Finite Group Databasebeta

← #10596112⋊C3 #13152949

Subgroups — #10596112⋊C3 #13152949

Order 1728.

816 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×C22⋊Q8C211yesmaximal, characteristic, complemented
2#64155⋊C3C211yesmaximal, characteristic, complemented
3Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4C4×#638⋊C3C211yesmaximal, characteristic
5#64135⋊C3C211yesmaximal, characteristic, complemented
6#64107⋊C3C212yesmaximal, complemented
7#64101⋊C3C211yesmaximal, characteristic, complemented
8#64097⋊C3C211yesmaximal, characteristic, complemented
9#64096⋊C3C212yesmaximal, complemented
10#64094⋊C3C212yesmaximal
11#64088⋊C3C212yesmaximal, complemented
12#10596112C311yesmaximal, characteristic, complemented
13C3×#5641—31nomaximal
14C2×Q8×H(3,𝔽3)C2211yescharacteristic
15C22×C4×H(3,𝔽3)C2211yescharacteristic
16C2×#638⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#634⋊C3C2212yes—
19C4⋊C4×H(3,𝔽3)C2212yes—
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C22⋊C4×H(3,𝔽3)C2212yes—
22#3757⋊C3C2212yes—
23#3757⋊C3C2211yescharacteristic
24#3752⋊C3C2211yescharacteristic
25#3752⋊C3C2212yes—
26#3752⋊C3C2212yes—
27C4⋊((C3⋊Dic3)⋊C3)C2212yes—
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
30#3750⋊C3C2212yes—
31#3750⋊C3C2211yescharacteristic
32#3750⋊C3C2212yes—
33C4×(C3⋊Dic3)⋊C3C2212yes—
34C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
35C4×(C3⋊Dic3)⋊C3C2212yes—
36C4×(C3⋊Dic3)⋊C3C2212yes—
37C32×C22⋊Q8—21no—
38C32×C22⋊Q8D311yescharacteristic, complemented
39C32×C22⋊Q8C2×C311yescharacteristic, complemented
40#64155C2×C311yescharacteristic, complemented
41Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
42C4×#638C2×C311yescharacteristic
43#64135C2×C311yescharacteristic, complemented
44#64107C2×C312yescomplemented
45#64101C2×C311yescharacteristic, complemented
46#64097C2×C311yescharacteristic, complemented
47#64096C2×C312yescomplemented
48#64094C2×C312yes—
49#64088C2×C312yescomplemented
50C3×#946—31no—
51C3×Q8×Dic3—31no—
52C3×C4×#67—31no—
53C3×#926—31no—
54C3×#898—32no—
55C3×#892—31no—
56C3×#888—31no—
57C3×#887—32no—
58C3×#885—32no—
59C3×#879—32no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62Q8×H(3,𝔽3)—22no—
63C2×C4×H(3,𝔽3)—22no—
64C2×C4×H(3,𝔽3)C2312yes—
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×C4×H(3,𝔽3)C2312yes—
67C2×C4×H(3,𝔽3)C2311yescharacteristic
68#638⋊C3—22no—
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74C4×(C3⋊D3)⋊C3—22no—
75#634⋊C3—24no—
76C3×#556—91no—
77#5641—31no—
78#5641—91no—
79C2×C32×Q8—21no—
80C2×C32×Q8C2×D311yescharacteristic
81C2×C32×Q8C22×C311yescharacteristic
82C22×C32×C4—21no—
83C22×C32×C4C2×D311yescharacteristic
