Finite Group Databasebeta

← #10596068⋊C3 #13152840

Subgroups — #10596068⋊C3 #13152840

Order 1728.

1123 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3830⋊C3C212yesmaximal, complemented
2H(3,𝔽3)×#121C211yesmaximal, characteristic, complemented
3#64148⋊C3C212yesmaximal, complemented
4#64148⋊C3C212yesmaximal, complemented
5#64139⋊C3C211yesmaximal, characteristic, complemented
6C2×#3750⋊C3C211yesmaximal, characteristic, complemented
7#64105⋊C3C212yesmaximal, complemented
8#64093⋊C3C212yesmaximal, complemented
9#64087⋊C3C212yesmaximal, complemented
10#10596068C311yesmaximal, characteristic, complemented
11C3×#5597—31nomaximal
12C2×D4×H(3,𝔽3)C2212yes—
13C2×D4×H(3,𝔽3)C2211yescharacteristic
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#638⋊C3C2212yes—
16C2×#638⋊C3C2212yes—
17C2×#638⋊C3C2212yes—
18C22×(C3⋊Dic3)⋊C3C2212yes—
19C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
21#3830⋊C3—28no—
22C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
23C2×C4×(C3⋊D3)⋊C3C2212yes—
24C2×#634⋊C3C2212yes—
25C4⋊C4×H(3,𝔽3)C2211yescharacteristic
26C22⋊C4×H(3,𝔽3)C2212yes—
27#3757⋊C3C2212yes—
28#3752⋊C3C2212yes—
29#3752⋊C3C2212yes—
30C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
31#3750⋊C3C2212yes—
32#3750⋊C3C2212yes—
33#3750⋊C3C2212yes—
34C2×#3830C2×C312yescomplemented
35C2×C3×#237—32no—
36C32×#121—21no—
37C32×#121D311yescharacteristic, complemented
38C32×#121C2×C311yescharacteristic, complemented
39#64148C2×C312yescomplemented
40#64148C2×C312yescomplemented
41#64139C2×C311yescharacteristic, complemented
42C2×#3750C2×C311yescharacteristic, complemented
43#64105C2×C312yescomplemented
44#64093C2×C312yescomplemented
45#64087C2×C312yescomplemented
46C3×#939—32no—
47C3×#939—32no—
48C3×#930—31no—
49C2×C3×Dic3⋊C4—31no—
50C3×#896—32no—
51C3×#884—32no—
52C3×#878—32no—
53C23×H(3,𝔽3)C2312yes—
54C23×H(3,𝔽3)C2311yescharacteristic
55C22×(C3⋊D3)⋊C3C2312yes—
56D4×H(3,𝔽3)—24no—
57D4×H(3,𝔽3)—41no—
58C2×C4×H(3,𝔽3)—41no—
59C2×C4×H(3,𝔽3)C2312yes—
60C2×C4×H(3,𝔽3)C2311yescharacteristic
61C2×C4×H(3,𝔽3)C2311yescharacteristic
62#638⋊C3—24no—
63#638⋊C3—24no—
64#638⋊C3—42no—
65C2×(C3⋊Dic3)⋊C3—24no—
66C2×(C3⋊Dic3)⋊C3—41no—
67C2×(C3⋊Dic3)⋊C3D414yescomplemented
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C4⋊((C3⋊D3)⋊C3)—41no—
72C4×(C3⋊D3)⋊C3—24no—
73#634⋊C3—24no—
74C3×#537—91no—
75#5597—31no—
76#5597—91no—
77C2×C32×D4—21no—
78C2×C32×D4—22no—
79C2×C32×D4C2×D312yes—
80C2×C32×D4C2×D311yescharacteristic
81C2×C32×D4C22×C312yes—
82C2×C32×D4C22×C311yescharacteristic
83C22×C32×C4—21no—
84C22×C32×C4C2×D311yescharacteristic
85C22×C32×C4C22×C311yescharacteristic
86C2×#638C22×C312yes—
87C2×#638C22×C312yes—
88C2×#638C22×C312yes—
89C22×C3⋊Dic3C22×C312yes—
