Finite Group Databasebeta

← #13154251

Subgroups — #13154251

Order 1728.

1375 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#126C211yesmaximal, characteristic, complemented
2#11721766C211yesmaximal, characteristic, complemented
3#11721764C211yesmaximal, characteristic, complemented
4#11721763C211yesmaximal, characteristic, complemented
5#11721763C211yesmaximal, characteristic, complemented
6#11721762C211yesmaximal, characteristic, complemented
7#11721761C211yesmaximal, characteristic, complemented
8#11721741C211yesmaximal, characteristic, complemented
9#11721740C211yesmaximal, characteristic, complemented
10#11721739C211yesmaximal, characteristic, complemented
11#11721738C211yesmaximal, characteristic, complemented
12#11721738C211yesmaximal, characteristic, complemented
13C22⋊C4×#272C211yesmaximal, characteristic, complemented
14#11721721C211yesmaximal, characteristic, complemented
15C4×C4⋊#272C211yesmaximal, characteristic, complemented
16C3×#5704—34nomaximal
17C23×#272C2211yescharacteristic
18C2×#6235C2211yescharacteristic
19C2×#6235C2211yescharacteristic
20C2×C4⋊#272C2211yescharacteristic
21C2×C4⋊#272C2211yescharacteristic
22C2×C4⋊#272C2211yescharacteristic
23C2×C4⋊#272C2211yescharacteristic
24C2×C4⋊#272C2211yescharacteristic, complemented
25C2×C4×#272C2211yescharacteristic
26C2×C4×#272C2211yescharacteristic
27C2×C4×#272C2211yescharacteristic, complemented
28C2×C4×#272C2211yescharacteristic, complemented
29C2×#6229C2211yescharacteristic, complemented
30C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
31C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
32C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
33C22⋊C4×H(3,𝔽3)C2211yescharacteristic
34C22⋊C4×H(3,𝔽3)C2211yescharacteristic
35C22⋊C4×H(3,𝔽3)C2211yescharacteristic
36C42×H(3,𝔽3)C2211yescharacteristic, complemented
37#88552C2211yescharacteristic
38#88542C2211yescharacteristic
39#88542C2211yescharacteristic
40#88542C2211yescharacteristic
41#88542C2211yescharacteristic
42#88542C2211yescharacteristic
43#88542C2211yescharacteristic
44#88542C2211yescharacteristic, complemented
45#88542C2211yescharacteristic, complemented
46#88542C2211yescharacteristic, complemented
47#88542C2211yescharacteristic, complemented
48C4⋊#1084C2211yescharacteristic, complemented
49#88540C2211yescharacteristic, complemented
50#88540C2211yescharacteristic, complemented
51C4×#1084C2211yescharacteristic, complemented
52C32×#126D314yescomplemented
53C3×#898—34no—
54C3×#896—34no—
55C3×#895—34no—
56C3×#895—34no—
57C3×#894—34no—
58C3×#893—34no—
59C3×#885—34no—
60C3×#884—34no—
61C3×#883—34no—
62C3×#882—34no—
63C3×#882—34no—
64C3×D3×C22⋊C4—34no—
65C3×#875—34no—
66C3×C4×D12—34no—
67C23×H(3,𝔽3)C2311yescharacteristic
68C22×#272—22no—
69C22×#272—41no—
70C22×#272—41no—
71C22×#272C2311yescharacteristic
72C22×#272C2311yescharacteristic
