Finite Group Databasebeta

← #10596152⋊C3 #13153113

Subgroups — #10596152⋊C3 #13153113

Order 1728.

902 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#124C211yesmaximal, characteristic, complemented
2#64156⋊C3C211yesmaximal, characteristic, complemented
3Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4#64149⋊C3C211yesmaximal, characteristic
5D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
6#64095⋊C3C212yesmaximal, complemented
7#64094⋊C3C212yesmaximal, complemented
8#64092⋊C3C212yesmaximal, complemented
9#64090⋊C3C212yesmaximal, complemented
10C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
11C4×#634⋊C3C211yesmaximal, characteristic, complemented
12#10596152C311yesmaximal, characteristic, complemented
13C3×#5681—31nomaximal
14C2×Q8×H(3,𝔽3)C2211yescharacteristic
15C2×D4×H(3,𝔽3)C2211yescharacteristic
16C2×#638⋊C3C2212yes—
17C2×#638⋊C3C2212yes—
18C22×(C3⋊Dic3)⋊C3C2212yes—
19C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
20C2×C4×(C3⋊D3)⋊C3C2212yes—
21C2×#634⋊C3C2211yescharacteristic
22C22⋊C4×H(3,𝔽3)C2212yes—
23C22⋊C4×H(3,𝔽3)C2212yes—
24C42×H(3,𝔽3)C2211yescharacteristic
25#3757⋊C3C2212yes—
26#3752⋊C3C2212yes—
27#3752⋊C3C2212yes—
28#3752⋊C3C2212yes—
29C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
30C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
31C4⋊((C3⋊Dic3)⋊C3)C2212yes—
32#3750⋊C3C2211yescharacteristic
33#3750⋊C3C2211yescharacteristic
34C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
35C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
36C4×(C3⋊Dic3)⋊C3C2212yes—
37C32×#124—21no—
38C32×#124D311yescharacteristic, complemented
39C32×#124C2×C311yescharacteristic, complemented
40#64156C2×C311yescharacteristic, complemented
41Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
42#64149C2×C311yescharacteristic
43D4×C3⋊Dic3C2×C311yescharacteristic, complemented
44#64095C2×C312yescomplemented
45#64094C2×C312yescomplemented
46#64092C2×C312yescomplemented
47#64090C2×C312yescomplemented
48C4×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
49C4×#634C2×C311yescharacteristic, complemented
50C3×#947—31no—
51C3×Q8×Dic3—31no—
52C3×#940—31no—
53C3×D4×Dic3—31no—
54C3×#886—32no—
55C3×#885—32no—
56C3×#883—32no—
57C3×#67⋊C4—32no—
58C3×C4×D12—31no—
59C3×C4×Dic6—31no—
60C23×H(3,𝔽3)C2312yes—
61C22×(C3⋊D3)⋊C3C2312yes—
62Q8×H(3,𝔽3)—22no—
63D4×H(3,𝔽3)—22no—
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×C4×H(3,𝔽3)C2312yes—
67C2×C4×H(3,𝔽3)C2311yescharacteristic
68#638⋊C3—24no—
69#638⋊C3—42no—
70C2×(C3⋊Dic3)⋊C3—42no—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
75C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
76C4⋊((C3⋊D3)⋊C3)—22no—
77C4×(C3⋊D3)⋊C3—42no—
78#634⋊C3—22no—
79C3×#559—91no—
80#5681—31no—
81#5681—91no—
82C2×C32×Q8—21no—
83C2×C32×Q8C2×D311yescharacteristic
