Finite Group Databasebeta

← #10596005⋊C3 #13152525

Subgroups — #10596005⋊C3 #13152525

Order 1728.

832 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic
3#64133⋊C3C211yesmaximal, characteristic, complemented
4#64107⋊C3C212yesmaximal, complemented
5#64105⋊C3C212yesmaximal, complemented
6#64094⋊C3C212yesmaximal, complemented
7#64092⋊C3C212yesmaximal, complemented
8#64084⋊C3C211yesmaximal, characteristic, complemented
9C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
10#64078⋊C3C211yesmaximal, characteristic, complemented
11C4×#634⋊C3C211yesmaximal, characteristic, complemented
12#10596005C311yesmaximal, characteristic, complemented
13C3×#5534—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#638⋊C3C2212yes—
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2212yes—
18C2×#634⋊C3C2211yescharacteristic
19C2×#634⋊C3C2212yes—
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C4⋊C4×H(3,𝔽3)C2211yescharacteristic
22C22⋊C4×H(3,𝔽3)C2211yescharacteristic
23C22⋊C4×H(3,𝔽3)C2211yescharacteristic
24C42×H(3,𝔽3)C2211yescharacteristic
25C42×H(3,𝔽3)C2211yescharacteristic
26#3757⋊C3C2212yes—
27#3752⋊C3C2212yes—
28#3752⋊C3C2212yes—
29#3752⋊C3C2212yes—
30#3752⋊C3C2212yes—
31C4⋊((C3⋊Dic3)⋊C3)C2212yes—
32C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
33C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
34#3750⋊C3C2212yes—
35#3750⋊C3C2212yes—
36C4×(C3⋊Dic3)⋊C3C2212yes—
37C32×#117—21no—
38C32×#117D311yescharacteristic, complemented
39C32×#117C2×C311yescharacteristic, complemented
40#64139C2×C311yescharacteristic
41#64133C2×C311yescharacteristic, complemented
42#64107C2×C312yescomplemented
43#64105C2×C312yescomplemented
44#64094C2×C312yescomplemented
45#64092C2×C312yescomplemented
46#64084C2×C311yescharacteristic, complemented
47C4×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
48#64078C2×C311yescharacteristic, complemented
49C4×#634C2×C311yescharacteristic, complemented
50C3×#930—31no—
51C3×#924—31no—
52C3×#898—32no—
53C3×#896—32no—
54C3×#885—32no—
55C3×#883—32no—
56C3×#875—31no—
57C3×C4×D12—31no—
58C3×#869—31no—
59C3×C4×Dic6—31no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2312yes—
62C2×C4×H(3,𝔽3)—22no—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67C2×C4×H(3,𝔽3)C2311yescharacteristic
68C2×C4×H(3,𝔽3)C2311yescharacteristic
69#638⋊C3—42no—
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C4⋊((C3⋊D3)⋊C3)—22no—
74C4×(C3⋊D3)⋊C3—42no—
75#634⋊C3—22no—
76#634⋊C3—42no—
77C3×#542—91no—
78#5534—31no—
79#5534—91no—
80C22×C32×C4—21no—
81C22×C32×C4C2×D311yescharacteristic
82C22×C32×C4C22×C311yescharacteristic
83C2×#638C22×C312yes—
84C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
85C2×C4×C3⋊D3C22×C312yes—
86C2×#634C22×C311yescharacteristic
87C2×#634C22×C312yes—
88C2×C3×#67—32no—