73C22×#272C2311yescharacteristic
74C22×#272C2311yescharacteristic
75C2×C4×H(3,𝔽3)C2311yescharacteristic
76C2×C4×H(3,𝔽3)C2311yescharacteristic
77C2×C4×H(3,𝔽3)C2311yescharacteristic
78C2×C4×H(3,𝔽3)C2311yescharacteristic
79C2×C4×H(3,𝔽3)C2311yescharacteristic
80C2×C4×H(3,𝔽3)C2311yescharacteristic
81#6235—41no—
82#6235—41no—
83C2×#1084C2311yescharacteristic
84C2×#1084C2311yescharacteristic
85C2×#1084C2311yescharacteristic
86C2×#1084C2311yescharacteristic
87C4⋊#272—22no—
88C4⋊#272—22no—
89C4⋊#272—41no—
90C4⋊#272—41no—
91C4⋊#272—41no—
92C4×#272—41no—
93C4×#272—41no—
94C4×#272—41no—
95C4×#272—41no—
96#6229—41no—
97C3×#538—91no—
98#5704—34no—
99C23×C3×D3—34no—
100C2×C3×#67—34no—
101C2×C3×#67—34no—
102C2×C3×D12—34no—
103C2×C3×D12—34no—
104C2×C3×D12—34no—
105C2×C3×D12—34no—
106C2×C3×D12—34no—
107C2×C3×C4×D3—34no—
108C2×C3×C4×D3—34no—
109C2×C3×C4×D3—34no—
110C2×C3×C4×D3—34no—
111C2×C3×Dic6—34no—
112C32×C4⋊C4C2×D314yescomplemented
113C32×C4⋊C4C2×D314yescomplemented
114C32×C4⋊C4C2×D314yescomplemented
115C32×C22⋊C4C2×D314yes—
116C32×C22⋊C4C2×D314yes—
117C32×C22⋊C4C2×D314yes—
118C32×C42C2×D314yescomplemented
119C3×#217—34no—
120C3×(C2×D3)⋊C4—34no—
121C3×(C2×D3)⋊C4—34no—
122C3×(C2×D3)⋊C4—34no—
123C3×(C2×D3)⋊C4—34no—
124C3×(C2×D3)⋊C4—34no—
125C3×(C2×D3)⋊C4—34no—
126C3×(C2×D3)⋊C4—34no—
127C3×(C2×D3)⋊C4—34no—
128C3×(C2×D3)⋊C4—34no—
129C3×(C2×D3)⋊C4—34no—
130C3×C4⋊Dic3—34no—
131C3×Dic3⋊C4—34no—
132C3×Dic3⋊C4—34no—
133C3×C4×Dic3—34no—
134C22×H(3,𝔽3)—21no—
135C22×H(3,𝔽3)—21no—
136C22×H(3,𝔽3)—21no—
137C22×H(3,𝔽3)C2411yescharacteristic
138C2×#272—21no—
139C2×#272—21no—
140C2×#272—21no—
141C2×#272—21no—
142C2×#272—21no—
143C2×#272—21no—
144C2×#272—21no—
145C2×#272—21no—
146C2×#272—21no—
147C2×#272—21no—
148C2×#272—21no—
149C2×#272—44no—
150C2×#272#4112yes—
151C4×H(3,𝔽3)—21no—
152C4×H(3,𝔽3)—21no—
153C4×H(3,𝔽3)—21no—
154C4×H(3,𝔽3)—21no—
155C4×H(3,𝔽3)—21no—
156C4×H(3,𝔽3)—21no—
157#1084—21no—
158#1084—21no—
159#1084—21no—
160#1084—21no—
161C3×#126—34no—
162C3×#126—91no—
163C3×#126C3⋊D311yescharacteristic
164C3×#124—91no—
165C3×#124—91no—
166C3×#123—91no—
167C3×#123—91no—
168C3×C22⋊Q8—91no—
169C3×#121—91no—
170C3×#121—91no—
171C3×#121—91no—
172C3×#120—91no—
173C3×#120—91no—
174C3×C4×D4—91no—
175C3×C4×D4—91no—
176C2×C3×C22⋊C4—91no—
177#898—34no—
178#896—34no—
179#895—34no—
180#895—34no—
181#894—34no—
182#893—34no—
183#885—34no—
184#884—34no—
185#883—34no—
186#882—34no—
187#882—34no—
188D3×C22⋊C4—34no—
189#875—34no—
190C4×D12—34no—
191C23×C32C22×D314yes—
192C22×C3×D3—34no—
193C22×C3×D3—34no—
194C22×C3×D3—34no—
195C22×C3×D3—34no—
196C22×C3×D3—68no—
197C22×C3×D3—124no—
198C22×C3×D3—124no—
199C2×C32×C4C22×D314yes—
200C2×C32×C4C22×D314yes—
201C2×C32×C4C22×D314yes—
202C2×C32×C4C22×D314yes—
203C2×C32×C4C22×D314yes—
204C2×C32×C4C22×D314yes—
205C3×#67—124no—
206C3×#67—124no—
207C2×C3×Dic3—34no—
208C2×C3×Dic3—34no—
