Finite Group Databasebeta

← #10592374⋊C3 #13141362

Subgroups — #10592374⋊C3 #13141362

Order 1728.

671 conjugacy classes of subgroups.

The subgroup lattice has 582 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×#3750⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#95C211yesmaximal, characteristic, complemented
6#63658⋊C3C211yesmaximal, characteristic, complemented
7#63658⋊C3C211yesmaximal, characteristic, complemented
8#10592374C311yesmaximal, characteristic, complemented
9C3×#4675—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⋊C3—22no—
20#3750⋊C3—22no—
21C2×#3752C2×C311yescharacteristic
22C2×#3752C2×C311yescharacteristic
23C2×#3752C2×C311yescharacteristic
24C2×#3750C2×C311yescharacteristic, complemented
25C2×C3×(C2×D3)⋊C4—31no—
26C2×C3×(C2×D3)⋊C4—31no—
27C2×C3×(C2×D3)⋊C4—31no—
28C2×C3×Dic3⋊C4—31no—
29C32×#95—21no—
30C32×#95D311yescharacteristic, complemented
31C32×#95C2×C311yescharacteristic, complemented
32#63658C2×C311yescharacteristic, complemented
33#63658C2×C311yescharacteristic, complemented
34C3×C3⋊#95—31no—
35C3×C3⋊#95—31no—
36C23×H(3,𝔽3)C2311yescharacteristic
37C22×(C3⋊D3)⋊C3—21no—
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)⋊C3D412yes—
44C2×C4×H(3,𝔽3)—21no—
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)—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—
62C3×#397—91no—
63#4675—31no—
64#4675—91no—
65C23×C3⋊D3C22×C311yescharacteristic
66C23×C3×D3—31no—
67C22×C32×C4—21no—
68C22×C32×C4—21no—
69C22×C32×C4—21no—
70C22×C32×C4C2×D311yescharacteristic
71C22×C32×C4C2×D311yescharacteristic
72C22×C32×C4C2×D311yescharacteristic
73C22×C32×C4C22×C311yescharacteristic
74C22×C32×C4C22×C311yescharacteristic
75C22×C32×C4C22×C311yescharacteristic
76C22×C3⋊Dic3C22×C311yescharacteristic
77C22×C3⋊Dic3C22×C311yescharacteristic
78C22×C3⋊Dic3C22×C311yescharacteristic
79C22×C3×Dic3—31no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82#3752—22no—
83#3752—22no—
84#3752—22no—
85#3750—22no—
86C3×(C2×D3)⋊C4—62no—
87C3×(C2×D3)⋊C4—62no—
88C3×(C2×D3)⋊C4—62no—
89C3×Dic3⋊C4—62no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C22×H(3,𝔽3)#4111yescharacteristic
93C22×H(3,𝔽3)#4111yescharacteristic
94C22×H(3,𝔽3)#4111yescharacteristic
95C22×H(3,𝔽3)#4111yescharacteristic
96C22×H(3,𝔽3)#4111yescharacteristic
97C2×(C3⋊D3)⋊C3—22no—
98C2×(C3⋊D3)⋊C3—22no—
99C2×(C3⋊D3)⋊C3—22no—
100C2×(C3⋊D3)⋊C3—41no—
101C2×(C3⋊D3)⋊C3—41no—
102C2×(C3⋊D3)⋊C3—41no—
103C2×(C3⋊D3)⋊C3—41no—
104C4×H(3,𝔽3)—41no—
105C4×H(3,𝔽3)—41no—
106C4×H(3,𝔽3)—41no—
107(C3⋊Dic3)⋊C3—22no—
108(C3⋊Dic3)⋊C3—41no—
109(C3⋊Dic3)⋊C3—41no—
110C2×C3×C4⋊C4—91no—
111C2×C3×C22⋊C4—91no—
112C2×C3×C22⋊C4—91no—
113C2×C3×C22⋊C4—91no—
114C2×(C2×D3)⋊C4—31no—
115C2×(C2×D3)⋊C4—31no—
116C2×(C2×D3)⋊C4—31no—
117C2×(C2×D3)⋊C4—91no—
118C2×(C2×D3)⋊C4—91no—
119C2×(C2×D3)⋊C4—91no—
120C2×Dic3⋊C4—31no—
121C2×Dic3⋊C4—91no—
122C3×#95—31no—
123C3×#95—31no—
124C3×#95—61no—
125C3×#95—91no—
126C3×#95—91no—
127C3×#95C3×D311yescharacteristic
128C3⋊#95—31no—
129C3⋊#95—31no—
130C3⋊#95—91no—
131C3⋊#95—91no—
132C23×C32—21no—
