Finite Group Databasebeta

← #10596162⋊C3 #13153182

Subgroups — #10596162⋊C3 #13153182

Order 1728.

875 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#125C211yesmaximal, characteristic, complemented
2#64107⋊C3C212yesmaximal, complemented
3#64106⋊C3C211yesmaximal, characteristic, complemented
4#64104⋊C3C211yesmaximal, characteristic
5#64103⋊C3C212yesmaximal, complemented
6#64102⋊C3C212yesmaximal, complemented
7#64102⋊C3C211yesmaximal, characteristic, complemented
8#64101⋊C3C212yesmaximal, complemented
9#64096⋊C3C211yesmaximal, characteristic, complemented
10#64084⋊C3C211yesmaximal, characteristic, complemented
11C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
12#10596162C311yesmaximal, characteristic, complemented
13C3×#5691—31nomaximal
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
19C2×C4×(C3⋊D3)⋊C3C2212yes—
20C2×#634⋊C3C2211yescharacteristic
21C4⋊C4×H(3,𝔽3)C2212yes—
22C4⋊C4×H(3,𝔽3)C2211yescharacteristic
23C4⋊C4×H(3,𝔽3)C2212yes—
24C4⋊C4×H(3,𝔽3)C2211yescharacteristic
25C42×H(3,𝔽3)C2211yescharacteristic
26#3752⋊C3C2212yes—
27#3752⋊C3C2212yes—
28#3752⋊C3C2212yes—
29#3752⋊C3C2212yes—
30#3752⋊C3C2212yes—
31C4⋊((C3⋊Dic3)⋊C3)C2212yes—
32#3750⋊C3C2212yes—
33C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
34C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
35C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
36C4×(C3⋊Dic3)⋊C3C2212yes—
37C32×#125—21no—
38C32×#125D311yescharacteristic, complemented
39C32×#125C2×C311yescharacteristic, complemented
40#64107C2×C312yescomplemented
41#64106C2×C311yescharacteristic, complemented
42#64104C2×C311yescharacteristic
43#64103C2×C312yescomplemented
44#64102C2×C311yescharacteristic, complemented
45#64102C2×C312yescomplemented
46#64101C2×C312yescomplemented
47#64096C2×C311yescharacteristic, complemented
48#64084C2×C311yescharacteristic, complemented
49C42×C3⋊D3C2×C311yescharacteristic, complemented
50C3×#898—32no—
51C3×#897—31no—
52C3×#895—31no—
53C3×#894—32no—
54C3×#893—32no—
55C3×#893—31no—
56C3×#892—32no—
57C3×#887—31no—
58C3×#875—31no—
59C3×C42×D3—31no—
60C22×(C3⋊D3)⋊C3C2312yes—
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2312yes—
65C2×C4×H(3,𝔽3)C2312yes—
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C4⋊((C3⋊D3)⋊C3)—24no—
72C4⋊((C3⋊D3)⋊C3)—22no—
73C4×(C3⋊D3)⋊C3—22no—
74C4×(C3⋊D3)⋊C3—22no—
75C4×(C3⋊D3)⋊C3—22no—
76C4×(C3⋊D3)⋊C3—24no—
77#634⋊C3—22no—
78C3×#529—91no—
79#5691—31no—
80#5691—91no—
81C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
82C2×C4⋊(C3⋊D3)C22×C312yes—
83C2×C4×C3⋊D3C22×C311yescharacteristic
84C2×C4×C3⋊D3C22×C311yescharacteristic
85C2×C4×C3⋊D3C22×C311yescharacteristic
86C2×C4×C3⋊D3C22×C312yes—