90C22×C3⋊Dic3C22×C311yescharacteristic
91C22×C3⋊Dic3C22×C311yescharacteristic
92#3830—28no—
93C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
94C2×C4×C3⋊D3C22×C312yes—
95C2×#634C22×C312yes—
96C2×C3×#67—32no—
97C2×C3×#67—32no—
98C2×C3×#67—32no—
99C22×C3×Dic3—31no—
100C22×C3×Dic3—32no—
101C22×C3×Dic3—31no—
102C3×#237—68no—
103C2×C3×D12—31no—
104C2×C3×C4×D3—32no—
105C2×C3×Dic6—32no—
106C32×C4⋊C4—21no—
107C32×C4⋊C4C2×D311yescharacteristic
108C32×C4⋊C4C22×C311yescharacteristic
109C32×C22⋊C4—22no—
110C32×C22⋊C4C2×D312yes—
111C32×C22⋊C4C22×C312yes—
112#3757C22×C312yes—
113#3752C22×C312yes—
114#3752C22×C312yes—
115C4⋊(C3⋊Dic3)C22×C311yescharacteristic
116#3750C22×C312yes—
117#3750C22×C312yes—
118#3750C22×C312yes—
119C3×#217—32no—
120C3×(C2×D3)⋊C4—32no—
121C3×(C2×D3)⋊C4—32no—
122C3×C4⋊Dic3—31no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C22×H(3,𝔽3)—21no—
127C22×H(3,𝔽3)—21no—
128C22×H(3,𝔽3)—22no—
129C22×H(3,𝔽3)—22no—
130C22×H(3,𝔽3)—22no—
131C22×H(3,𝔽3)C2×D412yes—
132C22×H(3,𝔽3)C2411yescharacteristic
133C2×(C3⋊D3)⋊C3—22no—
134C2×(C3⋊D3)⋊C3—22no—
135C2×(C3⋊D3)⋊C3—22no—
136C4×H(3,𝔽3)—22no—
137C4×H(3,𝔽3)—21no—
138C4×H(3,𝔽3)—21no—
139(C3⋊Dic3)⋊C3—22no—
140(C3⋊Dic3)⋊C3—22no—
141(C3⋊Dic3)⋊C3C2×D414yes—
142C2×C3×#41—92no—
143C2×#237—32no—
144C2×#237—92no—
145C3×C22⋊Q8—92no—
146C3×C22⋊Q8—92no—
147C3×#121—31no—
148C3×#121—31no—
149C3×#121—61no—
150C3×#121—92no—
151C3×#121—91no—
152C3×#121—92no—
153C3×#121—92no—
154C3×#121C3×D311yescharacteristic
155C2×C3×C4⋊C4—91no—
156#939—32no—
157#939—32no—
158#939—92no—
159#939—92no—
160#930—31no—
161#930—91no—
162C2×Dic3⋊C4—31no—
163C2×Dic3⋊C4—91no—
164#896—32no—
165#896—92no—
166#884—32no—
167#884—92no—
168#878—32no—
169#878—92no—
170C23×C32—21no—
171C23×C32—22no—
172C23×C32C22×D312yes—
173C23×C32C22×D311yescharacteristic
174C23×C32C23×C312yes—
175C23×C32C23×C311yescharacteristic
176C22×C3⋊D3C23×C312yes—
177C22×C3×D3—32no—
178C32×D4—24no—
179C32×D4—24no—
180C32×D4—44no—
181C32×D4—41no—
182C32×D4—41no—
183C32×D4—42no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—22no—
187C2×C32×C4—41no—
188C2×C32×C4—41no—
189C2×C32×C4—42no—
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D312yes—
193C2×C32×C4C23×C311yescharacteristic
194C2×C32×C4C23×C311yescharacteristic
195C2×C32×C4C23×C312yes—
196#638—24no—
197#638—24no—
198#638—42no—
199C2×C3⋊Dic3—24no—
200C2×C3⋊Dic3—41no—
201C2×C3⋊Dic3C3×D414yescomplemented
202C2×C3⋊Dic3C23×C312yes—
203C2×C3⋊Dic3C23×C312yes—
204C2×C3⋊Dic3C23×C312yes—
205C4⋊(C3⋊D3)—41no—
206C4×C3⋊D3—24no—
207#634—24no—
208C3×#67—64no—
209C3×#67—64no—
210C3×#67—122no—
211C2×C3×Dic3—34no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—32no—
215C2×C3×Dic3—64no—
