Finite Group Databasebeta

← #13155678

Subgroups — #13155678

Order 1728.

714 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723501C211yesmaximal, characteristic
2#11723397C211yesmaximal, characteristic, complemented
3C2×C3×#3717C211yesmaximal, characteristic
4#11722075C211yesmaximal, characteristic
5#11722021C211yesmaximal, characteristic, complemented
6C3×#63575C211yesmaximal, characteristic, complemented
7C3×#63565C211yesmaximal, characteristic, complemented
8#10591572—31nomaximal
9#10590779—31nomaximal
10#10590484—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×Dic3×C3⋊D3—21no—
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20#88803—21no—
21Dic3×C3⋊Dic3—21no—
22C3×#3717C414yescomplemented
23Q8×C3⋊Dic3—31no—
24#64102—31no—
25C2×C3×#215—31no—
26C2×C3×#204—31no—
27#63898—31no—
28#63898—31no—
29#63898—61no—
30#63843—31no—
31#63843—31no—
32#63843—61no—
33C2×#3717D311yescharacteristic
34#63642—31no—
35C3×#835—31no—
36C3×#818—31no—
37C3×#810—31no—
38C3×#807—31no—
39#63575D311yescharacteristic, complemented
40#63565D311yescharacteristic, complemented
41#63564—31no—
42#63563—31no—
43#63562—31no—
44#63555—31no—
45#63555—31no—
46#63555—61no—
47C22×C3×C3⋊D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)D411yescharacteristic
50C2×C3⋊(C3⋊Dic3)D411yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)C2×C412yes—
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C32×Dic6—21no—
57C32×Dic6C2×C412yescomplemented
58Dic3×C3⋊D3—42no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8C2×C412yescomplemented
61#5164—91no—
62#4889—91no—
63#4817—91no—
64C2×C32×Q8—31no—
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×C4×C3⋊D3—61no—
67C2×C3×D12—31no—
68C2×C3×D12—31no—
69C2×C3×D12—61no—
70C2×C3×Dic6—21no—
71C2×C3×Dic6C2×D311yescharacteristic
72C2×C3×Dic6C2×D311yescharacteristic
73C2×D3×Dic3—61no—
74C2×D3×Dic3—61no—
75C2×D3×Dic3—62no—
76C2×C32×C8—31no—
77C32×C4⋊C4—31no—
78#3752—61no—
79C4⋊(C3⋊Dic3)—61no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81C4×C3⋊Dic3—31no—
82C4×C3⋊Dic3—31no—
83C4×C3⋊Dic3—31no—
84C4×C3⋊Dic3—61no—
85C4×C3⋊Dic3—61no—
86C4×C3⋊Dic3—61no—
87C4×C3⋊Dic3—61no—
88C4×C3⋊Dic3—121no—
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90C3×#215—34no—
91C3×C4⋊Dic3—21no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—61no—
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×C4⋊Dic3C2×D311yescharacteristic
97C2×C3×C3⋊C8—21no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—61no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C3×#204—34no—
104Dic3⋊Dic3—61no—
105Dic3⋊Dic3—61no—
106Dic3⋊Dic3—62no—
107(C2×D3)⋊Dic3—61no—
108(C2×D3)⋊Dic3—61no—
109(C2×D3)⋊Dic3—62no—
110Dic32—61no—
111Dic32—61no—
112Dic32—62no—
113#3717Dic314yescomplemented
114C22×C33C2×D411yescharacteristic
115C2×C3×C3⋊D3—22no—
116C2×C3×C3⋊D3—41no—
117C2×C3×C3⋊D3—41no—
118C33×C4C22×C411yescharacteristic
119C33×C4#4111yescharacteristic
120C3⋊(C3⋊Dic3)—21no—
121C3⋊(C3⋊Dic3)—21no—
122C3×C3⋊Dic3—41no—
123C32×Dic3—22no—
124C32×Dic3—41no—
125C2×C3×SD16—91no—
