Finite Group Databasebeta

← #10845259

Subgroups — #10845259

Order 768.

872 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#489C211yesmaximal, characteristic, complemented
2#83260C211yesmaximal, characteristic, complemented
3#82955C211yesmaximal, characteristic, complemented
4C2×#5057C211yesmaximal, characteristic
5#71780C211yesmaximal, characteristic, complemented
6#70280C211yesmaximal, characteristic, complemented
7#70240C211yesmaximal, characteristic, complemented
8#70234C211yesmaximal, characteristic, complemented
9#70222C211yesmaximal, characteristic, complemented
10#70212C211yesmaximal, characteristic
11#70172C211yesmaximal, characteristic
12#70166C211yesmaximal, characteristic, complemented
13#70154C211yesmaximal, characteristic, complemented
14#69836C211yesmaximal, characteristic, complemented
15#69812C211yesmaximal, characteristic, complemented
16#25335—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#124C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C22×#213C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C3×#489—24no—
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#97C2211yescharacteristic
30#5231—21no—
31#5231—21no—
32#5229C2211yescharacteristic
33C2×#835C2211yescharacteristic
34#5221C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#821C2211yescharacteristic
37C2×#821C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40#5057—24no—
41#4988C2211yescharacteristic
42#4988C2211yescharacteristic
43#4987—21no—
44#4987—21no—
45#4977C2211yescharacteristic
46#4977C2211yescharacteristic
47#4951—21no—
48#4951—21no—
49#4950—21no—
50Dic3×C22⋊C4—21no—
51C4×C4⋊Dic3C2211yescharacteristic
52C4×C4⋊Dic3C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#489D311yescharacteristic, complemented
63C2×#463—31no—
64#2563—31no—
65#2367—31no—
66#2074—31no—
67#2073—31no—
68#2055—31no—
69#2054—31no—
70#1988—31no—
71#1940—31no—
72#1880—31no—
73#1879—31no—
74#1805—31no—
75#1804—31no—
76#1762—31no—
77C22×C3×Q8—21no—
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C23×Dic3—21no—
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4⋊Q8—22no—
86C3×C4⋊Q8—22no—
87C3×#124—22no—
88C3×#124D414yes—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98Q8×Dic3—22no—
99Q8×Dic3—41no—
100C2×#215—22no—
101C2×#215—22no—
102C2×#215—41no—
103D4×Dic3—22no—
104D4×Dic3—41no—
105C2×#213—22no—
106C2×#213—22no—
107C2×#213—41no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3D411yescharacteristic
115C2×C4⋊Dic3D411yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3C2311yescharacteristic
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C4×D12—22no—
126C4×D12—22no—
127C3×#103—22no—
128C3×#103—22no—
129C3×#102—22no—
130C3×#102—22no—
131C3×#97—22no—
132#835—22no—
133#832—22no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95D411yescharacteristic
142C3⋊#95D411yescharacteristic
143#821—22no—
144#821—22no—
145Dic3⋊C8—41no—
146Dic3⋊C8—41no—
147C4⋊(C3⋊C8)—22no—
148C22×SD16—31no—
149C22×D8—31no—
150C2×C4⋊Q8C2×D311yescharacteristic
151C2×#124C2×D311yescharacteristic
152C2×C4×Q8—31no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155#489—24no—
156#463—64no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103C2×D311yescharacteristic
162C2×#103C2×D311yescharacteristic
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102C2×D311yescharacteristic
167C2×#102C2×D311yescharacteristic
168C2×#97C2×D311yescharacteristic
169#393—61no—
170#393—61no—
171#391—31no—
172#390—31no—
173#390—31no—
174#390—31no—
175#388—61no—
176#388—61no—
177#385—61no—
178#382—31no—
179#382—31no—
180#380—61no—
181#380—61no—
182C4×C4⋊C4—31no—
183C4×C4⋊C4—31no—
184C4×C22⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C23×D3—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—42no—
201C2×C3×Q8—81no—
202C2×C3×Q8—81no—
203C2×C3×Q8C2×D412yes—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4C2×D412yes—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—42no—
220C22×Dic3—42no—
221C22×Dic3—81no—
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12#4112yes—
232C2×C4×D3—42no—
233C2×C4×D3—42no—
234C2×C4×D3—42no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—42no—
246C3×C22⋊C4—24no—
247C3×C22⋊C4—24no—
248C3×C42C2×D412yes—
249C3×C42#4112yes—
250#217—42no—
251#217—42no—
252#217—81no—
253#215—44no—
254#213—44no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—81no—
257(C2×D3)⋊C4—81no—
258(C2×D3)⋊C4—81no—
259(C2×D3)⋊C4—81no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—41no—
266C4⋊Dic3—41no—
267C4⋊Dic3—42no—
268C4⋊Dic3#4112yes—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C4×Dic3—21no—
273C4×Dic3—42no—
274C4×Dic3—81no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C22×Q8—21no—
282C22×Q8C22×D311yescharacteristic
283C22×D4—31no—
284C22×D4C22×D311yescharacteristic
285C23×C4—61no—
286C23×C4—61no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—121no—
290C2×D8—62no—
291C2×D8—62no—
292C2×D8—121no—
293C22×C8—31no—
294C22×C8—31no—
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299#124—22no—
300#124#6714yes—
301C4×Q8—62no—
302C4×Q8—121no—
303C4×D4—62no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—121no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—61no—
329C2×C42—61no—
330C2×C42C22×D311yescharacteristic
331C4⋊C8—62no—