209C2×C3×Dic3—34no—
210C2×C3×Dic3—34no—
211C3×D12—68no—
212C3×D12—68no—
213C3×D12—124no—
214C3×D12—124no—
215C3×D12—124no—
216C3×C4×D3—124no—
217C3×C4×D3—124no—
218C3×C4×D3—124no—
219C3×C4×D3—124no—
220C3×Dic6—124no—
221#538—91noSylow
222C2×H(3,𝔽3)—41no—
223C2×H(3,𝔽3)C2×#4111yescharacteristic
224C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
225C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
226#272—22no—
227#272—41no—
228#272—41no—
229#272—41no—
230C24×C3—91no—
231C23×D3—34no—
232C2×C3×Q8—91no—
233C2×C3×D4—91no—
234C2×C3×D4—91no—
235C2×C3×D4—91no—
236C2×C3×D4—91no—
237C2×C3×D4—91no—
238C2×C3×D4—91no—
239C2×C3×D4—91no—
240C22×C3×C4—91no—
241C22×C3×C4—91no—
242C22×C3×C4—91no—
243C22×C3×C4—91no—
244C2×#67—34no—
245C2×#67—34no—
246C2×D12—34no—
247C2×D12—34no—
248C2×D12—34no—
249C2×D12—34no—
250C2×D12—34no—
251C2×C4×D3—34no—
252C2×C4×D3—34no—
253C2×C4×D3—34no—
254C2×C4×D3—34no—
255C2×Dic6—34no—
256C3×C4⋊C4—34no—
257C3×C4⋊C4—34no—
258C3×C4⋊C4—34no—
259C3×C4⋊C4—91no—
260C3×C4⋊C4—91no—
261C3×C4⋊C4—91no—
262C3×C4⋊C4C2×C3⋊D311yescharacteristic
263C3×C4⋊C4C2×C3⋊D311yescharacteristic
264C3×C4⋊C4C2×C3⋊D311yescharacteristic
265C3×C22⋊C4—34no—
266C3×C22⋊C4—34no—
267C3×C22⋊C4—34no—
268C3×C22⋊C4—91no—
269C3×C22⋊C4—91no—
270C3×C22⋊C4—91no—
271C3×C22⋊C4—91no—
272C3×C22⋊C4—91no—
273C3×C22⋊C4—91no—
274C3×C22⋊C4—91no—
275C3×C22⋊C4—91no—
276C3×C22⋊C4—91no—
277C3×C22⋊C4—91no—
278C3×C22⋊C4—91no—
279C3×C22⋊C4C2×C3⋊D311yescharacteristic
280C3×C22⋊C4C2×C3⋊D311yescharacteristic
281C3×C22⋊C4C2×C3⋊D311yescharacteristic
282C3×C42—34no—
283C3×C42—91no—
284C3×C42C2×C3⋊D311yescharacteristic
285#217—34no—
286(C2×D3)⋊C4—34no—
287(C2×D3)⋊C4—34no—
288(C2×D3)⋊C4—34no—
289(C2×D3)⋊C4—34no—
290(C2×D3)⋊C4—34no—
291(C2×D3)⋊C4—34no—
292(C2×D3)⋊C4—34no—
293(C2×D3)⋊C4—34no—
294(C2×D3)⋊C4—34no—
295(C2×D3)⋊C4—34no—
296C4⋊Dic3—34no—
297Dic3⋊C4—34no—
298Dic3⋊C4—34no—
299C4×Dic3—34no—
300C22×C32—24no—
301C22×C32—24no—
302C22×C32—24no—
303C22×C32C23×D314yes—
304C2×C3×D3—38no—
305C2×C3×D3—64no—
306C2×C3×D3—64no—
307C2×C3×D3—64no—
308C2×C3×D3—64no—
309C2×C3×D3—64no—
310C2×C3×D3—64no—
311C2×C3×D3—64no—
312C2×C3×D3—64no—
313C2×C3×D3—64no—
314C2×C3×D3—64no—
315C2×C3×D3—64no—
316C2×C3×D3—1216no—
317C32×C4—24no—
318C32×C4—24no—
319C32×C4—24no—
320C32×C4—24no—
321C32×C4—24no—
322C32×C4—24no—
323C3×Dic3—64no—
324C3×Dic3—64no—
325C3×Dic3—64no—
326C3×Dic3—64no—
327#126—91no—
328#126#27211yescharacteristic, complemented
329#124—91no—
330#124—91no—
331#123—91no—
332#123—91no—
333C22⋊Q8—91no—
334#121—91no—
335#121—91no—
336#121—91no—
337#120—91no—
338#120—91no—
339C4×D4—91no—
340C4×D4—91no—
341C2×C22⋊C4—91no—
342H(3,𝔽3)#53811yescharacteristic, complemented, Sylow
343C23×C3—34no—
344C23×C3—91no—
345C23×C3—91no—
346C23×C3—91no—
347C23×C3—91no—
348C23×C3—182no—
349C23×C3—361no—
350C23×C3—361no—
351C23×C3C22×C3⋊D311yescharacteristic
