Finite Group Databasebeta

← #10592369⋊C3 #13141358

Subgroups — #10592369⋊C3 #13141358

Order 1728.

741 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3752⋊C3C211yesmaximal, characteristic
2C2×#3752⋊C3C211yesmaximal, characteristic
3C2×#3752⋊C3C211yesmaximal, characteristic
4C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
5C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
6H(3,𝔽3)×#95C211yesmaximal, characteristic, complemented
7#63658⋊C3C211yesmaximal, characteristic, complemented
8#10592369C311yesmaximal, characteristic, complemented
9C3×#4670—31nomaximal
10C23×(C3⋊D3)⋊C3C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17#3752⋊C3—22no—
18#3752⋊C3—22no—
19#3752⋊C3C414yescomplemented
20C4⋊((C3⋊Dic3)⋊C3)—22no—
21C4×(C3⋊Dic3)⋊C3—22no—
22C2×#3752C2×C311yescharacteristic
23C2×#3752C2×C311yescharacteristic
24C2×#3752C2×C311yescharacteristic
25C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
26C2×C4×C3⋊Dic3C2×C311yescharacteristic, complemented
27C2×C3×(C2×D3)⋊C4—31no—
28C2×C3×(C2×D3)⋊C4—31no—
29C2×C3×(C2×D3)⋊C4—31no—
30C2×C3×C4⋊Dic3—31no—
31C2×C3×C4×Dic3—31no—
32C32×#95—21no—
33C32×#95D311yescharacteristic, complemented
34C32×#95C2×C311yescharacteristic, complemented
35#63658C2×C311yescharacteristic, complemented
36C3×C3⋊#95—31no—
37C23×H(3,𝔽3)C2311yescharacteristic
38C22×(C3⋊D3)⋊C3—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3—21no—
41C22×(C3⋊D3)⋊C3—21no—
42C22×(C3⋊D3)⋊C3—21no—
43C22×(C3⋊D3)⋊C3—21no—
44C22×(C3⋊D3)⋊C3C2×C412yes—
45C2×C4×H(3,𝔽3)—21no—
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)—21no—
52C2×C4×H(3,𝔽3)C2×C412yes—
53C2×C4×H(3,𝔽3)D412yes—
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)⋊C3C2×C412yes—
62C2×(C3⋊Dic3)⋊C3D412yes—
63C3×#388—91no—
64#4670—31no—
65#4670—91no—
66C23×C3⋊D3C22×C311yescharacteristic
67C23×C3×D3—31no—
68C22×C32×C4—21no—
69C22×C32×C4—21no—
70C22×C32×C4—21no—
71C22×C32×C4C2×D311yescharacteristic
72C22×C32×C4C2×D311yescharacteristic
73C22×C32×C4C2×D311yescharacteristic
74C22×C32×C4C22×C311yescharacteristic
75C22×C32×C4C22×C311yescharacteristic
76C22×C32×C4C22×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—
83#3752—22no—
84#3752—22no—
85#3752C3×C414yescomplemented
86C4⋊(C3⋊Dic3)—22no—
87C4×C3⋊Dic3—22no—
88C3×(C2×D3)⋊C4—34no—
89C3×(C2×D3)⋊C4—62no—
90C3×(C2×D3)⋊C4—62no—
91C3×C4⋊Dic3—62no—
92C3×C4×Dic3—62no—
93C22×H(3,𝔽3)C22×C411yescharacteristic
94C22×H(3,𝔽3)C2×D411yescharacteristic
95C22×H(3,𝔽3)C2×D411yescharacteristic
96C22×H(3,𝔽3)#4111yescharacteristic
97C22×H(3,𝔽3)#4111yescharacteristic
98C22×H(3,𝔽3)#4111yescharacteristic
99C22×H(3,𝔽3)#4111yescharacteristic
