Finite Group Databasebeta

← #13141158

Subgroups — #13141158

Order 1728.

727 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88552C211yesmaximal, characteristic
2C2×#3757⋊C3C211yesmaximal, characteristic
3C2×#3757⋊C3C211yesmaximal, characteristic
4C2×#88448C211yesmaximal, characteristic, complemented
5#11720334C211yesmaximal, characteristic, complemented
6#11720334C211yesmaximal, characteristic, complemented
7#11720334C211yesmaximal, characteristic, complemented
8#10591717—31nomaximal
9#10591694—31nomaximal
10C24×H(3,𝔽3)C2211yescharacteristic
11C22×#1084C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17#88552—22no—
18#3757⋊C3—22no—
19#3757⋊C3—22no—
20#88448—22no—
21C2×#3757D311yescharacteristic
22C2×#3757D311yescharacteristic
23C2×C3×#217—31no—
24C2×C3×#217—31no—
25C2×C3×#217—31no—
26C2×C3×#217—31no—
27C2×C3×#217—61no—
28C2×#3724—31no—
29C2×Dic3⋊Dic3—31no—
30#63579—31no—
31#63579—31no—
32#63579—31no—
33#63579—31no—
34#63579—31no—
35#63579—31no—
36C23×H(3,𝔽3)—21no—
37C23×H(3,𝔽3)—21no—
38C23×H(3,𝔽3)—21no—
39C23×H(3,𝔽3)—21no—
40C23×H(3,𝔽3)—21no—
41C23×H(3,𝔽3)—21no—
42C23×H(3,𝔽3)D412yes—
43C23×H(3,𝔽3)C2311yescharacteristic
44C2×#1084—21no—
45C2×#1084—21no—
46C2×#1084—21no—
47C2×#1084—21no—
48C2×#1084—21no—
49C2×#1084—21no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C2×(C3⋊Dic3)⋊C3—21no—
52C2×(C3⋊Dic3)⋊C3—21no—
53C2×(C3⋊Dic3)⋊C3—21no—
54C2×(C3⋊Dic3)⋊C3—21no—
55C2×(C3⋊Dic3)⋊C3—21no—
56C2×(C3⋊Dic3)⋊C3—21no—
57C2×(C3⋊Dic3)⋊C3—21no—
58C2×(C3⋊Dic3)⋊C3—21no—
59C2×(C3⋊Dic3)⋊C3—21no—
60C2×(C3⋊Dic3)⋊C3—21no—
61C2×(C3⋊Dic3)⋊C3D412yes—
62#4953—91no—
63#4952—91no—
64#4952—91no—
65C24×C32—21no—
66C24×C32C2×D311yescharacteristic
67C24×C32C2×D311yescharacteristic
68C22×C3⋊Dic3C2×D311yescharacteristic
69C22×C3⋊Dic3C2×D311yescharacteristic
70C22×C3⋊Dic3C2×D311yescharacteristic
71C22×C3⋊Dic3C2×D311yescharacteristic
72C22×C3×Dic3—31no—
73C22×C3×Dic3—31no—
74C22×C3×Dic3—31no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3—31no—
77C22×C3×Dic3—31no—
78C22×C3×Dic3—31no—
79C22×C3×Dic3—31no—
80C22×C3×Dic3—61no—
81C22×C3×Dic3—61no—
82#3757—22no—
83#3757—22no—
84C3×#217—62no—
85C3×#217—64no—
86C3×#217—62no—
87C3×#217—62no—
88C3×#217—62no—
89#3724—62no—
90Dic3⋊Dic3—62no—
91C22×H(3,𝔽3)—22no—
92C22×H(3,𝔽3)—22no—
93C22×H(3,𝔽3)—22no—
94C22×H(3,𝔽3)—41no—
95C22×H(3,𝔽3)—41no—
96C22×H(3,𝔽3)—41no—
97C22×H(3,𝔽3)—41no—
98C22×H(3,𝔽3)C2×D411yescharacteristic
99C22×H(3,𝔽3)C2×D411yescharacteristic
100C22×H(3,𝔽3)#4111yescharacteristic
101C22×H(3,𝔽3)#4111yescharacteristic
102C22×H(3,𝔽3)#4111yescharacteristic
103C22×H(3,𝔽3)#4111yescharacteristic
104C22×H(3,𝔽3)#4111yescharacteristic
105#1084—41no—
106#1084—41no—
107(C3⋊Dic3)⋊C3—22no—
108(C3⋊Dic3)⋊C3—41no—
109(C3⋊Dic3)⋊C3—41no—
110(C3⋊Dic3)⋊C3—41no—
111C2×C3×C22⋊C4—91no—
112C2×C3×C22⋊C4—91no—
113C2×C3×C22⋊C4—91no—
114C2×#217—31no—
115C2×#217—31no—
116C2×#217—31no—
117C2×#217—31no—
118C2×#217—61no—
119C2×#217—91no—
120C2×#217—91no—
121C2×C4⋊Dic3—91no—
122C2×Dic3⋊C4—91no—
123C2×Dic3⋊C4—91no—
124C3⋊#95—91no—
125C3⋊#95—91no—
126C3⋊#95—91no—
127C3⋊#95—91no—
128C3⋊#95—91no—
129C3⋊#95—91no—