352C22×D3—34no—
353C22×D3—34no—
354C22×D3—34no—
355C22×D3—34no—
356C22×D3—68no—
357C22×D3—124no—
358C22×D3—124no—
359C3×Q8—361no—
360C3×D4—182no—
361C3×D4—182no—
362C3×D4—361no—
363C3×D4—361no—
364C3×D4—361no—
365C3×D4—361no—
366C3×D4—361no—
367C2×C3×C4—34no—
368C2×C3×C4—34no—
369C2×C3×C4—34no—
370C2×C3×C4—34no—
371C2×C3×C4—34no—
372C2×C3×C4—34no—
373C2×C3×C4—91no—
374C2×C3×C4—91no—
375C2×C3×C4—91no—
376C2×C3×C4—91no—
377C2×C3×C4—361no—
378C2×C3×C4—361no—
379C2×C3×C4—361no—
380C2×C3×C4—361no—
381C2×C3×C4C22×C3⋊D311yescharacteristic
382C2×C3×C4C22×C3⋊D311yescharacteristic
383C2×C3×C4C22×C3⋊D311yescharacteristic
384C2×C3×C4C22×C3⋊D311yescharacteristic
385C2×C3×C4C22×C3⋊D311yescharacteristic
386C2×C3×C4C22×C3⋊D311yescharacteristic
387#67—124no—
388#67—124no—
389C2×Dic3—34no—
390C2×Dic3—34no—
391C2×Dic3—34no—
392C2×Dic3—34no—
393D12—68no—
394D12—68no—
395D12—124no—
396D12—124no—
397D12—124no—
398C4×D3—124no—
399C4×D3—124no—
400C4×D3—124no—
401C4×D3—124no—
402Dic6—124no—
403C2×C32—44no—
404C2×C32D3×#4114yes—
405C2×C32#100914yes—
406C2×C32#100914yes—
407C3×D3—68no—
408C3×D3—124no—
409C3×D3—124no—
410C3×D3—124no—
411C24—91no—
412C2×Q8—91no—
413C2×D4—91no—
414C2×D4—91no—
415C2×D4—91no—
416C2×D4—91no—
417C2×D4—91no—
418C2×D4—91no—
419C2×D4—91no—
420C22×C4—91no—
421C22×C4—91no—
422C22×C4—91no—
423C22×C4—91no—
424C4⋊C4—91no—
425C4⋊C4—91no—
426C4⋊C4—91no—
427C4⋊C4C2×#27211yescharacteristic, complemented
428C4⋊C4C2×#27211yescharacteristic, complemented
429C4⋊C4C2×#27211yescharacteristic, complemented
430C22⋊C4—91no—
431C22⋊C4—91no—
432C22⋊C4—91no—
433C22⋊C4—91no—
434C22⋊C4—91no—
435C22⋊C4—91no—
436C22⋊C4—91no—
437C22⋊C4—91no—
438C22⋊C4—91no—
439C22⋊C4—91no—
440C22⋊C4—91no—
441C22⋊C4C2×#27211yescharacteristic
442C22⋊C4C2×#27211yescharacteristic
443C22⋊C4C2×#27211yescharacteristic
444C42—91no—
445C42C2×#27211yescharacteristic, complemented
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—34no—
450C22×C3—64no—
451C22×C3—64no—
452C22×C3—64no—
453C22×C3—92no—
454C22×C3—181no—
455C22×C3—181no—
456C22×C3—181no—
457C22×C3—181no—
458C22×C3—181no—
459C22×C3—181no—
460C22×C3—181no—
461C22×C3—181no—
462C22×C3—181no—
463C22×C3—181no—
464C22×C3—181no—
465C22×C3—364no—
466C22×C3C23×C3⋊D311yescharacteristic
467C2×D3—38no—
468C2×D3—64no—
469C2×D3—64no—
470C2×D3—64no—
471C2×D3—64no—
472C2×D3—64no—
473C2×D3—64no—
474C2×D3—64no—
475C2×D3—64no—
476C2×D3—64no—
477C2×D3—64no—
478C2×D3—64no—
479C2×D3—1216no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—64no—
487C3×C4—64no—
488C3×C4—64no—
489C3×C4—64no—
490C3×C4—64no—
491C3×C4—64no—
492C3×C4—181no—
493C3×C4—181no—
494C3×C4—181no—
495C3×C4—181no—
496Dic3—64no—
497Dic3—64no—
498Dic3—64no—
499Dic3—64no—
500C32#570414yescomplemented
501C23—91no—
502C23—91no—
503C23—91no—
504C23—91no—
505C23—182no—
506C23—361no—
507C23—361no—