133C23×C32C22×D311yescharacteristic
134C23×C32C23×C311yescharacteristic
135C22×C3⋊D3—21no—
136C22×C3⋊D3—21no—
137C22×C3⋊D3—21no—
138C22×C3⋊D3—21no—
139C22×C3⋊D3—21no—
140C22×C3⋊D3—21no—
141C22×C3⋊D3C3×D412yes—
142C22×C3×D3—32no—
143C22×C3×D3—61no—
144C22×C3×D3—61no—
145C22×C3×D3—61no—
146C22×C3×D3—61no—
147C22×C3×D3—61no—
148C22×C3×D3—61no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
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—21no—
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—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—41no—
171C2×C32×C4—41no—
172C2×C32×C4—41no—
173C2×C32×C4—41no—
174C2×C32×C4—41no—
175C2×C32×C4—41no—
176C2×C3⋊Dic3—21no—
177C2×C3⋊Dic3—21no—
178C2×C3⋊Dic3—21no—
179C2×C3⋊Dic3—21no—
180C2×C3⋊Dic3—21no—
181C2×C3⋊Dic3—21no—
182C2×C3⋊Dic3—21no—
183C2×C3⋊Dic3—21no—
184C2×C3⋊Dic3C3×D412yes—
185C2×C3×Dic3—32no—
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—
194#397—91noSylow
195C2×H(3,𝔽3)#12111yescharacteristic
196C2×H(3,𝔽3)#12311yescharacteristic
197C2×H(3,𝔽3)#12311yescharacteristic
198C2×H(3,𝔽3)#12311yescharacteristic
199C2×H(3,𝔽3)#12411yescharacteristic
200C2×H(3,𝔽3)#12611yescharacteristic
201C2×H(3,𝔽3)#12611yescharacteristic
202(C3⋊D3)⋊C3—42no—
203C24×C3—91no—
204C23×D3—31no—
205C23×D3—91no—
206C22×C3×C4—31no—
207C22×C3×C4—31no—
208C22×C3×C4—31no—
209C22×C3×C4—31no—
210C22×C3×C4—31no—
211C22×C3×C4—31no—
212C22×C3×C4—61no—
213C22×C3×C4—61no—
214C22×C3×C4—61no—
215C22×C3×C4—91no—
216C22×C3×C4—91no—
217C22×C3×C4—91no—
218C22×C3×C4C2×C3×D311yescharacteristic
219C22×C3×C4C2×C3×D311yescharacteristic
220C22×C3×C4C2×C3×D311yescharacteristic
221C22×Dic3—31no—
222C22×Dic3—31no—
223C22×Dic3—31no—
224C22×Dic3—91no—
225C22×Dic3—91no—
226C22×Dic3—91no—
227C3×C4⋊C4—182no—
228C3×C22⋊C4—182no—
229C3×C22⋊C4—182no—
230C3×C22⋊C4—182no—
231(C2×D3)⋊C4—62no—
232(C2×D3)⋊C4—62no—
233(C2×D3)⋊C4—62no—
234(C2×D3)⋊C4—182no—
235(C2×D3)⋊C4—182no—
236(C2×D3)⋊C4—182no—
237Dic3⋊C4—62no—
238Dic3⋊C4—182no—
239C22×C32—21no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—21no—
246C22×C32#23511yescharacteristic
247C22×C32#23511yescharacteristic
248C22×C32#23511yescharacteristic
249C22×C32D3×D411yescharacteristic
250C22×C32D3×D411yescharacteristic
251C22×C32#23711yescharacteristic
252C22×C32#23911yescharacteristic
253C22×C32C2×C3×D411yescharacteristic
254C22×C32C2×C3×D411yescharacteristic
255C22×C32C3×#4111yescharacteristic
256C22×C32C3×#4111yescharacteristic
257C22×C32C3×#4111yescharacteristic
258C22×C32C3×#4111yescharacteristic
259C22×C32C3×#4111yescharacteristic
260C2×C3⋊D3—22no—
261C2×C3⋊D3—22no—
262C2×C3⋊D3—22no—
263C2×C3⋊D3—41no—
264C2×C3⋊D3—41no—
265C2×C3⋊D3—41no—
266C2×C3⋊D3—41no—
267C2×C3×D3—62no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—121no—
271C2×C3×D3—121no—
272C2×C3×D3—121no—
273C2×C3×D3—121no—
274C32×C4—42no—
275C32×C4—42no—
276C32×C4—42no—
277C32×C4—41no—
278C32×C4—41no—
279C32×C4—41no—
280C32×C4—41no—
281C32×C4—41no—
282C32×C4—41no—
283C3⋊Dic3—22no—
284C3⋊Dic3—41no—
285C3⋊Dic3—41no—
286C3×Dic3—62no—
287C3×Dic3—121no—
288C3×Dic3—121no—
