Finite Group Databasebeta

← #13155886

Subgroups — #13155886

Order 1728.

912 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89021C211yesmaximal, characteristic
2C2×#88799C211yesmaximal, characteristic, complemented
3C3×#63856C211yesmaximal, characteristic, complemented
4C2×C3×#3717C211yesmaximal, characteristic, complemented
5#11722075C211yesmaximal, characteristic, complemented
6#11722042C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10591655—31nomaximal
9#10591184—31nomaximal
10#10590960—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×#6290—21no—
16C2×Dic3×C3⋊D3—21no—
17#89021—22no—
18#89021—22no—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22#88799—22no—
23C3×#3724—21no—
24C3×(C2×D3)⋊Dic3—21no—
25C3×#3717—22no—
26C2×#3828—31no—
27C2×#3795—31no—
28C2×#3795—31no—
29C2×#3795—61no—
30C2×#3756—31no—
31C3×#946—31no—
32C2×C3×#215—31no—
33C3×#924—31no—
34C2×C3×#204—31no—
35#63856D311yescharacteristic, complemented
36C2×#3719—31no—
37C2×#3719—31no—
38C2×#3719—61no—
39C2×#3717D311yescharacteristic, complemented
40#63643—31no—
41C3×#830—31no—
42C3×(C2×D3)⋊C8—31no—
43#63567—31no—
44#63567—31no—
45#63567—61no—
46#63564—31no—
47#63563—31no—
48#63562—31no—
49(C2×D3)⋊(C3⋊C8)D311yescharacteristic, complemented
50C22×C3×C3⋊D3—21no—
51C22×C32×D3D411yescharacteristic
52C2×C33×C4C2311yescharacteristic
53C2×C3⋊(C3⋊Dic3)—21no—
54#6308—21no—
55#6308D412yes—
56C2×C3×C3⋊Dic3—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)D412yes—
59C2×C32×Dic3—21no—
60C2×C32×Dic3D411yescharacteristic
61C32×C4×D3—21no—
62C32×C4×D3—21no—
63C32×Dic6—21no—
64C32×Dic6—21no—
65#6290—24no—
66Dic3×C3⋊D3—24no—
67C3×C3⋊(C3⋊C8)—21no—
68C32×C3⋊C8—21no—
69#5173—91no—
70#5047—91no—
71#4733—91no—
72C2×C32×Q8—31no—
73C22×C32×C4—31no—
74C2×#638—61no—
75#3828—62no—
76#3828—62no—
77C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
78C2×C4×C3⋊D3—61no—
79C2×#634—31no—
80C2×#634—31no—
81C2×#634—31no—
82C2×#634—61no—
83C2×#634—61no—
84C2×#634—61no—
85C2×#634—121no—
86C2×C3×D12—31no—
87C2×C3×D12—31no—
88C2×C3×D12—61no—
89C2×C3×C4×D3—21no—
90C2×C3×C4×D3C2×D311yescharacteristic
91C2×C3×C4×D3C2×D311yescharacteristic
92C2×C3×Dic6—21no—
93C2×C3×Dic6C2×D311yescharacteristic
94C2×C3×Dic6C2×D311yescharacteristic
95C2×#625—62no—
96C2×#625—61no—
97C2×#625—61no—
98C2×D3×Dic3—62no—
99C2×D3×Dic3—61no—
100C2×D3×Dic3—61no—
101#3795—62no—
102#3795—62no—
103#3795—62no—
104#3795—62no—
105#3795—64no—
106#3795—64no—
107C2×C32×C8—31no—
108#3756—62no—
109C4⋊(C3⋊Dic3)C2×D311yescharacteristic
110C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
111C3×#217—61no—
112C3×#215—62no—
113C3×(C2×D3)⋊C4—61no—
114C3×C4⋊Dic3—31no—
115C3×C4⋊Dic3—31no—
116C3×C4⋊Dic3—61no—
117C3×Dic3⋊C4—61no—
118C3×Dic3⋊C4—61no—
119C2×C3×C3⋊C8—21no—
120C2×C3×C3⋊C8—31no—
