Finite Group Databasebeta

← #10596071⋊C3 #13152843

Subgroups — #10596071⋊C3 #13152843

Order 1728.

913 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)×#121C211yesmaximal, characteristic
2#64145⋊C3C211yesmaximal, characteristic, complemented
3#64144⋊C3C212yesmaximal, complemented
4D4×(C3⋊Dic3)⋊C3C212yesmaximal, complemented
5D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
6#64133⋊C3C211yesmaximal, characteristic, complemented
7C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
8#64099⋊C3C211yesmaximal, characteristic, complemented
9#64096⋊C3C211yesmaximal, characteristic, complemented
10#64088⋊C3C212yesmaximal, complemented
11#64086⋊C3C212yesmaximal, complemented
12#10596071C311yesmaximal, characteristic, complemented
13C3×#5600—31nomaximal
14C2×D4×H(3,𝔽3)C2212yes—
15C2×D4×H(3,𝔽3)C2211yescharacteristic
16C22×C4×H(3,𝔽3)C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C22×(C3⋊Dic3)⋊C3C2212yes—
20C2×#634⋊C3C2211yescharacteristic
21C4⋊C4×H(3,𝔽3)C2211yescharacteristic
22C22⋊C4×H(3,𝔽3)C2212yes—
23#3757⋊C3C2212yes—
24#3757⋊C3C2212yes—
25#3757⋊C3C2212yes—
26#3757⋊C3C2212yes—
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28C4⋊((C3⋊Dic3)⋊C3)C2212yes—
29#3750⋊C3C2212yes—
30#3750⋊C3C2212yes—
31#3750⋊C3C2212yes—
32C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
33C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
34C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
35C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
36C4×(C3⋊Dic3)⋊C3C2212yes—
37C32×#121—21no—
38C32×#121D311yescharacteristic
39C32×#121C2×C311yescharacteristic
40#64145C2×C311yescharacteristic, complemented
41#64144C2×C312yescomplemented
42D4×C3⋊Dic3C2×C312yescomplemented
43D4×C3⋊Dic3C2×C311yescharacteristic, complemented
44#64133C2×C311yescharacteristic, complemented
45C2×C4×C3⋊Dic3C2×C311yescharacteristic, complemented
46#64099C2×C311yescharacteristic, complemented
47#64096C2×C311yescharacteristic, complemented
48#64088C2×C312yescomplemented
49#64086C2×C312yescomplemented
50C3×#936—31no—
51C3×#935—32no—
52C3×D4×Dic3—32no—
53C3×D4×Dic3—31no—
54C3×#924—31no—
55C2×C3×C4×Dic3—31no—
56C3×#890—31no—
57C3×#887—31no—
58C3×#879—32no—
59C3×#877—32no—
60C23×H(3,𝔽3)C2312yes—
61C23×H(3,𝔽3)C2311yescharacteristic
62D4×H(3,𝔽3)—24no—
63D4×H(3,𝔽3)—22no—
64C2×C4×H(3,𝔽3)—22no—
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×C4×H(3,𝔽3)C2312yes—
67C2×C4×H(3,𝔽3)C2311yescharacteristic
68C2×(C3⋊Dic3)⋊C3—22no—
69C2×(C3⋊Dic3)⋊C3—22no—
70C2×(C3⋊Dic3)⋊C3—24no—
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
75C2×(C3⋊Dic3)⋊C3C2312yes—
76C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
77#634⋊C3—22no—
78C3×#529—91no—
79#5600—31no—
80#5600—91no—
81C2×C32×D4—21no—
82C2×C32×D4—22no—
83C2×C32×D4C2×D312yes—