130C3⋊#95—91no—
131C3⋊#95—91no—
132C3⋊#95—91no—
133C23×C32—22no—
134C23×C32—22no—
135C23×C32—21no—
136C23×C32—21no—
137C23×C32—21no—
138C23×C32—21no—
139C23×C32—21no—
140C23×C32—21no—
141C23×C32—21no—
142C23×C32—21no—
143C23×C32—21no—
144C23×C32—21no—
145C23×C32—21no—
146C23×C32—21no—
147C23×C32—22no—
148C23×C32—21no—
149C23×C32—41no—
150C23×C32—41no—
151C23×C32—41no—
152C23×C32—41no—
153C23×C32D1212yes—
154C23×C32#6712yes—
155C23×C32C22×D311yescharacteristic
156C23×C32C22×D311yescharacteristic
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—21no—
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3—21no—
164C2×C3⋊Dic3—21no—
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3—21no—
167C2×C3⋊Dic3—21no—
168C2×C3⋊Dic3#6712yes—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—62no—
173C2×C3×Dic3—62no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—121no—
196C2×C3×Dic3—121no—
197C2×C3×Dic3—121no—
198C2×C3×Dic3—121no—
199#397—271noSylow
200C2×H(3,𝔽3)—42no—
201C2×H(3,𝔽3)#12111yescharacteristic
202C2×H(3,𝔽3)#12311yescharacteristic
203C2×H(3,𝔽3)#12311yescharacteristic
204C2×H(3,𝔽3)#12311yescharacteristic
205C2×H(3,𝔽3)#12411yescharacteristic
206C2×H(3,𝔽3)#12611yescharacteristic
207C2×H(3,𝔽3)#12611yescharacteristic
208C24×C3—31no—
209C24×C3—31no—
210C24×C3—61no—
211C24×C3D3211yescharacteristic
212C22×C3×C4—91no—
213C22×C3×C4—91no—
214C22×C3×C4—91no—
215C22×C3×C4—91no—
216C22×C3×C4—91no—
217C22×C3×C4—91no—
218C22×Dic3—31no—
219C22×Dic3—31no—
220C22×Dic3—31no—
221C22×Dic3—31no—
222C22×Dic3—31no—
223C22×Dic3—31no—
224C22×Dic3—31no—
225C22×Dic3—31no—
226C22×Dic3—61no—
227C22×Dic3—61no—
228C22×Dic3—91no—
229C22×Dic3—91no—
230C22×Dic3—91no—
231C22×Dic3—91no—
232C3×C22⋊C4—182no—
233C3×C22⋊C4—182no—
234C3×C22⋊C4—182no—
235#217—62no—
236#217—62no—
237#217—62no—
238#217—64no—
239#217—62no—
240#217—182no—
241#217—182no—
242C4⋊Dic3—182no—
243Dic3⋊C4—182no—
244Dic3⋊C4—182no—
245C22×C32—24no—
246C22×C32—22no—
247C22×C32—22no—
248C22×C32—22no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32—22no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—42no—
260C22×C32—42no—
261C22×C32—42no—
262C22×C32—42no—
263C22×C32—42no—
264C22×C32—42no—
265C22×C32—41no—
266C22×C32—41no—
267C22×C32—41no—
268C22×C32—41no—
269C22×C32—41no—
270C22×C32—41no—
271C22×C32—41no—
272C22×C32—41no—
273C22×C32C2×D1211yescharacteristic
274C22×C32#23511yescharacteristic
275C22×C32#23511yescharacteristic
276C22×C32#23511yescharacteristic
277C22×C32D3×D411yescharacteristic
278C22×C32#23711yescharacteristic
279C22×C32#23711yescharacteristic
280C22×C32#23711yescharacteristic
281C22×C32#23711yescharacteristic
282C22×C32#23711yescharacteristic
283C22×C32#23711yescharacteristic
284C22×C32#23711yescharacteristic
285C22×C32C2×#6711yescharacteristic
286C22×C32C2×#6711yescharacteristic
287C3⋊Dic3—22no—
288C3⋊Dic3—41no—
289C3⋊Dic3—41no—
290C3⋊Dic3—41no—
291C3×Dic3—62no—
292C3×Dic3—122no—
293C3×Dic3—122no—
294C3×Dic3—121no—
295C3×Dic3—121no—
296C3×Dic3—121no—
297C3×Dic3—121no—
298C3×Dic3—121no—
299C3×Dic3—121no—
300C3×Dic3—121no—
301C2×C4⋊C4—271no—