126Q8×Dic3—91no—
127Q8×Dic3—91no—
128Q8×Dic3—181no—
129C2×#215—31no—
130D4×Dic3—91no—
131C2×#204—31no—
132#893—91no—
133#893—91no—
134#893—181no—
135#887—91no—
136C3×#107—91no—
137C3×#103—91no—
138#835—31no—
139#832—91no—
140#821—91no—
141#818—31no—
142#815—91no—
143#815—91no—
144#815—91no—
145#815—181no—
146#810—31no—
147#809—91no—
148#807—31no—
149C22×C3⋊D3—21no—
150C22×C3×D3—62no—
151C22×C3×D3—61no—
152C22×C3×D3—61no—
153C32×Q8—32no—
154C32×Q8—61no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—41no—
159C2×C32×C4—61no—
160C2×C32×C4—61no—
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164C2×C3⋊Dic3—21no—
165C2×C3⋊Dic3—31no—
166C2×C3⋊Dic3—31no—
167C2×C3⋊Dic3—31no—
168C2×C3⋊Dic3—31no—
169C2×C3⋊Dic3—31no—
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—61no—
173C2×C3⋊Dic3—61no—
174C2×C3⋊Dic3—61no—
175C2×C3⋊Dic3—61no—
176C2×C3⋊Dic3—61no—
177C2×C3⋊Dic3—121no—
178C2×C3⋊Dic3—121no—
179C4⋊(C3⋊D3)—21no—
180C4⋊(C3⋊D3)C2×Dic312yes—
181C4×C3⋊D3—122no—
182C2×C3×Dic3—22no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—22no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—61no—
191C3×D12—32no—
192C3×D12—32no—
193C3×D12—62no—
194C3×D12—62no—
195C3×D12—61no—
196C3×D12—61no—
197C3×Dic6—22no—
198C3×Dic6—21no—
199C3×Dic6—21no—
200C3×Dic6—22no—
201C3×Dic6C4×D312yescomplemented
202C3×Dic6C2×Dic312yescomplemented
203D3×Dic3—124no—
204D3×Dic3—122no—
205D3×Dic3—122no—
206C32×C8—32no—
207C3⋊(C3⋊C8)—21no—
208C3×C3⋊C8—22no—
209C3×C3⋊C8—62no—
210C3×C3⋊C8—61no—
211C3×C3⋊C8—61no—
212C3×C3⋊C8C4×D312yescomplemented
213C3×C3⋊C8C2×Dic312yescomplemented
214#440—271noSylow
215C2×C33C4×D411yescharacteristic
216C2×C33AGL(1,ℤ/8)11yescharacteristic
217C2×C33#13711yescharacteristic
218C3×C3⋊D3—42no—
219C2×C3×Q8—31no—
220C2×C3×Q8—31no—
221C2×C3×Q8—61no—
222C2×C3×D4—91no—
223C22×Dic3—181no—
224C2×D12—31no—
225C2×D12—31no—
226C2×D12—61no—
227C2×C4×D3—181no—
228C2×C4×D3—181no—
229C2×C4×D3—182no—
230C2×Dic6D3211yescharacteristic
231C3×SD16—94no—
232C2×C3×C8—31no—
233C2×C3×C8—31no—
234C2×C3×C8—61no—
235C2×C3×C8—91no—
236C3×C4⋊C4—31no—
237C3×C4⋊C4—31no—
238C3×C4⋊C4—61no—
239C3×C4⋊C4—91no—
240#217—181no—
241#215—34no—
242(C2×D3)⋊C4—181no—
243(C2×D3)⋊C4—181no—
244(C2×D3)⋊C4—182no—
245C4⋊Dic3—31no—
246C4⋊Dic3—31no—
247C4⋊Dic3—61no—
248C4⋊Dic3—181no—
249C4⋊Dic3—181no—
250C4⋊Dic3—182no—
251C4⋊Dic3D3211yescharacteristic
252Dic3⋊C4—181no—
253C4×Dic3—91no—
254C4×Dic3—91no—
255C4×Dic3—91no—
256C4×Dic3—181no—
257C4×Dic3—181no—
258C4×Dic3—181no—
259C4×Dic3—181no—
260C4×Dic3—181no—
261C4×Dic3—181no—
262C4×Dic3—182no—
263C4×Dic3—361no—
264C2×C3⋊C8—31no—
265C2×C3⋊C8—31no—
266C2×C3⋊C8—61no—
267C2×C3⋊C8D3211yescharacteristic
268#204—34no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—41no—
273C22×C32D3×D411yescharacteristic
274C22×C32D3×D411yescharacteristic
275C22×C32D3×D411yescharacteristic
276C2×C3⋊D3—22no—
277C2×C3⋊D3—41no—
278C2×C3⋊D3—41no—
279C2×C3×D3—64no—
280C2×C3×D3—62no—
281C2×C3×D3—62no—
282C2×C3×D3—122no—
283C2×C3×D3—122no—
284C2×C3×D3—121no—
285C2×C3×D3—121no—
286C2×C3×D3—121no—
