Finite Group Databasebeta

← #10596151⋊C3 #13153122

Subgroups — #10596151⋊C3 #13153122

Order 1728.

895 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#124C211yesmaximal, characteristic, complemented
2#64155⋊C3C211yesmaximal, characteristic, complemented
3Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4#64147⋊C3C211yesmaximal, characteristic
5D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
6#64092⋊C3C212yesmaximal, complemented
7#64090⋊C3C212yesmaximal, complemented
8#64088⋊C3C212yesmaximal, complemented
9#64086⋊C3C212yesmaximal, complemented
10C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
11#64079⋊C3C211yesmaximal, characteristic, complemented
12#10596151C311yesmaximal, characteristic, complemented
13C3×#5680—31nomaximal
14C2×Q8×H(3,𝔽3)C2211yescharacteristic
15C2×D4×H(3,𝔽3)C2211yescharacteristic
16C2×#638⋊C3C2212yes—
17C22×(C3⋊Dic3)⋊C3C2212yes—
18C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
19C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
20C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
21C22⋊C4×H(3,𝔽3)C2212yes—
22C22⋊C4×H(3,𝔽3)C2212yes—
23C42×H(3,𝔽3)C2211yescharacteristic
24#3757⋊C3C2212yes—
25#3757⋊C3C2212yes—
26#3752⋊C3C2212yes—
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29C4⋊((C3⋊Dic3)⋊C3)C2212yes—
30#3750⋊C3C2212yes—
31#3750⋊C3C2212yes—
32#3750⋊C3C2212yes—
33C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
34C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
35C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
36C4×(C3⋊Dic3)⋊C3C2212yes—
37C32×#124—21no—
38C32×#124D311yescharacteristic, complemented
39C32×#124C2×C311yescharacteristic, complemented
40#64155C2×C311yescharacteristic, complemented
41Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
42#64147C2×C311yescharacteristic
43D4×C3⋊Dic3C2×C311yescharacteristic, complemented
44#64092C2×C312yescomplemented
45#64090C2×C312yescomplemented
46#64088C2×C312yescomplemented
47#64086C2×C312yescomplemented
48C42×C3⋊D3C2×C311yescharacteristic, complemented
49#64079C2×C311yescharacteristic, complemented
50C3×#946—31no—
51C3×Q8×Dic3—31no—
52C3×#938—31no—
53C3×D4×Dic3—31no—
54C3×#883—32no—
55C3×#67⋊C4—32no—
56C3×#879—32no—
57C3×#877—32no—
58C3×C42×D3—31no—
59C3×#870—31no—
60C23×H(3,𝔽3)C2312yes—
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
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—
69C2×(C3⋊Dic3)⋊C3—24no—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2312yes—
74C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
75C4×(C3⋊D3)⋊C3—22no—
76C4×(C3⋊D3)⋊C3—22no—
77C4×(C3⋊D3)⋊C3—22no—
78C3×#529—91no—
79#5680—31no—
80#5680—91no—
81C2×C32×Q8—21no—
82C2×C32×Q8C2×D311yescharacteristic
83C2×C32×Q8C22×C311yescharacteristic
