Finite Group Databasebeta

← #10855235

Subgroups — #10855235

Order 768.

918 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#460C211yesmaximal, characteristic
2C2×Dic3×SD16C211yesmaximal, characteristic, complemented
3#83176C211yesmaximal, characteristic
4Q8×Dic3⋊C4C211yesmaximal, characteristic, complemented
5#72048C211yesmaximal, characteristic, complemented
6#72048C211yesmaximal, characteristic, complemented
7#72048C211yesmaximal, characteristic, complemented
8#71762C211yesmaximal, characteristic
9#71629C211yesmaximal, characteristic, complemented
10#70464C211yesmaximal, characteristic, complemented
11#70447C211yesmaximal, characteristic, complemented
12#70186C211yesmaximal, characteristic, complemented
13#70184C211yesmaximal, characteristic, complemented
14#70169C211yesmaximal, characteristic, complemented
15#70165C211yesmaximal, characteristic, complemented
16#26163—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C22×C3×SD16C2211yescharacteristic
19C2×C3×#127C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×Q8×Dic3C2211yescharacteristic
23C2×#935—21no—
24C2×D4×Dic3C2211yescharacteristic
25C3×#460—24no—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31#5222—21no—
32#5222—21no—
33#5221C2211yescharacteristic
34#5221C2211yescharacteristic
35#5221C2211yescharacteristic
36#5220—21no—
37C2×#832C2211yescharacteristic
38C2×#832C2211yescharacteristic
39C2×#832C2211yescharacteristic
40C2×#832C2211yescharacteristic
41Dic3×SD16—24no—
42C2×#817C2211yescharacteristic
43C2×C8×Dic3C2211yescharacteristic
44#4975—21no—
45#4975C2211yescharacteristic
46#4972—21no—
47#4972C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C4×Dic3⋊C4—21no—
50C4×Dic3⋊C4C2211yescharacteristic
51#4928—21no—
52#4928C2211yescharacteristic
53#4928C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#460D311yescharacteristic
63C2×C4×SD16—31no—
64#2445—31no—
65Q8×C4⋊C4—31no—
66#2094—31no—
67#2088—31no—
68#2071—31no—
69#2041—31no—
70#1934—31no—
71#1916—31no—
72#1880—31no—
73#1880—31no—
74#1880—31no—
75#1865—31no—
76#1864—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4—21no—
79C22×C3×D4—21no—
80C22×C3×D4C2311yescharacteristic
81C22×C3×D4C2311yescharacteristic
82C23×Dic3—21no—
83C2×C3×SD16—22no—
84C2×C3×SD16—22no—
85C2×C3×SD16—22no—
86C2×C3×SD16—22no—
87C2×C3×SD16—22no—
88C2×C3×SD16D414yes—
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×#127—22no—
92C3×#127—41no—
93C3×C4×Q8—22no—
94C3×C4×Q8—41no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104C2×#217—21no—
105C2×#217—21no—
106Q8×Dic3—22no—
107Q8×Dic3—22no—
108Q8×Dic3—22no—
109Q8×Dic3—41no—
110#935—44no—
111D4×Dic3—22no—
112D4×Dic3—41no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×Dic3⋊C4—21no—
123C2×Dic3⋊C4—21no—
124C2×Dic3⋊C4D411yescharacteristic
125C2×Dic3⋊C4D411yescharacteristic
126C2×C4×Dic3—21no—
127C2×C4×Dic3—21no—
128C2×C4×Dic3C2311yescharacteristic
129C2×C4×Dic3C2311yescharacteristic
130C22×C3⋊C8C2311yescharacteristic
131C22×C3⋊C8C2311yescharacteristic
132C3×C4⋊C8—22no—
133C3×#102—22no—
134C3×#102—22no—
135#835—22no—
136#835—22no—
137#832—22no—
138#832—22no—
139#832—22no—
140#832—41no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95D411yescharacteristic
145C3⋊#95D411yescharacteristic
146#817—22no—
147C8×Dic3—22no—
148#808—41no—
149C22×SD16C2×D311yescharacteristic
150C22×SD16C2×D311yescharacteristic
151C2×#127C2×D311yescharacteristic
152C2×#123—61no—
153C2×C4×Q8—31no—
154C2×C4×Q8—31no—
155C2×C4×Q8C2×D311yescharacteristic
156C2×C4×D4—31no—
157#460—24no—
158C4×SD16—64no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8C2×D311yescharacteristic
162C2×#103—31no—
163C2×#103—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102C2×D311yescharacteristic
169C2×#102C2×D311yescharacteristic
170C2×C4×C8—31no—
171#394—61no—
172#394—61no—
173#390—31no—
174#390—31no—
175#390—31no—
