Finite Group Databasebeta

← #10592647⋊C3 #13142501

Subgroups — #10592647⋊C3 #13142501

Order 1728.

783 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#3757⋊C3C211yesmaximal, characteristic
3C2×#3757⋊C3C211yesmaximal, characteristic
4C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
5C2×#3750⋊C3C211yesmaximal, characteristic, complemented
6C2×#3750⋊C3C211yesmaximal, characteristic, complemented
7#63658⋊C3C211yesmaximal, characteristic, complemented
8#10592647C311yesmaximal, characteristic, complemented
9C3×#4948—31nomaximal
10C24×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22⋊C4×H(3,𝔽3)—22no—
18#3757⋊C3—22no—
19#3757⋊C3—22no—
20C4⋊((C3⋊Dic3)⋊C3)—22no—
21#3750⋊C3—22no—
22#3750⋊C3—22no—
23C2×C32×C22⋊C4—21no—
24C2×C32×C22⋊C4D311yescharacteristic
25C2×C32×C22⋊C4C2×C311yescharacteristic
26C2×#3757C2×C311yescharacteristic
27C2×#3757C2×C311yescharacteristic
28C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
29C2×#3750C2×C311yescharacteristic, complemented
30C2×#3750C2×C311yescharacteristic, complemented
31C2×C3×#217—31no—
32C2×C3×#217—31no—
33C2×C3×C4⋊Dic3—31no—
34C2×C3×Dic3⋊C4—31no—
35C2×C3×Dic3⋊C4—31no—
36#63658C2×C311yescharacteristic, complemented
37C3×C3⋊#95—31no—
38C23×H(3,𝔽3)—21no—
39C23×H(3,𝔽3)—21no—
40C23×H(3,𝔽3)—21no—
41C23×H(3,𝔽3)—21no—
42C23×H(3,𝔽3)—21no—
43C23×H(3,𝔽3)—21no—
44C23×H(3,𝔽3)Q812yes—
45C23×H(3,𝔽3)C2311yescharacteristic
46C2×C4×H(3,𝔽3)—21no—
47C2×C4×H(3,𝔽3)—21no—
48C2×C4×H(3,𝔽3)—21no—
49C2×C4×H(3,𝔽3)—21no—
50C2×C4×H(3,𝔽3)—21no—
51C2×C4×H(3,𝔽3)D412yes—
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)⋊C3—21no—
62C2×(C3⋊Dic3)⋊C3D412yes—
63C2×(C3⋊Dic3)⋊C3D412yes—
64C3×#396—91no—
65#4948—31no—
66#4948—91no—
67C24×C32—21no—
68C24×C32C2×D311yescharacteristic
69C24×C32C22×C311yescharacteristic
70C22×C32×C4—21no—
71C22×C32×C4—21no—
72C22×C32×C4C2×D311yescharacteristic
73C22×C32×C4C2×D311yescharacteristic
74C22×C32×C4C22×C311yescharacteristic
75C22×C32×C4C22×C311yescharacteristic
76C22×C3⋊Dic3C22×C311yescharacteristic
77C22×C3⋊Dic3C22×C311yescharacteristic
78C22×C3⋊Dic3C22×C311yescharacteristic
79C22×C3⋊Dic3C22×C311yescharacteristic
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—31no—
83C22×C3×Dic3—31no—
84C32×C22⋊C4—24no—
85C32×C22⋊C4—22no—
86C32×C22⋊C4—22no—
87#3757—22no—
88#3757—22no—
89C4⋊(C3⋊Dic3)—22no—
90#3750—22no—
91#3750—22no—
92C3×#217—62no—
93C3×#217—62no—
94C3×C4⋊Dic3—62no—
95C3×Dic3⋊C4—62no—
96C3×Dic3⋊C4—62no—
97C22×H(3,𝔽3)—22no—
98C22×H(3,𝔽3)—22no—
99C22×H(3,𝔽3)—22no—
100C22×H(3,𝔽3)—41no—
101C22×H(3,𝔽3)—41no—
102C22×H(3,𝔽3)—41no—
103C22×H(3,𝔽3)—41no—
104C22×H(3,𝔽3)C2×D411yescharacteristic
105C22×H(3,𝔽3)C2×D411yescharacteristic