84C2×C32×D4—21no—
85C2×C32×D4C2×D311yescharacteristic
86C2×C32×D4C22×C311yescharacteristic
87C2×#638C22×C312yes—
88C22×C3⋊Dic3C22×C312yes—
89C2×C4×C3⋊D3C22×C311yescharacteristic
90C2×C4×C3⋊D3C22×C311yescharacteristic
91C2×C4×C3⋊D3C22×C311yescharacteristic
92C2×C3×#67—32no—
93C22×C3×Dic3—32no—
94C2×C3×C4×D3—31no—
95C2×C3×C4×D3—31no—
96C2×C3×C4×D3—31no—
97C32×C22⋊C4—22no—
98C32×C22⋊C4—22no—
99C32×C22⋊C4C2×D312yes—
100C32×C22⋊C4C2×D312yes—
101C32×C22⋊C4C22×C312yes—
102C32×C22⋊C4C22×C312yes—
103C32×C42—21no—
104C32×C42C2×D311yescharacteristic
105C32×C42C22×C311yescharacteristic
106#3757C22×C312yes—
107#3757C22×C312yes—
108#3752C22×C312yes—
109C4⋊(C3⋊Dic3)C22×C311yescharacteristic
110C4⋊(C3⋊Dic3)C22×C311yescharacteristic
111C4⋊(C3⋊Dic3)C22×C312yes—
112#3750C22×C312yes—
113#3750C22×C312yes—
114#3750C22×C312yes—
115C4×C3⋊Dic3C22×C311yescharacteristic
116C4×C3⋊Dic3C22×C311yescharacteristic
117C4×C3⋊Dic3C22×C311yescharacteristic
118C4×C3⋊Dic3C22×C312yes—
119C3×#217—32no—
120C3×#217—32no—
121C3×(C2×D3)⋊C4—32no—
122C3×C4⋊Dic3—32no—
123C3×C4⋊Dic3—31no—
124C3×C4⋊Dic3—31no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—32no—
128C3×C4×Dic3—31no—
129C3×C4×Dic3—32no—
130C3×C4×Dic3—31no—
131C3×C4×Dic3—31no—
132C22×H(3,𝔽3)—22no—
133C22×H(3,𝔽3)—22no—
134C22×H(3,𝔽3)—22no—
135C22×H(3,𝔽3)C2411yescharacteristic
136C2×(C3⋊D3)⋊C3—21no—
137C2×(C3⋊D3)⋊C3—21no—
138C2×(C3⋊D3)⋊C3#4112yes—
139C4×H(3,𝔽3)—21no—
140C4×H(3,𝔽3)—21no—
141C4×H(3,𝔽3)—22no—
142C4×H(3,𝔽3)#4112yes—
143(C3⋊Dic3)⋊C3—21no—
144(C3⋊Dic3)⋊C3—21no—
145(C3⋊Dic3)⋊C3—22no—
146(C3⋊Dic3)⋊C3—22no—
147(C3⋊Dic3)⋊C3#4112yes—
148C3×#126—92no—
149C3×#125—91no—
150C3×#124—31no—
151C3×#124—31no—
152C3×#124—61no—
153C3×#124C3×D311yescharacteristic
154C3×#123—92no—
155C3×C22⋊Q8—91no—
156C3×#121—91no—
157C3×C4×Q8—91no—
158C3×C4×D4—92no—
159C3×C4×D4—91no—
160C3×#117—92no—
161C2×C3×C42—91no—
162#946—31no—
163#946—91no—
164Q8×Dic3—31no—
165Q8×Dic3—91no—
166#938—31no—
167#938—91no—
168D4×Dic3—31no—
169D4×Dic3—91no—
170#883—32no—
171#883—92no—
172#67⋊C4—32no—
173#67⋊C4—92no—
174#879—32no—
175#879—92no—
176#877—32no—
177#877—92no—
178C42×D3—31no—
179C42×D3—91no—
180#870—31no—
181#870—91no—
182C23×C32—22no—
183C23×C32C22×D312yes—
184C23×C32C23×C312yes—
185C22×C3⋊D3C23×C311yescharacteristic
186C22×C3×D3—31no—
187C32×Q8—22no—
188C32×Q8—22no—
189C32×Q8—42no—
190C32×D4—22no—
191C32×D4—22no—
192C32×D4—42no—
193C2×C32×C4—21no—
194C2×C32×C4—21no—
195C2×C32×C4—21no—
196C2×C32×C4—22no—
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D312yes—
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C23×C311yescharacteristic
202C2×C32×C4C23×C311yescharacteristic
203C2×C32×C4C23×C312yes—
204C2×C32×C4C23×C311yescharacteristic
205#638—24no—
206C2×C3⋊Dic3—24no—