176#389—31no—
177#389—31no—
178#389—31no—
179#389—61no—
180#389—61no—
181#386—61no—
182#384—31no—
183#384—61no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—61no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#336—31no—
191#336—31no—
192#336—31no—
193#336—31no—
194C24×C3—21no—
195C24×C3—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8C2×D412yes—
200C2×C3×Q8#4112yes—
201C2×C3×Q8#4112yes—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—42no—
214C2×C3×D4—42no—
215C2×C3×D4—42no—
216C2×C3×D4—42no—
217C2×C3×D4—81no—
218C2×C3×D4—81no—
219C2×C3×D4C2×D412yes—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2411yescharacteristic
225C22×Dic3—21no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—42no—
230C22×Dic3—42no—
231C22×Dic3—81no—
232C22×Dic3C2×D411yescharacteristic
233C22×Dic3C2×D411yescharacteristic
234C22×Dic3C2×D411yescharacteristic
235C22×Dic3C2×D411yescharacteristic
236C3×SD16—28no—
237C3×SD16—44no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8C2×D412yes—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—42no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—42no—
250#217—42no—
251#217—42no—
252#217—42no—
253#217—42no—
254#217—81no—
255#217—81no—
256#217—81no—
257C4⋊Dic3—24no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—42no—
266C4⋊Dic3#4112yes—
267Dic3⋊C4—41no—
268Dic3⋊C4—42no—
269Dic3⋊C4—42no—
270Dic3⋊C4—81no—
271Dic3⋊C4—81no—
272Dic3⋊C4—81no—
273Dic3⋊C4SD1614yescomplemented
274C4×Dic3—24no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277C4×Dic3—21no—
278C4×Dic3—21no—
279C4×Dic3—42no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C22×Q8C22×D311yescharacteristic
287C22×D4—21no—
288C22×D4—21no—
289C22×D4C22×D311yescharacteristic
290C22×D4C22×D311yescharacteristic
291C23×C4—61no—
292C2×SD16—22no—
293C2×SD16—22no—
294C2×SD16—22no—
295C2×SD16—22no—
296C2×SD16—22no—
297C2×SD16#6714yes—
298C22×C8—31no—
299C22×C8—31no—
300C22×C8C22×D311yescharacteristic
301C22×C8C22×D311yescharacteristic
302#127—22no—
303#127—41no—
304#123—124no—
305C4×Q8—22no—
306C4×Q8—41no—
307C4×Q8—62no—
308C4×Q8—62no—
309C4×Q8—62no—
310C4×Q8—121no—
311C4×D4—62no—
312C4×D4—121no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C42—21no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—61no—
339C2×C42—61no—
340C2×C42C22×D311yescharacteristic
341#106—62no—
342#106—121no—
343C4⋊C8—22no—
344#103—62no—
345#103—62no—
346#102—22no—
347#102—22no—
348#102—62no—
349#102—62no—
350#102—62no—
351#102—121no—
352C4×C8—62no—
353#95—31no—
354#95—31no—
355#95—61no—
356#95—61no—
357#95—61no—
358C23×C3—22no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3C22×D411yescharacteristic
373C3×Q8—22no—
374C3×Q8—22no—
375C3×Q8—22no—
376C3×Q8#12314yescomplemented
377C3×D4—22no—
378C3×D4—22no—
379C3×D4—41no—
380C3×D4—41no—
381C3×D4—41no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C3×D4—84no—
386C3×D4—84no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—21no—
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4C2×SD1612yes—
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4H(5,𝔽2)11yescharacteristic
404C2×C3×C42−1+411yescharacteristic
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—22no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—84no—
428C2×Dic3C2×SD1612yes—
429C2×Dic3C2×SD1612yes—
430C3×C8—22no—
431C3×C8—41no—
432C3⋊C8—41no—
433C3⋊C8—41no—
434C24—21no—
435C24—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8#23712yes—
440C2×Q8#23712yes—
441C2×Q8C2×#6712yes—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—81no—
458C2×D4—81no—
459C2×D4C2×#6712yes—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4—241no—
473C22×C4D3×D411yescharacteristic
474C22×C4D3×D411yescharacteristic
475C22×C4C23×D311yescharacteristic
476SD16—28no—
477SD16—44no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8C2×#6712yes—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—34no—