216C2×C3×Dic3—121no—
217C3×D12—121no—
218C3×C4×D3—64no—
219C3×Dic6—64no—
220#537—91noSylow
221C2×H(3,𝔽3)—22no—
222C2×H(3,𝔽3)—42no—
223C2×H(3,𝔽3)C22×D411yescharacteristic
224C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
225C2×H(3,𝔽3)2−1+411yescharacteristic
226(C3⋊D3)⋊C3—42no—
227C3×#41—188no—
228C2×C3×Q8—92no—
229C2×C3×D4—32no—
230C2×C3×D4—32no—
231C2×C3×D4—31no—
232C2×C3×D4—31no—
233C2×C3×D4—62no—
234C2×C3×D4—61no—
235C2×C3×D4—91no—
236C2×C3×D4—92no—
237C2×C3×D4—92no—
238C2×C3×D4—92no—
239C2×C3×D4C2×C3×D312yes—
240C2×C3×D4C2×C3×D311yescharacteristic
241C22×C3×C4—31no—
242C22×C3×C4—31no—
243C22×C3×C4—61no—
244C22×C3×C4—92no—
245C22×C3×C4—91no—
246C22×C3×C4—92no—
247C22×C3×C4—91no—
248C22×C3×C4C2×C3×D311yescharacteristic
249C2×#67—32no—
250C2×#67—32no—
251C2×#67—32no—
252C2×#67—92no—
253C2×#67—92no—
254C2×#67—92no—
255C22×Dic3—31no—
256C22×Dic3—32no—
257C22×Dic3—31no—
258C22×Dic3—91no—
259C22×Dic3—92no—
260C22×Dic3—91no—
261#237—68no—
262#237—188no—
263C2×D12—31no—
264C2×D12—91no—
265C2×C4×D3—32no—
266C2×C4×D3—92no—
267C2×Dic6—32no—
268C2×Dic6—92no—
269C3×C4⋊C4—31no—
270C3×C4⋊C4—31no—
271C3×C4⋊C4—61no—
272C3×C4⋊C4—92no—
273C3×C4⋊C4—92no—
274C3×C4⋊C4—92no—
275C3×C4⋊C4—91no—
276C3×C4⋊C4C2×C3×D311yescharacteristic
277C3×C22⋊C4—32no—
278C3×C22⋊C4—32no—
279C3×C22⋊C4—62no—
280C3×C22⋊C4—92no—
281C3×C22⋊C4—92no—
282C3×C22⋊C4—92no—
283C3×C22⋊C4C2×C3×D312yes—
284#217—32no—
285#217—92no—
286(C2×D3)⋊C4—32no—
287(C2×D3)⋊C4—32no—
288(C2×D3)⋊C4—92no—
289(C2×D3)⋊C4—92no—
290C4⋊Dic3—31no—
291C4⋊Dic3—91no—
292Dic3⋊C4—32no—
293Dic3⋊C4—32no—
294Dic3⋊C4—32no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298C22×C32—21no—
299C22×C32—21no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—22no—
303C22×C32—22no—
304C22×C32—22no—
305C22×C32—22no—
306C22×C32—22no—
307C22×C32—22no—
308C22×C32—21no—
309C22×C32—22no—
310C22×C32—41no—
311C22×C32—41no—
312C22×C32—42no—
313C22×C32—42no—
314C22×C32—42no—
315C22×C32D3×D412yes—
316C22×C32C2×C3×D412yes—
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—22no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—22no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—41no—
333C32×C4—42no—
334C3⋊Dic3—22no—
335C3⋊Dic3—22no—
336C3⋊Dic3C2×C3×D414yes—
337C3×Dic3—34no—
338C3×Dic3—62no—
339C3×Dic3—62no—
340C2×#41—92no—
341C22⋊Q8—92no—
342C22⋊Q8—92no—
343#121—92no—
344#121—92no—
345#121—91no—
346#121—92no—
347#121(C3⋊D3)⋊C311yescharacteristic, complemented
348C2×C4⋊C4—91no—
349H(3,𝔽3)#53711yescharacteristic, complemented, Sylow
350C23×C3—31no—
351C23×C3—31no—
352C23×C3—32no—
353C23×C3—32no—
354C23×C3—61no—
355C23×C3—62no—
356C23×C3—92no—
357C23×C3C22×C3×D312yes—
358C23×C3C22×C3×D311yescharacteristic