207C2×C3⋊Dic3C23×C311yescharacteristic
208C2×C3⋊Dic3C23×C311yescharacteristic
209C2×C3⋊Dic3C23×C312yes—
210C2×C3⋊Dic3C23×C312yes—
211C2×C3⋊Dic3C23×C311yescharacteristic
212C4×C3⋊D3—22no—
213C4×C3⋊D3—22no—
214C4×C3⋊D3—22no—
215C3×#67—64no—
216C2×C3×Dic3—31no—
217C2×C3×Dic3—31no—
218C2×C3×Dic3—31no—
219C2×C3×Dic3—32no—
220C2×C3×Dic3—32no—
221C2×C3×Dic3—64no—
222C3×C4×D3—62no—
223C3×C4×D3—62no—
224C3×C4×D3—62no—
225#529—91noSylow
226C2×H(3,𝔽3)—42no—
227C2×H(3,𝔽3)C2×#4111yescharacteristic
228C2×H(3,𝔽3)C2×#4111yescharacteristic
229C2×H(3,𝔽3)C2×#4111yescharacteristic
230(C3⋊D3)⋊C3—22no—
231C2×C3×Q8—31no—
232C2×C3×Q8—31no—
233C2×C3×Q8—61no—
234C2×C3×Q8C2×C3×D311yescharacteristic
235C2×C3×D4—31no—
236C2×C3×D4—31no—
237C2×C3×D4—61no—
238C2×C3×D4—92no—
239C2×C3×D4C2×C3×D311yescharacteristic
240C22×C3×C4—91no—
241C22×C3×C4—91no—
242C22×C3×C4—91no—
243C22×C3×C4—92no—
244C2×#67—32no—
245C2×#67—92no—
246C22×Dic3—32no—
247C22×Dic3—92no—
248C2×C4×D3—31no—
249C2×C4×D3—31no—
250C2×C4×D3—31no—
251C2×C4×D3—91no—
252C2×C4×D3—91no—
253C2×C4×D3—91no—
254C3×C4⋊C4—92no—
255C3×C4⋊C4—92no—
256C3×C4⋊C4—92no—
257C3×C4⋊C4—91no—
258C3×C4⋊C4—91no—
259C3×C4⋊C4—92no—
260C3×C22⋊C4—32no—
261C3×C22⋊C4—32no—
262C3×C22⋊C4—32no—
263C3×C22⋊C4—32no—
264C3×C22⋊C4—62no—
265C3×C22⋊C4—62no—
266C3×C22⋊C4—92no—
267C3×C22⋊C4—92no—
268C3×C22⋊C4—92no—
269C3×C22⋊C4C2×C3×D312yes—
270C3×C22⋊C4C2×C3×D312yes—
271C3×C42—31no—
272C3×C42—31no—
273C3×C42—61no—
274C3×C42—91no—
275C3×C42—91no—
276C3×C42—91no—
277C3×C42—92no—
278C3×C42C2×C3×D311yescharacteristic
279#217—32no—
280#217—32no—
281#217—92no—
282#217—92no—
283(C2×D3)⋊C4—32no—
284(C2×D3)⋊C4—92no—
285C4⋊Dic3—31no—
286C4⋊Dic3—31no—
287C4⋊Dic3—32no—
288C4⋊Dic3—91no—
289C4⋊Dic3—91no—
290C4⋊Dic3—92no—
291Dic3⋊C4—32no—
292Dic3⋊C4—32no—
293Dic3⋊C4—32no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297C4×Dic3—31no—
298C4×Dic3—31no—
299C4×Dic3—31no—
300C4×Dic3—32no—
301C4×Dic3—91no—
302C4×Dic3—91no—
303C4×Dic3—91no—
304C4×Dic3—92no—
305C22×C32—22no—
306C22×C32—22no—
307C22×C32—22no—
308C22×C32—22no—
309C22×C32—22no—
310C22×C32—22no—
311C22×C32—21no—
312C22×C32—42no—
313C22×C32—42no—
314C22×C32—42no—
315C22×C32C23×D311yescharacteristic
316C22×C32C24×C311yescharacteristic
317C2×C3⋊D3—21no—
318C2×C3⋊D3—21no—
319C2×C3⋊D3C3×#4112yes—
320C2×C3×D3—32no—
321C2×C3×D3—61no—
322C2×C3×D3—61no—
323C32×C4—22no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—22no—
329C32×C4—22no—
330C32×C4—41no—
331C32×C4—41no—
332C32×C4—42no—
333C32×C4#23712yes—
334C32×C4C3×#4112yes—
335C3⋊Dic3—21no—
336C3⋊Dic3—21no—
337C3⋊Dic3—22no—
338C3⋊Dic3—22no—
339C3⋊Dic3C3×#4112yes—
340C3×Dic3—32no—
341C3×Dic3—61no—
342C3×Dic3—61no—
343C3×Dic3—62no—
344C3×Dic3—62no—
345#126—92no—
346#125—91no—