493C4⋊C4—32no—
494C4⋊C4—42no—
495C4⋊C4—64no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—121no—
504C4⋊C4—122no—
505C4⋊C4—122no—
506C4⋊C4—122no—
507C4⋊C4—241no—
508C4⋊C4—241no—
509C4⋊C4—241no—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C22⋊C4—241no—
515C22⋊C4—241no—
516C22⋊C4—241no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—64no—
521C42—61no—
522C42—61no—
523C42—61no—
524C42—61no—
525C42—122no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3—81no—
532C22×C3—81no—
533C22×C3—81no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3—81no—
538C22×C3—81no—
539C22×C3—81no—
540C22×C3#52511yescharacteristic
541C22×C3D4211yescharacteristic
542C22×C3#54811yescharacteristic
543C22×C3C22×SD1611yescharacteristic
544C22×C3#57511yescharacteristic
545C22×C3#57611yescharacteristic
546C22×C3#57711yescharacteristic
547C3×C4—24no—
548C3×C4—22no—
549C3×C4—41no—
550C3×C4—81no—
551C3×C4C2×#12311yescharacteristic
552C3×C4#53911yescharacteristic
553C3×C4#54711yescharacteristic
554C3×C4#55311yescharacteristic
555Dic3—22no—
556Dic3—22no—
557Dic3—41no—
558Dic3—42no—
559Dic3—41no—
560Dic3—81no—
561Dic3—81no—
562Dic3—81no—
563C23—22no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23C2×D3×D411yescharacteristic
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8#93514yescomplemented
582D4—22no—
583D4—22no—
584D4—41no—
585D4—41no—
586D4—41no—
587D4—41no—
588D4—41no—
589D4—41no—
590D4—84no—
591D4—84no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—32no—
601C2×C4—32no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—62no—
615C2×C4—61no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—244no—
628C2×C4D3×SD1612yes—
629C2×C4C2×#23711yescharacteristic
630C2×C4C2×#23711yescharacteristic
631C2×C4#100411yescharacteristic
632C2×C4D3×#4111yescharacteristic
633C2×C4#101011yescharacteristic
634C2×C4C22×#6711yescharacteristic
635C8—22no—
636C8—41no—
637C8—121no—
638C8—121no—
639C2×C3—82no—
640C2×C3—161no—
641C2×C3#244311yescharacteristic
642C2×C3#310311yescharacteristic
643C2×C3#311911yescharacteristic
644C2×C3#326811yescharacteristic
645C2×C3D4×SD1611yescharacteristic
646C2×C3#336211yescharacteristic
647C2×C3#337111yescharacteristic
648C22—42no—
649C22—42no—
650C22—42no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22#559811yescharacteristic
663C22#560611yescharacteristic
664C22C2×D3×SD1611yescharacteristic
665C22#576211yescharacteristic
666C22#577311yescharacteristic
667C22#577411yescharacteristic
668C22D4×#6711yescharacteristic
669C4—22no—
670C4—24no—
671C4—41no—
672C4—62no—
673C4—62no—
674C4—81no—
675C4—121no—
676C4—122no—
677C4—121no—
678C4—241no—
679C4—241no—
680C4—241no—
681C4#560111yescharacteristic
682C4#560211yescharacteristic
683C4C2×#93511yescharacteristic
684C4#582411yescharacteristic
685C3#2616311yescharacteristic, complemented, Sylow
686C2—82no—
687C2—161no—
688C2#8036011yescharacteristic
689C2#8036411yescharacteristic
690C2#8036911yescharacteristic
691C2#8037311yescharacteristic
692C2#8265511yescharacteristic
693C2SD16×#6711yescharacteristic
694C2#8395511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#460C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenterC2×#460D3
hypocenterC3#26163
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×SD16C2semidirect
4. Q8×Dic3⋊C4C2semidirect
5. #72048C2semidirect
6. #72048C2semidirect
7. #72048C2semidirect
9. #71629C2semidirect
10. #70464C2semidirect
11. #70447C2semidirect
12. #70186C2semidirect
13. #70184C2semidirect
14. #70169C2semidirect
15. #70165C2semidirect
273. Dic3⋊C4SD16semidirect
376. C3×Q8#123semidirect
581. Q8#935semidirect
685. C3#26163semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4602
C2×Dic3×SD162
#831762
Q8×Dic3⋊C42
#720482
#717622
#716292
#704642
#704472
#701862
#701842
#701692
#701652
#261633