84C22×C32×C4C22×C311yescharacteristic
85C2×#638C22×C311yescharacteristic
86C2×C4×C3⋊D3C22×C311yescharacteristic
87C2×#634C22×C312yes—
88C2×C3×#67—31no—
89C2×C3×C4×D3—31no—
90C2×C3×Dic6—32no—
91C32×C4⋊C4—21no—
92C32×C4⋊C4—22no—
93C32×C4⋊C4C2×D312yes—
94C32×C4⋊C4C2×D311yescharacteristic
95C32×C4⋊C4C22×C312yes—
96C32×C4⋊C4C22×C311yescharacteristic
97C32×C22⋊C4—22no—
98C32×C22⋊C4C2×D312yes—
99C32×C22⋊C4C22×C312yes—
100#3757C22×C311yescharacteristic
101#3757C22×C312yes—
102#3752C22×C312yes—
103#3752C22×C312yes—
104#3752C22×C311yescharacteristic
105C4⋊(C3⋊Dic3)C22×C312yes—
106C4⋊(C3⋊Dic3)C22×C311yescharacteristic
107C4⋊(C3⋊Dic3)C22×C311yescharacteristic
108#3750C22×C312yes—
109#3750C22×C312yes—
110#3750C22×C311yescharacteristic
111C4×C3⋊Dic3C22×C312yes—
112C4×C3⋊Dic3C22×C312yes—
113C4×C3⋊Dic3C22×C312yes—
114C4×C3⋊Dic3C22×C311yescharacteristic
115C3×#217—31no—
116C3×#217—32no—
117C3×(C2×D3)⋊C4—32no—
118C3×(C2×D3)⋊C4—32no—
119C3×(C2×D3)⋊C4—31no—
120C3×C4⋊Dic3—32no—
121C3×C4⋊Dic3—31no—
122C3×C4⋊Dic3—31no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—31no—
126C3×C4×Dic3—32no—
127C3×C4×Dic3—32no—
128C3×C4×Dic3—32no—
129C3×C4×Dic3—31no—
130C22×H(3,𝔽3)—21no—
131C22×H(3,𝔽3)—21no—
132C22×H(3,𝔽3)—21no—
133C22×H(3,𝔽3)C2411yescharacteristic
134C2×(C3⋊D3)⋊C3—21no—
135C2×(C3⋊D3)⋊C3—21no—
136C2×(C3⋊D3)⋊C3—21no—
137C4×H(3,𝔽3)—22no—
138C4×H(3,𝔽3)—21no—
139C4×H(3,𝔽3)—22no—
140C4×H(3,𝔽3)#4112yes—
141(C3⋊Dic3)⋊C3—22no—
142(C3⋊Dic3)⋊C3—21no—
143(C3⋊Dic3)⋊C3—22no—
144(C3⋊Dic3)⋊C3—21no—
145(C3⋊Dic3)⋊C3#4112yes—
146C3×C4⋊Q8—91no—
147C3×#126—92no—
148C3×#126—92no—
149C3×#124—92no—
150C3×C22⋊Q8—31no—
151C3×C22⋊Q8—31no—
152C3×C22⋊Q8—61no—
153C3×C22⋊Q8—91no—
154C3×C22⋊Q8C3×D311yescharacteristic
155C3×C4×Q8—92no—
156C3×C4×Q8—91no—
157C3×C4×D4—91no—
158C3×#117—91no—
159C3×#117—91no—
160#946—31no—
161#946—91no—
162Q8×Dic3—31no—
163Q8×Dic3—91no—
164C4×#67—31no—
165C4×#67—91no—
166#926—31no—
167#926—91no—
168#898—32no—
169#898—92no—
170#892—31no—
171#892—91no—
172#888—31no—
173#888—91no—
174#887—32no—
175#887—92no—
176#885—32no—
177#885—92no—
178#879—32no—
179#879—92no—
180C23×C32—21no—
181C23×C32C22×D311yescharacteristic
182C23×C32C23×C311yescharacteristic
183C22×C3⋊D3C23×C311yescharacteristic
184C22×C3×D3—31no—
185C32×Q8—22no—
186C32×Q8—22no—
187C32×Q8—42no—
188C2×C32×C4—22no—
189C2×C32×C4—22no—
190C2×C32×C4—21no—
191C2×C32×C4—22no—
192C2×C32×C4—22no—
193C2×C32×C4—21no—
194C2×C32×C4—42no—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D312yes—
197C2×C32×C4C22×D312yes—
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C23×C311yescharacteristic
200C2×C32×C4C23×C312yes—
201C2×C32×C4C23×C312yes—
202C2×C32×C4C23×C311yescharacteristic