359C22×D3—32no—
360C22×D3—92no—
361C3×Q8—184no—
362C3×D4—24no—
363C3×D4—41no—
364C3×D4—64no—
365C3×D4—64no—
366C3×D4—124no—
367C3×D4—121no—
368C3×D4—121no—
369C3×D4—122no—
370C3×D4—184no—
371C3×D4—184no—
372C3×D4—361no—
373C3×D4—362no—
374C2×C3×C4—31no—
375C2×C3×C4—32no—
376C2×C3×C4—31no—
377C2×C3×C4—31no—
378C2×C3×C4—31no—
379C2×C3×C4—32no—
380C2×C3×C4—41no—
381C2×C3×C4—61no—
382C2×C3×C4—61no—
383C2×C3×C4—62no—
384C2×C3×C4—92no—
385C2×C3×C4—94no—
386C2×C3×C4—92no—
387C2×C3×C4—92no—
388C2×C3×C4—121no—
389C2×C3×C4—121no—
390C2×C3×C4—122no—
391C2×C3×C4—184no—
392C2×C3×C4—184no—
393C2×C3×C4—361no—
394C2×C3×C4C22×C3×D311yescharacteristic
395C2×C3×C4C22×C3×D311yescharacteristic
396C2×C3×C4C22×C3×D312yes—
397#67—64no—
398#67—64no—
399#67—122no—
400#67—184no—
401#67—184no—
402#67—362no—
403C2×Dic3—32no—
404C2×Dic3—34no—
405C2×Dic3—32no—
406C2×Dic3—32no—
407C2×Dic3—64no—
408C2×Dic3—94no—
409C2×Dic3—92no—
410C2×Dic3—92no—
411C2×Dic3—92no—
412C2×Dic3—121no—
413C2×Dic3—184no—
414C2×Dic3—361no—
415D12—121no—
416D12—361no—
417C4×D3—64no—
418C4×D3—184no—
419Dic6—64no—
420Dic6—184no—
421C2×C32—22no—
422C2×C32—22no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—42no—
427C2×C32—42no—
428C2×C32—42no—
429C2×C32—82no—
430C2×C32C2×D3×D411yescharacteristic
431C2×C32#100411yescharacteristic
432C2×C32#101011yescharacteristic
433C2×C32C22×C3×D411yescharacteristic
434C2×C32C3×H(5,𝔽2)11yescharacteristic
435C2×C32C3×2−1+411yescharacteristic
436C3⋊D3—42no—
437C3×D3—122no—
438#41—188no—
439C2×Q8—92no—
440C2×D4—92no—
441C2×D4—92no—
442C2×D4—92no—
443C2×D4—91no—
444C2×D4C2×(C3⋊D3)⋊C312yes—
445C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
446C22×C4—91no—
447C22×C4—92no—
448C22×C4—91no—
449C22×C4—92no—
450C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
451C4⋊C4—92no—
452C4⋊C4—92no—
453C4⋊C4—92no—
454C4⋊C4—91no—
455C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
456C22⋊C4—92no—
457C22⋊C4—92no—
458C22⋊C4—92no—
459C22⋊C4C2×(C3⋊D3)⋊C312yes—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—31no—
466C22×C3—31no—
467C22×C3—32no—
468C22×C3—32no—
469C22×C3—61no—
470C22×C3—61no—
471C22×C3—61no—
472C22×C3—61no—
473C22×C3—62no—
474C22×C3—62no—
475C22×C3—62no—
476C22×C3—62no—
477C22×C3—62no—
478C22×C3—62no—
479C22×C3—61no—
480C22×C3—62no—
481C22×C3—121no—
482C22×C3—121no—
483C22×C3—122no—
484C22×C3—122no—
485C22×C3—122no—
486C22×C3—182no—
487C22×C3—182no—
488C22×C3—182no—
489C22×C3C3×D3×D412yes—
490C22×C3C23×C3×D311yescharacteristic
491C2×D3—62no—
492C2×D3—62no—
493C2×D3—62no—
494C2×D3—182no—
495C2×D3—182no—
496C2×D3—182no—
497C3×C4—22no—
498C3×C4—21no—
499C3×C4—21no—
500C3×C4—62no—
501C3×C4—61no—
502C3×C4—61no—
503C3×C4—61no—
504C3×C4—61no—