302C2×C22⋊C4—91no—
303C2×C22⋊C4—91no—
304C2×C22⋊C4—91no—
305#95—271no—
306#95—271no—
307#95—271no—
308H(3,𝔽3)#39711yescharacteristic, complemented, Sylow
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—32no—
316C23×C3—32no—
317C23×C3—31no—
318C23×C3—31no—
319C23×C3—61no—
320C23×C3—61no—
321C23×C3—61no—
322C23×C3—61no—
323C23×C3—61no—
324C23×C3—61no—
325C23×C3—61no—
326C23×C3—61no—
327C23×C3—61no—
328C23×C3—61no—
329C23×C3—61no—
330C23×C3—61no—
331C23×C3—62no—
332C23×C3—62no—
333C23×C3—62no—
334C23×C3—61no—
335C23×C3—121no—
336C23×C3—121no—
337C23×C3—121no—
338C23×C3—121no—
339C23×C3#62612yes—
340C23×C3C2×D3211yescharacteristic
341C2×C3×C4—92no—
342C2×C3×C4—181no—
343C2×C3×C4—181no—
344C2×C3×C4—181no—
345C2×C3×C4—181no—
346C2×C3×C4—181no—
347C2×C3×C4—181no—
348C2×C3×C4—181no—
349C2×C3×C4—181no—
350C2×C3×C4—181no—
351C2×C3×C4—181no—
352C2×C3×C4—181no—
353C2×C3×C4—181no—
354C2×C3×C4—181no—
355C2×C3×C4—181no—
356C2×C3×C4—181no—
357C2×C3×C4—181no—
358C2×C3×C4—181no—
359C2×Dic3—32no—
360C2×Dic3—61no—
361C2×Dic3—61no—
362C2×Dic3—61no—
363C2×Dic3—61no—
364C2×Dic3—61no—
365C2×Dic3—61no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—61no—
371C2×Dic3—61no—
372C2×Dic3—61no—
373C2×Dic3—61no—
374C2×Dic3—62no—
375C2×Dic3—62no—
376C2×Dic3—61no—
377C2×Dic3—61no—
378C2×Dic3—61no—
379C2×Dic3—61no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—61no—
383C2×Dic3—61no—
384C2×Dic3—61no—
385C2×Dic3—92no—
386C2×Dic3—121no—
387C2×Dic3—121no—
388C2×Dic3—121no—
389C2×Dic3—121no—
390C2×Dic3—181no—
391C2×Dic3—181no—
392C2×Dic3—181no—
393C2×Dic3—181no—
394C2×Dic3—181no—
395C2×Dic3—181no—
396C2×Dic3—181no—
397C2×Dic3—181no—
398C2×Dic3—181no—
399C2×Dic3—181no—
400C2×Dic3—181no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—44no—
409C2×C32—42no—
410C2×C32—42no—
411C2×C32#87911yescharacteristic
412C2×C32#87911yescharacteristic
413C2×C32#87911yescharacteristic
414C2×C32#87911yescharacteristic
415C2×C32#88311yescharacteristic
416C2×C32#88511yescharacteristic
417C2×C32#88611yescharacteristic
418C2×C32#88611yescharacteristic
419C2×C32#92911yescharacteristic
420C2×C32#93011yescharacteristic
421C2×C32#93511yescharacteristic
422C2×C32#93511yescharacteristic
423C2×C32#93611yescharacteristic
424C2×C32#93911yescharacteristic
425C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
426C22×C4—91no—
427C22×C4—91no—
428C22×C4—91no—
429C22×C4—91no—
430C22×C4—91no—
431C22×C4—91no—
432C4⋊C4—542no—
433C22⋊C4—182no—
434C22⋊C4—182no—
435C22⋊C4—182no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—31no—
440C22×C3—31no—
441C22×C3—31no—
442C22×C3—31no—
443C22×C3—31no—
444C22×C3—31no—
445C22×C3—31no—
446C22×C3—31no—
447C22×C3—31no—
448C22×C3—31no—
449C22×C3—31no—
450C22×C3—31no—
451C22×C3—31no—
452C22×C3—31no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3—62no—
458C22×C3—62no—
459C22×C3—62no—
460C22×C3—62no—
461C22×C3—62no—
462C22×C3—62no—
463C22×C3—64no—
464C22×C3—61no—
465C22×C3—61no—
466C22×C3—61no—
467C22×C3—61no—
468C22×C3—61no—
469C22×C3—61no—
470C22×C3—61no—
471C22×C3—122no—
472C22×C3—122no—
473C22×C3—121no—
474C22×C3—121no—