347#124(C3⋊D3)⋊C311yescharacteristic, complemented
348#123—92no—
349C22⋊Q8—91no—
350#121—91no—
351C4×Q8—91no—
352C4×D4—91no—
353C4×D4—92no—
354#117—92no—
355C2×C42—91no—
356H(3,𝔽3)#52911yescharacteristic, complemented, Sylow
357C23×C3—32no—
358C23×C3—32no—
359C23×C3—62no—
360C23×C3—91no—
361C23×C3C22×C3×D312yes—
362C22×D3—31no—
363C22×D3—91no—
364C3×Q8—22no—
365C3×Q8—62no—
366C3×Q8—62no—
367C3×Q8—122no—
368C3×D4—22no—
369C3×D4—62no—
370C3×D4—62no—
371C3×D4—122no—
372C3×D4—184no—
373C2×C3×C4—31no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—31no—
377C2×C3×C4—32no—
378C2×C3×C4—31no—
379C2×C3×C4—31no—
380C2×C3×C4—32no—
381C2×C3×C4—61no—
382C2×C3×C4—61no—
383C2×C3×C4—61no—
384C2×C3×C4—62no—
385C2×C3×C4—91no—
386C2×C3×C4—91no—
387C2×C3×C4—91no—
388C2×C3×C4—92no—
389C2×C3×C4—92no—
390C2×C3×C4—182no—
391C2×C3×C4—182no—
392C2×C3×C4—182no—
393C2×C3×C4—184no—
394C2×C3×C4C22×C3×D311yescharacteristic
395C2×C3×C4C22×C3×D311yescharacteristic
396C2×C3×C4C22×C3×D312yes—
397C2×C3×C4C22×C3×D311yescharacteristic
398#67—64no—
399#67—184no—
400C2×Dic3—31no—
401C2×Dic3—31no—
402C2×Dic3—31no—
403C2×Dic3—32no—
404C2×Dic3—32no—
405C2×Dic3—64no—
406C2×Dic3—91no—
407C2×Dic3—91no—
408C2×Dic3—91no—
409C2×Dic3—92no—
410C2×Dic3—92no—
411C2×Dic3—184no—
412C4×D3—62no—
413C4×D3—62no—
414C4×D3—62no—
415C4×D3—182no—
416C4×D3—182no—
417C4×D3—182no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—42no—
422C2×C32—42no—
423C2×C32—82no—
424C2×C32C2×#23711yescharacteristic
425C2×C32D3×#4111yescharacteristic
426C2×C32D3×#4111yescharacteristic
427C2×C32C2×C3×#4111yescharacteristic
428C2×C32C2×C3×#4111yescharacteristic
429C2×C32C2×C3×#4111yescharacteristic
430C3⋊D3—22no—
431C3×D3—62no—
432C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
433C2×D4—92no—
434C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
435C22×C4—92no—
436C22×C4—91no—
437C22×C4—91no—
438C22×C4—91no—
439C4⋊C4—92no—
440C4⋊C4—92no—
441C4⋊C4—91no—
442C4⋊C4—92no—
443C4⋊C4—91no—
444C4⋊C4—92no—
445C22⋊C4—92no—
446C22⋊C4—92no—
447C22⋊C4—92no—
448C22⋊C4C2×(C3⋊D3)⋊C312yes—
449C22⋊C4C2×(C3⋊D3)⋊C312yes—
450C42—91no—
451C42—91no—
452C42—91no—
453C42—92no—
454C42C2×(C3⋊D3)⋊C311yescharacteristic
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—31no—
459C22×C3—31no—
460C22×C3—62no—
461C22×C3—62no—
462C22×C3—62no—
463C22×C3—62no—
464C22×C3—62no—
465C22×C3—62no—
466C22×C3—61no—
467C22×C3—92no—
468C22×C3—122no—
469C22×C3—122no—
470C22×C3—122no—
471C22×C3—181no—
472C22×C3—181no—
473C22×C3C23×C3×D311yescharacteristic
474C2×D3—32no—
475C2×D3—61no—
476C2×D3—61no—
477C2×D3—92no—
478C2×D3—181no—
479C2×D3—181no—
480C3×C4—22no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—32no—
484C3×C4—32no—
485C3×C4—61no—
486C3×C4—61no—