87C2×#634C22×C311yescharacteristic
88C2×C3×D12—32no—
89C2×C3×D12—31no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3—31no—
92C2×C3×C4×D3—31no—
93C2×C3×C4×D3—31no—
94C2×C3×Dic6—31no—
95C32×C4⋊C4—21no—
96C32×C4⋊C4—21no—
97C32×C4⋊C4—22no—
98C32×C4⋊C4—22no—
99C32×C4⋊C4C2×D311yescharacteristic
100C32×C4⋊C4C2×D312yes—
101C32×C4⋊C4C2×D311yescharacteristic
102C32×C4⋊C4C2×D312yes—
103C32×C4⋊C4C22×C311yescharacteristic
104C32×C4⋊C4C22×C312yes—
105C32×C4⋊C4C22×C311yescharacteristic
106C32×C4⋊C4C22×C312yes—
107C32×C42—21no—
108C32×C42C2×D311yescharacteristic
109C32×C42C22×C311yescharacteristic
110#3752C22×C312yes—
111#3752C22×C312yes—
112#3752C22×C312yes—
113#3752C22×C312yes—
114#3752C22×C312yes—
115C4⋊(C3⋊Dic3)C22×C312yes—
116#3750C22×C312yes—
117C4×C3⋊Dic3C22×C311yescharacteristic
118C4×C3⋊Dic3C22×C311yescharacteristic
119C4×C3⋊Dic3C22×C311yescharacteristic
120C4×C3⋊Dic3C22×C312yes—
121C3×(C2×D3)⋊C4—32no—
122C3×(C2×D3)⋊C4—32no—
123C3×(C2×D3)⋊C4—32no—
124C3×(C2×D3)⋊C4—32no—
125C3×(C2×D3)⋊C4—32no—
126C3×C4⋊Dic3—32no—
127C3×Dic3⋊C4—32no—
128C3×C4×Dic3—32no—
129C3×C4×Dic3—31no—
130C3×C4×Dic3—31no—
131C3×C4×Dic3—31no—
132C22×H(3,𝔽3)C2411yescharacteristic
133C2×(C3⋊D3)⋊C3—21no—
134C2×(C3⋊D3)⋊C3—21no—
135C2×(C3⋊D3)⋊C3—22no—
136C2×(C3⋊D3)⋊C3—22no—
137C2×(C3⋊D3)⋊C3—22no—
138C2×(C3⋊D3)⋊C3#4112yes—
139C4×H(3,𝔽3)—21no—
140C4×H(3,𝔽3)—21no—
141C4×H(3,𝔽3)—22no—
142C4×H(3,𝔽3)—22no—
143C4×H(3,𝔽3)#4112yes—
144(C3⋊Dic3)⋊C3—22no—
145(C3⋊Dic3)⋊C3—21no—
146(C3⋊Dic3)⋊C3—21no—
147(C3⋊Dic3)⋊C3#4112yes—
148C3×#126—92no—
149C3×#125—31no—
150C3×#125—31no—
151C3×#125—61no—
152C3×#125C3×D311yescharacteristic
153C3×#124—91no—
154C3×#123—92no—
155C3×C22⋊Q8—91no—
156C3×#121—91no—
157C3×C4×Q8—91no—
158C3×C4×D4—91no—
159C3×C4×D4—92no—
160C3×#117—92no—
161C2×C3×C42—91no—
162#898—32no—
163#898—92no—
164#897—31no—
165#897—91no—
166#895—31no—
167#895—91no—
168#894—32no—
169#894—92no—
170#893—31no—
171#893—32no—
172#893—91no—
173#893—92no—
174#892—32no—
175#892—92no—
176#887—31no—
177#887—91no—
178#875—31no—
179#875—91no—
180C42×D3—31no—
181C42×D3—91no—
182C22×C3⋊D3C23×C312yes—
183C22×C3⋊D3C23×C311yescharacteristic
184C22×C3×D3—31no—
185C22×C3×D3—32no—
186C2×C32×C4—21no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4—22no—
190C2×C32×C4—22no—
191C2×C32×C4C22×D312yes—
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D312yes—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C23×C312yes—
197C2×C32×C4C23×C311yescharacteristic
198C2×C32×C4C23×C311yescharacteristic
199C2×C32×C4C23×C312yes—
200C2×C32×C4C23×C311yescharacteristic
201C2×C3⋊Dic3C23×C311yescharacteristic
202C2×C3⋊Dic3C23×C311yescharacteristic
203C2×C3⋊Dic3C23×C312yes—