332C4⋊C8—121no—
333C4⋊C8—121no—
334#103—22no—
335#103—22no—
336#103—62no—
337#102—22no—
338#102—22no—
339#102—62no—
340#102—62no—
341#102—62no—
342#97—22no—
343#95—31no—
344#95—31no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352C23×C3—22no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3C22×D411yescharacteristic
360C22×D3—22no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C3×Q8—22no—
368C3×Q8—22no—
369C3×Q8—41no—
370C3×Q8—41no—
371C3×Q8—84no—
372C3×D4—22no—
373C3×D4—22no—
374C3×D4—41no—
375C3×D4—41no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—22no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4#12112yes—
390C2×C3×C4#12112yes—
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4H(5,𝔽2)11yescharacteristic
396C2×C3×C42−1+411yescharacteristic
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—22no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—84no—
419D12—22no—
420D12—22no—
421D12—41no—
422D12—41no—
423C4×D3—84no—
424C3×C8—41no—
425C3×C8—41no—
426C3⋊C8—41no—
427C3⋊C8—41no—
428C24—21no—
429C24—61no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—42no—
436C2×Q8—81no—
437C2×Q8—81no—
438C2×Q8C2×#6712yes—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—32no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4C2×#6712yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—122no—
462C22×C4—122no—
463C22×C4—122no—
464C22×C4—122no—
465C22×C4—122no—
466C22×C4—241no—
467C22×C4D3×D411yescharacteristic
468C22×C4D3×D411yescharacteristic
469C22×C4C23×D311yescharacteristic
470SD16—124no—
471D8—124no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—32no—
489C4⋊C4—42no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—121no—
496C4⋊C4—121no—
497C4⋊C4—122no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—241no—
504C22⋊C4—241no—
505C22⋊C4—241no—
506C22⋊C4—241no—
507C22⋊C4—241no—
508C42—61no—
509C42—61no—
510C42—61no—
511C42—61no—
512C42—122no—
513C42—241no—
514C42#23712yes—
515C42C2×#6712yes—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3C2×#12111yescharacteristic
524C22×C3#54611yescharacteristic
525C22×C3#54811yescharacteristic
526C22×C3#57511yescharacteristic
527C22×C3#57511yescharacteristic
528C22×C3#57611yescharacteristic
529C22×C3#57711yescharacteristic
530C2×D3—42no—
531C2×D3—42no—
532C2×D3—42no—
533C2×D3—81no—
534C2×D3—81no—
535C2×D3—81no—
536C2×D3—81no—
537C3×C4—22no—
538C3×C4—22no—
539C3×C4—42no—
540C3×C4—81no—
541C3×C4—81no—
542C3×C4#52911yescharacteristic
543C3×C4#54211yescharacteristic
544C3×C4#54611yescharacteristic
545C3×C4#54711yescharacteristic
546Dic3—41no—
547Dic3—41no—
548Dic3—42no—
549Dic3—41no—
550Dic3—41no—
551Dic3—81no—
552Dic3—81no—
553C23—22no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—62no—
561C23—121no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23C2×D3×D411yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—84no—
573D4—22no—
574D4—22no—
575D4—41no—
576D4—41no—
577D4—62no—
578D4—62no—
579D4—121no—
580D4—121no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—62no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—244no—
616C2×C4—244no—
617C2×C4#93912yes—
618C2×C4#93912yes—
619C2×C4C2×#23711yescharacteristic
620C2×C4D3×#4111yescharacteristic
621C2×C4D3×#4111yescharacteristic
622C2×C4#100911yescharacteristic
623C2×C4#101011yescharacteristic
624C2×C4C22×#6711yescharacteristic
625C8—41no—
626C8—41no—
627C8—121no—
628C8—121no—
629C2×C3—82no—
630C2×C3#242711yescharacteristic
631C2×C3#304011yescharacteristic
632C2×C3#304511yescharacteristic
633C2×C3#331911yescharacteristic
634C2×C3#332111yescharacteristic
635C2×C3#337011yescharacteristic
636C2×C3#337611yescharacteristic
637D3—82no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—122no—
646C22—122no—
647C22—122no—
648C22—241no—
649C22—241no—
650C22—241no—
651C22—241no—
652C22#567611yescharacteristic
653C22#567711yescharacteristic
654C22#577411yescharacteristic
655C22#577511yescharacteristic
656C22#577711yescharacteristic
657C22#577711yescharacteristic
658C22C2×#93911yescharacteristic
659C4—22no—
660C4—22no—
661C4—42no—
662C4—81no—
663C4—81no—
664C4—121no—
665C4—121no—
666C4—122no—
667C4—121no—
668C4—121no—
669C4—241no—
670C4—241no—
671C4#568011yescharacteristic
672C4#568211yescharacteristic
673C4#582411yescharacteristic
674C4#582911yescharacteristic
675C3#2533511yescharacteristic, complemented, Sylow
676C2—82no—
677C2—242no—
678C2#8167211yescharacteristic
679C2#8167611yescharacteristic
680C2#8170211yescharacteristic
681C2#8170311yescharacteristic
682C2#8299611yescharacteristic
683C2#8393611yescharacteristic
684C2#8396811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#489C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenterC2×#489D3
hypocenterC3#25335
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
1. C2×C3×#489C2semidirect
2. #83260C2semidirect
3. #82955C2semidirect
5. #71780C2semidirect
6. #70280C2semidirect
7. #70240C2semidirect
8. #70234C2semidirect
9. #70222C2semidirect
12. #70166C2semidirect
13. #70154C2semidirect
14. #69836C2semidirect
15. #69812C2semidirect
62. C2×#489D3semidirect
675. C3#25335semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4892
#832602
#829552
C2×#50572
#717802
#702802
#702402
#702342
#702222
#702122
#701722
#701662
#701542
#698362
#698122
#253353