100C2×(C3⋊D3)⋊C3—22no—
101C2×(C3⋊D3)⋊C3—22no—
102C2×(C3⋊D3)⋊C3—22no—
103C2×(C3⋊D3)⋊C3—41no—
104C2×(C3⋊D3)⋊C3—41no—
105C2×(C3⋊D3)⋊C3—41no—
106C2×(C3⋊D3)⋊C3—41no—
107C4×H(3,𝔽3)—22no—
108C4×H(3,𝔽3)—22no—
109C4×H(3,𝔽3)—41no—
110(C3⋊Dic3)⋊C3—22no—
111(C3⋊Dic3)⋊C3—22no—
112(C3⋊Dic3)⋊C3—41no—
113C2×C3×C4⋊C4—91no—
114C2×C3×C22⋊C4—91no—
115C2×C3×C22⋊C4—91no—
116C2×C3×C22⋊C4—91no—
117C2×C3×C42—91no—
118C2×(C2×D3)⋊C4—31no—
119C2×(C2×D3)⋊C4—31no—
120C2×(C2×D3)⋊C4—31no—
121C2×(C2×D3)⋊C4—91no—
122C2×(C2×D3)⋊C4—91no—
123C2×(C2×D3)⋊C4—91no—
124C2×C4⋊Dic3—31no—
125C2×C4⋊Dic3—91no—
126C2×C4×Dic3—31no—
127C2×C4×Dic3—91no—
128C3×#95—31no—
129C3×#95—31no—
130C3×#95—61no—
131C3×#95—91no—
132C3×#95C3×D311yescharacteristic
133C3⋊#95—31no—
134C3⋊#95—91no—
135C23×C32—21no—
136C23×C32C22×D311yescharacteristic
137C23×C32C23×C311yescharacteristic
138C22×C3⋊D3—21no—
139C22×C3⋊D3—21no—
140C22×C3⋊D3—21no—
141C22×C3⋊D3—21no—
142C22×C3⋊D3—21no—
143C22×C3⋊D3—21no—
144C22×C3⋊D3C2×C3×C412yes—
145C22×C3×D3—32no—
146C22×C3×D3—61no—
147C22×C3×D3—61no—
148C22×C3×D3—61no—
149C22×C3×D3—61no—
150C22×C3×D3—61no—
151C22×C3×D3—61no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—22no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—22no—
168C2×C32×C4—22no—
169C2×C32×C4—41no—
170C2×C32×C4—41no—
171C2×C32×C4—41no—
172C2×C32×C4—41no—
173C2×C32×C4—41no—
174C2×C32×C4—41no—
175C2×C32×C4C4×D312yes—
176C2×C32×C4D1212yes—
177C2×C32×C4C2×C3×C412yes—
178C2×C32×C4C3×D412yes—
179C2×C3⋊Dic3—21no—
180C2×C3⋊Dic3—21no—
181C2×C3⋊Dic3—21no—
182C2×C3⋊Dic3—21no—
183C2×C3⋊Dic3—21no—
184C2×C3⋊Dic3—21no—
185C2×C3⋊Dic3—21no—
186C2×C3⋊Dic3C2×C3×C412yes—
187C2×C3⋊Dic3C3×D412yes—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197#388—91noSylow
198C2×H(3,𝔽3)#11711yescharacteristic
199C2×H(3,𝔽3)C4×D411yescharacteristic
200C2×H(3,𝔽3)C4×D411yescharacteristic
201C2×H(3,𝔽3)#12111yescharacteristic
202C2×H(3,𝔽3)#12311yescharacteristic
203C2×H(3,𝔽3)#12411yescharacteristic
204C2×H(3,𝔽3)#12611yescharacteristic
205(C3⋊D3)⋊C3—42no—
206C24×C3—91no—
207C23×D3—31no—
208C23×D3—91no—
209C22×C3×C4—31no—
210C22×C3×C4—31no—
211C22×C3×C4—31no—
212C22×C3×C4—31no—
213C22×C3×C4—31no—
214C22×C3×C4—31no—
215C22×C3×C4—61no—
216C22×C3×C4—61no—
217C22×C3×C4—61no—
218C22×C3×C4—91no—
219C22×C3×C4—91no—
220C22×C3×C4—91no—
221C22×C3×C4C2×C3×D311yescharacteristic
222C22×C3×C4C2×C3×D311yescharacteristic
223C22×C3×C4C2×C3×D311yescharacteristic
224C22×Dic3—31no—
225C22×Dic3—31no—
226C22×Dic3—31no—
227C22×Dic3—91no—
228C22×Dic3—91no—
229C22×Dic3—91no—
230C3×C4⋊C4—182no—
231C3×C22⋊C4—94no—
232C3×C22⋊C4—182no—