289C2×C4⋊C4—91no—
290C2×C22⋊C4—91no—
291C2×C22⋊C4—91no—
292C2×C22⋊C4—91no—
293#95—91no—
294#95—91no—
295#95(C3⋊D3)⋊C311yescharacteristic, complemented
296H(3,𝔽3)#39711yescharacteristic, complemented, Sylow
297C23×C3—31no—
298C23×C3—31no—
299C23×C3—61no—
300C23×C3—92no—
301C23×C3—181no—
302C23×C3—181no—
303C23×C3—181no—
304C23×C3—181no—
305C23×C3—181no—
306C23×C3—181no—
307C23×C3C22×C3×D311yescharacteristic
308C22×D3—32no—
309C22×D3—61no—
310C22×D3—61no—
311C22×D3—61no—
312C22×D3—61no—
313C22×D3—61no—
314C22×D3—61no—
315C22×D3—92no—
316C22×D3—181no—
317C22×D3—181no—
318C22×D3—181no—
319C22×D3—181no—
320C22×D3—181no—
321C22×D3—181no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
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—62no—
350C2×C3×C4—62no—
351C2×C3×C4—62no—
352C2×C3×C4—92no—
353C2×C3×C4—121no—
354C2×C3×C4—121no—
355C2×C3×C4—121no—
356C2×C3×C4—121no—
357C2×C3×C4—121no—
358C2×C3×C4—121no—
359C2×C3×C4—181no—
360C2×C3×C4—181no—
361C2×C3×C4—181no—
362C2×C3×C4—181no—
363C2×C3×C4—181no—
364C2×C3×C4—181no—
365C2×C3×C4—181no—
366C2×C3×C4—181no—
367C2×Dic3—32no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—61no—
371C2×Dic3—61no—
372C2×Dic3—61no—
373C2×Dic3—61no—
374C2×Dic3—61no—
375C2×Dic3—61no—
376C2×Dic3—92no—
377C2×Dic3—181no—
378C2×Dic3—181no—
379C2×Dic3—181no—
380C2×Dic3—181no—
381C2×Dic3—181no—
382C2×Dic3—181no—
383C2×Dic3—181no—
384C2×Dic3—181no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32#87611yescharacteristic
393C2×C32#88311yescharacteristic
394C2×C32#88411yescharacteristic
395C2×C32#88511yescharacteristic
396C2×C32#89411yescharacteristic
397C2×C32#89411yescharacteristic
398C2×C32#89811yescharacteristic
399C2×C32C3×#12111yescharacteristic
400C2×C32C3×#12311yescharacteristic
401C2×C32C3×#12311yescharacteristic
402C2×C32C3×#12311yescharacteristic
403C2×C32C3×#12411yescharacteristic
404C2×C32C3×#12611yescharacteristic
405C2×C32C3×#12611yescharacteristic
406C3⋊D3—42no—
407C3×D3—122no—
408C24—91no—
409C22×C4—91no—
410C22×C4—91no—
411C22×C4—91no—
412C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
413C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
414C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
415C4⋊C4—182no—
416C22⋊C4—182no—
417C22⋊C4—182no—
418C22⋊C4—182no—
419C22×C3—31no—
420C22×C3—31no—
421C22×C3—31no—
422C22×C3—31no—
423C22×C3—31no—
424C22×C3—31no—
425C22×C3—31no—
426C22×C3—31no—
427C22×C3—31no—
428C22×C3—31no—
429C22×C3—31no—
430C22×C3—31no—
431C22×C3—31no—
432C22×C3—31no—
433C22×C3—61no—
434C22×C3—61no—
435C22×C3—61no—
436C22×C3—61no—
437C22×C3—61no—
438C22×C3—61no—
439C22×C3—61no—
440C22×C3—182no—
441C22×C3—182no—
442C22×C3—182no—
443C22×C3—361no—
444C22×C3—361no—
445C22×C3—361no—
446C22×C3—361no—
447C22×C3C3×#23511yescharacteristic
448C22×C3C3×#23511yescharacteristic
449C22×C3C3×#23511yescharacteristic
450C22×C3C3×D3×D411yescharacteristic
451C22×C3C3×D3×D411yescharacteristic
452C22×C3C3×#23711yescharacteristic
453C22×C3C3×#23911yescharacteristic
454C2×D3—62no—
455C2×D3—62no—
456C2×D3—62no—
457C2×D3—121no—
