Finite Group Databasebeta

← #13154253

Subgroups — #13154253

Order 1728.

1287 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#126C211yesmaximal, characteristic, complemented
2#11721764C211yesmaximal, characteristic, complemented
3#11721762C211yesmaximal, characteristic, complemented
4#11721761C211yesmaximal, characteristic, complemented
5#11721760C211yesmaximal, characteristic, complemented
6#11721757C211yesmaximal, characteristic, complemented
7#11721755C211yesmaximal, characteristic, complemented
8#11721741C211yesmaximal, characteristic, complemented
9#11721740C211yesmaximal, characteristic
10#11721739C211yesmaximal, characteristic, complemented
11#11721737C211yesmaximal, characteristic, complemented
12#11721737C211yesmaximal, characteristic, complemented
13#11721733C211yesmaximal, characteristic, complemented
14#11721722C211yesmaximal, characteristic, complemented
15C42×#272C211yesmaximal, characteristic, complemented
16C3×#5706—34nomaximal
17C2×#6235C2211yescharacteristic
18C2×#6235C2211yescharacteristic
19C22×#1084C2211yescharacteristic
20C2×C4⋊#272C2211yescharacteristic
21C2×C4×#272C2211yescharacteristic
22C2×C4×#272C2211yescharacteristic
23C2×C4×#272C2211yescharacteristic
24C2×C4×#272C2211yescharacteristic
25C2×#6229C2211yescharacteristic
26C4⋊C4×H(3,𝔽3)C2211yescharacteristic
27C4⋊C4×H(3,𝔽3)C2211yescharacteristic
28C4⋊C4×H(3,𝔽3)C2211yescharacteristic
29C22⋊C4×H(3,𝔽3)C2211yescharacteristic
30C22⋊C4×H(3,𝔽3)C2211yescharacteristic
31C22⋊C4×H(3,𝔽3)C2211yescharacteristic
32C42×H(3,𝔽3)C2211yescharacteristic
33#88552C2211yescharacteristic
34#88542C2211yescharacteristic
35#88542C2211yescharacteristic
36#88542C2211yescharacteristic
37#88542C2211yescharacteristic
38#88542C2211yescharacteristic
39#88542C2211yescharacteristic
40C4⋊#1084C2211yescharacteristic
41#88540C2211yescharacteristic
42#88540C2211yescharacteristic
43#88540C2211yescharacteristic
44#88540C2211yescharacteristic
45#88540C2211yescharacteristic
46#88540C2211yescharacteristic
47C4×#1084C2211yescharacteristic
48C4×#1084C2211yescharacteristic
49C4×#1084C2211yescharacteristic
50C4×#1084C2211yescharacteristic
51C4×#1084C2211yescharacteristic
52C32×#126D314yescomplemented
53C3×#896—34no—
54C3×#894—34no—
55C3×#893—34no—
56C3×#892—34no—
57C3×#889—34no—
58C3×#887—34no—
59C3×#885—34no—
60C3×#884—34no—
61C3×#883—34no—
62C3×#67⋊C4—34no—
63C3×#67⋊C4—34no—
64C3×#877—34no—
65C3×#876—34no—
66C3×C42×D3—34no—
67C23×H(3,𝔽3)C2311yescharacteristic
68C22×#272C2311yescharacteristic
69C22×#272C2311yescharacteristic
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×C4×H(3,𝔽3)C2311yescharacteristic
76#6235—22no—
77#6235—22no—
78C2×#1084—22no—
79C2×#1084C2311yescharacteristic
80C2×#1084C2311yescharacteristic
81C2×#1084C2311yescharacteristic
82C2×#1084C2311yescharacteristic
83C2×#1084C2311yescharacteristic
84C2×#1084C2311yescharacteristic
85C4⋊#272—22no—
86C4×#272—22no—
87C4×#272—22no—
88C4×#272—22no—
89C4×#272—22no—
90#6229—22no—
91C3×#529—91no—
92#5706—34no—
93C2×C3×#67—34no—
