Finite Group Databasebeta

← C32×#5629 #13179010

Subgroups — C32×#5629 #13179010

Order 1728.

2060 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×C32×#239C211yesmaximal, characteristic, complemented
2C2×C32×#235C211yesmaximal, characteristic, complemented
3C33×C22⋊Q8C211yesmaximal, characteristic, complemented
4C32×#946C211yesmaximal, characteristic, complemented
5C32×#930C211yesmaximal, characteristic, complemented
6C32×#896C212yesmaximal, complemented
7C32×#895C211yesmaximal, characteristic, complemented
8C32×#895C212yesmaximal, complemented
9C32×D3×C4⋊C4C211yesmaximal, characteristic, complemented
10C32×#884C212yesmaximal, complemented
11C32×#884C212yesmaximal, complemented
12C32×#537—31nomaximal
13C3×#5629C314yesmaximal, direct product factor
14C2×C33×Q8C2211yescharacteristic
15C22×C33×C4C2211yescharacteristic
16C2×C32×#67C2211yescharacteristic
17C2×C32×#67C2211yescharacteristic
18C2×C32×#67C2212yes—
19C32×#239—24no—
20C32×#235—24no—
21C2×C32×D12C2211yescharacteristic
22C2×C32×D12C2211yescharacteristic
23C2×C32×D12C2212yes—
24C2×C32×D12C2212yes—
25C2×C32×C4×D3C2211yescharacteristic
26C2×C32×C4×D3C2212yes—
27C2×C32×C4×D3C2212yes—
28C2×C32×C4×D3C2211yescharacteristic
29C2×C32×Dic6C2211yescharacteristic
30C33×C4⋊C4C2211yescharacteristic
31C33×C4⋊C4C2212yes—
32C33×C22⋊C4C2212yes—
33C32×(C2×D3)⋊C4C2212yes—
34C32×(C2×D3)⋊C4C2212yes—
35C32×(C2×D3)⋊C4C2212yes—
36C32×C4⋊Dic3C2211yescharacteristic
37C32×Dic3⋊C4C2212yes—
38C32×Dic3⋊C4C2212yes—
39C2×C32×#41—31no—
40C2×C32×#41—31no—
41C2×C3×#239C2×C314yescomplemented
42C2×C3×#235C2×C314yescomplemented
43C32×C22⋊Q8—24no—
44C32×C22⋊Q8—31no—
45C32×C22⋊Q8—32no—
46C32×C22⋊Q8D311yescharacteristic, complemented
47C32×C22⋊Q8C2×C314yescomplemented
48C32×#121—31no—
49C32×#121—31no—
50C32×#121—32no—
51C32×#121—32no—
52C32×#121—32no—
53C2×C32×C4⋊C4—31no—
54C3×#946C2×C314yescomplemented
55C3×#930C2×C314yescomplemented
56C3×#896C2×C318yescomplemented
57C3×#895C2×C314yescomplemented
58C3×#895C2×C318yescomplemented
59C3×D3×C4⋊C4C2×C314yescomplemented
60C3×#884C2×C318yescomplemented
61C3×#884C2×C318yescomplemented
62C23×C33C2311yescharacteristic
63C22×C32×D3C2312yes—
64C22×C32×D3C2311yescharacteristic
65C22×C32×D3C2311yescharacteristic
66C33×Q8—22no—
67C2×C33×C4—22no—
68C2×C33×C4C2312yes—
69C2×C33×C4C2312yes—
70C2×C33×C4C2311yescharacteristic
71C2×C33×C4C2311yescharacteristic
72C32×#67—22no—
73C32×#67—22no—
74C32×#67—42no—
75C2×C32×Dic3C2311yescharacteristic
76C2×C32×Dic3C2312yes—