84C2×C32×Q8C22×C311yescharacteristic
85C2×C32×D4—21no—
86C2×C32×D4C2×D311yescharacteristic
87C2×C32×D4C22×C311yescharacteristic
88C2×#638C22×C312yes—
89C2×#638C22×C312yes—
90C22×C3⋊Dic3C22×C312yes—
91C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
92C2×C4×C3⋊D3C22×C312yes—
93C2×#634C22×C311yescharacteristic
94C2×C3×#67—32no—
95C2×C3×#67—32no—
96C22×C3×Dic3—32no—
97C2×C3×D12—31no—
98C2×C3×C4×D3—32no—
99C2×C3×Dic6—31no—
100C32×C22⋊C4—22no—
101C32×C22⋊C4—22no—
102C32×C22⋊C4C2×D312yes—
103C32×C22⋊C4C2×D312yes—
104C32×C22⋊C4C22×C312yes—
105C32×C22⋊C4C22×C312yes—
106C32×C42—21no—
107C32×C42C2×D311yescharacteristic
108C32×C42C22×C311yescharacteristic
109#3757C22×C312yes—
110#3752C22×C312yes—
111#3752C22×C312yes—
112#3752C22×C312yes—
113C4⋊(C3⋊Dic3)C22×C311yescharacteristic
114C4⋊(C3⋊Dic3)C22×C311yescharacteristic
115C4⋊(C3⋊Dic3)C22×C312yes—
116#3750C22×C311yescharacteristic
117#3750C22×C311yescharacteristic
118C4×C3⋊Dic3C22×C311yescharacteristic
119C4×C3⋊Dic3C22×C311yescharacteristic
120C4×C3⋊Dic3C22×C312yes—
121C3×#217—32no—
122C3×(C2×D3)⋊C4—32no—
123C3×(C2×D3)⋊C4—32no—
124C3×(C2×D3)⋊C4—32no—
125C3×C4⋊Dic3—32no—
126C3×C4⋊Dic3—31no—
127C3×C4⋊Dic3—31no—
128C3×Dic3⋊C4—31no—
129C3×Dic3⋊C4—31no—
130C3×C4×Dic3—32no—
131C3×C4×Dic3—31no—
132C3×C4×Dic3—31no—
133C22×H(3,𝔽3)—22no—
134C22×H(3,𝔽3)—22no—
135C22×H(3,𝔽3)—22no—
136C22×H(3,𝔽3)C2411yescharacteristic
137C2×(C3⋊D3)⋊C3—22no—
138C2×(C3⋊D3)⋊C3—22no—
139C2×(C3⋊D3)⋊C3—22no—
140C4×H(3,𝔽3)—21no—
141C4×H(3,𝔽3)—22no—
142C4×H(3,𝔽3)—21no—
143C4×H(3,𝔽3)#4112yes—
144(C3⋊Dic3)⋊C3—21no—
145(C3⋊Dic3)⋊C3—22no—
146(C3⋊Dic3)⋊C3—21no—
147(C3⋊Dic3)⋊C3—21no—
148(C3⋊Dic3)⋊C3#4112yes—
149C3×#127—91no—
150C3×#124—31no—
151C3×#124—31no—
152C3×#124—61no—
153C3×#124—92no—
154C3×#124—91no—
155C3×#124C3×D311yescharacteristic
156C3×#123—92no—
157C3×#123—92no—
158C3×C4×Q8—91no—
159C3×C4×Q8—91no—
160C3×C4×D4—91no—
161C3×C4×D4—92no—
162C3×C4×D4—91no—
163#947—31no—
164#947—91no—
165Q8×Dic3—31no—
166Q8×Dic3—91no—
167#940—31no—
168#940—91no—
169D4×Dic3—31no—
170D4×Dic3—91no—
171#886—32no—
172#886—92no—
173#885—32no—
174#885—92no—
175#883—32no—
176#883—92no—
177#67⋊C4—32no—
178#67⋊C4—92no—
179C4×D12—31no—
180C4×D12—91no—
181C4×Dic6—31no—
182C4×Dic6—91no—
183C23×C32—22no—
184C23×C32C22×D312yes—
185C23×C32C23×C312yes—
186C22×C3⋊D3C23×C312yes—
187C22×C3×D3—32no—
188C32×Q8—22no—
189C32×Q8—22no—
190C32×Q8—42no—
191C32×D4—22no—
192C32×D4—22no—
193C32×D4—42no—
194C2×C32×C4—21no—
195C2×C32×C4—21no—
196C2×C32×C4—21no—
197C2×C32×C4—22no—
198C2×C32×C4C22×D312yes—
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C22×D311yescharacteristic
202C2×C32×C4C23×C312yes—
203C2×C32×C4C23×C311yescharacteristic
204C2×C32×C4C23×C311yescharacteristic
205C2×C32×C4C23×C311yescharacteristic
206#638—24no—
207#638—42no—
208C2×C3⋊Dic3—42no—