458C2×D3—121no—
459C2×D3—121no—
460C2×D3—121no—
461C2×D3—182no—
462C2×D3—182no—
463C2×D3—182no—
464C2×D3—361no—
465C2×D3—361no—
466C2×D3—361no—
467C2×D3—361no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—121no—
472C3×C4—121no—
473C3×C4—121no—
474C3×C4—121no—
475C3×C4—121no—
476C3×C4—121no—
477C3×C4—122no—
478C3×C4—122no—
479C3×C4—122no—
480C3×C4—182no—
481C3×C4—361no—
482C3×C4—361no—
483Dic3—62no—
484Dic3—121no—
485Dic3—121no—
486Dic3—182no—
487Dic3—361no—
488Dic3—361no—
489C32—21no—
490C32#467511yescharacteristic, complemented
491C32C3×#39711yescharacteristic, complemented
492C23—92no—
493C23—181no—
494C23—181no—
495C23—181no—
496C23—181no—
497C23—181no—
498C23—181no—
499C23C22×(C3⋊D3)⋊C311yescharacteristic
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—92no—
510C2×C4—181no—
511C2×C4—181no—
512C2×C4—181no—
513C2×C4—181no—
514C2×C4—181no—
515C2×C4—181no—
516C2×C4—181no—
517C2×C4—181no—
518C2×C3—31no—
519C2×C3—31no—
520C2×C3—31no—
521C2×C3—31no—
522C2×C3—31no—
523C2×C3—31no—
524C2×C3—31no—
525C2×C3—31no—
526C2×C3—31no—
527C2×C3—31no—
528C2×C3—31no—
529C2×C3—31no—
530C2×C3—31no—
531C2×C3—31no—
532C2×C3—61no—
533C2×C3—61no—
534C2×C3—61no—
535C2×C3—61no—
536C2×C3—61no—
537C2×C3—61no—
538C2×C3—61no—
539C2×C3—362no—
540C2×C3C3×#87611yescharacteristic
541C2×C3C3×#88311yescharacteristic
542C2×C3C3×#88411yescharacteristic
543C2×C3C3×#88511yescharacteristic
544C2×C3C3×#89411yescharacteristic
545C2×C3C3×#89411yescharacteristic
546C2×C3C3×#89811yescharacteristic
547D3—122no—
548D3—362no—
549C22—182no—
550C22—182no—
551C22—182no—
552C22—361no—
553C22—361no—
554C22—361no—
555C22—361no—
556C22#3828⋊C311yescharacteristic
557C22#3828⋊C311yescharacteristic
558C22#3828⋊C311yescharacteristic
559C22D4×(C3⋊D3)⋊C311yescharacteristic
560C22D4×(C3⋊D3)⋊C311yescharacteristic
561C22#3830⋊C311yescharacteristic
562C22#8872211yescharacteristic
563C4—41no—
564C4—41no—
565C4—41no—
566C4—182no—
567C4—361no—
568C4—361no—
569C3—31no—
570C3—31no—
571C3—61no—
572C3C3×#467511yescharacteristic
573C2—362no—
574C2#1172138711yescharacteristic
575C2#64092⋊C311yescharacteristic
576C2#64093⋊C311yescharacteristic
577C2#64094⋊C311yescharacteristic
578C2#64103⋊C311yescharacteristic
579C2#64103⋊C311yescharacteristic
580C2#64107⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC23×C32C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingH(3,𝔽3)×#95C2
prime radicalC23×(C3⋊D3)⋊C3C22
hypercenter#95(C3⋊D3)⋊C3
hypocenterC32C3×#397
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
4. C2×#3750⋊C3C2semidirect
5. H(3,𝔽3)×#95C2semidirect
6. #63658⋊C3C2semidirect
7. #63658⋊C3C2semidirect
8. #10592374C3semidirect
24. C2×#3750C2×C3semidirect
30. C32×#95D3semidirect
31. C32×#95C2×C3semidirect
32. #63658C2×C3semidirect
33. #63658C2×C3semidirect
295. #95(C3⋊D3)⋊C3semidirect
296. H(3,𝔽3)#397semidirect
490. C32#4675semidirect
491. C32C3×#397semidirect

Maximal subgroups

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

subgroupindex
C2×#3752⋊C32
C2×#3750⋊C32
H(3,𝔽3)×#952
#63658⋊C32
#105923743
C3×#46753