77C2×C32×Dic3C2311yescharacteristic
78C32×D12—22no—
79C32×D12—24no—
80C32×D12—22no—
81C32×D12—42no—
82C32×C4×D3—22no—
83C32×C4×D3—24no—
84C32×C4×D3—42no—
85C32×C4×D3D414yescomplemented
86C32×Dic6—22no—
87C3×#537—34no—
88#5629C3211yescharacteristic, direct product factor
89C32×#41—64no—
90C32×#41—64no—
91C2×C32×Q8—24no—
92C2×C32×Q8—31no—
93C2×C32×Q8C2×D311yescharacteristic
94C2×C32×Q8C22×C314yes—
95C2×C32×D4—31no—
96C2×C32×D4—31no—
97C2×C32×D4—32no—
98C2×C32×D4—31no—
99C2×C32×D4—32no—
100C2×C32×D4—31no—
101C2×C32×D4—32no—
102C22×C32×C4—24no—
103C22×C32×C4—31no—
104C22×C32×C4—32no—
105C22×C32×C4—31no—
106C22×C32×C4—32no—
107C22×C32×C4C2×D311yescharacteristic
108C22×C32×C4C22×C314yes—
109C2×C3×#67C22×C314yes—
110C2×C3×#67C22×C314yes—
111C2×C3×#67C22×C318yes—
112C3×#239—216no—
113C3×#235—216no—
114C2×C3×D12C22×C318yes—
115C2×C3×D12C22×C314yes—
116C2×C3×D12C22×C314yes—
117C2×C3×D12C22×C318yes—
118C2×C3×C4×D3C22×C318yes—
119C2×C3×C4×D3C22×C314yes—
120C2×C3×C4×D3C22×C318yes—
121C2×C3×C4×D3C22×C314yes—
122C2×C3×Dic6C22×C314yes—
123C32×C4⋊C4—24no—
124C32×C4⋊C4—28no—
125C32×C4⋊C4—32no—
126C32×C4⋊C4—31no—
127C32×C4⋊C4—32no—
128C32×C4⋊C4C2×D311yescharacteristic
129C32×C4⋊C4C2×D312yes—
130C32×C4⋊C4C22×C314yes—
131C32×C4⋊C4C22×C318yes—
132C32×C22⋊C4—28no—
133C32×C22⋊C4—32no—
134C32×C22⋊C4—32no—
135C32×C22⋊C4—32no—
136C32×C22⋊C4C2×D312yes—
137C32×C22⋊C4C22×C318yes—
138C3×(C2×D3)⋊C4C22×C318yes—
139C3×(C2×D3)⋊C4C22×C318yes—
140C3×(C2×D3)⋊C4C22×C318yes—
141C3×C4⋊Dic3C22×C314yes—
142C3×Dic3⋊C4C22×C318yes—
143C3×Dic3⋊C4C22×C318yes—
144C22×C33—21no—
145C22×C33—21no—
146C22×C33—21no—
147C22×C33C2411yescharacteristic
148C2×C32×D3—21no—
149C2×C32×D3—21no—
150C2×C32×D3—22no—
151C2×C32×D3—21no—
152C2×C32×D3—21no—
153C2×C32×D3—22no—
154C2×C32×D3—22no—
155C2×C32×D3—21no—
156C2×C32×D3C2×D412yes—
157C33×C4—21no—
158C33×C4—22no—
159C33×C4—22no—
160C33×C4C2×D412yes—
161C32×Dic3—21no—
162C32×Dic3—22no—
163C32×Dic3C2×D412yes—
164C2×C3×#41—34no—
165C2×C3×#41—34no—
166C2×#239C2×C3211yescharacteristic, complemented
167C2×#235C2×C3211yescharacteristic, complemented
168C3×C22⋊Q8—24no—
169C3×C22⋊Q8—34no—
170C3×C22⋊Q8—38no—
171C3×C22⋊Q8C3×D314yescomplemented
172C3×C22⋊Q8C2×C3211yescharacteristic, complemented
173C3×#121—34no—
174C3×#121—34no—
175C3×#121—38no—
176C3×#121—38no—
177C3×#121—38no—
178C2×C3×C4⋊C4—34no—
179#946C2×C3211yescharacteristic, complemented
180#930C2×C3211yescharacteristic, complemented
181#896C2×C3212yescomplemented
182#895C2×C3211yescharacteristic, complemented
