—
308C2×C3×C4—91no—
309C2×C3×C4—91no—
310C2×C3×C4—91no—
311C2×C3×C4—91no—
312C2×C3×C4—91no—
313C2×C3×C4—91no—
314C2×C3×C4—91no—
315C2×C3×C4—182no—
316C2×C3×C4C22×C3⋊D311yescharacteristic
317C2×C3×C4C22×C3⋊D311yescharacteristic
318C2×C3×C4C22×C3⋊D311yescharacteristic
319C2×C3×C4C22×C3⋊D311yescharacteristic
320C2×C3×C4C22×C3⋊D311yescharacteristic
321C2×C3×C4C22×C3⋊D311yescharacteristic
322C2×C3×C4C22×C3⋊D311yescharacteristic
323C2×Dic3—34no—
324C2×Dic3—34no—
325C2×Dic3—34no—
326C2×Dic3—34no—
327C2×Dic3—34no—
328C2×Dic3—34no—
329C2×Dic3—34no—
330C4×D3—68no—
331Dic6—68no—
332Dic6—68no—
333C2×C32#100714yes—
334C2×C32D3×#4114yes—
335C2×C32#101014yes—
336C3×D3—124no—
337C2×Q8—91no—
338C2×Q8—91no—
339C22×C4—91no—
340C4⋊C4—91no—
341C4⋊C4—91no—
342C4⋊C4—91no—
343C4⋊C4—91no—
344C4⋊C4—91no—
345C4⋊C4—91no—
346C4⋊C4—91no—
347C4⋊C4—91no—
348C4⋊C4—91no—
349C4⋊C4—91no—
350C4⋊C4—91no—
351C4⋊C4—91no—
352C4⋊C4—91no—
353C4⋊C4—91no—
354C4⋊C4C2×#27211yescharacteristic
355C4⋊C4C2×#27211yescharacteristic
356C4⋊C4C2×#27211yescharacteristic
357C4⋊C4C2×#27211yescharacteristic
358C4⋊C4C2×#27211yescharacteristic
359C4⋊C4C2×#27211yescharacteristic
360C22⋊C4—91no—
361C22⋊C4—91no—
362C22⋊C4—91no—
363C22⋊C4—91no—
364C22⋊C4—91no—
365C22⋊C4—91no—
366C42—91no—
367C42—91no—
368C42—91no—
369C42—91no—
370C42—91no—
371C42C2×#27211yescharacteristic
372C22×C3—34no—
373C22×C3—181no—
374C22×C3—181no—
375C22×C3—181no—
376C22×C3C23×C3⋊D311yescharacteristic
377C2×D3—64no—
378C2×D3—64no—
379C2×D3—64no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—64no—
388C3×C4—64no—
389C3×C4—64no—
390C3×C4—64no—
391C3×C4—64no—
392C3×C4—64no—
393C3×C4—64no—
394C3×C4—92no—
395C3×C4—181no—
396C3×C4—181no—
397C3×C4—181no—
398C3×C4—181no—
399C3×C4—181no—
400C3×C4—181no—
401Dic3—38no—
402Dic3—64no—
403Dic3—64no—
404Dic3—64no—
405Dic3—64no—
406Dic3—64no—
407Dic3—64no—
408C32#569614yescomplemented
409C23—91no—
410Q8—182no—
411Q8—182no—
412C2×C4—91no—
413C2×C4—91no—
414C2×C4—91no—
415C2×C4—91no—
416C2×C4—91no—
417C2×C4—91no—
418C2×C4—91no—
419C2×C4—182no—
420C2×C4C22×#27211yescharacteristic
421C2×C4C22×#27211yescharacteristic
422C2×C4C22×#27211yescharacteristic
423C2×C4C22×#27211yescharacteristic
424C2×C4C22×#27211yescharacteristic
425C2×C4C22×#27211yescharacteristic
426C2×C4C22×#27211yescharacteristic
427C2×C3—34no—
428C2×C3—34no—
429C2×C3—34no—
430C2×C3—361no—
431C2×C3#6436411yescharacteristic
432C2×C3#41×C3⋊D311yescharacteristic
433C2×C3#6436711yescharacteristic
434D3—124no—
435C22—181no—
436C22—181no—
437C22—181no—
438C22C23×#27211yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—92no—
447C4—181no—
448C4—181no—
449C4—181no—
450C4—181no—
451C4—181no—
452C4—181no—
453C3—34no—
454C3#1059616711yescharacteristic
455C2—361no—
456C2#1172267011yescharacteristic
457C2#41×#27211yescharacteristic
458C2#1172267611yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#125C2
prime radicalC22×#272C23
hypercenterC3×#125C3⋊D3
hypocenterH(3,𝔽3)#541
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
8. #11721758C2semidirect
9. #11721757C2semidirect
10. #11721757C2semidirect
11. #11721757C2semidirect
12. #11721756C2semidirect
13. #11721755C2semidirect
15. C4×#6229C2semidirect
52. C32×#125D3semidirect
290. #125#272semidirect
296. H(3,𝔽3)#541semidirect
408. C32#5696semidirect

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
#117217642
#117217602
#117217582
#117217572
#117217562
#117217552
#117217222
C4×#62292
C3×#56963