89C2×C3×D12—31no—
90C2×C3×C4×D3—32no—
91C2×C3×Dic6—31no—
92C2×C3×Dic6—32no—
93C32×C4⋊C4—21no—
94C32×C4⋊C4—21no—
95C32×C4⋊C4C2×D311yescharacteristic
96C32×C4⋊C4C2×D311yescharacteristic
97C32×C4⋊C4C22×C311yescharacteristic
98C32×C4⋊C4C22×C311yescharacteristic
99C32×C22⋊C4—21no—
100C32×C22⋊C4—21no—
101C32×C22⋊C4C2×D311yescharacteristic
102C32×C22⋊C4C2×D311yescharacteristic
103C32×C22⋊C4C22×C311yescharacteristic
104C32×C22⋊C4C22×C311yescharacteristic
105C32×C42—21no—
106C32×C42—21no—
107C32×C42C2×D311yescharacteristic
108C32×C42C2×D311yescharacteristic
109C32×C42C22×C311yescharacteristic
110C32×C42C22×C311yescharacteristic
111#3757C22×C312yes—
112#3752C22×C312yes—
113#3752C22×C312yes—
114#3752C22×C312yes—
115#3752C22×C312yes—
116C4⋊(C3⋊Dic3)C22×C311yescharacteristic
117C4⋊(C3⋊Dic3)C22×C311yescharacteristic
118C4⋊(C3⋊Dic3)C22×C312yes—
119#3750C22×C312yes—
120#3750C22×C312yes—
121C4×C3⋊Dic3C22×C312yes—
122C3×#217—32no—
123C3×(C2×D3)⋊C4—32no—
124C3×(C2×D3)⋊C4—32no—
125C3×(C2×D3)⋊C4—32no—
126C3×(C2×D3)⋊C4—32no—
127C3×C4⋊Dic3—32no—
128C3×C4⋊Dic3—31no—
129C3×C4⋊Dic3—31no—
130C3×Dic3⋊C4—32no—
131C3×Dic3⋊C4—32no—
132C3×C4×Dic3—32no—
133C22×H(3,𝔽3)—21no—
134C22×H(3,𝔽3)—21no—
135C22×H(3,𝔽3)—21no—
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)—21no—
142C4×H(3,𝔽3)—21no—
143C4×H(3,𝔽3)—21no—
144C4×H(3,𝔽3)—21no—
145C4×H(3,𝔽3)#4112yes—
146(C3⋊Dic3)⋊C3—22no—
147(C3⋊Dic3)⋊C3—22no—
148(C3⋊Dic3)⋊C3—22no—
149C3×C4⋊Q8—91no—
150C3×#126—92no—
151C3×#124—91no—
152C3×#124—92no—
153C3×#123—92no—
154C3×C22⋊Q8—92no—
155C3×C22⋊Q8—91no—
156C3×#121—91no—
157C3×C4×Q8—91no—
158C3×C4×D4—91no—
159C3×#117—31no—
160C3×#117—31no—
161C3×#117—61no—
162C3×#117C3×D311yescharacteristic
163#930—31no—
164#930—91no—
165#924—31no—
166#924—91no—
167#898—32no—
168#898—92no—
169#896—32no—
170#896—92no—
171#885—32no—
172#885—92no—
173#883—32no—
174#883—92no—
175#875—31no—
176#875—91no—
177C4×D12—31no—
178C4×D12—91no—
179#869—31no—
180#869—91no—
181C4×Dic6—31no—
182C4×Dic6—91no—
183C23×C32—21no—
184C23×C32C22×D311yescharacteristic
185C23×C32C23×C311yescharacteristic
186C22×C3⋊D3C23×C312yes—
187C22×C3×D3—32no—
188C2×C32×C4—24no—
189C2×C32×C4—22no—
190C2×C32×C4—22no—
191C2×C32×C4—21no—
192C2×C32×C4—21no—
193C2×C32×C4—21no—
194C2×C32×C4—21no—
195C2×C32×C4—21no—
196C2×C32×C4—21no—
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C22×D311yescharacteristic
202C2×C32×C4C22×D311yescharacteristic
203C2×C32×C4C23×C311yescharacteristic
204C2×C32×C4C23×C311yescharacteristic
205C2×C32×C4C23×C311yescharacteristic
206C2×C32×C4C23×C311yescharacteristic
207C2×C32×C4C23×C311yescharacteristic
208C2×C32×C4C23×C311yescharacteristic
209#638—42no—
210C2×C3⋊Dic3C23×C312yes—
211C2×C3⋊Dic3C23×C312yes—
212C2×C3⋊Dic3C23×C312yes—
213C4⋊(C3⋊D3)—22no—
214C4×C3⋊D3—42no—