84C2×C32×D4C2×D311yescharacteristic
85C2×C32×D4C22×C312yes—
86C2×C32×D4C22×C311yescharacteristic
87C22×C32×C4—21no—
88C22×C32×C4C2×D311yescharacteristic
89C22×C32×C4C22×C311yescharacteristic
90C22×C3⋊Dic3C22×C312yes—
91C22×C3⋊Dic3C22×C311yescharacteristic
92C22×C3⋊Dic3C22×C311yescharacteristic
93C2×#634C22×C311yescharacteristic
94C22×C3×Dic3—32no—
95C22×C3×Dic3—31no—
96C22×C3×Dic3—31no—
97C2×C3×Dic6—31no—
98C32×C4⋊C4—21no—
99C32×C4⋊C4C2×D311yescharacteristic
100C32×C4⋊C4C22×C311yescharacteristic
101C32×C22⋊C4—22no—
102C32×C22⋊C4C2×D312yes—
103C32×C22⋊C4C22×C312yes—
104#3757C22×C312yes—
105#3757C22×C312yes—
106#3757C22×C312yes—
107#3757C22×C312yes—
108C4⋊(C3⋊Dic3)C22×C312yes—
109C4⋊(C3⋊Dic3)C22×C311yescharacteristic
110#3750C22×C312yes—
111#3750C22×C312yes—
112#3750C22×C312yes—
113C4×C3⋊Dic3C22×C311yescharacteristic
114C4×C3⋊Dic3C22×C312yes—
115C4×C3⋊Dic3C22×C311yescharacteristic
116C4×C3⋊Dic3C22×C311yescharacteristic
117C4×C3⋊Dic3C22×C311yescharacteristic
118C3×#217—32no—
119C3×#217—32no—
120C3×#217—32no—
121C3×#217—32no—
122C3×C4⋊Dic3—32no—
123C3×C4⋊Dic3—31no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—32no—
127C3×C4×Dic3—31no—
128C3×C4×Dic3—32no—
129C3×C4×Dic3—31no—
130C3×C4×Dic3—31no—
131C3×C4×Dic3—31no—
132C22×H(3,𝔽3)—21no—
133C22×H(3,𝔽3)—21no—
134C22×H(3,𝔽3)—22no—
135C22×H(3,𝔽3)—22no—
136C22×H(3,𝔽3)—22no—
137C22×H(3,𝔽3)#4112yes—
138C22×H(3,𝔽3)C2411yescharacteristic
139C4×H(3,𝔽3)—21no—
140C4×H(3,𝔽3)—22no—
141C4×H(3,𝔽3)#4112yes—
142(C3⋊Dic3)⋊C3—22no—
143(C3⋊Dic3)⋊C3—21no—
144(C3⋊Dic3)⋊C3—21no—
145(C3⋊Dic3)⋊C3—21no—
146(C3⋊Dic3)⋊C3—22no—
147(C3⋊Dic3)⋊C3#4112yes—
148C3×#126—92no—
149C3×#125—91no—
150C3×#124—91no—
151C3×#123—92no—
152C3×C22⋊Q8—91no—
153C3×#121—31no—
154C3×#121—31no—
155C3×#121—61no—
156C3×#121C3×D311yescharacteristic
157C3×C4×Q8—91no—
158C3×C4×D4—91no—
159C3×C4×D4—92no—
160C3×#117—92no—
161C2×C3×C42—91no—
162#936—31no—
163#936—91no—
164#935—32no—
165#935—92no—
166D4×Dic3—32no—
167D4×Dic3—31no—
168D4×Dic3—92no—
169D4×Dic3—91no—
170#924—31no—
171#924—91no—
172C2×C4×Dic3—31no—
173C2×C4×Dic3—91no—
174#890—31no—
175#890—91no—
176#887—31no—
177#887—91no—
178#879—32no—
179#879—92no—
180#877—32no—
181#877—92no—
182C23×C32—21no—
183C23×C32—22no—
184C23×C32C22×D312yes—
185C23×C32C22×D311yescharacteristic
186C23×C32C23×C312yes—
187C23×C32C23×C311yescharacteristic
188C32×D4—24no—
189C32×D4—24no—
190C32×D4—22no—
191C32×D4—22no—
192C32×D4—42no—
193C32×D4—44no—
194C2×C32×C4—22no—
195C2×C32×C4—22no—
196C2×C32×C4—21no—
197C2×C32×C4—21no—
198C2×C32×C4—22no—
199C2×C32×C4—42no—
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C22×D312yes—
202C2×C32×C4C22×D311yescharacteristic
203C2×C32×C4C23×C311yescharacteristic
204C2×C32×C4C23×C312yes—
205C2×C32×C4C23×C311yescharacteristic
206C2×C3⋊Dic3—22no—