94C2×C3×#67—34no—
95C22×C3×Dic3—34no—
96C2×C3×D12—34no—
97C2×C3×C4×D3—34no—
98C2×C3×C4×D3—34no—
99C2×C3×C4×D3—34no—
100C2×C3×C4×D3—34no—
101C2×C3×Dic6—34no—
102C32×C4⋊C4C2×D314yes—
103C32×C4⋊C4C2×D314yes—
104C32×C4⋊C4C2×D314yes—
105C32×C22⋊C4C2×D314yes—
106C32×C22⋊C4C2×D314yes—
107C32×C22⋊C4C2×D314yes—
108C32×C42C2×D314yes—
109C3×#217—34no—
110C3×(C2×D3)⋊C4—34no—
111C3×(C2×D3)⋊C4—34no—
112C3×(C2×D3)⋊C4—34no—
113C3×(C2×D3)⋊C4—34no—
114C3×(C2×D3)⋊C4—34no—
115C3×(C2×D3)⋊C4—34no—
116C3×C4⋊Dic3—34no—
117C3×Dic3⋊C4—34no—
118C3×Dic3⋊C4—34no—
119C3×Dic3⋊C4—34no—
120C3×Dic3⋊C4—34no—
121C3×Dic3⋊C4—34no—
122C3×Dic3⋊C4—34no—
123C3×C4×Dic3—34no—
124C3×C4×Dic3—34no—
125C3×C4×Dic3—34no—
126C3×C4×Dic3—34no—
127C3×C4×Dic3—34no—
128C22×H(3,𝔽3)—21no—
129C22×H(3,𝔽3)—21no—
130C22×H(3,𝔽3)—21no—
131C22×H(3,𝔽3)C2411yescharacteristic
132C2×#272—21no—
133C2×#272—21no—
134C2×#272—21no—
135C2×#272—21no—
136C2×#272—21no—
137C2×#272#4112yes—
138C4×H(3,𝔽3)—21no—
139C4×H(3,𝔽3)—21no—
140C4×H(3,𝔽3)—21no—
141C4×H(3,𝔽3)—21no—
142C4×H(3,𝔽3)—21no—
143C4×H(3,𝔽3)—21no—
144#1084—21no—
145#1084—21no—
146#1084—21no—
147#1084—21no—
148#1084#4112yes—
149#1084#4112yes—
150C3×#126—34no—
151C3×#126—91no—
152C3×#126C3⋊D311yescharacteristic
153C3×#125—91no—
154C3×#124—91no—
155C3×#123—91no—
156C3×#123—91no—
157C3×C22⋊Q8—91no—
158C3×#121—91no—
159C3×C4×Q8—91no—
160C3×C4×D4—91no—
161C3×C4×D4—91no—
162C3×C4×D4—91no—
163C3×#117—91no—
164C3×#117—91no—
165C2×C3×C42—91no—
166#896—34no—
167#894—34no—
168#893—34no—
169#892—34no—
170#889—34no—
171#887—34no—
172#885—34no—
173#884—34no—
174#883—34no—
175#67⋊C4—34no—
176#67⋊C4—34no—
177#877—34no—
178#876—34no—
179C42×D3—34no—
180C23×C32C22×D314yes—
181C22×C3×D3—34no—
182C22×C3×D3—34no—
183C2×C32×C4C22×D314yes—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C22×D314yes—
186C2×C32×C4C22×D314yes—
187C2×C32×C4C22×D314yes—
188C2×C32×C4C22×D314yes—
189C3×#67—68no—
190C3×#67—68no—
191C2×C3×Dic3—34no—
192C2×C3×Dic3—34no—
193C2×C3×Dic3—34no—
194C2×C3×Dic3—34no—
195C2×C3×Dic3—34no—
196C2×C3×Dic3—34no—
197C2×C3×Dic3—68no—
198C3×D12—68no—
199C3×C4×D3—68no—
200C3×C4×D3—68no—
201C3×C4×D3—68no—
202C3×C4×D3—68no—
203C3×Dic6—68no—
204#529—91noSylow
205C2×H(3,𝔽3)—41no—
206C2×H(3,𝔽3)C2×#4111yescharacteristic
207C2×H(3,𝔽3)C2×#4111yescharacteristic
208C2×H(3,𝔽3)C2×#4111yescharacteristic
209#272—22no—
210#272—41no—
211C2×C3×Q8—91no—
212C2×C3×D4—91no—
213C2×C3×D4—91no—
214C2×C3×D4—91no—
215C22×C3×C4—91no—
216C22×C3×C4—91no—
217C22×C3×C4—91no—
218C22×C3×C4—91no—
219C22×C3×C4—91no—
220C2×#67—34no—
221C2×#67—34no—
222C22×Dic3—34no—
223C2×D12—34no—
224C2×C4×D3—34no—
225C2×C4×D3—34no—
226C2×C4×D3—34no—
227C2×C4×D3—34no—
228C2×Dic6—34no—
229C3×C4⋊C4—34no—
230C3×C4⋊C4—34no—
231C3×C4⋊C4—34no—
232C3×C4⋊C4—91no—
233C3×C4⋊C4—91no—
234C3×C4⋊C4—91no—
235C3×C4⋊C4—91no—
236C3×C4⋊C4—91no—
237C3×C4⋊C4—91no—
238C3×C4⋊C4—91no—
239C3×C4⋊C4C2×C3⋊D311yescharacteristic
240C3×C4⋊C4C2×C3⋊D311yescharacteristic
241C3×C4⋊C4C2×C3⋊D311yescharacteristic
242C3×C22⋊C4—34no—
243C3×C22⋊C4—34no—