121C2×C3×C3⋊C8—31no—
122C2×C3×C3⋊C8—61no—
123C2×C3×C3⋊C8C2×D311yescharacteristic
124C2×C3×C3⋊C8C2×D311yescharacteristic
125C3×#204—62no—
126#3724—21no—
127(C2×D3)⋊Dic3—21no—
128#3719—62no—
129#3719—62no—
130#3719—64no—
131#3717—22no—
132C22×C33C2×D411yescharacteristic
133C2×C3×C3⋊D3—22no—
134C2×C3×C3⋊D3—41no—
135C2×C3×C3⋊D3—41no—
136C2×C32×D3—21no—
137C2×C32×D3—21no—
138C2×C32×D3—21no—
139C33×C4C2×D411yescharacteristic
140C33×C4C2×D411yescharacteristic
141C3⋊(C3⋊Dic3)—22no—
142C3×C3⋊Dic3—41no—
143C32×Dic3—21no—
144C32×Dic3—41no—
145C2×#237—91no—
146C2×#235—91no—
147C2×#235—91no—
148C2×#235—181no—
149C2×C3×SD16—91no—
150C3×C22⋊Q8—91no—
151#946—31no—
152C2×#216—91no—
153C2×#216—91no—
154C2×#216—181no—
155C2×#215—31no—
156#924—31no—
157C2×Dic12—91no—
158C2×#204—31no—
159C3×C22⋊C8—91no—
160#832—91no—
161#830—31no—
162#821—91no—
163(C2×D3)⋊C8—31no—
164#816—91no—
165#816—91no—
166#816—181no—
167#810—91no—
168#809—91no—
169C23×C32—31no—
170C22×C3⋊D3—21no—
171C22×C3×D3—21no—
172C22×C3×D3—62no—
173C22×C3×D3—61no—
174C22×C3×D3—61no—
175C22×C3×D3D1211yescharacteristic
176C22×C3×D3#6711yescharacteristic
177C32×Q8—61no—
178C32×Q8—61no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—31no—
183C2×C32×C4—41no—
184C2×C32×C4—61no—
185C2×C32×C4—61no—
186C2×C32×C4—61no—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190#638—64no—
191C2×C3⋊Dic3—21no—
192C2×C3⋊Dic3—61no—
193C2×C3⋊Dic3—62no—
194C2×C3⋊Dic3—61no—
195C2×C3⋊Dic3—61no—
196C2×C3⋊Dic3—121no—
197C2×C3⋊Dic3—121no—
198C2×C3⋊Dic3—122no—
199C4⋊(C3⋊D3)—21no—
200C4⋊(C3⋊D3)#6712yes—
201C4×C3⋊D3—64no—
202#634—32no—
203#634—32no—
204#634—32no—
205#634—62no—
206#634—62no—
207#634—62no—
208#634—61no—
209#634—62no—
210#634—61no—
211#634—61no—
212#634—122no—
213#634—121no—
214#634—121no—
215#634—122no—
216C2×C3×Dic3—21no—
217C2×C3×Dic3—21no—
218C2×C3×Dic3—21no—
219C2×C3×Dic3—41no—
220C2×C3×Dic3—61no—
221C2×C3×Dic3—61no—
222C2×C3×Dic3—121no—
223C2×C3×Dic3D1211yescharacteristic
224C2×C3×Dic3#6711yescharacteristic
225C3×D12—32no—
226C3×D12—32no—
227C3×D12—62no—
228C3×D12—62no—
229C3×D12—61no—
230C3×D12—61no—
231C3×C4×D3—22no—
232C3×C4×D3—22no—
233C3×C4×D3—21no—
234C3×C4×D3—21no—
235C3×C4×D3—21no—
236C3×C4×D3—21no—
237C3×Dic6—22no—
238C3×Dic6—21no—
239C3×Dic6—21no—
240C3×Dic6—21no—
241C3×Dic6—21no—
242C3×Dic6—41no—
243#625—68no—
244#625—64no—
245#625—64no—
246D3×Dic3—68no—
247D3×Dic3—64no—
248D3×Dic3—64no—
249C32×C8—61no—
250C3⋊(C3⋊C8)—21no—
251C3×C3⋊C8—21no—
252C3×C3⋊C8—21no—
253C3×C3⋊C8—41no—
254C3×C3⋊C8—62no—
255C3×C3⋊C8—61no—
256C3×C3⋊C8—61no—
257#452—271noSylow
258C2×C33#12011yescharacteristic
259C2×C33#13511yescharacteristic
260C2×C33#13711yescharacteristic
261C3×C3⋊D3—42no—
262C32×D3—41no—
263C2×C3×Q8—31no—
264C2×C3×Q8—31no—
265C2×C3×Q8—61no—
266C2×C3×D4—91no—
267C22×C3×C4—31no—