204C2×C3⋊Dic3C23×C311yescharacteristic
205C4⋊(C3⋊D3)—24no—
206C4⋊(C3⋊D3)—22no—
207C4×C3⋊D3—22no—
208C4×C3⋊D3—22no—
209C4×C3⋊D3—22no—
210C4×C3⋊D3—24no—
211#634—22no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—32no—
216C3×D12—64no—
217C3×D12—62no—
218C3×C4×D3—62no—
219C3×C4×D3—64no—
220C3×C4×D3—62no—
221C3×C4×D3—62no—
222C3×Dic6—62no—
223#529—91noSylow
224C2×H(3,𝔽3)C2×#4111yescharacteristic
225C2×H(3,𝔽3)C2×#4111yescharacteristic
226C2×H(3,𝔽3)C2×#4111yescharacteristic
227(C3⋊D3)⋊C3—22no—
228(C3⋊D3)⋊C3—42no—
229C2×C3×Q8—91no—
230C2×C3×D4—92no—
231C2×C3×D4—91no—
232C22×C3×C4—91no—
233C22×C3×C4—91no—
234C22×C3×C4—91no—
235C22×C3×C4—92no—
236C2×D12—31no—
237C2×D12—32no—
238C2×D12—91no—
239C2×D12—92no—
240C2×C4×D3—31no—
241C2×C4×D3—31no—
242C2×C4×D3—31no—
243C2×C4×D3—32no—
244C2×C4×D3—91no—
245C2×C4×D3—91no—
246C2×C4×D3—91no—
247C2×C4×D3—92no—
248C2×Dic6—31no—
249C2×Dic6—91no—
250C3×C4⋊C4—31no—
251C3×C4⋊C4—32no—
252C3×C4⋊C4—31no—
253C3×C4⋊C4—32no—
254C3×C4⋊C4—32no—
255C3×C4⋊C4—31no—
256C3×C4⋊C4—32no—
257C3×C4⋊C4—31no—
258C3×C4⋊C4—62no—
259C3×C4⋊C4—61no—
260C3×C4⋊C4—62no—
261C3×C4⋊C4—61no—
262C3×C4⋊C4—92no—
263C3×C4⋊C4—92no—
264C3×C4⋊C4C2×C3×D311yescharacteristic
265C3×C4⋊C4C2×C3×D312yes—
266C3×C4⋊C4C2×C3×D311yescharacteristic
267C3×C4⋊C4C2×C3×D312yes—
268C3×C22⋊C4—92no—
269C3×C22⋊C4—92no—
270C3×C22⋊C4—92no—
271C3×C22⋊C4—92no—
272C3×C22⋊C4—92no—
273C3×C42—31no—
274C3×C42—31no—
275C3×C42—61no—
276C3×C42—91no—
277C3×C42—91no—
278C3×C42—91no—
279C3×C42—92no—
280C3×C42C2×C3×D311yescharacteristic
281(C2×D3)⋊C4—32no—
282(C2×D3)⋊C4—32no—
283(C2×D3)⋊C4—32no—
284(C2×D3)⋊C4—32no—
285(C2×D3)⋊C4—32no—
286(C2×D3)⋊C4—92no—
287(C2×D3)⋊C4—92no—
288(C2×D3)⋊C4—92no—
289(C2×D3)⋊C4—92no—
290(C2×D3)⋊C4—92no—
291C4⋊Dic3—32no—
292C4⋊Dic3—92no—
293Dic3⋊C4—32no—
294Dic3⋊C4—92no—
295C4×Dic3—31no—
296C4×Dic3—31no—
297C4×Dic3—31no—
298C4×Dic3—32no—
299C4×Dic3—91no—
300C4×Dic3—91no—
301C4×Dic3—91no—
302C4×Dic3—92no—
303C22×C32—21no—
304C22×C32C23×D311yescharacteristic
305C22×C32C24×C311yescharacteristic
306C2×C3⋊D3—21no—
307C2×C3⋊D3—21no—
308C2×C3⋊D3—22no—
309C2×C3⋊D3—22no—
310C2×C3⋊D3—22no—
311C2×C3⋊D3C3×#4112yes—
312C2×C3×D3—32no—
313C2×C3×D3—61no—
314C2×C3×D3—61no—
315C2×C3×D3—62no—
316C2×C3×D3—62no—
317C2×C3×D3—62no—
318C32×C4—21no—
319C32×C4—22no—
320C32×C4—21no—
321C32×C4—22no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—22no—
325C32×C4—22no—
326C32×C4—22no—
327C32×C4—41no—
328C32×C4—41no—
329C32×C4—42no—
330C32×C4—42no—
331C32×C4#23912yes—
332C32×C4C3×#4112yes—
333C3⋊Dic3—22no—
334C3⋊Dic3—21no—
335C3⋊Dic3—21no—
336C3⋊Dic3C3×#4112yes—
337C3×Dic3—32no—
338C3×Dic3—62no—
339C3×Dic3—61no—
340C3×Dic3—61no—
341#126—92no—