209C2×C3⋊Dic3C23×C311yescharacteristic
210C2×C3⋊Dic3C23×C311yescharacteristic
211C2×C3⋊Dic3C23×C311yescharacteristic
212C2×C3⋊Dic3C23×C312yes—
213C2×C3⋊Dic3C23×C311yescharacteristic
214C4⋊(C3⋊D3)—22no—
215C4×C3⋊D3—42no—
216#634—22no—
217C3×#67—64no—
218C3×#67—122no—
219C2×C3×Dic3—31no—
220C2×C3×Dic3—31no—
221C2×C3×Dic3—31no—
222C2×C3×Dic3—31no—
223C2×C3×Dic3—32no—
224C2×C3×Dic3—122no—
225C3×D12—62no—
226C3×C4×D3—122no—
227C3×Dic6—62no—
228#559—91noSylow
229C2×H(3,𝔽3)—42no—
230C2×H(3,𝔽3)C2×#4111yescharacteristic
231C2×H(3,𝔽3)C2×#4111yescharacteristic
232C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
233(C3⋊D3)⋊C3—42no—
234C2×C3×Q8—31no—
235C2×C3×Q8—31no—
236C2×C3×Q8—61no—
237C2×C3×Q8—91no—
238C2×C3×Q8C2×C3×D311yescharacteristic
239C2×C3×D4—31no—
240C2×C3×D4—31no—
241C2×C3×D4—61no—
242C2×C3×D4—91no—
243C2×C3×D4—92no—
244C2×C3×D4—92no—
245C2×C3×D4C2×C3×D311yescharacteristic
246C22×C3×C4—92no—
247C22×C3×C4—92no—
248C2×#67—32no—
249C2×#67—32no—
250C2×#67—92no—
251C2×#67—92no—
252C22×Dic3—32no—
253C22×Dic3—92no—
254C2×D12—31no—
255C2×D12—91no—
256C2×C4×D3—32no—
257C2×C4×D3—92no—
258C2×Dic6—31no—
259C2×Dic6—91no—
260C3×C4⋊C4—91no—
261C3×C4⋊C4—91no—
262C3×C4⋊C4—91no—
263C3×C4⋊C4—91no—
264C3×C4⋊C4—92no—
265C3×C22⋊C4—32no—
266C3×C22⋊C4—32no—
267C3×C22⋊C4—32no—
268C3×C22⋊C4—32no—
269C3×C22⋊C4—62no—
270C3×C22⋊C4—62no—
271C3×C22⋊C4—92no—
272C3×C22⋊C4—92no—
273C3×C22⋊C4—92no—
274C3×C22⋊C4—92no—
275C3×C22⋊C4C2×C3×D312yes—
276C3×C22⋊C4C2×C3×D312yes—
277C3×C42—31no—
278C3×C42—31no—
279C3×C42—61no—
280C3×C42—91no—
281C3×C42—92no—
282C3×C42—91no—
283C3×C42C2×C3×D311yescharacteristic
284#217—32no—
285#217—92no—
286(C2×D3)⋊C4—32no—
287(C2×D3)⋊C4—32no—
288(C2×D3)⋊C4—32no—
289(C2×D3)⋊C4—92no—
290(C2×D3)⋊C4—92no—
291(C2×D3)⋊C4—92no—
292C4⋊Dic3—31no—
293C4⋊Dic3—31no—
294C4⋊Dic3—32no—
295C4⋊Dic3—91no—
296C4⋊Dic3—91no—
297C4⋊Dic3—92no—
298Dic3⋊C4—31no—
299Dic3⋊C4—31no—
300Dic3⋊C4—91no—
301Dic3⋊C4—91no—
302C4×Dic3—31no—
303C4×Dic3—31no—
304C4×Dic3—32no—
305C4×Dic3—91no—
306C4×Dic3—91no—
307C4×Dic3—92no—
308C22×C32—22no—
309C22×C32—22no—
310C22×C32—22no—
311C22×C32—22no—
312C22×C32—22no—
313C22×C32—22no—
314C22×C32—21no—
315C22×C32—42no—
316C22×C32—42no—
317C22×C32—42no—
318C22×C32C23×D311yescharacteristic
319C22×C32C24×C311yescharacteristic
320C2×C3⋊D3—22no—
321C2×C3⋊D3—22no—
322C2×C3⋊D3—22no—
323C2×C3×D3—62no—
324C2×C3×D3—62no—
325C2×C3×D3—62no—
326C32×C4—22no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—22no—
331C32×C4—21no—
332C32×C4—22no—
333C32×C4—41no—
334C32×C4—41no—
335C32×C4—42no—
336C32×C4#23712yes—
337C32×C4C3×#4112yes—
338C3⋊Dic3—21no—
339C3⋊Dic3—22no—
340C3⋊Dic3—21no—
341C3⋊Dic3—21no—
342C3⋊Dic3C3×#4112yes—
343C3×Dic3—32no—
344C3×Dic3—61no—
345C3×Dic3—62no—
346C3×Dic3—61no—
347C3×Dic3—61no—
348#127—91no—
349#124—91no—