244C3×C22⋊C4—34no—
245C3×C22⋊C4—91no—
246C3×C22⋊C4—91no—
247C3×C22⋊C4—91no—
248C3×C22⋊C4—91no—
249C3×C22⋊C4—91no—
250C3×C22⋊C4—91no—
251C3×C22⋊C4—91no—
252C3×C22⋊C4C2×C3⋊D311yescharacteristic
253C3×C22⋊C4C2×C3⋊D311yescharacteristic
254C3×C22⋊C4C2×C3⋊D311yescharacteristic
255C3×C42—34no—
256C3×C42—91no—
257C3×C42—91no—
258C3×C42—91no—
259C3×C42—91no—
260C3×C42—91no—
261C3×C42C2×C3⋊D311yescharacteristic
262#217—34no—
263(C2×D3)⋊C4—34no—
264(C2×D3)⋊C4—34no—
265(C2×D3)⋊C4—34no—
266(C2×D3)⋊C4—34no—
267(C2×D3)⋊C4—34no—
268(C2×D3)⋊C4—34no—
269C4⋊Dic3—34no—
270Dic3⋊C4—34no—
271Dic3⋊C4—34no—
272Dic3⋊C4—34no—
273Dic3⋊C4—34no—
274Dic3⋊C4—34no—
275Dic3⋊C4—34no—
276C4×Dic3—34no—
277C4×Dic3—34no—
278C4×Dic3—34no—
279C4×Dic3—34no—
280C4×Dic3—34no—
281C22×C32—24no—
282C22×C32—24no—
283C22×C32—24no—
284C22×C32C23×D314yes—
285C2×C3×D3—38no—
286C2×C3×D3—64no—
287C2×C3×D3—64no—
288C2×C3×D3—64no—
289C2×C3×D3—64no—
290C2×C3×D3—64no—
291C32×C4—24no—
292C32×C4—24no—
293C32×C4—24no—
294C32×C4—24no—
295C32×C4—24no—
296C32×C4—24no—
297C3×Dic3—38no—
298C3×Dic3—38no—
299C3×Dic3—64no—
300C3×Dic3—64no—
301C3×Dic3—64no—
302C3×Dic3—64no—
303#126—91no—
304#126#27211yescharacteristic, complemented
305#125—91no—
306#124—91no—
307#123—91no—
308#123—91no—
309C22⋊Q8—91no—
310#121—91no—
311C4×Q8—91no—
312C4×D4—91no—
313C4×D4—91no—
314C4×D4—91no—
315#117—91no—
316#117—91no—
317C2×C42—91no—
318H(3,𝔽3)#52911yescharacteristic, complemented, Sylow
319C23×C3—34no—
320C23×C3—91no—
321C23×C3—91no—
322C23×C3C22×C3⋊D311yescharacteristic
323C22×D3—34no—
324C22×D3—34no—
325C3×Q8—182no—
326C3×D4—182no—
327C3×D4—182no—
328C3×D4—182no—
329C2×C3×C4—34no—
330C2×C3×C4—34no—
331C2×C3×C4—34no—
332C2×C3×C4—34no—
333C2×C3×C4—34no—
334C2×C3×C4—34no—
335C2×C3×C4—91no—
336C2×C3×C4—91no—
337C2×C3×C4—91no—
338C2×C3×C4—91no—
339C2×C3×C4—91no—
340C2×C3×C4—91no—
341C2×C3×C4—182no—
342C2×C3×C4—182no—
343C2×C3×C4—182no—
344C2×C3×C4—182no—
345C2×C3×C4—182no—
346C2×C3×C4C22×C3⋊D311yescharacteristic
347C2×C3×C4C22×C3⋊D311yescharacteristic
348C2×C3×C4C22×C3⋊D311yescharacteristic
349C2×C3×C4C22×C3⋊D311yescharacteristic
350C2×C3×C4C22×C3⋊D311yescharacteristic
351C2×C3×C4C22×C3⋊D311yescharacteristic
352#67—68no—
353#67—68no—
354C2×Dic3—34no—
355C2×Dic3—34no—
356C2×Dic3—34no—
357C2×Dic3—34no—
358C2×Dic3—34no—
359C2×Dic3—34no—
360C2×Dic3—68no—
361D12—68no—
362C4×D3—68no—
363C4×D3—68no—
364C4×D3—68no—
365C4×D3—68no—
366Dic6—68no—
367C2×C32—44no—
368C2×C32D3×#4114yes—
369C2×C32D3×#4114yes—
370C2×C32D3×#4114yes—
371C3×D3—68no—
372C3×D3—124no—
373C2×Q8—91no—
374C2×D4—91no—
375C2×D4—91no—
376C2×D4—91no—
377C22×C4—91no—
378C22×C4—91no—
379C22×C4—91no—
380C22×C4—91no—
381C22×C4—91no—
382C4⋊C4—91no—
383C4⋊C4—91no—
384C4⋊C4—91no—
385C4⋊C4—91no—
386C4⋊C4—91no—
387C4⋊C4—91no—
388C4⋊C4—91no—
389C4⋊C4C2×#27211yescharacteristic
390C4⋊C4C2×#27211yescharacteristic
391C4⋊C4C2×#27211yescharacteristic
392C22⋊C4—91no—
393C22⋊C4—91no—
394C22⋊C4—91no—
395C22⋊C4—91no—
396C22⋊C4—91no—
397C22⋊C4—91no—
398C22⋊C4—91no—