268C22×C3×C4—31no—
269C22×C3×C4—61no—
270C2×#67—181no—
271C2×#67—181no—
272C2×#67—182no—
273C2×#67—181no—
274C22×Dic3—181no—
275#237—182no—
276#237—182no—
277#235—182no—
278#235—182no—
279#235—182no—
280#235—182no—
281#235—184no—
282#235—184no—
283C2×D12—31no—
284C2×D12—31no—
285C2×D12—61no—
286C2×C4×D3—181no—
287C2×C4×D3—181no—
288C2×C4×D3—182no—
289C2×C4×D3D3211yescharacteristic
290C2×Dic6—91no—
291C2×Dic6—91no—
292C2×Dic6—91no—
293C2×Dic6—181no—
294C2×Dic6—181no—
295C2×Dic6—181no—
296C2×Dic6—361no—
297C2×Dic6D3211yescharacteristic
298C3×SD16—182no—
299C2×C3×C8—31no—
300C2×C3×C8—31no—
301C2×C3×C8—61no—
302C2×C3×C8—91no—
303C3×C4⋊C4—91no—
304C3×C4⋊C4—181no—
305C3×C22⋊C4—181no—
306#217—61no—
307#216—182no—
308#216—182no—
309#216—184no—
310#215—62no—
311(C2×D3)⋊C4—61no—
312C4⋊Dic3—31no—
313C4⋊Dic3—31no—
314C4⋊Dic3—61no—
315Dic3⋊C4—61no—
316Dic3⋊C4—61no—
317C2×C3⋊C8—31no—
318C2×C3⋊C8—31no—
319C2×C3⋊C8—61no—
320C2×C3⋊C8D3211yescharacteristic
321Dic12—182no—
322#204—62no—
323C22×C32—21no—
324C22×C32—21no—
325C22×C32—21no—
326C22×C32—41no—
327C22×C32—61no—
328C22×C32—61no—
329C22×C32—61no—
330C22×C32C2×D1211yescharacteristic
331C22×C32D3×D411yescharacteristic
332C22×C32C2×#6711yescharacteristic
333C2×C3⋊D3—22no—
334C2×C3⋊D3—41no—
335C2×C3⋊D3—41no—
336C2×C3×D3—22no—
337C2×C3×D3—21no—
338C2×C3×D3—21no—
339C2×C3×D3—21no—
340C2×C3×D3—21no—
341C2×C3×D3—21no—
342C2×C3×D3—21no—
343C2×C3×D3—41no—
344C2×C3×D3—41no—
345C2×C3×D3—64no—
346C2×C3×D3—62no—
347C2×C3×D3—62no—
348C2×C3×D3—122no—
349C2×C3×D3—122no—
350C2×C3×D3—121no—
351C2×C3×D3—121no—
352C2×C3×D3—121no—
353C2×C3×D3—121no—
354C32×C4—21no—
355C32×C4—21no—
356C32×C4—21no—
357C32×C4—21no—
358C32×C4—21no—
359C32×C4—21no—
360C32×C4—41no—
361C32×C4—41no—
362C32×C4—61no—
363C32×C4—121no—
364C32×C4D3×D411yescharacteristic
365C32×C4D3×D411yescharacteristic
366C32×C4D3×D411yescharacteristic
367C32×C4D3×D411yescharacteristic
368C32×C4D3×D411yescharacteristic
369C32×C4C2×#6711yescharacteristic
370C3⋊Dic3—41no—
371C3⋊Dic3—62no—
372C3⋊Dic3—64no—
373C3⋊Dic3—62no—
374C3⋊Dic3—62no—
375C3⋊Dic3—122no—
376C3⋊Dic3—122no—
377C3⋊Dic3—124no—
378C3×Dic3—22no—
379C3×Dic3—21no—
380C3×Dic3—21no—
381C3×Dic3—42no—
382C3×Dic3—41no—
383C3×Dic3—41no—
384C3×Dic3—121no—
385C3×Dic3—121no—
386C3×Dic3—241no—
387C2×#41—271no—
388C2×Q16—271no—
389C2×SD16—91no—
390C22⋊Q8—91no—
391#103—271no—
392#102—271no—
393C22⋊C8—91no—
394C33#45211yescharacteristic, complemented, Sylow
395C23×C3—31no—
396C23×C3—31no—
397C23×C3—61no—
398C23×C3—181no—
399C22×D3—62no—
400C22×D3—61no—
401C22×D3—61no—
402C22×D3#62611yescharacteristic
403C3×Q8—62no—
404C3×Q8—61no—
405C3×Q8—61no—
406C3×Q8—61no—
407C3×Q8—61no—
408C3×Q8—121no—
409C3×D4—92no—
410C3×D4—181no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—31no—