287C2×C3×D3—121no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—41no—
295C32×C4—41no—
296C32×C4—62no—
297C32×C4—121no—
298C32×C4C2×C4×D311yescharacteristic
299C32×C4C2×C4×D311yescharacteristic
300C32×C4#23711yescharacteristic
301C32×C4#23711yescharacteristic
302C32×C4#23911yescharacteristic
303C32×C4C22×Dic311yescharacteristic
304C3⋊Dic3—41no—
305C3⋊Dic3—61no—
306C3⋊Dic3—61no—
307C3⋊Dic3—62no—
308C3⋊Dic3—62no—
309C3⋊Dic3—61no—
310C3⋊Dic3—61no—
311C3⋊Dic3—61no—
312C3⋊Dic3—61no—
313C3⋊Dic3—121no—
314C3⋊Dic3—121no—
315C3⋊Dic3—121no—
316C3⋊Dic3—121no—
317C3⋊Dic3—122no—
318C3⋊Dic3—122no—
319C3×Dic3—24no—
320C3×Dic3—22no—
321C3×Dic3—22no—
322C3×Dic3—42no—
323C3×Dic3—41no—
324C3×Dic3—41no—
325C3×Dic3—122no—
326C3×Dic3—121no—
327C3×Dic3—121no—
328C2×SD16—91no—
329C4×Q8—271no—
330C4×D4—271no—
331#107—91no—
332#103—91no—
333#102—271no—
334#97—271no—
335C33#44011yescharacteristic, complemented, Sylow
336C23×C3—181no—
337C22×D3—62no—
338C22×D3—61no—
339C22×D3—61no—
340C3×Q8—32no—
341C3×Q8—32no—
342C3×Q8—62no—
343C3×Q8—62no—
344C3×Q8—61no—
345C3×Q8—61no—
346C3×D4—92no—
347C3×D4—181no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—41no—
352C2×C3×C4—62no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—62no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—181no—
359C2×C3×C4C2×D3211yescharacteristic
360C2×C3×C4C2×D3211yescharacteristic
361C2×C3×C4C2×D3211yescharacteristic
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—62no—
365C2×Dic3—61no—
366C2×Dic3—61no—
367C2×Dic3—91no—
368C2×Dic3—91no—
369C2×Dic3—91no—
370C2×Dic3—91no—
371C2×Dic3—91no—
372C2×Dic3—91no—
373C2×Dic3—181no—
374C2×Dic3—181no—
375C2×Dic3—181no—
376C2×Dic3—181no—
377C2×Dic3—181no—
378C2×Dic3—181no—
379C2×Dic3—361no—
380C2×Dic3—361no—
381C2×Dic3—362no—
382D12—32no—
383D12—32no—
384D12—62no—
385D12—62no—
386D12—61no—
387D12—61no—
388C4×D3—364no—
389C4×D3—362no—
390C4×D3—362no—
391Dic6—21no—
392Dic6D3×Dic312yescomplemented
393C3×C8—32no—
394C3×C8—32no—
395C3×C8—62no—
396C3×C8—181no—
397C3⋊C8—62no—
398C3⋊C8—61no—
399C3⋊C8—61no—
400C3⋊C8D3×Dic312yescomplemented
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—41no—
411C2×C32—41no—
412C2×C32—41no—
413C2×C32#67⋊C411yescharacteristic
414C2×C32#89311yescharacteristic
415C2×C32#91111yescharacteristic
416C2×C32#91411yescharacteristic
417C2×C32#91411yescharacteristic
418C2×C32#91511yescharacteristic
419C2×C32#91811yescharacteristic
420C2×C32#91811yescharacteristic
421C2×C32D4×Dic311yescharacteristic
422C3⋊D3—42no—
423C3×D3—124no—
424C3×D3—122no—
425C3×D3—122no—
426C2×Q8—31no—
427C2×D4—91no—
428C22×C4—541no—
429SD16—94no—
430C2×C8—31no—
431C2×C8—91no—
432C4⋊C4—31no—
433C4⋊C4—91no—
434C4⋊C4—541no—
435C22⋊C4—541no—
436C42—271no—
437C42—541no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—41no—
442C22×C3—182no—
443C22×C3—361no—
444C22×C3—361no—
445C22×C3#380211yescharacteristic
446C22×C3#381111yescharacteristic
447C22×C3#381111yescharacteristic
448C2×D3—64no—
449C2×D3—62no—
450C2×D3—62no—
451C2×D3—122no—
452C2×D3—122no—
453C2×D3—121no—
454C2×D3—121no—