487C3×C4—61no—
488C3×C4—61no—
489C3×C4—62no—
490C3×C4—62no—
491C3×C4—62no—
492C3×C4—92no—
493C3×C4—122no—
494C3×C4—121no—
495C3×C4—121no—
496C3×C4—181no—
497C3×C4—182no—
498C3×C4—181no—
499C3×C4—182no—
500C3×C4C3×#23712yes—
501Dic3—32no—
502Dic3—61no—
503Dic3—61no—
504Dic3—62no—
505Dic3—62no—
506Dic3—92no—
507Dic3—181no—
508Dic3—181no—
509Dic3—182no—
510Dic3—182no—
511C32—21no—
512C32#568011yescharacteristic, complemented
513C32C3×#52911yescharacteristic, complemented
514C23—91no—
515C23C22×(C3⋊D3)⋊C312yes—
516Q8—22no—
517D4—22no—
518D4—184no—
519C2×C4—91no—
520C2×C4—91no—
521C2×C4—91no—
522C2×C4—92no—
523C2×C4—92no—
524C2×C4—182no—
525C2×C4—182no—
526C2×C4—184no—
527C2×C4—182no—
528C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
529C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
530C2×C4C22×(C3⋊D3)⋊C312yes—
531C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
532C2×C3—31no—
533C2×C3—31no—
534C2×C3—31no—
535C2×C3—31no—
536C2×C3—31no—
537C2×C3—31no—
538C2×C3—42no—
539C2×C3—61no—
540C2×C3—61no—
541C2×C3—61no—
542C2×C3—122no—
543C2×C3—122no—
544C2×C3—182no—
545C2×C3—242no—
546C2×C3C2×C3×#23711yescharacteristic
547C2×C3C3×D3×#4111yescharacteristic
548C2×C3C3×D3×#4111yescharacteristic
549D3—62no—
550D3—182no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—92no—
555C22—181no—
556C22—181no—
557C22C23×(C3⋊D3)⋊C311yescharacteristic
558C4—22no—
559C4—21no—
560C4—21no—
561C4—92no—
562C4—181no—
563C4—182no—
564C4—181no—
565C4—182no—
566C4#3830⋊C312yes—
567C3—31no—
568C3—31no—
569C3—61no—
570C3C3×#568011yescharacteristic
571C2—42no—
572C2—182no—
573C2C2×#3830⋊C311yescharacteristic
574C2#41×(C3⋊D3)⋊C311yescharacteristic
575C2#41×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#124C2
prime radical#64147⋊C3C2
hypercenter#124(C3⋊D3)⋊C3
hypocenterC32C3×#529
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. #64155⋊C3C2semidirect
3. Q8×(C3⋊Dic3)⋊C3C2semidirect
5. D4×(C3⋊Dic3)⋊C3C2semidirect
6. #64092⋊C3C2semidirect
7. #64090⋊C3C2semidirect
8. #64088⋊C3C2semidirect
9. #64086⋊C3C2semidirect
10. C42×(C3⋊D3)⋊C3C2semidirect
11. #64079⋊C3C2semidirect
12. #10596151C3semidirect
38. C32×#124D3semidirect
39. C32×#124C2×C3semidirect
40. #64155C2×C3semidirect
41. Q8×C3⋊Dic3C2×C3semidirect
43. D4×C3⋊Dic3C2×C3semidirect
44. #64092C2×C3semidirect
45. #64090C2×C3semidirect
46. #64088C2×C3semidirect
47. #64086C2×C3semidirect
48. C42×C3⋊D3C2×C3semidirect
49. #64079C2×C3semidirect
347. #124(C3⋊D3)⋊C3semidirect
356. H(3,𝔽3)#529semidirect
512. C32#5680semidirect
513. 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)×#1242
#64155⋊C32
Q8×(C3⋊Dic3)⋊C32
#64147⋊C32
D4×(C3⋊Dic3)⋊C32
#64092⋊C32
#64090⋊C32
#64088⋊C32
#64086⋊C32
C42×(C3⋊D3)⋊C32
#64079⋊C32
#105961513
C3×#56803