183#895C2×C3212yescomplemented
184D3×C4⋊C4C2×C3211yescharacteristic, complemented
185#884C2×C3212yescomplemented
186#884C2×C3212yescomplemented
187C23×C32—24no—
188C23×C32—31no—
189C23×C32—31no—
190C23×C32—32no—
191C23×C32C22×D311yescharacteristic
192C23×C32C23×C314yes—
193C22×C3×D3C23×C318yes—
194C22×C3×D3C23×C314yes—
195C22×C3×D3C23×C314yes—
196C32×Q8—22no—
197C32×Q8—28no—
198C32×Q8—48no—
199C32×Q8—62no—
200C32×D4—62no—
201C32×D4—64no—
202C32×D4—62no—
203C32×D4—62no—
204C32×D4—62no—
205C32×D4—122no—
206C32×D4—122no—
207C2×C32×C4—22no—
208C2×C32×C4—24no—
209C2×C32×C4—28no—
210C2×C32×C4—28no—
211C2×C32×C4—24no—
212C2×C32×C4—28no—
213C2×C32×C4—31no—
214C2×C32×C4—34no—
215C2×C32×C4—32no—
216C2×C32×C4—31no—
217C2×C32×C4—48no—
218C2×C32×C4—64no—
219C2×C32×C4—62no—
220C2×C32×C4—122no—
221C2×C32×C4C22×D311yescharacteristic
222C2×C32×C4C22×D312yes—
223C2×C32×C4C22×D312yes—
224C2×C32×C4C22×D311yescharacteristic
225C2×C32×C4C23×C318yes—
226C2×C32×C4C23×C318yes—
227C2×C32×C4C23×C314yes—
228C2×C32×C4C23×C314yes—
229C3×#67—28no—
230C3×#67—28no—
231C3×#67—48no—
232C2×C3×Dic3C23×C318yes—
233C2×C3×Dic3C23×C314yes—
234C2×C3×Dic3C23×C314yes—
235C3×D12—28no—
236C3×D12—216no—
237C3×D12—28no—
238C3×D12—48no—
239C3×C4×D3—28no—
240C3×C4×D3—216no—
241C3×C4×D3—48no—
242C3×C4×D3C3×D4116yescomplemented
243C3×Dic6—28no—
244#537—31noSylow
245C2×C33—41no—
246C2×C33C22×D411yescharacteristic
247C2×C33H(5,𝔽2)11yescharacteristic
248C2×C332−1+411yescharacteristic
249C32×D3—22no—
250C32×D3—42no—
251C32×D3—41no—
252C3×#41—616no—
253C3×#41—616no—
254C2×C3×Q8—24no—
255C2×C3×Q8—34no—
256C2×C3×Q8C2×C3×D314yes—
257C2×C3×Q8C22×C3211yescharacteristic
258C2×C3×D4—34no—
259C2×C3×D4—34no—
260C2×C3×D4—38no—
261C2×C3×D4—34no—
262C2×C3×D4—38no—
263C2×C3×D4—34no—
264C2×C3×D4—38no—
265C22×C3×C4—24no—
266C22×C3×C4—34no—
267C22×C3×C4—38no—
268C22×C3×C4—34no—
269C22×C3×C4—38no—
270C22×C3×C4C2×C3×D314yes—
271C22×C3×C4C22×C3211yescharacteristic
272C2×#67C22×C3211yescharacteristic
273C2×#67C22×C3211yescharacteristic
274C2×#67C22×C3212yes—
275#239—24no—
276#235—24no—
277C2×D12C22×C3212yes—
278C2×D12C22×C3211yescharacteristic
279C2×D12C22×C3211yescharacteristic
280C2×D12C22×C3212yes—
281C2×C4×D3C22×C3212yes—
282C2×C4×D3C22×C3211yescharacteristic
283C2×C4×D3C22×C3212yes—
284C2×C4×D3C22×C3211yescharacteristic
285C2×Dic6C22×C3211yescharacteristic
286C3×C4⋊C4—24no—
287C3×C4⋊C4—28no—
288C3×C4⋊C4—38no—
289C3×C4⋊C4—38no—
290C3×C4⋊C4—34no—
291C3×C4⋊C4C2×C3×D314yes—