106C22×H(3,𝔽3)C2×D411yescharacteristic
107C22×H(3,𝔽3)C2×Q811yescharacteristic
108C22×H(3,𝔽3)#4111yescharacteristic
109C22×H(3,𝔽3)#4111yescharacteristic
110C22×H(3,𝔽3)#4111yescharacteristic
111C4×H(3,𝔽3)—22no—
112C4×H(3,𝔽3)—41no—
113(C3⋊Dic3)⋊C3—22no—
114(C3⋊Dic3)⋊C3—22no—
115(C3⋊Dic3)⋊C3—41no—
116(C3⋊Dic3)⋊C3—41no—
117C2×C3×C4⋊C4—91no—
118C2×C3×C4⋊C4—91no—
119C2×C3×C4⋊C4—91no—
120C2×C3×C22⋊C4—31no—
121C2×C3×C22⋊C4—31no—
122C2×C3×C22⋊C4—61no—
123C2×C3×C22⋊C4—91no—
124C2×C3×C22⋊C4—91no—
125C2×C3×C22⋊C4C3×D311yescharacteristic
126C2×#217—31no—
127C2×#217—31no—
128C2×#217—91no—
129C2×#217—91no—
130C2×C4⋊Dic3—31no—
131C2×C4⋊Dic3—91no—
132C2×Dic3⋊C4—31no—
133C2×Dic3⋊C4—31no—
134C2×Dic3⋊C4—91no—
135C2×Dic3⋊C4—91no—
136C3×#95—91no—
137C3⋊#95—31no—
138C3⋊#95—91no—
139C23×C32—22no—
140C23×C32—22no—
141C23×C32—21no—
142C23×C32—21no—
143C23×C32—21no—
144C23×C32—21no—
145C23×C32—21no—
146C23×C32—21no—
147C23×C32—21no—
148C23×C32—21no—
149C23×C32—21no—
150C23×C32—21no—
151C23×C32—21no—
152C23×C32—21no—
153C23×C32—22no—
154C23×C32—21no—
155C23×C32—41no—
156C23×C32—41no—
157C23×C32—41no—
158C23×C32—41no—
159C23×C32Dic612yes—
160C23×C32C3×Q812yes—
161C23×C32C22×D311yescharacteristic
162C23×C32C23×C311yescharacteristic
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—22no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—22no—
175C2×C32×C4—41no—
176C2×C32×C4—41no—
177C2×C32×C4—41no—
178C2×C32×C4—41no—
179C2×C32×C4#6712yes—
180C2×C32×C4C3×D412yes—
181C2×C3⋊Dic3—21no—
182C2×C3⋊Dic3—21no—
183C2×C3⋊Dic3—21no—
184C2×C3⋊Dic3—21no—
185C2×C3⋊Dic3—21no—
186C2×C3⋊Dic3—21no—
187C2×C3⋊Dic3—21no—
188C2×C3⋊Dic3—21no—
189C2×C3⋊Dic3—21no—
190C2×C3⋊Dic3—21no—
191C2×C3⋊Dic3C3×D412yes—
192C2×C3⋊Dic3C3×D412yes—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—61no—
205#396—91noSylow
206C2×H(3,𝔽3)—42no—
207C2×H(3,𝔽3)#12111yescharacteristic
208C2×H(3,𝔽3)#12111yescharacteristic
209C2×H(3,𝔽3)#12111yescharacteristic
210C2×H(3,𝔽3)C22⋊Q811yescharacteristic
211C2×H(3,𝔽3)C22⋊Q811yescharacteristic
212C2×H(3,𝔽3)C22⋊Q811yescharacteristic
213C2×H(3,𝔽3)#12611yescharacteristic
214C24×C3—31no—
215C24×C3—31no—
216C24×C3—61no—
217C24×C3C2×C3×D311yescharacteristic
218C22×C3×C4—31no—
219C22×C3×C4—31no—
220C22×C3×C4—31no—
221C22×C3×C4—31no—
222C22×C3×C4—61no—
223C22×C3×C4—61no—
224C22×C3×C4—91no—
225C22×C3×C4—91no—
226C22×C3×C4—91no—
227C22×C3×C4—91no—
228C22×C3×C4C2×C3×D311yescharacteristic
229C22×C3×C4C2×C3×D311yescharacteristic
230C22×Dic3—31no—
231C22×Dic3—31no—
232C22×Dic3—31no—
233C22×Dic3—31no—
234C22×Dic3—91no—
235C22×Dic3—91no—
236C22×Dic3—91no—
237C22×Dic3—91no—
238C3×C4⋊C4—182no—
239C3×C4⋊C4—182no—
240C3×C4⋊C4—182no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—62no—