207C2×C3⋊Dic3—22no—
208C2×C3⋊Dic3—24no—
209C2×C3⋊Dic3C23×C311yescharacteristic
210C2×C3⋊Dic3C23×C311yescharacteristic
211C2×C3⋊Dic3C23×C311yescharacteristic
212C2×C3⋊Dic3C23×C312yes—
213C2×C3⋊Dic3C23×C312yes—
214C2×C3⋊Dic3C23×C311yescharacteristic
215#634—22no—
216C2×C3×Dic3—31no—
217C2×C3×Dic3—31no—
218C2×C3×Dic3—31no—
219C2×C3×Dic3—31no—
220C2×C3×Dic3—32no—
221C2×C3×Dic3—32no—
222C2×C3×Dic3—62no—
223C2×C3×Dic3—64no—
224C2×C3×Dic3—62no—
225C3×Dic6—62no—
226#529—91noSylow
227C2×H(3,𝔽3)—22no—
228C2×H(3,𝔽3)—42no—
229C2×H(3,𝔽3)C2×#4111yescharacteristic
230C2×H(3,𝔽3)C2×#4111yescharacteristic
231C2×H(3,𝔽3)C2×#4111yescharacteristic
232C2×C3×Q8—91no—
233C2×C3×D4—32no—
234C2×C3×D4—31no—
235C2×C3×D4—32no—
236C2×C3×D4—31no—
237C2×C3×D4—62no—
238C2×C3×D4—61no—
239C2×C3×D4C2×C3×D312yes—
240C2×C3×D4C2×C3×D311yescharacteristic
241C22×C3×C4—31no—
242C22×C3×C4—31no—
243C22×C3×C4—61no—
244C22×C3×C4—91no—
245C22×C3×C4—91no—
246C22×C3×C4—92no—
247C22×C3×C4C2×C3×D311yescharacteristic
248C22×Dic3—32no—
249C22×Dic3—31no—
250C22×Dic3—31no—
251C22×Dic3—92no—
252C22×Dic3—91no—
253C22×Dic3—91no—
254C2×Dic6—31no—
255C2×Dic6—91no—
256C3×C4⋊C4—31no—
257C3×C4⋊C4—31no—
258C3×C4⋊C4—61no—
259C3×C4⋊C4—92no—
260C3×C4⋊C4—92no—
261C3×C4⋊C4—92no—
262C3×C4⋊C4—91no—
263C3×C4⋊C4—92no—
264C3×C4⋊C4C2×C3×D311yescharacteristic
265C3×C22⋊C4—32no—
266C3×C22⋊C4—32no—
267C3×C22⋊C4—62no—
268C3×C22⋊C4—92no—
269C3×C22⋊C4—92no—
270C3×C22⋊C4—92no—
271C3×C22⋊C4—92no—
272C3×C22⋊C4C2×C3×D312yes—
273C3×C42—91no—
274C3×C42—91no—
275C3×C42—91no—
276C3×C42—91no—
277C3×C42—92no—
278#217—32no—
279#217—32no—
280#217—32no—
281#217—32no—
282#217—92no—
283#217—92no—
284#217—92no—
285#217—92no—
286C4⋊Dic3—32no—
287C4⋊Dic3—31no—
288C4⋊Dic3—92no—
289C4⋊Dic3—91no—
290Dic3⋊C4—32no—
291Dic3⋊C4—32no—
292Dic3⋊C4—32no—
293Dic3⋊C4—92no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296C4×Dic3—31no—
297C4×Dic3—32no—
298C4×Dic3—31no—
299C4×Dic3—31no—
300C4×Dic3—31no—
301C4×Dic3—91no—
302C4×Dic3—92no—
303C4×Dic3—91no—
304C4×Dic3—91no—
305C4×Dic3—91no—
306C22×C32—21no—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—22no—
311C22×C32—22no—
312C22×C32—22no—
313C22×C32—22no—
314C22×C32—22no—
315C22×C32—22no—
316C22×C32—21no—
317C22×C32—22no—
318C22×C32—41no—
319C22×C32—41no—
320C22×C32—42no—
321C22×C32—42no—
322C22×C32—42no—
323C22×C32#23712yes—
324C22×C32C3×#4112yes—
325C22×C32C23×D311yescharacteristic
326C22×C32C24×C311yescharacteristic
327C32×C4—21no—
328C32×C4—22no—
329C32×C4—21no—
330C32×C4—22no—
331C32×C4—22no—
332C32×C4—41no—
333C32×C4—42no—
334C32×C4#23712yes—
335C32×C4C3×#4112yes—
336C3⋊Dic3—22no—
337C3⋊Dic3—21no—
338C3⋊Dic3—21no—
339C3⋊Dic3—21no—
340C3⋊Dic3—22no—
341C3⋊Dic3C3×#4112yes—
342C3×Dic3—32no—
343C3×Dic3—62no—
344C3×Dic3—61no—
345C3×Dic3—61no—
346C3×Dic3—61no—