233C3×C22⋊C4—182no—
234C3×C42—182no—
235(C2×D3)⋊C4—34no—
236(C2×D3)⋊C4—62no—
237(C2×D3)⋊C4—62no—
238(C2×D3)⋊C4—94no—
239(C2×D3)⋊C4—182no—
240(C2×D3)⋊C4—182no—
241C4⋊Dic3—62no—
242C4⋊Dic3—182no—
243C4×Dic3—62no—
244C4×Dic3—182no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32C2×C4×D311yescharacteristic
253C22×C32C2×D1211yescharacteristic
254C22×C32#23511yescharacteristic
255C22×C32D3×D411yescharacteristic
256C22×C32#23711yescharacteristic
257C22×C32#23711yescharacteristic
258C22×C32#23911yescharacteristic
259C22×C32C22×C3×C411yescharacteristic
260C22×C32C2×C3×D411yescharacteristic
261C22×C32C2×C3×D411yescharacteristic
262C22×C32C3×#4111yescharacteristic
263C22×C32C3×#4111yescharacteristic
264C22×C32C3×#4111yescharacteristic
265C22×C32C3×#4111yescharacteristic
266C2×C3⋊D3—22no—
267C2×C3⋊D3—22no—
268C2×C3⋊D3—22no—
269C2×C3⋊D3—41no—
270C2×C3⋊D3—41no—
271C2×C3⋊D3—41no—
272C2×C3⋊D3—41no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C2×C3×D3—121no—
277C2×C3×D3—121no—
278C2×C3×D3—121no—
279C2×C3×D3—121no—
280C32×C4—22no—
281C32×C4—22no—
282C32×C4—22no—
283C32×C4—22no—
284C32×C4—42no—
285C32×C4—42no—
286C32×C4—42no—
287C32×C4—41no—
288C32×C4—41no—
289C3⋊Dic3—22no—
290C3⋊Dic3—22no—
291C3⋊Dic3—41no—
292C3×Dic3—62no—
293C3×Dic3—62no—
294C3×Dic3—121no—
295C2×C4⋊C4—91no—
296C2×C22⋊C4—91no—
297C2×C22⋊C4—91no—
298C2×C22⋊C4—91no—
299C2×C42—91no—
300#95—91no—
301#95(C3⋊D3)⋊C311yescharacteristic, complemented
302H(3,𝔽3)#38811yescharacteristic, complemented, Sylow
303C23×C3—31no—
304C23×C3—31no—
305C23×C3—61no—
306C23×C3—92no—
307C23×C3—181no—
308C23×C3—181no—
309C23×C3—181no—
310C23×C3—181no—
311C23×C3—181no—
312C23×C3—181no—
313C23×C3C22×C3×D311yescharacteristic
314C22×D3—32no—
315C22×D3—61no—
316C22×D3—61no—
317C22×D3—61no—
318C22×D3—61no—
319C22×D3—61no—
320C22×D3—61no—
321C22×D3—92no—
322C22×D3—181no—
323C22×D3—181no—
324C22×D3—181no—
325C22×D3—181no—
326C22×D3—181no—
327C22×D3—181no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—32no—
336C2×C3×C4—32no—
337C2×C3×C4—32no—
338C2×C3×C4—32no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—61no—
352C2×C3×C4—61no—
353C2×C3×C4—62no—
354C2×C3×C4—62no—
355C2×C3×C4—62no—
356C2×C3×C4—92no—
357C2×C3×C4—92no—
358C2×C3×C4—121no—
359C2×C3×C4—121no—
360C2×C3×C4—121no—
361C2×C3×C4—121no—
362C2×C3×C4—121no—
363C2×C3×C4—121no—
364C2×C3×C4—181no—
365C2×C3×C4—181no—
366C2×C3×C4—181no—
367C2×C3×C4—181no—
368C2×C3×C4—181no—
369C2×C3×C4—181no—
370C2×C3×C4—181no—
371C2×C3×C4C3×C4×D312yes—
372C2×C3×C4C3×D1212yes—
373C2×Dic3—32no—
374C2×Dic3—32no—
375C2×Dic3—61no—
376C2×Dic3—61no—
377C2×Dic3—61no—
378C2×Dic3—61no—
