Finite Group Databasebeta

← #10830102

Subgroups — #10830102

Order 768.

1084 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5567C211yesmaximal, characteristic, complemented
2C2×#5162C211yesmaximal, characteristic
3C2×#4757C211yesmaximal, characteristic
4#76364C211yesmaximal, characteristic
5C3×#1880C211yesmaximal, characteristic
6#72057C211yesmaximal, characteristic
7#71628C211yesmaximal, characteristic, complemented
8#70246C211yesmaximal, characteristic, complemented
9#70228C211yesmaximal, characteristic, complemented
10#70213C211yesmaximal, characteristic
11#69661C211yesmaximal, characteristic
12#68906C211yesmaximal, characteristic, complemented
13#68905C211yesmaximal, characteristic, complemented
14#68631C211yesmaximal, characteristic, complemented
15#68620C211yesmaximal, characteristic, complemented
16#24199—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×#944C2211yescharacteristic
20C2×#944C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C22×#214C2211yescharacteristic
23C22×#214C2211yescharacteristic
24#5567—24no—
25#5567—24no—
26C2×#889—21no—
27C2×#889—21no—
28C2×#887C2211yescharacteristic
29C2×#887C2211yescharacteristic
30C2×#870—21no—
31C2×#869C2211yescharacteristic
32C2×C4×Dic6—21no—
33C2×C4×Dic6C2211yescharacteristic
34C2×C3×#103C2211yescharacteristic
35C2×C3×#102C2211yescharacteristic
36C3×#390C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#835C2211yescharacteristic
39#5221C2211yescharacteristic
40#5162—24no—
41C2×#816C2211yescharacteristic
42C2×#816C2211yescharacteristic
43C2×#816C2211yescharacteristic
44C2×#816C2211yescharacteristic
45C2×#815C2211yescharacteristic
46C2×Dic3⋊C8C2211yescharacteristic
47#4984C2211yescharacteristic
48#4977C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#810C2211yescharacteristic
51C2×#807C2211yescharacteristic
52#4950—21no—
53#4949—21no—
54#4947—21no—
55#4946—21no—
56#4928C2211yescharacteristic
57#4757—24no—
58C3×#340C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#557—31no—
65C2×#461—31no—
66C2×#440—31no—
67#2580—31no—
68#2043—31no—
69#2043—31no—
70#2013—31no—
71#2013—31no—
72#1985—31no—
73#1960—31no—
74#1880—31no—
75#1880D311yescharacteristic
76#1863—31no—
77#1863—31no—
78#1805—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×Dic3—21no—
82C22×Dic6D411yescharacteristic
83C22×Dic6C2311yescharacteristic
84C22×Dic6C2311yescharacteristic
85C2×C3×SD16—22no—
86C2×C3×SD16—22no—
87C2×C3×SD16—41no—
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×C4×Q8—22no—
91C3×C4×Q8—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102#944—22no—
103#944—22no—
104#944—22no—
105#944D414yes—
106D4×Dic3—22no—
107D4×Dic3—22no—
108C2×#214—22no—
109C2×#214—22no—
110C2×#214—22no—
111C2×#214—22no—
112C2×#214—22no—
113C2×#214—41no—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3—21no—
116C2×C4⋊Dic3—21no—
117C2×C4⋊Dic3D411yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119C2×C4⋊Dic3C2311yescharacteristic
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×Dic3⋊C4—21no—
123C2×Dic3⋊C4—21no—
124C2×Dic3⋊C4—21no—