292C3×C4⋊C4C2×C3×D318yes—
293C3×C4⋊C4C22×C3211yescharacteristic
294C3×C4⋊C4C22×C3212yes—
295C3×C22⋊C4—28no—
296C3×C22⋊C4—38no—
297C3×C22⋊C4—38no—
298C3×C22⋊C4—38no—
299C3×C22⋊C4C2×C3×D318yes—
300C3×C22⋊C4C22×C3212yes—
301(C2×D3)⋊C4C22×C3212yes—
302(C2×D3)⋊C4C22×C3212yes—
303(C2×D3)⋊C4C22×C3212yes—
304C4⋊Dic3C22×C3211yescharacteristic
305Dic3⋊C4C22×C3212yes—
306Dic3⋊C4C22×C3212yes—
307C22×C32—28no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—21no—
311C22×C32—24no—
312C22×C32—24no—
313C22×C32—24no—
314C22×C32—24no—
315C22×C32—32no—
316C22×C32—44no—
317C22×C32—44no—
318C22×C32—61no—
319C22×C32—61no—
320C22×C32—62no—
321C22×C32—61no—
322C22×C32—61no—
323C22×C32—62no—
324C22×C32—61no—
325C22×C32—62no—
326C22×C32C23×D311yescharacteristic
327C22×C32C24×C314yes—
328C2×C3×D3—24no—
329C2×C3×D3—24no—
330C2×C3×D3—28no—
331C2×C3×D3—24no—
332C2×C3×D3—24no—
333C2×C3×D3—28no—
334C2×C3×D3—28no—
335C2×C3×D3—24no—
336C2×C3×D3C2×C3×D418yes—
337C32×C4—21no—
338C32×C4—22no—
339C32×C4—22no—
340C32×C4—28no—
341C32×C4—24no—
342C32×C4—28no—
343C32×C4—28no—
344C32×C4—32no—
345C32×C4—48no—
346C32×C4—44no—
347C32×C4—48no—
348C32×C4—61no—
349C32×C4—62no—
350C32×C4D3×D412yes—
351C32×C4C2×C3×D418yes—
352C3×Dic3—24no—
353C3×Dic3—28no—
354C3×Dic3C2×C3×D418yes—
355C2×#41—31no—
356C2×#41—31no—
357C22⋊Q8—31no—
358C22⋊Q8—32no—
359C22⋊Q8C32×D311yescharacteristic, complemented
360#121—31no—
361#121—31no—
362#121—32no—
363#121—32no—
364#121—32no—
365C2×C4⋊C4—31no—
366C33#53711yescharacteristic, complemented, Sylow
367C23×C3—24no—
368C23×C3—34no—
369C23×C3—34no—
370C23×C3—38no—
371C23×C3C22×C3×D314yes—
372C23×C3C23×C3211yescharacteristic
373C22×D3C23×C3212yes—
374C22×D3C23×C3211yescharacteristic
375C22×D3C23×C3211yescharacteristic
376C3×Q8—28no—
377C3×Q8—22no—
378C3×Q8—48no—
379C3×Q8—68no—
380C3×D4—68no—
381C3×D4—616no—
382C3×D4—68no—
383C3×D4—68no—
384C3×D4—68no—
385C3×D4—128no—
386C3×D4—128no—
387C2×C3×C4—28no—
388C2×C3×C4—22no—
389C2×C3×C4—24no—
390C2×C3×C4—28no—
391C2×C3×C4—28no—
392C2×C3×C4—24no—
393C2×C3×C4—34no—
394C2×C3×C4—316no—
395C2×C3×C4—38no—
396C2×C3×C4—34no—
397C2×C3×C4—48no—
398C2×C3×C4—616no—
399C2×C3×C4—68no—
400C2×C3×C4—128no—
401C2×C3×C4C22×C3×D314yes—
402C2×C3×C4C22×C3×D318yes—
403C2×C3×C4C22×C3×D318yes—
404C2×C3×C4C22×C3×D314yes—
405C2×C3×C4C23×C3212yes—
406C2×C3×C4C23×C3212yes—
407C2×C3×C4C23×C3211yescharacteristic
408C2×C3×C4C23×C3211yescharacteristic
409#67—22no—
410#67—22no—
411#67—42no—
412C2×Dic3C23×C3212yes—
413C2×Dic3C23×C3211yescharacteristic