203#638—22no—
204C2×C3⋊Dic3C23×C312yes—
205C2×C3⋊Dic3C23×C311yescharacteristic
206C2×C3⋊Dic3C23×C311yescharacteristic
207C2×C3⋊Dic3C23×C312yes—
208C2×C3⋊Dic3C23×C311yescharacteristic
209C4×C3⋊D3—22no—
210#634—24no—
211C3×#67—62no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—31no—
216C2×C3×Dic3—32no—
217C3×C4×D3—62no—
218C3×Dic6—64no—
219#556—91noSylow
220C2×H(3,𝔽3)—41no—
221C2×H(3,𝔽3)C2×#4111yescharacteristic
222C2×H(3,𝔽3)C2×#4111yescharacteristic
223C2×H(3,𝔽3)2−1+411yescharacteristic
224(C3⋊D3)⋊C3—41no—
225C2×C3×Q8—31no—
226C2×C3×Q8—31no—
227C2×C3×Q8—61no—
228C2×C3×Q8—92no—
229C2×C3×Q8C2×C3×D311yescharacteristic
230C2×C3×D4—91no—
231C22×C3×C4—31no—
232C22×C3×C4—31no—
233C22×C3×C4—61no—
234C22×C3×C4—91no—
235C22×C3×C4C2×C3×D311yescharacteristic
236C2×#67—31no—
237C2×#67—91no—
238C2×C4×D3—31no—
239C2×C4×D3—91no—
240C2×Dic6—32no—
241C2×Dic6—92no—
242C3×C4⋊C4—32no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—32no—
246C3×C4⋊C4—61no—
247C3×C4⋊C4—62no—
248C3×C4⋊C4—91no—
249C3×C4⋊C4—92no—
250C3×C4⋊C4—92no—
251C3×C4⋊C4—91no—
252C3×C4⋊C4—91no—
253C3×C4⋊C4—92no—
254C3×C4⋊C4C2×C3×D312yes—
255C3×C4⋊C4C2×C3×D311yescharacteristic
256C3×C22⋊C4—32no—
257C3×C22⋊C4—32no—
258C3×C22⋊C4—62no—
259C3×C22⋊C4—92no—
260C3×C22⋊C4—91no—
261C3×C22⋊C4—92no—
262C3×C22⋊C4—92no—
263C3×C22⋊C4—91no—
264C3×C22⋊C4C2×C3×D312yes—
265C3×C42—92no—
266C3×C42—92no—
267C3×C42—91no—
268C3×C42—92no—
269#217—31no—
270#217—32no—
271#217—91no—
272#217—92no—
273(C2×D3)⋊C4—32no—
274(C2×D3)⋊C4—32no—
275(C2×D3)⋊C4—31no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—91no—
279C4⋊Dic3—32no—
280C4⋊Dic3—31no—
281C4⋊Dic3—31no—
282C4⋊Dic3—92no—
283C4⋊Dic3—91no—
284C4⋊Dic3—91no—
285Dic3⋊C4—32no—
286Dic3⋊C4—32no—
287Dic3⋊C4—31no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—91no—
291C4×Dic3—32no—
292C4×Dic3—32no—
293C4×Dic3—32no—
294C4×Dic3—31no—
295C4×Dic3—92no—
296C4×Dic3—92no—
297C4×Dic3—92no—
298C4×Dic3—91no—
299C22×C32—22no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—21no—
305C22×C32—21no—
306C22×C32—21no—
307C22×C32—41no—
308C22×C32—41no—
309C22×C32C23×D311yescharacteristic
310C22×C32C24×C311yescharacteristic
311C2×C3⋊D3—21no—
312C2×C3⋊D3—21no—
313C2×C3⋊D3—21no—
314C2×C3×D3—61no—
315C2×C3×D3—61no—
316C2×C3×D3—61no—
317C32×C4—21no—
318C32×C4—22no—
319C32×C4—22no—
320C32×C4—22no—
321C32×C4—21no—
322C32×C4—22no—
323C32×C4—22no—
324C32×C4—42no—
325C32×C4—41no—
326C32×C4—42no—
327C32×C4#23712yes—
328C32×C4C3×#4112yes—
329C3⋊Dic3—22no—
330C3⋊Dic3—21no—
331C3⋊Dic3—22no—
332C3⋊Dic3—21no—
333C3⋊Dic3C3×#4112yes—
334C3×Dic3—32no—
335C3×Dic3—61no—
336C3×Dic3—62no—
337C3×Dic3—61no—
338C3×Dic3—62no—
339C4⋊Q8—91no—