415C2×C3×C4—31no—
416C2×C3×C4—41no—
417C2×C3×C4—61no—
418C2×C3×C4—62no—
419C2×C3×C4—62no—
420C2×C3×C4—61no—
421C2×C3×C4—61no—
422C2×C3×C4—61no—
423C2×C3×C4—61no—
424C2×C3×C4—61no—
425C2×C3×C4—61no—
426C2×C3×C4—121no—
427C2×C3×C4—181no—
428C2×C3×C4C2×D3211yescharacteristic
429C2×C3×C4C2×D3211yescharacteristic
430C2×C3×C4C2×D3211yescharacteristic
431#67—184no—
432#67—188no—
433#67—184no—
434#67—184no—
435C2×Dic3—21no—
436C2×Dic3—61no—
437C2×Dic3—61no—
438C2×Dic3—121no—
439C2×Dic3—184no—
440C2×Dic3—182no—
441C2×Dic3—181no—
442C2×Dic3—181no—
443C2×Dic3—181no—
444C2×Dic3—361no—
445C2×Dic3—361no—
446C2×Dic3—362no—
447C2×Dic3#62611yescharacteristic
448D12—32no—
449D12—32no—
450D12—62no—
451D12—62no—
452D12—61no—
453D12—61no—
454C4×D3—21no—
455C4×D3—21no—
456C4×D3—188no—
457C4×D3—184no—
458C4×D3—184no—
459Dic6—21no—
460Dic6—21no—
461Dic6—92no—
462Dic6—92no—
463Dic6—92no—
464Dic6—182no—
465Dic6—182no—
466Dic6—182no—
467Dic6—182no—
468Dic6—181no—
469Dic6—181no—
470Dic6—181no—
471Dic6—362no—
472Dic6—361no—
473Dic6—361no—
474Dic6—362no—
475C3×C8—61no—
476C3×C8—61no—
477C3×C8—121no—
478C3×C8—181no—
479C3⋊C8—21no—
480C3⋊C8—62no—
481C3⋊C8—61no—
482C3⋊C8—61no—
483C2×C32—21no—
484C2×C32—21no—
485C2×C32—21no—
486C2×C32—21no—
487C2×C32—21no—
488C2×C32—21no—
489C2×C32—21no—
490C2×C32—21no—
491C2×C32—21no—
492C2×C32—41no—
493C2×C32—41no—
494C2×C32—41no—
495C2×C32—121no—
496C2×C32#88211yescharacteristic
497C2×C32#90411yescharacteristic
498C2×C32#90911yescharacteristic
499C2×C32#91511yescharacteristic
500C2×C32#91611yescharacteristic
501C2×C32#93711yescharacteristic
502C2×C32#93711yescharacteristic
503C2×C32#94211yescharacteristic
504C2×C32#95011yescharacteristic
505C3⋊D3—42no—
506C3×D3—42no—
507C3×D3—41no—
508C3×D3—41no—
509C3×D3—124no—
510C3×D3—122no—
511C3×D3—122no—
512#41—542no—
513#41—542no—
514C2×Q8—31no—
515C2×Q8—271no—
516C2×D4—91no—
517C2×D4—541no—
518C22×C4—31no—
519C22×C4—541no—
520Q16—542no—
521SD16—182no—
522C2×C8—31no—
523C2×C8—91no—
524C4⋊C4—91no—
525C4⋊C4—181no—
526C22⋊C4—181no—
527C22×C3—21no—
528C22×C3—21no—
529C22×C3—21no—
530C22×C3—41no—
531C22×C3—62no—
532C22×C3—61no—
533C22×C3—61no—
534C22×C3—61no—
535C22×C3—61no—
536C22×C3—61no—
537C22×C3—61no—
538C22×C3—121no—
539C22×C3—121no—
540C22×C3—182no—
541C22×C3—361no—
542C22×C3—361no—
543C22×C3D3×D1211yescharacteristic
544C22×C3C2×#62611yescharacteristic
545C22×C3D3×#6711yescharacteristic
546C2×D3—21no—
547C2×D3—21no—
548C2×D3—21no—
549C2×D3—64no—
550C2×D3—62no—
551C2×D3—62no—
552C2×D3—122no—
553C2×D3—122no—
554C2×D3—121no—
555C2×D3—121no—
556C2×D3—121no—
557C2×D3—121no—
558C3×C4—21no—
559C3×C4—21no—
560C3×C4—21no—
561C3×C4—21no—
562C3×C4—21no—
563C3×C4—21no—
564C3×C4—41no—
565C3×C4—41no—
566C3×C4—62no—
567C3×C4—61no—
568C3×C4—61no—
569C3×C4—122no—
570C3×C4—121no—