414C2×Dic3C23×C3211yescharacteristic
415D12—22no—
416D12—24no—
417D12—22no—
418D12—42no—
419C4×D3—22no—
420C4×D3—24no—
421C4×D3—42no—
422C4×D3C32×D414yescomplemented
423Dic6—22no—
424C2×C32—24no—
425C2×C32—24no—
426C2×C32—24no—
427C2×C32—48no—
428C2×C32—44no—
429C2×C32—41no—
430C2×C32—62no—
431C2×C32—122no—
432C2×C32—121no—
433C2×C32C2×D3×D411yescharacteristic
434C2×C32#100711yescharacteristic
435C2×C32#100911yescharacteristic
436C2×C32C22×C3×D414yes—
437C2×C32C3×H(5,𝔽2)14yes—
438C2×C32C3×2−1+414yes—
439C3×D3—28no—
440C3×D3—48no—
441C3×D3—44no—
442#41—64no—
443#41—64no—
444C2×Q8—31no—
445C2×Q8C2×C32×D311yescharacteristic
446C2×D4—31no—
447C2×D4—31no—
448C2×D4—32no—
449C2×D4—31no—
450C2×D4—32no—
451C2×D4—31no—
452C2×D4—32no—
453C22×C4—31no—
454C22×C4—32no—
455C22×C4—31no—
456C22×C4—32no—
457C22×C4C2×C32×D311yescharacteristic
458C4⋊C4—32no—
459C4⋊C4—31no—
460C4⋊C4—32no—
461C4⋊C4C2×C32×D311yescharacteristic
462C4⋊C4C2×C32×D312yes—
463C22⋊C4—32no—
464C22⋊C4—32no—
465C22⋊C4—32no—
466C22⋊C4C2×C32×D312yes—
467C22×C3—28no—
468C22×C3—24no—
469C22×C3—24no—
470C22×C3—24no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—24no—
475C22×C3—38no—
476C22×C3—44no—
477C22×C3—44no—
478C22×C3—64no—
479C22×C3—64no—
480C22×C3—68no—
481C22×C3—64no—
482C22×C3—64no—
483C22×C3—68no—
484C22×C3—64no—
485C22×C3—68no—
486C22×C3C23×C3×D314yes—
487C22×C3C24×C3211yescharacteristic
488C2×D3—21no—
489C2×D3—21no—
490C2×D3—22no—
491C2×D3—21no—
492C2×D3—21no—
493C2×D3—22no—
494C2×D3—22no—
495C2×D3—21no—
496C2×D3C2×C32×D412yes—
497C3×C4—24no—
498C3×C4—28no—
499C3×C4—28no—
500C3×C4—21no—
501C3×C4—22no—
502C3×C4—22no—
503C3×C4—28no—
504C3×C4—38no—
505C3×C4—44no—
506C3×C4—48no—
507C3×C4—48no—
508C3×C4—64no—
509C3×C4—68no—
510C3×C4C3×D3×D418yes—
511C3×C4C2×C32×D412yes—
512Dic3—21no—
513Dic3—22no—
514Dic3C2×C32×D412yes—
515C32—24no—
516C32#562911yescharacteristic, direct product factor
517C32C3×#53714yescomplemented
518C23—31no—
519C23—31no—
520C23—32no—
521C23C22×C32×D311yescharacteristic
522Q8—22no—
523Q8—62no—
524D4—62no—
525D4—64no—
526D4—62no—
527D4—62no—
528D4—62no—
529D4—122no—
530D4—122no—
531C2×C4—22no—
532C2×C4—31no—
533C2×C4—34no—
534C2×C4—32no—
535C2×C4—31no—
536C2×C4—64no—
537C2×C4—62no—
538C2×C4—122no—
539C2×C4C22×C32×D311yescharacteristic
540C2×C4C22×C32×D312yes—
541C2×C4C22×C32×D312yes—
542C2×C4C22×C32×D311yescharacteristic
543C2×C3—24no—
544C2×C3—24no—
545C2×C3—24no—
546C2×C3—48no—
547C2×C3—44no—
548C2×C3—41no—
549C2×C3—68no—
550C2×C3—128no—