399C22⋊C4C2×#27211yescharacteristic
400C22⋊C4C2×#27211yescharacteristic
401C22⋊C4C2×#27211yescharacteristic
402C42—91no—
403C42—91no—
404C42—91no—
405C42—91no—
406C42—91no—
407C42C2×#27211yescharacteristic
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—34no—
412C22×C3—64no—
413C22×C3—64no—
414C22×C3—64no—
415C22×C3—92no—
416C22×C3—181no—
417C22×C3—181no—
418C22×C3—181no—
419C22×C3—181no—
420C22×C3—181no—
421C22×C3C23×C3⋊D311yescharacteristic
422C2×D3—38no—
423C2×D3—64no—
424C2×D3—64no—
425C2×D3—64no—
426C2×D3—64no—
427C2×D3—64no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—64no—
435C3×C4—64no—
436C3×C4—64no—
437C3×C4—64no—
438C3×C4—64no—
439C3×C4—64no—
440C3×C4—92no—
441C3×C4—92no—
442C3×C4—181no—
443C3×C4—181no—
444C3×C4—181no—
445C3×C4—181no—
446Dic3—38no—
447Dic3—38no—
448Dic3—64no—
449Dic3—64no—
450Dic3—64no—
451Dic3—64no—
452C32#570614yescomplemented
453C23—91no—
454C23—91no—
455C23C22×#27211yescharacteristic
456Q8—182no—
457D4—182no—
458D4—182no—
459D4—182no—
460C2×C4—91no—
461C2×C4—91no—
462C2×C4—91no—
463C2×C4—91no—
464C2×C4—91no—
465C2×C4—91no—
466C2×C4—182no—
467C2×C4—182no—
468C2×C4—182no—
469C2×C4—182no—
470C2×C4—182no—
471C2×C4C22×#27211yescharacteristic
472C2×C4C22×#27211yescharacteristic
473C2×C4C22×#27211yescharacteristic
474C2×C4C22×#27211yescharacteristic
475C2×C4C22×#27211yescharacteristic
476C2×C4C22×#27211yescharacteristic
477C2×C3—34no—
478C2×C3—34no—
479C2×C3—34no—
480C2×C3—41no—
481C2×C3—124no—
482C2×C3—182no—
483C2×C3—361no—
484C2×C3#41×C3⋊D311yescharacteristic
485C2×C3#41×C3⋊D311yescharacteristic
486C2×C3#41×C3⋊D311yescharacteristic
487D3—68no—
488D3—124no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—92no—
493C22—181no—
494C22—181no—
495C22—181no—
496C22—181no—
497C22—181no—
498C22C23×#27211yescharacteristic
499C4—21no—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—21no—
505C4—92no—
506C4—92no—
507C4—181no—
508C4—181no—
509C4—181no—
510C4—181no—
511C3—34no—
512C3#1059617711yescharacteristic
513C2—41no—
514C2—182no—
515C2—361no—
516C2#41×#27211yescharacteristic
517C2#41×#27211yescharacteristic
518C2#41×#27211yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#126C2
prime radical#11721740C2
hypercenterC3×#126C3⋊D3
hypocenterH(3,𝔽3)#529
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#126

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#126C2semidirect
2. #11721764C2semidirect
3. #11721762C2semidirect
4. #11721761C2semidirect
5. #11721760C2semidirect
6. #11721757C2semidirect
7. #11721755C2semidirect
8. #11721741C2semidirect
10. #11721739C2semidirect
11. #11721737C2semidirect
12. #11721737C2semidirect
13. #11721733C2semidirect
14. #11721722C2semidirect
15. C42×#272C2semidirect
52. C32×#126D3semidirect
304. #126#272semidirect
318. H(3,𝔽3)#529semidirect
452. C32#5706semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1262
#117217642
#117217622
#117217612
#117217602
#117217572
#117217552
#117217412
#117217402
#117217392
#117217372
#117217332
#117217222
C42×#2722
C3×#57063