350#124—92no—
351#124(C3⋊D3)⋊C311yescharacteristic, complemented
352#123—92no—
353#123—92no—
354C4×Q8—91no—
355C4×Q8—91no—
356C4×D4—91no—
357C4×D4—92no—
358C4×D4—91no—
359H(3,𝔽3)#55911yescharacteristic, complemented, Sylow
360C23×C3—32no—
361C23×C3—32no—
362C23×C3—62no—
363C23×C3—92no—
364C23×C3C22×C3×D312yes—
365C22×D3—32no—
366C22×D3—92no—
367C3×Q8—22no—
368C3×Q8—62no—
369C3×Q8—62no—
370C3×Q8—122no—
371C3×Q8—182no—
372C3×D4—22no—
373C3×D4—62no—
374C3×D4—62no—
375C3×D4—122no—
376C3×D4—182no—
377C3×D4—184no—
378C3×D4—362no—
379C2×C3×C4—31no—
380C2×C3×C4—31no—
381C2×C3×C4—31no—
382C2×C3×C4—31no—
383C2×C3×C4—32no—
384C2×C3×C4—32no—
385C2×C3×C4—31no—
386C2×C3×C4—31no—
387C2×C3×C4—61no—
388C2×C3×C4—61no—
389C2×C3×C4—61no—
390C2×C3×C4—62no—
391C2×C3×C4—91no—
392C2×C3×C4—91no—
393C2×C3×C4—91no—
394C2×C3×C4—92no—
395C2×C3×C4—91no—
396C2×C3×C4—362no—
397C2×C3×C4—362no—
398C2×C3×C4C22×C3×D312yes—
399C2×C3×C4C22×C3×D311yescharacteristic
400C2×C3×C4C22×C3×D311yescharacteristic
401C2×C3×C4C22×C3×D311yescharacteristic
402#67—64no—
403#67—122no—
404#67—184no—
405#67—362no—
406C2×Dic3—31no—
407C2×Dic3—31no—
408C2×Dic3—31no—
409C2×Dic3—31no—
410C2×Dic3—32no—
411C2×Dic3—91no—
412C2×Dic3—91no—
413C2×Dic3—91no—
414C2×Dic3—91no—
415C2×Dic3—92no—
416C2×Dic3—122no—
417C2×Dic3—362no—
418D12—62no—
419D12—182no—
420C4×D3—122no—
421C4×D3—362no—
422Dic6—62no—
423Dic6—182no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—42no—
428C2×C32—42no—
429C2×C32—82no—
430C2×C32C2×#23711yescharacteristic
431C2×C32D3×#4111yescharacteristic
432C2×C32#100911yescharacteristic
433C2×C32C2×C3×#4111yescharacteristic
434C2×C32C2×C3×#4111yescharacteristic
435C2×C32C3×H(5,𝔽2)11yescharacteristic
436C3⋊D3—42no—
437C3×D3—122no—
438C2×Q8—91no—
439C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
440C2×D4—92no—
441C2×D4—92no—
442C2×D4—91no—
443C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
444C22×C4—92no—
445C22×C4—92no—
446C4⋊C4—91no—
447C4⋊C4—91no—
448C4⋊C4—91no—
449C4⋊C4—92no—
450C4⋊C4—91no—
451C22⋊C4—92no—
452C22⋊C4—92no—
453C22⋊C4—92no—
454C22⋊C4—92no—
455C22⋊C4C2×(C3⋊D3)⋊C312yes—
456C22⋊C4C2×(C3⋊D3)⋊C312yes—
457C42—92no—
458C42—91no—
459C42—91no—
460C42C2×(C3⋊D3)⋊C311yescharacteristic
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—31no—
465C22×C3—31no—
466C22×C3—62no—
467C22×C3—62no—
468C22×C3—62no—
469C22×C3—62no—
470C22×C3—62no—
471C22×C3—62no—
472C22×C3—61no—
473C22×C3—122no—
474C22×C3—122no—
475C22×C3—122no—
476C22×C3—182no—
477C22×C3—182no—
478C22×C3—182no—
479C22×C3C23×C3×D311yescharacteristic
480C2×D3—62no—
481C2×D3—62no—
482C2×D3—62no—
483C2×D3—182no—
484C2×D3—182no—
485C2×D3—182no—
486C3×C4—22no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—32no—
490C3×C4—32no—
491C3×C4—61no—
492C3×C4—61no—
493C3×C4—61no—
494C3×C4—62no—
495C3×C4—62no—