215#634—22no—
216#634—42no—
217C3×#67—122no—
218C2×C3×Dic3—32no—
219C2×C3×Dic3—32no—
220C2×C3×Dic3—32no—
221C3×D12—62no—
222C3×C4×D3—122no—
223C3×Dic6—62no—
224C3×Dic6—122no—
225#542—91noSylow
226C2×H(3,𝔽3)—41no—
227C2×H(3,𝔽3)C2×#4111yescharacteristic
228C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
229C2×H(3,𝔽3)2−1+411yescharacteristic
230(C3⋊D3)⋊C3—42no—
231C2×C3×Q8—91no—
232C2×C3×Q8—92no—
233C2×C3×D4—91no—
234C2×C3×D4—92no—
235C22×C3×C4—31no—
236C22×C3×C4—31no—
237C22×C3×C4—61no—
238C22×C3×C4—92no—
239C22×C3×C4C2×C3×D311yescharacteristic
240C2×#67—32no—
241C2×#67—92no—
242C2×D12—31no—
243C2×D12—91no—
244C2×C4×D3—32no—
245C2×C4×D3—92no—
246C2×Dic6—31no—
247C2×Dic6—32no—
248C2×Dic6—91no—
249C2×Dic6—92no—
250C3×C4⋊C4—31no—
251C3×C4⋊C4—31no—
252C3×C4⋊C4—31no—
253C3×C4⋊C4—31no—
254C3×C4⋊C4—61no—
255C3×C4⋊C4—61no—
256C3×C4⋊C4—92no—
257C3×C4⋊C4—92no—
258C3×C4⋊C4—91no—
259C3×C4⋊C4—91no—
260C3×C4⋊C4—92no—
261C3×C4⋊C4C2×C3×D311yescharacteristic
262C3×C4⋊C4C2×C3×D311yescharacteristic
263C3×C22⋊C4—31no—
264C3×C22⋊C4—31no—
265C3×C22⋊C4—31no—
266C3×C22⋊C4—31no—
267C3×C22⋊C4—61no—
268C3×C22⋊C4—61no—
269C3×C22⋊C4—92no—
270C3×C22⋊C4—92no—
271C3×C22⋊C4—92no—
272C3×C22⋊C4—92no—
273C3×C22⋊C4—92no—
274C3×C22⋊C4C2×C3×D311yescharacteristic
275C3×C22⋊C4C2×C3×D311yescharacteristic
276C3×C42—31no—
277C3×C42—31no—
278C3×C42—31no—
279C3×C42—31no—
280C3×C42—61no—
281C3×C42—61no—
282C3×C42—92no—
283C3×C42C2×C3×D311yescharacteristic
284C3×C42C2×C3×D311yescharacteristic
285#217—32no—
286#217—92no—
287(C2×D3)⋊C4—32no—
288(C2×D3)⋊C4—32no—
289(C2×D3)⋊C4—32no—
290(C2×D3)⋊C4—32no—
291(C2×D3)⋊C4—92no—
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295C4⋊Dic3—31no—
296C4⋊Dic3—31no—
297C4⋊Dic3—32no—
298C4⋊Dic3—91no—
299C4⋊Dic3—91no—
300C4⋊Dic3—92no—
301Dic3⋊C4—32no—
302Dic3⋊C4—32no—
303Dic3⋊C4—92no—
304Dic3⋊C4—92no—
305C4×Dic3—32no—
306C4×Dic3—92no—
307C22×C32—22no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—21no—
311C22×C32—21no—
312C22×C32—21no—
313C22×C32—21no—
314C22×C32—21no—
315C22×C32—41no—
316C22×C32—41no—
317C22×C32C23×D311yescharacteristic
318C22×C32C24×C311yescharacteristic
319C2×C3⋊D3—22no—
320C2×C3⋊D3—22no—
321C2×C3⋊D3—22no—
322C2×C3×D3—62no—
323C2×C3×D3—62no—
324C2×C3×D3—62no—
325C32×C4—22no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—21no—
336C32×C4—22no—
337C32×C4—41no—
338C32×C4—41no—
339C32×C4—41no—
340C32×C4—41no—
341C32×C4#23512yes—
342C32×C4C3×#4112yes—
343C3⋊Dic3—22no—
344C3⋊Dic3—22no—
345C3⋊Dic3—22no—
346C3×Dic3—62no—
347C3×Dic3—62no—
348C3×Dic3—62no—
349C4⋊Q8—91no—
350#126—92no—
351#124—92no—
352#124—91no—
353#123—92no—
354C22⋊Q8—91no—
355C22⋊Q8—92no—
356#121—91no—
357C4×Q8—91no—
358C4×D4—91no—
359#117(C3⋊D3)⋊C311yescharacteristic, complemented
360H(3,𝔽3)#54211yescharacteristic, complemented, Sylow