347C3×Dic3—62no—
348#126—92no—
349#125—91no—
350#124—91no—
351#123—92no—
352C22⋊Q8—91no—
353#121(C3⋊D3)⋊C311yescharacteristic
354C4×Q8—91no—
355C4×D4—91no—
356C4×D4—92no—
357#117—92no—
358C2×C42—91no—
359H(3,𝔽3)#52911yescharacteristic, complemented, Sylow
360C23×C3—31no—
361C23×C3—31no—
362C23×C3—32no—
363C23×C3—32no—
364C23×C3—61no—
365C23×C3—62no—
366C23×C3C22×C3×D312yes—
367C23×C3C22×C3×D311yescharacteristic
368C3×Q8—182no—
369C3×D4—24no—
370C3×D4—22no—
371C3×D4—62no—
372C3×D4—62no—
373C3×D4—64no—
374C3×D4—64no—
375C3×D4—122no—
376C3×D4—124no—
377C2×C3×C4—22no—
378C2×C3×C4—31no—
379C2×C3×C4—31no—
380C2×C3×C4—31no—
381C2×C3×C4—32no—
382C2×C3×C4—31no—
383C2×C3×C4—32no—
384C2×C3×C4—62no—
385C2×C3×C4—62no—
386C2×C3×C4—61no—
387C2×C3×C4—61no—
388C2×C3×C4—62no—
389C2×C3×C4—91no—
390C2×C3×C4—91no—
391C2×C3×C4—91no—
392C2×C3×C4—91no—
393C2×C3×C4—92no—
394C2×C3×C4—92no—
395C2×C3×C4—122no—
396C2×C3×C4—182no—
397C2×C3×C4—184no—
398C2×C3×C4—182no—
399C2×C3×C4C22×C3×D311yescharacteristic
400C2×C3×C4C22×C3×D312yes—
401C2×C3×C4C22×C3×D311yescharacteristic
402C2×Dic3—31no—
403C2×Dic3—31no—
404C2×Dic3—31no—
405C2×Dic3—31no—
406C2×Dic3—32no—
407C2×Dic3—32no—
408C2×Dic3—62no—
409C2×Dic3—64no—
410C2×Dic3—62no—
411C2×Dic3—91no—
412C2×Dic3—91no—
413C2×Dic3—91no—
414C2×Dic3—91no—
415C2×Dic3—92no—
416C2×Dic3—92no—
417C2×Dic3—182no—
418C2×Dic3—184no—
419C2×Dic3—182no—
420Dic6—62no—
421Dic6—182no—
422C2×C32—22no—
423C2×C32—22no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—42no—
428C2×C32—42no—
429C2×C32—42no—
430C2×C32—82no—
431C2×C32C2×#23711yescharacteristic
432C2×C32C2×#23711yescharacteristic
433C2×C32D3×#4111yescharacteristic
434C2×C32C2×C3×#4111yescharacteristic
435C2×C32C2×C3×#4111yescharacteristic
436C2×C32C2×C3×#4111yescharacteristic
437C2×Q8—91no—
438C2×D4C2×(C3⋊D3)⋊C312yes—
439C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
440C22×C4—91no—
441C22×C4—91no—
442C22×C4—92no—
443C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
444C4⋊C4—92no—
445C4⋊C4—92no—
446C4⋊C4—91no—
447C4⋊C4—92no—
448C4⋊C4—92no—
449C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
450C22⋊C4—92no—
451C22⋊C4—92no—
452C22⋊C4—92no—
453C22⋊C4—92no—
454C22⋊C4C2×(C3⋊D3)⋊C312yes—
455C42—91no—
456C42—91no—
457C42—91no—
458C42—91no—
459C42—92no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—31no—
466C22×C3—31no—
467C22×C3—32no—
468C22×C3—32no—
469C22×C3—61no—
470C22×C3—61no—
471C22×C3—61no—
472C22×C3—61no—
473C22×C3—62no—
474C22×C3—62no—
475C22×C3—62no—
476C22×C3—62no—
477C22×C3—62no—
478C22×C3—62no—
479C22×C3—61no—
480C22×C3—62no—
481C22×C3—121no—
482C22×C3—121no—
483C22×C3—122no—
484C22×C3—122no—
485C22×C3—122no—
486C22×C3C3×#23712yes—
487C22×C3C23×C3×D311yescharacteristic
488C3×C4—21no—
489C3×C4—22no—
490C3×C4—32no—
491C3×C4—32no—