551C2×C3—124no—
552C2×C3C2×C3×D3×D414yes—
553C2×C3C3×#100714yes—
554C2×C3C3×#100914yes—
555C2×C3C22×C32×D411yescharacteristic
556C2×C3C32×H(5,𝔽2)11yescharacteristic
557C2×C3C32×2−1+411yescharacteristic
558D3—22no—
559D3—42no—
560D3—41no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—32no—
565C22—61no—
566C22—61no—
567C22—62no—
568C22—61no—
569C22—61no—
570C22—62no—
571C22—61no—
572C22—62no—
573C22C23×C32×D311yescharacteristic
574C4—21no—
575C4—22no—
576C4—22no—
577C4—32no—
578C4—61no—
579C4—62no—
580C4C32×D3×D412yes—
581C3—24no—
582C3C3×#562914yesdirect product factor
583C3C32×#53711yescharacteristic, complemented
584C2—41no—
585C2—62no—
586C2—122no—
587C2—121no—
588C2C2×C32×D3×D411yescharacteristic
589C2C32×#100711yescharacteristic
590C2C32×#100911yescharacteristic
subgroupG/H
centerC22×C32C23×D3
commutator, cosocleC22×C3C24×C32
FrattiniC22C23×C32×D3
FittingC33×C22⋊Q8C2
socleC22×C33C24
hypercenterC32×C22⋊Q8D3
hypocenterC3C32×#537
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C32×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C32×#239C2semidirect
2. C2×C32×#235C2semidirect
3. C33×C22⋊Q8C2semidirect
4. C32×#946C2semidirect
5. C32×#930C2semidirect
6. C32×#896C2semidirect
7. C32×#895C2semidirect
8. C32×#895C2semidirect
9. C32×D3×C4⋊C4C2semidirect
10. C32×#884C2semidirect
11. C32×#884C2semidirect
13. C3×#5629C3direct (N × H)
41. C2×C3×#239C2×C3semidirect
42. C2×C3×#235C2×C3semidirect
46. C32×C22⋊Q8D3semidirect
47. C32×C22⋊Q8C2×C3semidirect
54. C3×#946C2×C3semidirect
55. C3×#930C2×C3semidirect
56. C3×#896C2×C3semidirect
57. C3×#895C2×C3semidirect
58. C3×#895C2×C3semidirect
59. C3×D3×C4⋊C4C2×C3semidirect
60. C3×#884C2×C3semidirect
61. C3×#884C2×C3semidirect
85. C32×C4×D3D4semidirect
88. #5629C32direct (N × H)
166. C2×#239C2×C32semidirect
167. C2×#235C2×C32semidirect
171. C3×C22⋊Q8C3×D3semidirect
172. C3×C22⋊Q8C2×C32semidirect
179. #946C2×C32semidirect
180. #930C2×C32semidirect
181. #896C2×C32semidirect
182. #895C2×C32semidirect
183. #895C2×C32semidirect
184. D3×C4⋊C4C2×C32semidirect
185. #884C2×C32semidirect
186. #884C2×C32semidirect
242. C3×C4×D3C3×D4semidirect
359. C22⋊Q8C32×D3semidirect
366. C33#537semidirect
422. C4×D3C32×D4semidirect
516. C32#5629direct (N × H)
517. C32C3×#537semidirect
582. C3C3×#5629direct (N × H)
583. C3C32×#537semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2392
C2×C32×#2352
C33×C22⋊Q82
C32×#9462
C32×#9302
C32×#8962
C32×#8952
C32×D3×C4⋊C42
C32×#8842
C32×#5373
C3×#56293