379C2×Dic3—61no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—92no—
383C2×Dic3—92no—
384C2×Dic3—181no—
385C2×Dic3—181no—
386C2×Dic3—181no—
387C2×Dic3—181no—
388C2×Dic3—181no—
389C2×Dic3—181no—
390C2×Dic3—181no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32C4×D1211yescharacteristic
399C2×C32#67⋊C411yescharacteristic
400C2×C32#88511yescharacteristic
401C2×C32#88611yescharacteristic
402C2×C32#89211yescharacteristic
403C2×C32#89511yescharacteristic
404C2×C32#89811yescharacteristic
405C2×C32C3×#11711yescharacteristic
406C2×C32C3×C4×D411yescharacteristic
407C2×C32C3×C4×D411yescharacteristic
408C2×C32C3×#12111yescharacteristic
409C2×C32C3×#12311yescharacteristic
410C2×C32C3×#12411yescharacteristic
411C2×C32C3×#12611yescharacteristic
412C3⋊D3—42no—
413C3×D3—122no—
414C24—91no—
415C22×C4—91no—
416C22×C4—91no—
417C22×C4—91no—
418C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
419C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
420C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
421C4⋊C4—182no—
422C22⋊C4—94no—
423C22⋊C4—182no—
424C22⋊C4—182no—
425C42—182no—
426C22×C3—31no—
427C22×C3—31no—
428C22×C3—31no—
429C22×C3—31no—
430C22×C3—31no—
431C22×C3—31no—
432C22×C3—31no—
433C22×C3—31no—
434C22×C3—31no—
435C22×C3—31no—
436C22×C3—31no—
437C22×C3—31no—
438C22×C3—31no—
439C22×C3—31no—
440C22×C3—61no—
441C22×C3—61no—
442C22×C3—61no—
443C22×C3—61no—
444C22×C3—61no—
445C22×C3—61no—
446C22×C3—61no—
447C22×C3—182no—
448C22×C3—182no—
449C22×C3—182no—
450C22×C3—361no—
451C22×C3—361no—
452C22×C3—361no—
453C22×C3—361no—
454C22×C3C2×C3×C4×D311yescharacteristic
455C22×C3C2×C3×D1211yescharacteristic
456C22×C3C3×#23511yescharacteristic
457C22×C3C3×D3×D411yescharacteristic
458C22×C3C3×#23711yescharacteristic
459C22×C3C3×#23711yescharacteristic
460C22×C3C3×#23911yescharacteristic
461C2×D3—62no—
462C2×D3—62no—
463C2×D3—62no—
464C2×D3—121no—
465C2×D3—121no—
466C2×D3—121no—
467C2×D3—121no—
468C2×D3—182no—
469C2×D3—182no—
470C2×D3—182no—
471C2×D3—361no—
472C2×D3—361no—
473C2×D3—361no—
474C2×D3—361no—
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—41no—
478C3×C4—62no—
479C3×C4—62no—
480C3×C4—62no—
481C3×C4—62no—
482C3×C4—122no—
483C3×C4—122no—
484C3×C4—121no—
485C3×C4—121no—
486C3×C4—122no—
487C3×C4—182no—
488C3×C4—182no—
489C3×C4—361no—
490Dic3—62no—
491Dic3—62no—
492Dic3—121no—
493Dic3—182no—
494Dic3—182no—
495Dic3—361no—
496C32—21no—
497C32#467011yescharacteristic, complemented
498C32C3×#38811yescharacteristic, complemented
499C23—92no—
500C23—181no—
501C23—181no—
502C23—181no—
503C23—181no—
504C23—181no—
505C23—181no—