342#125(C3⋊D3)⋊C311yescharacteristic, complemented
343#124—91no—
344#123—92no—
345C22⋊Q8—91no—
346#121—91no—
347C4×Q8—91no—
348C4×D4—91no—
349C4×D4—92no—
350#117—92no—
351C2×C42—91no—
352H(3,𝔽3)#52911yescharacteristic, complemented, Sylow
353C23×C3—91no—
354C23×C3—92no—
355C22×D3—31no—
356C22×D3—32no—
357C22×D3—91no—
358C22×D3—92no—
359C3×Q8—182no—
360C3×D4—182no—
361C3×D4—184no—
362C2×C3×C4—31no—
363C2×C3×C4—31no—
364C2×C3×C4—31no—
365C2×C3×C4—31no—
366C2×C3×C4—32no—
367C2×C3×C4—32no—
368C2×C3×C4—32no—
369C2×C3×C4—31no—
370C2×C3×C4—31no—
371C2×C3×C4—32no—
372C2×C3×C4—61no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—62no—
376C2×C3×C4—62no—
377C2×C3×C4—91no—
378C2×C3×C4—91no—
379C2×C3×C4—91no—
380C2×C3×C4—92no—
381C2×C3×C4—182no—
382C2×C3×C4—182no—
383C2×C3×C4—184no—
384C2×C3×C4—182no—
385C2×C3×C4C22×C3×D312yes—
386C2×C3×C4C22×C3×D311yescharacteristic
387C2×C3×C4C22×C3×D311yescharacteristic
388C2×C3×C4C22×C3×D312yes—
389C2×C3×C4C22×C3×D311yescharacteristic
390C2×Dic3—31no—
391C2×Dic3—31no—
392C2×Dic3—31no—
393C2×Dic3—32no—
394C2×Dic3—91no—
395C2×Dic3—91no—
396C2×Dic3—91no—
397C2×Dic3—92no—
398D12—64no—
399D12—62no—
400D12—182no—
401D12—184no—
402C4×D3—62no—
403C4×D3—64no—
404C4×D3—62no—
405C4×D3—62no—
406C4×D3—182no—
407C4×D3—184no—
408C4×D3—182no—
409C4×D3—182no—
410Dic6—62no—
411Dic6—182no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32C2×#23911yescharacteristic
416C2×C32D3×#4111yescharacteristic
417C2×C32D3×#4111yescharacteristic
418C2×C32C2×C3×#4111yescharacteristic
419C2×C32C2×C3×#4111yescharacteristic
420C2×C32C2×C3×#4111yescharacteristic
421C3⋊D3—22no—
422C3⋊D3—42no—
423C3×D3—62no—
424C3×D3—122no—
425C2×Q8—91no—
426C2×D4—92no—
427C2×D4—91no—
428C22×C4—91no—
429C22×C4—91no—
430C22×C4—91no—
431C22×C4—92no—
432C4⋊C4—92no—
433C4⋊C4—92no—
434C4⋊C4C2×(C3⋊D3)⋊C312yes—
435C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
436C4⋊C4C2×(C3⋊D3)⋊C312yes—
437C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
438C22⋊C4—92no—
439C22⋊C4—92no—
440C22⋊C4—92no—
441C22⋊C4—92no—
442C22⋊C4—92no—
443C42—91no—
444C42—91no—
445C42—91no—
446C42—92no—
447C42C2×(C3⋊D3)⋊C311yescharacteristic
448C22×C3—31no—
449C22×C3—31no—
450C22×C3—61no—
451C22×C3—92no—
452C22×C3—181no—
453C22×C3—181no—
454C22×C3—182no—
455C22×C3—182no—
456C22×C3—182no—
457C22×C3C23×C3×D311yescharacteristic
458C2×D3—32no—
459C2×D3—61no—
460C2×D3—61no—
461C2×D3—62no—
462C2×D3—62no—
463C2×D3—62no—
464C2×D3—92no—
465C2×D3—181no—
466C2×D3—181no—
467C2×D3—182no—
468C2×D3—182no—
469C2×D3—182no—
470C3×C4—21no—
471C3×C4—22no—
472C3×C4—21no—
473C3×C4—22no—
474C3×C4—32no—
475C3×C4—32no—
476C3×C4—61no—
477C3×C4—62no—
478C3×C4—61no—