340#126—92no—
341#126—92no—
342#124—92no—
343C22⋊Q8—91no—
344C22⋊Q8(C3⋊D3)⋊C311yescharacteristic, complemented
345C4×Q8—91no—
346C4×Q8—92no—
347C4×D4—91no—
348#117—91no—
349#117—91no—
350H(3,𝔽3)#55611yescharacteristic, complemented, Sylow
351C23×C3—31no—
352C23×C3—31no—
353C23×C3—61no—
354C23×C3—91no—
355C23×C3C22×C3×D311yescharacteristic
356C22×D3—31no—
357C22×D3—91no—
358C3×Q8—22no—
359C3×Q8—62no—
360C3×Q8—62no—
361C3×Q8—122no—
362C3×Q8—184no—
363C3×D4—182no—
364C2×C3×C4—22no—
365C2×C3×C4—31no—
366C2×C3×C4—31no—
367C2×C3×C4—31no—
368C2×C3×C4—32no—
369C2×C3×C4—32no—
370C2×C3×C4—31no—
371C2×C3×C4—32no—
372C2×C3×C4—32no—
373C2×C3×C4—62no—
374C2×C3×C4—62no—
375C2×C3×C4—61no—
376C2×C3×C4—62no—
377C2×C3×C4—62no—
378C2×C3×C4—61no—
379C2×C3×C4—91no—
380C2×C3×C4—92no—
381C2×C3×C4—91no—
382C2×C3×C4—91no—
383C2×C3×C4—92no—
384C2×C3×C4—122no—
385C2×C3×C4—182no—
386C2×C3×C4C22×C3×D311yescharacteristic
387C2×C3×C4C22×C3×D312yes—
388C2×C3×C4C22×C3×D312yes—
389C2×C3×C4C22×C3×D311yescharacteristic
390#67—62no—
391#67—182no—
392C2×Dic3—31no—
393C2×Dic3—32no—
394C2×Dic3—31no—
395C2×Dic3—31no—
396C2×Dic3—32no—
397C2×Dic3—91no—
398C2×Dic3—92no—
399C2×Dic3—91no—
400C2×Dic3—91no—
401C2×Dic3—92no—
402C4×D3—62no—
403C4×D3—182no—
404Dic6—64no—
405Dic6—184no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—42no—
410C2×C32—41no—
411C2×C32—41no—
412C2×C32C2×#23711yescharacteristic
413C2×C32#100711yescharacteristic
414C2×C32D3×#4111yescharacteristic
415C2×C32C2×C3×#4111yescharacteristic
416C2×C32C2×C3×#4111yescharacteristic
417C2×C32C3×2−1+411yescharacteristic
418C3⋊D3—41no—
419C3×D3—121no—
420C2×Q8—92no—
421C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
422C2×D4—91no—
423C22×C4—91no—
424C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
425C4⋊C4—91no—
426C4⋊C4—91no—
427C4⋊C4—91no—
428C4⋊C4—92no—
429C4⋊C4—92no—
430C4⋊C4—92no—
431C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
432C4⋊C4C2×(C3⋊D3)⋊C312yes—
433C22⋊C4—92no—
434C22⋊C4—91no—
435C22⋊C4—92no—
436C22⋊C4—91no—
437C22⋊C4—92no—
438C22⋊C4C2×(C3⋊D3)⋊C312yes—
439C42—92no—
440C42—92no—
441C42—91no—
442C42—92no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—31no—
447C22×C3—31no—
448C22×C3—61no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—61no—
452C22×C3—61no—
453C22×C3—61no—
454C22×C3—62no—
455C22×C3—61no—
456C22×C3—121no—
457C22×C3—121no—
458C22×C3—181no—
459C22×C3—181no—
460C22×C3—181no—
461C22×C3C23×C3×D311yescharacteristic
462C2×D3—61no—
463C2×D3—61no—
464C2×D3—61no—
465C2×D3—181no—
466C2×D3—181no—
467C2×D3—181no—
468C3×C4—21no—
469C3×C4—22no—
470C3×C4—22no—
471C3×C4—32no—
472C3×C4—32no—
473C3×C4—62no—
474C3×C4—61no—
475C3×C4—61no—