492C3×C4—61no—
493C3×C4—61no—
494C3×C4—62no—
495C3×C4—62no—
496C3×C4—62no—
497C3×C4—92no—
498C3×C4—121no—
499C3×C4—122no—
500C3×C4—181no—
501C3×C4—181no—
502C3×C4—181no—
503C3×C4—182no—
504C3×C4—182no—
505C3×C4C3×#23712yes—
506Dic3—32no—
507Dic3—62no—
508Dic3—61no—
509Dic3—61no—
510Dic3—61no—
511Dic3—62no—
512Dic3—92no—
513Dic3—182no—
514Dic3—181no—
515Dic3—181no—
516Dic3—181no—
517Dic3—182no—
518C32—21no—
519C32#560011yescharacteristic, complemented
520C32C3×#52911yescharacteristic, complemented
521C23C22×(C3⋊D3)⋊C312yes—
522C23C22×(C3⋊D3)⋊C311yescharacteristic
523Q8—182no—
524D4—24no—
525D4—22no—
526C2×C4—22no—
527C2×C4—91no—
528C2×C4—91no—
529C2×C4—91no—
530C2×C4—91no—
531C2×C4—92no—
532C2×C4—92no—
533C2×C4—182no—
534C2×C4—184no—
535C2×C4—182no—
536C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
537C2×C4C22×(C3⋊D3)⋊C312yes—
538C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
539C2×C3—22no—
540C2×C3—31no—
541C2×C3—31no—
542C2×C3—31no—
543C2×C3—31no—
544C2×C3—31no—
545C2×C3—31no—
546C2×C3—42no—
547C2×C3—62no—
548C2×C3—62no—
549C2×C3—61no—
550C2×C3—61no—
551C2×C3—61no—
552C2×C3—122no—
553C2×C3—122no—
554C2×C3—122no—
555C2×C3—242no—
556C2×C3C2×C3×#23711yescharacteristic
557C2×C3C2×C3×#23711yescharacteristic
558C2×C3C3×D3×#4111yescharacteristic
559C22—21no—
560C22—21no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22#3830⋊C312yes—
565C22C23×(C3⋊D3)⋊C311yescharacteristic
566C4—21no—
567C4—22no—
568C4—92no—
569C4—181no—
570C4—181no—
571C4—182no—
572C4—181no—
573C4—182no—
574C4#3830⋊C312yes—
575C3—31no—
576C3—31no—
577C3—61no—
578C3C3×#560011yescharacteristic
579C2—22no—
580C2—42no—
581C2C2×#3830⋊C311yescharacteristic
582C2C2×#3830⋊C311yescharacteristic
583C2#41×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
Fitting, prime radicalH(3,𝔽3)×#121C2
hypercenter#121(C3⋊D3)⋊C3
hypocenterC32C3×#529
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
2. #64145⋊C3C2semidirect
3. #64144⋊C3C2semidirect
4. D4×(C3⋊Dic3)⋊C3C2semidirect
5. D4×(C3⋊Dic3)⋊C3C2semidirect
6. #64133⋊C3C2semidirect
7. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
8. #64099⋊C3C2semidirect
9. #64096⋊C3C2semidirect
10. #64088⋊C3C2semidirect
11. #64086⋊C3C2semidirect
12. #10596071C3semidirect
40. #64145C2×C3semidirect
41. #64144C2×C3semidirect
42. D4×C3⋊Dic3C2×C3semidirect
43. D4×C3⋊Dic3C2×C3semidirect
44. #64133C2×C3semidirect
45. C2×C4×C3⋊Dic3C2×C3semidirect
46. #64099C2×C3semidirect
47. #64096C2×C3semidirect
48. #64088C2×C3semidirect
49. #64086C2×C3semidirect
359. H(3,𝔽3)#529semidirect
519. C32#5600semidirect
520. C32C3×#529semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1212
#64145⋊C32
#64144⋊C32
D4×(C3⋊Dic3)⋊C32
#64133⋊C32
C2×C4×(C3⋊Dic3)⋊C32
#64099⋊C32
#64096⋊C32
#64088⋊C32
#64086⋊C32
#105960713
C3×#56003