243C3×C22⋊C4—62no—
244C3×C22⋊C4—64no—
245C3×C22⋊C4—182no—
246C3×C22⋊C4—182no—
247#217—62no—
248#217—62no—
249#217—182no—
250#217—182no—
251C4⋊Dic3—62no—
252C4⋊Dic3—182no—
253Dic3⋊C4—62no—
254Dic3⋊C4—62no—
255Dic3⋊C4—182no—
256Dic3⋊C4—182no—
257C22×C32—24no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32—22no—
261C22×C32—22no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32—21no—
267C22×C32—21no—
268C22×C32—21no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—42no—
272C22×C32—42no—
273C22×C32—42no—
274C22×C32—42no—
275C22×C32—42no—
276C22×C32—42no—
277C22×C32—41no—
278C22×C32—41no—
279C22×C32—41no—
280C22×C32—41no—
281C22×C32—41no—
282C22×C32—41no—
283C22×C32—41no—
284C22×C32—41no—
285C22×C32C2×Dic611yescharacteristic
286C22×C32#23511yescharacteristic
287C22×C32D3×D411yescharacteristic
288C22×C32D3×D411yescharacteristic
289C22×C32#23711yescharacteristic
290C22×C32#23711yescharacteristic
291C22×C32C2×#6711yescharacteristic
292C22×C32C2×C3×D411yescharacteristic
293C22×C32C2×C3×D411yescharacteristic
294C22×C32C2×C3×D411yescharacteristic
295C22×C32C2×C3×Q811yescharacteristic
296C22×C32C3×#4111yescharacteristic
297C22×C32C3×#4111yescharacteristic
298C22×C32C3×#4111yescharacteristic
299C32×C4—22no—
300C32×C4—22no—
301C32×C4—42no—
302C32×C4—42no—
303C32×C4—41no—
304C32×C4—41no—
305C3⋊Dic3—22no—
306C3⋊Dic3—22no—
307C3⋊Dic3—41no—
308C3⋊Dic3—41no—
309C3×Dic3—62no—
310C3×Dic3—62no—
311C3×Dic3—121no—
312C3×Dic3—121no—
313C2×C4⋊C4—91no—
314C2×C4⋊C4—91no—
315C2×C4⋊C4—91no—
316C2×C22⋊C4—91no—
317C2×C22⋊C4—91no—
318C2×C22⋊C4(C3⋊D3)⋊C311yescharacteristic
319#95—91no—
320H(3,𝔽3)#39611yescharacteristic, complemented, Sylow
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—32no—
328C23×C3—32no—
329C23×C3—31no—
330C23×C3—31no—
331C23×C3—61no—
332C23×C3—61no—
333C23×C3—61no—
334C23×C3—61no—
335C23×C3—61no—
336C23×C3—61no—
337C23×C3—61no—
338C23×C3—61no—
339C23×C3—61no—
340C23×C3—61no—
341C23×C3—61no—
342C23×C3—61no—
343C23×C3—62no—
344C23×C3—62no—
345C23×C3—62no—
346C23×C3—61no—
347C23×C3—121no—
348C23×C3—121no—
349C23×C3—121no—
350C23×C3—121no—
351C23×C3C3×Dic612yes—
352C23×C3C22×C3×D311yescharacteristic
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—32no—
359C2×C3×C4—32no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—61no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—62no—
371C2×C3×C4—62no—
372C2×C3×C4—92no—
373C2×C3×C4—92no—
374C2×C3×C4—121no—
375C2×C3×C4—121no—
376C2×C3×C4—121no—
377C2×C3×C4—121no—
378C2×C3×C4—181no—
379C2×C3×C4—181no—
380C2×C3×C4—181no—
381C2×C3×C4—181no—
382C2×C3×C4—181no—
383C2×C3×C4—181no—
384C2×C3×C4—181no—
385C2×C3×C4—181no—