479C3×C4—62no—
480C3×C4—61no—
481C3×C4—62no—
482C3×C4—61no—
483C3×C4—62no—
484C3×C4—62no—
485C3×C4—92no—
486C3×C4—121no—
487C3×C4—121no—
488C3×C4—122no—
489C3×C4—122no—
490C3×C4—181no—
491C3×C4—181no—
492C3×C4—182no—
493C3×C4C3×#23912yes—
494Dic3—32no—
495Dic3—62no—
496Dic3—61no—
497Dic3—61no—
498Dic3—92no—
499Dic3—182no—
500Dic3—181no—
501Dic3—181no—
502C32—21no—
503C32#569111yescharacteristic, complemented
504C32C3×#52911yescharacteristic, complemented
505C23—91no—
506C23—92no—
507Q8—182no—
508D4—184no—
509D4—182no—
510C2×C4—91no—
511C2×C4—91no—
512C2×C4—91no—
513C2×C4—92no—
514C2×C4—182no—
515C2×C4—184no—
516C2×C4—182no—
517C2×C4—182no—
518C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
519C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
520C2×C4C22×(C3⋊D3)⋊C312yes—
521C2×C4C22×(C3⋊D3)⋊C312yes—
522C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
523C2×C3—31no—
524C2×C3—31no—
525C2×C3—31no—
526C2×C3—31no—
527C2×C3—31no—
528C2×C3—31no—
529C2×C3—61no—
530C2×C3—61no—
531C2×C3—61no—
532C2×C3—182no—
533C2×C3—362no—
534C2×C3C2×C3×#23911yescharacteristic
535C2×C3C3×D3×#4111yescharacteristic
536C2×C3C3×D3×#4111yescharacteristic
537D3—62no—
538D3—122no—
539D3—182no—
540D3—362no—
541C22—92no—
542C22—181no—
543C22—181no—
544C22—182no—
545C22—182no—
546C22—182no—
547C22C23×(C3⋊D3)⋊C311yescharacteristic
548C4—21no—
549C4—22no—
550C4—21no—
551C4—22no—
552C4—92no—
553C4—181no—
554C4—182no—
555C4—181no—
556C4#8872212yes—
557C3—31no—
558C3—31no—
559C3—61no—
560C3C3×#569111yescharacteristic
561C2—182no—
562C2—362no—
563C2C2×#8872211yescharacteristic
564C2#41×(C3⋊D3)⋊C311yescharacteristic
565C2#41×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#125C2
prime radical#64104⋊C3C2
hypercenter#125(C3⋊D3)⋊C3
hypocenterC32C3×#529
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#125C2semidirect
2. #64107⋊C3C2semidirect
3. #64106⋊C3C2semidirect
5. #64103⋊C3C2semidirect
6. #64102⋊C3C2semidirect
7. #64102⋊C3C2semidirect
8. #64101⋊C3C2semidirect
9. #64096⋊C3C2semidirect
10. #64084⋊C3C2semidirect
11. C42×(C3⋊D3)⋊C3C2semidirect
12. #10596162C3semidirect
38. C32×#125D3semidirect
39. C32×#125C2×C3semidirect
40. #64107C2×C3semidirect
41. #64106C2×C3semidirect
43. #64103C2×C3semidirect
44. #64102C2×C3semidirect
45. #64102C2×C3semidirect
46. #64101C2×C3semidirect
47. #64096C2×C3semidirect
48. #64084C2×C3semidirect
49. C42×C3⋊D3C2×C3semidirect
342. #125(C3⋊D3)⋊C3semidirect
352. H(3,𝔽3)#529semidirect
503. C32#5691semidirect
504. 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)×#1252
#64107⋊C32
#64106⋊C32
#64104⋊C32
#64103⋊C32
#64102⋊C32
#64101⋊C32
#64096⋊C32
#64084⋊C32
C42×(C3⋊D3)⋊C32
#105961623
C3×#56913