571C3×C4—121no—
572C3×C4—361no—
573C3×C4#380211yescharacteristic
574C3×C4D3×#6711yescharacteristic
575C3×C4D3×#6711yescharacteristic
576C3×C4#381111yescharacteristic
577C3×C4#381111yescharacteristic
578C3×C4#381111yescharacteristic
579Dic3—21no—
580Dic3—41no—
581Dic3—121no—
582Dic3—121no—
583Dic3—184no—
584Dic3—182no—
585Dic3—182no—
586Dic3—182no—
587Dic3—241no—
588Dic3—362no—
589Dic3—362no—
590Dic3—364no—
591C32—21no—
592C32—21no—
593C32—21no—
594C32—41no—
595C32#473311yescharacteristic, complemented
596C32#504711yescharacteristic, complemented
597C32#517311yescharacteristic, complemented
598C23—31no—
599C23—181no—
600Q8—61no—
601Q8—61no—
602Q8—272no—
603Q8—541no—
604D4—92no—
605D4—181no—
606D4—544no—
607C2×C4—31no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—181no—
612C2×C4—544no—
613C2×C4—541no—
614C2×C4D3311yescharacteristic
615C8—61no—
616C8—181no—
617C2×C3—21no—
618C2×C3—21no—
619C2×C3—21no—
620C2×C3—21no—
621C2×C3—21no—
622C2×C3—21no—
623C2×C3—21no—
624C2×C3—21no—
625C2×C3—21no—
626C2×C3—41no—
627C2×C3—41no—
628C2×C3—41no—
629C2×C3—122no—
630C2×C3—121no—
631C2×C3—121no—
632C2×C3—362no—
633C2×C3#6380011yescharacteristic
634C2×C3#6380611yescharacteristic
635C2×C3#6382911yescharacteristic
636C2×C3#6383311yescharacteristic
637C2×C3#6392211yescharacteristic
638C2×C3#6394611yescharacteristic
639C2×C3#6394711yescharacteristic
640C2×C3#6398111yescharacteristic
641C2×C3#6398311yescharacteristic
642D3—41no—
643D3—124no—
644D3—122no—
645D3—122no—
646C22—61no—
647C22—61no—
648C22—61no—
649C22—182no—
650C22—361no—
651C22—361no—
652C22D3×#62611yescharacteristic
653C4—61no—
654C4—121no—
655C4—361no—
656C4—542no—
657C4#8895211yescharacteristic
658C4#8895411yescharacteristic
659C3—21no—
660C3—21no—
661C3—21no—
662C3—41no—
663C3#1059096011yescharacteristic, complemented
664C3#1059118411yescharacteristic, complemented
665C3#1059165511yescharacteristic, complemented
666C2—121no—
667C2—362no—
668C2#1172285511yescharacteristic
669C2#1172288411yescharacteristic
670C2#1172294911yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×#89021C2
hypocenterC33#452
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88799C2semidirect
3. C3×#63856C2semidirect
4. C2×C3×#3717C2semidirect
5. #11722075C2semidirect
6. #11722042C2semidirect
7. C3×(C2×D3)⋊(C3⋊C8)C2semidirect
35. #63856D3semidirect
39. C2×#3717D3semidirect
49. (C2×D3)⋊(C3⋊C8)D3semidirect
394. C33#452semidirect
595. C32#4733semidirect
596. C32#5047semidirect
597. C32#5173semidirect
663. C3#10590960semidirect
664. C3#10591184semidirect
665. C3#10591655semidirect

Maximal subgroups

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

subgroupindex
C2×#890212
C2×#887992
C3×#638562
C2×C3×#37172
#117220752
#117220422
C3×(C2×D3)⋊(C3⋊C8)2
#105916553
#105911843
#105909603