496C3×C4—61no—
497C3×C4—62no—
498C3×C4—92no—
499C3×C4—122no—
500C3×C4—121no—
501C3×C4—121no—
502C3×C4—182no—
503C3×C4—181no—
504C3×C4—181no—
505C3×C4—181no—
506C3×C4C3×#23712yes—
507Dic3—32no—
508Dic3—61no—
509Dic3—62no—
510Dic3—61no—
511Dic3—61no—
512Dic3—92no—
513Dic3—181no—
514Dic3—182no—
515Dic3—181no—
516Dic3—181no—
517C32—21no—
518C32#568111yescharacteristic, complemented
519C32C3×#55911yescharacteristic, complemented
520C23—92no—
521C23C22×(C3⋊D3)⋊C312yes—
522Q8—22no—
523Q8—182no—
524D4—22no—
525D4—184no—
526D4—182no—
527D4—362no—
528C2×C4—91no—
529C2×C4—91no—
530C2×C4—92no—
531C2×C4—91no—
532C2×C4—91no—
533C2×C4—362no—
534C2×C4—362no—
535C2×C4C22×(C3⋊D3)⋊C312yes—
536C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
537C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
538C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
539C2×C3—31no—
540C2×C3—31no—
541C2×C3—31no—
542C2×C3—31no—
543C2×C3—31no—
544C2×C3—31no—
545C2×C3—42no—
546C2×C3—61no—
547C2×C3—61no—
548C2×C3—61no—
549C2×C3—122no—
550C2×C3—122no—
551C2×C3—242no—
552C2×C3—362no—
553C2×C3C2×C3×#23711yescharacteristic
554C2×C3C3×D3×#4111yescharacteristic
555C2×C3C3×#100911yescharacteristic
556D3—122no—
557D3—362no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—182no—
562C22—182no—
563C22—182no—
564C22C23×(C3⋊D3)⋊C311yescharacteristic
565C4—22no—
566C4—21no—
567C4—21no—
568C4—92no—
569C4—181no—
570C4—182no—
571C4—181no—
572C4—181no—
573C4#3830⋊C312yes—
574C3—31no—
575C3—31no—
576C3—61no—
577C3C3×#568111yescharacteristic
578C2—42no—
579C2—362no—
580C2C2×#3830⋊C311yescharacteristic
581C2#41×(C3⋊D3)⋊C311yescharacteristic
582C2#1172263811yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#124C2
prime radical#64149⋊C3C2
hypercenter#124(C3⋊D3)⋊C3
hypocenterC32C3×#559
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#124C2semidirect
2. #64156⋊C3C2semidirect
3. Q8×(C3⋊Dic3)⋊C3C2semidirect
5. D4×(C3⋊Dic3)⋊C3C2semidirect
6. #64095⋊C3C2semidirect
7. #64094⋊C3C2semidirect
8. #64092⋊C3C2semidirect
9. #64090⋊C3C2semidirect
10. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
11. C4×#634⋊C3C2semidirect
12. #10596152C3semidirect
38. C32×#124D3semidirect
39. C32×#124C2×C3semidirect
40. #64156C2×C3semidirect
41. Q8×C3⋊Dic3C2×C3semidirect
43. D4×C3⋊Dic3C2×C3semidirect
44. #64095C2×C3semidirect
45. #64094C2×C3semidirect
46. #64092C2×C3semidirect
47. #64090C2×C3semidirect
48. C4×C4⋊(C3⋊D3)C2×C3semidirect
49. C4×#634C2×C3semidirect
351. #124(C3⋊D3)⋊C3semidirect
359. H(3,𝔽3)#559semidirect
518. C32#5681semidirect
519. C32C3×#559semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1242
#64156⋊C32
Q8×(C3⋊Dic3)⋊C32
#64149⋊C32
D4×(C3⋊Dic3)⋊C32
#64095⋊C32
#64094⋊C32
#64092⋊C32
#64090⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
C4×#634⋊C32
#105961523
C3×#56813