386C2×C3×C4—181no—
387C2×C3×C4—181no—
388C2×C3×C4C3×#6712yes—
389C2×Dic3—32no—
390C2×Dic3—32no—
391C2×Dic3—61no—
392C2×Dic3—61no—
393C2×Dic3—61no—
394C2×Dic3—61no—
395C2×Dic3—61no—
396C2×Dic3—61no—
397C2×Dic3—61no—
398C2×Dic3—61no—
399C2×Dic3—61no—
400C2×Dic3—61no—
401C2×Dic3—92no—
402C2×Dic3—92no—
403C2×Dic3—181no—
404C2×Dic3—181no—
405C2×Dic3—181no—
406C2×Dic3—181no—
407C2×Dic3—181no—
408C2×Dic3—181no—
409C2×Dic3—181no—
410C2×Dic3—181no—
411C2×Dic3—181no—
412C2×Dic3—181no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—44no—
421C2×C32—42no—
422C2×C32—42no—
423C2×C32#87811yescharacteristic
424C2×C32#87811yescharacteristic
425C2×C32#87911yescharacteristic
426C2×C32#88411yescharacteristic
427C2×C32#92411yescharacteristic
428C2×C32#93811yescharacteristic
429C2×C32#93911yescharacteristic
430C2×C32C3×#12111yescharacteristic
431C2×C32C3×#12111yescharacteristic
432C2×C32C3×#12111yescharacteristic
433C2×C32C3×C22⋊Q811yescharacteristic
434C2×C32C3×C22⋊Q811yescharacteristic
435C2×C32C3×C22⋊Q811yescharacteristic
436C2×C32C3×#12611yescharacteristic
437C24C2×(C3⋊D3)⋊C311yescharacteristic
438C22×C4—91no—
439C22×C4—91no—
440C22×C4—91no—
441C22×C4—91no—
442C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
443C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
444C4⋊C4—182no—
445C4⋊C4—182no—
446C4⋊C4—182no—
447C22⋊C4—22no—
448C22⋊C4—182no—
449C22⋊C4—182no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—31no—
454C22×C3—31no—
455C22×C3—31no—
456C22×C3—31no—
457C22×C3—31no—
458C22×C3—31no—
459C22×C3—31no—
460C22×C3—31no—
461C22×C3—31no—
462C22×C3—31no—
463C22×C3—31no—
464C22×C3—31no—
465C22×C3—31no—
466C22×C3—31no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3—62no—
472C22×C3—62no—
473C22×C3—62no—
474C22×C3—62no—
475C22×C3—62no—
476C22×C3—62no—
477C22×C3—64no—
478C22×C3—61no—
479C22×C3—61no—
480C22×C3—61no—
481C22×C3—61no—
482C22×C3—61no—
483C22×C3—61no—
484C22×C3—61no—
485C22×C3—122no—
486C22×C3—122no—
487C22×C3—121no—
488C22×C3—121no—
489C22×C3—121no—
490C22×C3—121no—
491C22×C3—121no—
492C22×C3—121no—
493C22×C3—121no—
494C22×C3—121no—
495C22×C3—122no—
496C22×C3—122no—
497C22×C3—122no—
498C22×C3—122no—
499C22×C3C2×C3×Dic611yescharacteristic
500C22×C3C3×#23511yescharacteristic
501C22×C3C3×D3×D411yescharacteristic
502C22×C3C3×D3×D411yescharacteristic
503C22×C3C3×#23711yescharacteristic
504C22×C3C3×#23711yescharacteristic
505C22×C3C2×C3×#6711yescharacteristic
506C3×C4—22no—
507C3×C4—41no—
508C3×C4—62no—
509C3×C4—62no—
510C3×C4—122no—
511C3×C4—121no—
512C3×C4—121no—
513C3×C4—122no—
514C3×C4—182no—
515C3×C4—182no—
516C3×C4—361no—
517C3×C4—361no—
518Dic3—62no—
519Dic3—62no—
520Dic3—121no—