506C23C22×(C3⋊D3)⋊C311yescharacteristic
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—92no—
515C2×C4—92no—
516C2×C4—181no—
517C2×C4—181no—
518C2×C4—181no—
519C2×C4—181no—
520C2×C4—181no—
521C2×C4—181no—
522C2×C4—181no—
523C2×C4C4×(C3⋊D3)⋊C312yes—
524C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
525C2×C3—31no—
526C2×C3—31no—
527C2×C3—31no—
528C2×C3—31no—
529C2×C3—31no—
530C2×C3—31no—
531C2×C3—31no—
532C2×C3—31no—
533C2×C3—31no—
534C2×C3—31no—
535C2×C3—31no—
536C2×C3—31no—
537C2×C3—31no—
538C2×C3—31no—
539C2×C3—61no—
540C2×C3—61no—
541C2×C3—61no—
542C2×C3—61no—
543C2×C3—61no—
544C2×C3—61no—
545C2×C3—61no—
546C2×C3—362no—
547C2×C3C3×C4×D1211yescharacteristic
548C2×C3C3×#67⋊C411yescharacteristic
549C2×C3C3×#88511yescharacteristic
550C2×C3C3×#88611yescharacteristic
551C2×C3C3×#89211yescharacteristic
552C2×C3C3×#89511yescharacteristic
553C2×C3C3×#89811yescharacteristic
554D3—122no—
555D3—362no—
556C22—182no—
557C22—182no—
558C22—182no—
559C22—361no—
560C22—361no—
561C22—361no—
562C22—361no—
563C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
564C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
565C22#3828⋊C311yescharacteristic
566C22D4×(C3⋊D3)⋊C311yescharacteristic
567C22#3830⋊C311yescharacteristic
568C22#3830⋊C311yescharacteristic
569C22#8872211yescharacteristic
570C4—22no—
571C4—22no—
572C4—41no—
573C4—182no—
574C4—182no—
575C4—361no—
576C3—31no—
577C3—31no—
578C3—61no—
579C3C3×#467011yescharacteristic
580C2—362no—
581C2C4×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
582C2#64090⋊C311yescharacteristic
583C2#64094⋊C311yescharacteristic
584C2#64095⋊C311yescharacteristic
585C2#64101⋊C311yescharacteristic
586C2#64104⋊C311yescharacteristic
587C2#64107⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
Frattini, socleC23×C3C22×C3×D3
FittingH(3,𝔽3)×#95C2
prime radicalC23×(C3⋊D3)⋊C3C22
cosocleC23×C32C23×C3
hypercenter#95(C3⋊D3)⋊C3
hypocenterC32C3×#388
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
6. H(3,𝔽3)×#95C2semidirect
7. #63658⋊C3C2semidirect
8. #10592369C3semidirect
19. #3752⋊C3C4semidirect
25. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
26. C2×C4×C3⋊Dic3C2×C3semidirect
33. C32×#95D3semidirect
34. C32×#95C2×C3semidirect
35. #63658C2×C3semidirect
85. #3752C3×C4semidirect
301. #95(C3⋊D3)⋊C3semidirect
302. H(3,𝔽3)#388semidirect
497. C32#4670semidirect
498. C32C3×#388semidirect

Maximal subgroups

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

subgroupindex
C2×#3752⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×C4×(C3⋊Dic3)⋊C32
H(3,𝔽3)×#952
#63658⋊C32
#105923693
C3×#46703