508C23C22×#27211yescharacteristic
509Q8—361no—
510D4—182no—
511D4—182no—
512D4—361no—
513D4—361no—
514D4—361no—
515D4—361no—
516D4—361no—
517C2×C4—91no—
518C2×C4—91no—
519C2×C4—91no—
520C2×C4—91no—
521C2×C4—361no—
522C2×C4—361no—
523C2×C4—361no—
524C2×C4—361no—
525C2×C4C22×#27211yescharacteristic
526C2×C4C22×#27211yescharacteristic
527C2×C4C22×#27211yescharacteristic
528C2×C4C22×#27211yescharacteristic
529C2×C4C22×#27211yescharacteristic
530C2×C4C22×#27211yescharacteristic
531C2×C3—34no—
532C2×C3—34no—
533C2×C3—34no—
534C2×C3—41no—
535C2×C3—124no—
536C2×C3—182no—
537C2×C3—361no—
538C2×C3—361no—
539C2×C3—361no—
540C2×C3#41×C3⋊D311yescharacteristic
541C2×C3#6436611yescharacteristic
542C2×C3#6436611yescharacteristic
543D3—68no—
544D3—124no—
545D3—124no—
546D3—124no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—92no—
551C22—181no—
552C22—181no—
553C22—181no—
554C22—181no—
555C22—181no—
556C22—181no—
557C22—181no—
558C22—181no—
559C22—181no—
560C22—181no—
561C22—181no—
562C22—364no—
563C22C23×#27211yescharacteristic
564C4—21no—
565C4—21no—
566C4—21no—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—181no—
571C4—181no—
572C4—181no—
573C4—181no—
574C3—34no—
575C3#1059617511yescharacteristic
576C2—41no—
577C2—182no—
578C2—361no—
579C2—361no—
580C2—361no—
581C2#41×#27211yescharacteristic
582C2#1172267511yescharacteristic
583C2#1172267511yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#126C2
hypercenterC3×#126C3⋊D3
hypocenterH(3,𝔽3)#538
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. #11721766C2semidirect
3. #11721764C2semidirect
4. #11721763C2semidirect
5. #11721763C2semidirect
6. #11721762C2semidirect
7. #11721761C2semidirect
8. #11721741C2semidirect
9. #11721740C2semidirect
10. #11721739C2semidirect
11. #11721738C2semidirect
12. #11721738C2semidirect
13. C22⋊C4×#272C2semidirect
14. #11721721C2semidirect
15. C4×C4⋊#272C2semidirect
24. C2×C4⋊#272C22semidirect
27. C2×C4×#272C22semidirect
28. C2×C4×#272C22semidirect
29. C2×#6229C22semidirect
30. C4⋊C4×H(3,𝔽3)C22semidirect
31. C4⋊C4×H(3,𝔽3)C22semidirect
32. C4⋊C4×H(3,𝔽3)C22semidirect
36. C42×H(3,𝔽3)C22semidirect
44. #88542C22semidirect
45. #88542C22semidirect
46. #88542C22semidirect
47. #88542C22semidirect
48. C4⋊#1084C22semidirect
49. #88540C22semidirect
50. #88540C22semidirect
51. C4×#1084C22semidirect
52. C32×#126D3semidirect
112. C32×C4⋊C4C2×D3semidirect
113. C32×C4⋊C4C2×D3semidirect
114. C32×C4⋊C4C2×D3semidirect
118. C32×C42C2×D3semidirect
328. #126#272semidirect
342. H(3,𝔽3)#538semidirect
427. C4⋊C4C2×#272semidirect
428. C4⋊C4C2×#272semidirect
429. C4⋊C4C2×#272semidirect
445. C42C2×#272semidirect
500. C32#5704semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1262
#117217662
#117217642
#117217632
#117217622
#117217612
#117217412
#117217402
#117217392
#117217382
C22⋊C4×#2722
#117217212
C4×C4⋊#2722
C3×#57043