505C3×C4—62no—
506C3×C4—62no—
507C3×C4—94no—
508C3×C4—121no—
509C3×C4—122no—
510C3×C4—182no—
511C3×C4—182no—
512Dic3—34no—
513Dic3—62no—
514Dic3—62no—
515Dic3—94no—
516Dic3—182no—
517Dic3—182no—
518C32—21no—
519C32#559711yescharacteristic, complemented
520C32C3×#53711yescharacteristic, complemented
521C23—92no—
522C23C22×(C3⋊D3)⋊C312yes—
523C23C22×(C3⋊D3)⋊C311yescharacteristic
524Q8—184no—
525D4—24no—
526D4—41no—
527D4—184no—
528D4—184no—
529D4—361no—
530D4—362no—
531C2×C4—41no—
532C2×C4—92no—
533C2×C4—94no—
534C2×C4—92no—
535C2×C4—92no—
536C2×C4—184no—
537C2×C4—184no—
538C2×C4—361no—
539C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
540C2×C4C22×(C3⋊D3)⋊C312yes—
541C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
542C2×C3—22no—
543C2×C3—31no—
544C2×C3—31no—
545C2×C3—31no—
546C2×C3—31no—
547C2×C3—31no—
548C2×C3—31no—
549C2×C3—42no—
550C2×C3—62no—
551C2×C3—62no—
552C2×C3—61no—
553C2×C3—61no—
554C2×C3—61no—
555C2×C3—122no—
556C2×C3—122no—
557C2×C3—122no—
558C2×C3—242no—
559C2×C3—362no—
560C2×C3C2×C3×D3×D411yescharacteristic
561C2×C3C3×#100411yescharacteristic
562C2×C3C3×#101011yescharacteristic
563D3—122no—
564D3—362no—
565C22—21no—
566C22—21no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—182no—
571C22—182no—
572C22—182no—
573C22D4×(C3⋊D3)⋊C312yes—
574C22C23×(C3⋊D3)⋊C311yescharacteristic
575C4—22no—
576C4—21no—
577C4—21no—
578C4—94no—
579C4—182no—
580C4—182no—
581C3—31no—
582C3—31no—
583C3—61no—
584C3C3×#559711yescharacteristic
585C2—22no—
586C2—42no—
587C2—362no—
588C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
589C2#64361⋊C311yescharacteristic
590C2#1172263911yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#121C2
hypercenter#121(C3⋊D3)⋊C3
hypocenterC32C3×#537
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#3830⋊C3C2semidirect
2. H(3,𝔽3)×#121C2semidirect
3. #64148⋊C3C2semidirect
4. #64148⋊C3C2semidirect
5. #64139⋊C3C2semidirect
6. C2×#3750⋊C3C2semidirect
7. #64105⋊C3C2semidirect
8. #64093⋊C3C2semidirect
9. #64087⋊C3C2semidirect
10. #10596068C3semidirect
34. C2×#3830C2×C3semidirect
37. C32×#121D3semidirect
38. C32×#121C2×C3semidirect
39. #64148C2×C3semidirect
40. #64148C2×C3semidirect
41. #64139C2×C3semidirect
42. C2×#3750C2×C3semidirect
43. #64105C2×C3semidirect
44. #64093C2×C3semidirect
45. #64087C2×C3semidirect
67. C2×(C3⋊Dic3)⋊C3D4semidirect
201. C2×C3⋊Dic3C3×D4semidirect
347. #121(C3⋊D3)⋊C3semidirect
349. H(3,𝔽3)#537semidirect
519. C32#5597semidirect
520. C32C3×#537semidirect

Maximal subgroups

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

subgroupindex
C2×#3830⋊C32
H(3,𝔽3)×#1212
#64148⋊C32
#64139⋊C32
C2×#3750⋊C32
#64105⋊C32
#64093⋊C32
#64087⋊C32
#105960683
C3×#55973