476C3×C4—62no—
477C3×C4—62no—
478C3×C4—62no—
479C3×C4—62no—
480C3×C4—92no—
481C3×C4—121no—
482C3×C4—122no—
483C3×C4—122no—
484C3×C4—181no—
485C3×C4—182no—
486C3×C4—182no—
487C3×C4—181no—
488C3×C4C3×#23712yes—
489Dic3—32no—
490Dic3—61no—
491Dic3—62no—
492Dic3—61no—
493Dic3—62no—
494Dic3—92no—
495Dic3—182no—
496Dic3—181no—
497Dic3—182no—
498Dic3—181no—
499C32—21no—
500C32#564111yescharacteristic, complemented
501C32C3×#55611yescharacteristic, complemented
502C23—91no—
503C23C22×(C3⋊D3)⋊C311yescharacteristic
504Q8—22no—
505Q8—184no—
506D4—182no—
507C2×C4—22no—
508C2×C4—91no—
509C2×C4—92no—
510C2×C4—91no—
511C2×C4—91no—
512C2×C4—92no—
513C2×C4—182no—
514C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
515C2×C4C22×(C3⋊D3)⋊C312yes—
516C2×C4C22×(C3⋊D3)⋊C312yes—
517C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
518C2×C3—31no—
519C2×C3—31no—
520C2×C3—31no—
521C2×C3—31no—
522C2×C3—31no—
523C2×C3—31no—
524C2×C3—41no—
525C2×C3—61no—
526C2×C3—61no—
527C2×C3—61no—
528C2×C3—121no—
529C2×C3—121no—
530C2×C3—122no—
531C2×C3—361no—
532C2×C3C2×C3×#23711yescharacteristic
533C2×C3C3×#100711yescharacteristic
534C2×C3C3×D3×#4111yescharacteristic
535D3—121no—
536D3—361no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—181no—
541C22—181no—
542C22—181no—
543C22C23×(C3⋊D3)⋊C311yescharacteristic
544C4—21no—
545C4—22no—
546C4—22no—
547C4—92no—
548C4—181no—
549C4—182no—
550C4—181no—
551C4—182no—
552C4#3830⋊C312yes—
553C3—31no—
554C3—31no—
555C3—61no—
556C3C3×#564111yescharacteristic
557C2—41no—
558C2—361no—
559C2C2×#3830⋊C311yescharacteristic
560C2#1172262711yescharacteristic
561C2#41×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×C22⋊Q8C2
prime radicalC2×#638⋊C3C22
hypercenterC22⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#556
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×C22⋊Q8C2semidirect
2. #64155⋊C3C2semidirect
3. Q8×(C3⋊Dic3)⋊C3C2semidirect
5. #64135⋊C3C2semidirect
6. #64107⋊C3C2semidirect
7. #64101⋊C3C2semidirect
8. #64097⋊C3C2semidirect
9. #64096⋊C3C2semidirect
11. #64088⋊C3C2semidirect
12. #10596112C3semidirect
38. C32×C22⋊Q8D3semidirect
39. C32×C22⋊Q8C2×C3semidirect
40. #64155C2×C3semidirect
41. Q8×C3⋊Dic3C2×C3semidirect
43. #64135C2×C3semidirect
44. #64107C2×C3semidirect
45. #64101C2×C3semidirect
46. #64097C2×C3semidirect
47. #64096C2×C3semidirect
49. #64088C2×C3semidirect
344. C22⋊Q8(C3⋊D3)⋊C3semidirect
350. H(3,𝔽3)#556semidirect
500. C32#5641semidirect
501. C32C3×#556semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C22⋊Q82
#64155⋊C32
Q8×(C3⋊Dic3)⋊C32
C4×#638⋊C32
#64135⋊C32
#64107⋊C32
#64101⋊C32
#64097⋊C32
#64096⋊C32
#64094⋊C32
#64088⋊C32
#105961123
C3×#56413