521Dic3—121no—
522Dic3—182no—
523Dic3—182no—
524Dic3—361no—
525Dic3—361no—
526C32—21no—
527C32#494811yescharacteristic, complemented
528C32C3×#39611yescharacteristic, complemented
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23#634⋊C312yes—
536C23C22×(C3⋊D3)⋊C311yescharacteristic
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—92no—
543C2×C4—92no—
544C2×C4—181no—
545C2×C4—181no—
546C2×C4—181no—
547C2×C4—181no—
548C2×C4—181no—
549C2×C4—181no—
550C2×C4—181no—
551C2×C4—181no—
552C2×C4—181no—
553C2×C4—181no—
554C2×C4#638⋊C312yes—
555C2×C3—31no—
556C2×C3—31no—
557C2×C3—31no—
558C2×C3—31no—
559C2×C3—31no—
560C2×C3—31no—
561C2×C3—31no—
562C2×C3—31no—
563C2×C3—31no—
564C2×C3—31no—
565C2×C3—31no—
566C2×C3—31no—
567C2×C3—31no—
568C2×C3—31no—
569C2×C3—42no—
570C2×C3—61no—
571C2×C3—61no—
572C2×C3—61no—
573C2×C3—61no—
574C2×C3—61no—
575C2×C3—61no—
576C2×C3—61no—
577C2×C3—122no—
578C2×C3—122no—
579C2×C3—124no—
580C2×C3C3×#87811yescharacteristic
581C2×C3C3×#87811yescharacteristic
582C2×C3C3×#87911yescharacteristic
583C2×C3C3×#88411yescharacteristic
584C2×C3C3×#92411yescharacteristic
585C2×C3C3×#93811yescharacteristic
586C2×C3C3×#93911yescharacteristic
587C22—22no—
588C22—22no—
589C22—22no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22C2×#634⋊C311yescharacteristic
595C22#3828⋊C311yescharacteristic
596C22D4×(C3⋊D3)⋊C311yescharacteristic
597C22D4×(C3⋊D3)⋊C311yescharacteristic
598C22#3830⋊C311yescharacteristic
599C22#3830⋊C311yescharacteristic
600C22C2×#638⋊C311yescharacteristic
601C4—22no—
602C4—41no—
603C4—182no—
604C4—182no—
605C4—361no—
606C4—361no—
607C3—31no—
608C3—31no—
609C3—61no—
610C3C3×#494811yescharacteristic
611C2—42no—
612C2#64087⋊C311yescharacteristic
613C2#64087⋊C311yescharacteristic
614C2#64088⋊C311yescharacteristic
615C2#64093⋊C311yescharacteristic
616C2#64133⋊C311yescharacteristic
617C2#64147⋊C311yescharacteristic
618C2#64148⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC23×C32C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingC2×C22⋊C4×H(3,𝔽3)C2
prime radicalC24×H(3,𝔽3)C22
hypercenterC2×C22⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#396
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. C2×#3750⋊C3C2semidirect
6. C2×#3750⋊C3C2semidirect
7. #63658⋊C3C2semidirect
8. #10592647C3semidirect
28. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
29. C2×#3750C2×C3semidirect
30. C2×#3750C2×C3semidirect
36. #63658C2×C3semidirect
320. H(3,𝔽3)#396semidirect
527. C32#4948semidirect
528. C32C3×#396semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×H(3,𝔽3)2
C2×#3757⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#3750⋊C32
#63658⋊C32
#105926473
C3×#49483