361C23×C3—31no—
362C23×C3—31no—
363C23×C3—61no—
364C23×C3—92no—
365C23×C3C22×C3×D311yescharacteristic
366C22×D3—32no—
367C22×D3—92no—
368C3×Q8—182no—
369C3×Q8—362no—
370C3×D4—182no—
371C3×D4—362no—
372C2×C3×C4—22no—
373C2×C3×C4—31no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—31no—
377C2×C3×C4—31no—
378C2×C3×C4—31no—
379C2×C3×C4—31no—
380C2×C3×C4—31no—
381C2×C3×C4—31no—
382C2×C3×C4—31no—
383C2×C3×C4—31no—
384C2×C3×C4—31no—
385C2×C3×C4—62no—
386C2×C3×C4—62no—
387C2×C3×C4—64no—
388C2×C3×C4—61no—
389C2×C3×C4—61no—
390C2×C3×C4—61no—
391C2×C3×C4—61no—
392C2×C3×C4—61no—
393C2×C3×C4—61no—
394C2×C3×C4—92no—
395C2×C3×C4—92no—
396C2×C3×C4—92no—
397C2×C3×C4—362no—
398C2×C3×C4C22×C3×D311yescharacteristic
399C2×C3×C4C22×C3×D311yescharacteristic
400C2×C3×C4C22×C3×D311yescharacteristic
401C2×C3×C4C22×C3×D311yescharacteristic
402C2×C3×C4C22×C3×D311yescharacteristic
403C2×C3×C4C22×C3×D311yescharacteristic
404#67—122no—
405#67—362no—
406C2×Dic3—32no—
407C2×Dic3—32no—
408C2×Dic3—32no—
409C2×Dic3—92no—
410C2×Dic3—92no—
411C2×Dic3—92no—
412D12—62no—
413D12—182no—
414C4×D3—122no—
415C4×D3—362no—
416Dic6—62no—
417Dic6—122no—
418Dic6—182no—
419Dic6—362no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—42no—
424C2×C32—41no—
425C2×C32—41no—
426C2×C32C2×#23511yescharacteristic
427C2×C32#100911yescharacteristic
428C2×C32#101011yescharacteristic
429C2×C32C2×C3×#4111yescharacteristic
430C2×C32C3×H(5,𝔽2)11yescharacteristic
431C2×C32C3×2−1+411yescharacteristic
432C3⋊D3—42no—
433C3×D3—122no—
434C2×Q8—91no—
435C2×Q8—92no—
436C2×D4—92no—
437C2×D4—91no—
438C22×C4—92no—
439C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
440C4⋊C4—91no—
441C4⋊C4—91no—
442C4⋊C4—92no—
443C4⋊C4—92no—
444C4⋊C4—92no—
445C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
446C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
447C22⋊C4—92no—
448C22⋊C4—92no—
449C22⋊C4—92no—
450C22⋊C4—92no—
451C22⋊C4—92no—
452C22⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
453C22⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
454C42—92no—
455C42C2×(C3⋊D3)⋊C311yescharacteristic
456C42C2×(C3⋊D3)⋊C311yescharacteristic
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—31no—
461C22×C3—31no—
462C22×C3—61no—
463C22×C3—61no—
464C22×C3—61no—
465C22×C3—61no—
466C22×C3—61no—
467C22×C3—61no—
468C22×C3—62no—
469C22×C3—61no—
470C22×C3—121no—
471C22×C3—121no—
472C22×C3—182no—
473C22×C3—182no—
474C22×C3—182no—
475C22×C3C23×C3×D311yescharacteristic
476C2×D3—62no—
477C2×D3—62no—
478C2×D3—62no—
479C2×D3—182no—
480C2×D3—182no—
481C2×D3—182no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—32no—
488C3×C4—32no—
489C3×C4—61no—
490C3×C4—61no—
491C3×C4—61no—
492C3×C4—61no—
493C3×C4—61no—
494C3×C4—61no—
495C3×C4—61no—
496C3×C4—61no—
497C3×C4—61no—