455C2×D3—121no—
456C2×D3—121no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—41no—
464C3×C4—41no—
465C3×C4—64no—
466C3×C4—62no—
467C3×C4—62no—
468C3×C4—122no—
469C3×C4—121no—
470C3×C4—121no—
471C3×C4—361no—
472C3×C4#379611yescharacteristic
473C3×C4#379611yescharacteristic
474C3×C4C4×D3211yescharacteristic
475C3×C4C2×D3×Dic311yescharacteristic
476C3×C4C2×D3×Dic311yescharacteristic
477C3×C4#380511yescharacteristic
478Dic3—22no—
479Dic3—41no—
480Dic3—122no—
481Dic3—121no—
482Dic3—121no—
483Dic3—182no—
484Dic3—182no—
485Dic3—181no—
486Dic3—181no—
487Dic3—181no—
488Dic3—181no—
489Dic3—181no—
490Dic3—181no—
491Dic3—361no—
492Dic3—361no—
493Dic3—361no—
494Dic3—361no—
495Dic3—362no—
496Dic3—362no—
497C32—21no—
498C32—21no—
499C32—21no—
500C32—41no—
501C32#481711yescharacteristic, complemented
502C32#488911yescharacteristic, complemented
503C32#516411yescharacteristic, complemented
504C23—181no—
505Q8—32no—
506Q8—61no—
507D4—92no—
508D4—181no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—181no—
512C2×C4—271no—
513C2×C4—271no—
514C2×C4—1082no—
515C2×C4D3311yescharacteristic
516C8—32no—
517C8—181no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—41no—
528C2×C3—41no—
529C2×C3—41no—
530C2×C3—362no—
531C2×C3#6379811yescharacteristic
532C2×C3#6381111yescharacteristic
533C2×C3#6388111yescharacteristic
534C2×C3#6389811yescharacteristic
535C2×C3#6393711yescharacteristic
536C2×C3#6394111yescharacteristic
537C2×C3#6394511yescharacteristic
538C2×C3#6395111yescharacteristic
539C2×C3#67⋊Dic311yescharacteristic
540D3—124no—
541D3—122no—
542D3—122no—
543C22—182no—
544C22—361no—
545C22—361no—
546C22#8895411yescharacteristic
547C4—62no—
548C4—121no—
549C4—361no—
550C4—541no—
551C4—541no—
552C4D32×Dic311yescharacteristic
553C4#8896411yescharacteristic
554C3—21no—
555C3—21no—
556C3—21no—
557C3—41no—
558C3#1059048411yescharacteristic, complemented
559C3#1059077911yescharacteristic, complemented
560C3#1059157211yescharacteristic, complemented
561C2—362no—
562C2#1172287811yescharacteristic
563C2#1172289411yescharacteristic
564C2#1172292911yescharacteristic
subgroupG/H
centerC22#88954
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#440
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723397C2semidirect
5. #11722021C2semidirect
6. C3×#63575C2semidirect
7. C3×#63565C2semidirect
22. C3×#3717C4semidirect
39. #63575D3semidirect
40. #63565D3semidirect
57. C32×Dic6C2×C4semidirect
60. C32×C3⋊C8C2×C4semidirect
113. #3717Dic3semidirect
201. C3×Dic6C4×D3semidirect
202. C3×Dic6C2×Dic3semidirect
212. C3×C3⋊C8C4×D3semidirect
213. C3×C3⋊C8C2×Dic3semidirect
335. C33#440semidirect
392. Dic6D3×Dic3semidirect
400. C3⋊C8D3×Dic3semidirect
501. C32#4817semidirect
502. C32#4889semidirect
503. C32#5164semidirect
558. C3#10590484semidirect
559. C3#10590779semidirect
560. C3#10591572semidirect

Maximal subgroups

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

subgroupindex
#117235012
#117233972
C2×C3×#37172
#117220752
#117220212
C3×#635752
C3×#635652
#105915723
#105907793
#105904843