475C22×C3—121no—
476C22×C3—121no—
477C22×C3—121no—
478C22×C3—121no—
479C22×C3—121no—
480C22×C3—121no—
481C22×C3—122no—
482C22×C3—122no—
483C22×C3—122no—
484C22×C3—122no—
485C22×C3#380411yescharacteristic
486C22×C3#380411yescharacteristic
487C22×C3#380411yescharacteristic
488C22×C3#380511yescharacteristic
489C22×C3#380511yescharacteristic
490C22×C3C2×#62611yescharacteristic
491C22×C3D3×#6711yescharacteristic
492C3×C4—182no—
493C3×C4—361no—
494C3×C4—361no—
495C3×C4—361no—
496C3×C4—361no—
497C3×C4—361no—
498Dic3—62no—
499Dic3—121no—
500Dic3—121no—
501Dic3—121no—
502Dic3—121no—
503Dic3—122no—
504Dic3—122no—
505Dic3—121no—
506Dic3—121no—
507Dic3—121no—
508Dic3—182no—
509Dic3—361no—
510Dic3—361no—
511Dic3—361no—
512C32—21no—
513C32#495211yescharacteristic, complemented
514C32#495311yescharacteristic, complemented
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23#620012yes—
522C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
523C2×C4—92no—
524C2×C4—181no—
525C2×C4—181no—
526C2×C4—181no—
527C2×C4—181no—
528C2×C4—181no—
529C2×C4—181no—
530C2×C4—181no—
531C2×C4—181no—
532C2×C4—181no—
533C2×C4—181no—
534C2×C4—181no—
535C2×C4—181no—
536C2×C4—181no—
537C2×C4—181no—
538C2×C4—181no—
539C2×C4—181no—
540C2×C4—181no—
541C2×C3—31no—
542C2×C3—31no—
543C2×C3—31no—
544C2×C3—31no—
545C2×C3—31no—
546C2×C3—31no—
547C2×C3—31no—
548C2×C3—31no—
549C2×C3—31no—
550C2×C3—31no—
551C2×C3—31no—
552C2×C3—31no—
553C2×C3—31no—
554C2×C3—31no—
555C2×C3—42no—
556C2×C3—61no—
557C2×C3—61no—
558C2×C3—61no—
559C2×C3—61no—
560C2×C3—61no—
561C2×C3—61no—
562C2×C3—61no—
563C2×C3—122no—
564C2×C3—122no—
565C2×C3—124no—
566C2×C3#6395611yescharacteristic
567C2×C3#6395611yescharacteristic
568C2×C3#6395911yescharacteristic
569C2×C3#6396211yescharacteristic
570C2×C3#6396611yescharacteristic
571C2×C3#6396911yescharacteristic
572C2×C3#6397811yescharacteristic
573C22—22no—
574C22—22no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22#8867111yescharacteristic
581C22#8867111yescharacteristic
582C22#8867111yescharacteristic
583C22#8867211yescharacteristic
584C22#8867211yescharacteristic
585C22C2×#620011yescharacteristic
586C22#8867611yescharacteristic
587C4—182no—
588C4—361no—
589C4—361no—
590C4—361no—
591C4—361no—
592C4—361no—
593C3—31no—
594C3—31no—
595C3—61no—
596C3#1059171711yescharacteristic
597C2—42no—
598C2#1172129711yescharacteristic
599C2#1172129711yescharacteristic
600C2#1172130111yescharacteristic
601C2#1172130411yescharacteristic
602C2#1172130711yescharacteristic
603C2#1172131111yescharacteristic
604C2#1172132211yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
Fitting, prime radicalC24×H(3,𝔽3)C22
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#397
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C2×#88448C2semidirect
5. #11720334C2semidirect
6. #11720334C2semidirect
7. #11720334C2semidirect
308. H(3,𝔽3)#397semidirect
513. C32#4952semidirect
514. C32#4953semidirect

Maximal subgroups

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

subgroupindex
C2×#885522
C2×#3757⋊C32
C2×#884482
#117203342
#105917173
#105916943