498C3×C4—61no—
499C3×C4—62no—
500C3×C4—62no—
501C3×C4—121no—
502C3×C4—121no—
503C3×C4—121no—
504C3×C4—121no—
505C3×C4—182no—
506C3×C4—182no—
507C3×C4—182no—
508C3×C4C3×#23512yes—
509Dic3—62no—
510Dic3—62no—
511Dic3—62no—
512Dic3—182no—
513Dic3—182no—
514Dic3—182no—
515C32—21no—
516C32#553411yescharacteristic, complemented
517C32C3×#54211yescharacteristic, complemented
518C23—92no—
519C23C22×(C3⋊D3)⋊C311yescharacteristic
520Q8—182no—
521Q8—362no—
522D4—182no—
523D4—362no—
524C2×C4—22no—
525C2×C4—92no—
526C2×C4—92no—
527C2×C4—92no—
528C2×C4—362no—
529C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
530C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
531C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
532C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
533C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
534C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
535C2×C3—31no—
536C2×C3—31no—
537C2×C3—31no—
538C2×C3—31no—
539C2×C3—31no—
540C2×C3—31no—
541C2×C3—41no—
542C2×C3—61no—
543C2×C3—61no—
544C2×C3—61no—
545C2×C3—121no—
546C2×C3—121no—
547C2×C3—122no—
548C2×C3—362no—
549C2×C3C2×C3×#23511yescharacteristic
550C2×C3C3×#100911yescharacteristic
551C2×C3C3×#101011yescharacteristic
552D3—122no—
553D3—362no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—182no—
558C22—182no—
559C22—182no—
560C22C23×(C3⋊D3)⋊C311yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—182no—
567C4—182no—
568C4—182no—
569C4#3828⋊C312yes—
570C3—31no—
571C3—31no—
572C3—61no—
573C3C3×#553411yescharacteristic
574C2—41no—
575C2—362no—
576C2C2×#3828⋊C311yescharacteristic
577C2#1172263811yescharacteristic
578C2#1172263911yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#117C2
prime radical#64139⋊C3C2
hypercenter#117(C3⋊D3)⋊C3
hypocenterC32C3×#542
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#117C2semidirect
3. #64133⋊C3C2semidirect
4. #64107⋊C3C2semidirect
5. #64105⋊C3C2semidirect
6. #64094⋊C3C2semidirect
7. #64092⋊C3C2semidirect
8. #64084⋊C3C2semidirect
9. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
10. #64078⋊C3C2semidirect
11. C4×#634⋊C3C2semidirect
12. #10596005C3semidirect
38. C32×#117D3semidirect
39. C32×#117C2×C3semidirect
41. #64133C2×C3semidirect
42. #64107C2×C3semidirect
43. #64105C2×C3semidirect
44. #64094C2×C3semidirect
45. #64092C2×C3semidirect
46. #64084C2×C3semidirect
47. C4×C4⋊(C3⋊D3)C2×C3semidirect
48. #64078C2×C3semidirect
49. C4×#634C2×C3semidirect
359. #117(C3⋊D3)⋊C3semidirect
360. H(3,𝔽3)#542semidirect
516. C32#5534semidirect
517. C32C3×#542semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1172
#64139⋊C32
#64133⋊C32
#64107⋊C32
#64105⋊C32
#64094⋊C32
#64092⋊C32
#64084⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
#64078⋊C32
C4×#634⋊C32
#105960053
C3×#55343