125C2×Dic3⋊C4—21no—
126C2×Dic3⋊C4—21no—
127C2×Dic3⋊C4D411yescharacteristic
128C2×Dic3⋊C4D411yescharacteristic
129C2×C4×Dic3—21no—
130C2×C4×Dic3—21no—
131C2×C4×Dic3C2311yescharacteristic
132C2×C4×Dic3C2311yescharacteristic
133C22×C3⋊C8C2311yescharacteristic
134C22×C3⋊C8C2311yescharacteristic
135#889—28no—
136#889—28no—
137#887—22no—
138#887—22no—
139#887—22no—
140#887D414yes—
141#870—28no—
142#869—22no—
143#869—22no—
144C4×Dic6—28no—
145C4×Dic6—22no—
146C4×Dic6—22no—
147C3×#103—41no—
148C3×#102—22no—
149#835—22no—
150#835—22no—
151C3⋊#95—21no—
152C3⋊#95—21no—
153C3⋊#95—21no—
154C3⋊#95—21no—
155#816—22no—
156#816—22no—
157#816—41no—
158#816—41no—
159#815—22no—
160Dic3⋊C8—22no—
161#810—22no—
162#810—41no—
163#807—22no—
164C22×SD16—31no—
165C22×SD16—31no—
166C22×SD16C2×D311yescharacteristic
167#557—64no—
168#557—64no—
169C2×C4⋊Q8—31no—
170C2×C4⋊Q8—31no—
171C2×C4⋊Q8—31no—
172C2×#125—61no—
173C2×#125—61no—
174C2×#125—61no—
175C2×C4×Q8—31no—
176C2×C4×Q8—31no—
177C2×C4×Q8—31no—
178C2×C4×Q8—61no—
179C2×C4×Q8C2×D311yescharacteristic
180C2×C4×D4—31no—
181#461—64no—
182#440—64no—
183C2×#107—31no—
184C2×C4⋊C8—31no—
185C2×#103—31no—
186C2×#103—31no—
187C2×#103—31no—
188C2×#103—31no—
189C2×#103—31no—
190C2×#103—31no—
191C2×#103—31no—
192C2×#103—31no—
193C2×#103C2×D311yescharacteristic
194C2×#102C2×D311yescharacteristic
195C2×#97—31no—
196#399—61no—
197#393—61no—
198#390—31no—
199#390C2×D311yescharacteristic
200#389—31no—
201#389—31no—
202#385—61no—
203#385—61no—
204#382—31no—
205#340—31no—
206#340C2×D311yescharacteristic
207#336—31no—
208#336—31no—
209#336—31no—
210#336C2×D311yescharacteristic
211C24×C3—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8C2×D412yes—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4#4112yes—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2411yescharacteristic
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3—21no—
234C22×Dic3—42no—
235C22×Dic3—81no—
236C22×Dic3—81no—
237C22×Dic3C2×D411yescharacteristic
238C22×Dic3C2×D411yescharacteristic
239C22×Dic3C2×D411yescharacteristic
240C22×Dic3C2×D411yescharacteristic
241C2×Dic6—22no—
242C2×Dic6—22no—
243C2×Dic6—22no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6—21no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6—21no—
253C2×Dic6—21no—
254C2×Dic6C2×D412yes—
255C2×Dic6C2×D412yes—
256C3×SD16—44no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C2×C3×C8—21no—
263C3×C4⋊C4—24no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4C2×D412yes—
268C3×C22⋊C4—42no—
269C3×C22⋊C4—81no—
270C3×C42—24no—
271C3×C42—21no—
272C3×C42—21no—
273#217—42no—
274#217—42no—
275#217—81no—
276#214—28no—
277#214—44no—
278C4⋊Dic3—24no—
279C4⋊Dic3—24no—
280C4⋊Dic3—22no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283C4⋊Dic3—21no—
284C4⋊Dic3—21no—
285C4⋊Dic3—81no—
286Dic3⋊C4—24no—
287Dic3⋊C4—24no—
288Dic3⋊C4—24no—
289Dic3⋊C4—22no—
290Dic3⋊C4—22no—
291Dic3⋊C4—24no—
292Dic3⋊C4—24no—
293Dic3⋊C4—24no—
294Dic3⋊C4—81no—
295C4×Dic3—24no—
296C4×Dic3—21no—
297C4×Dic3—21no—
298C4×Dic3—24no—
299C4×Dic3C2×D412yes—
300C4×Dic3C2×D412yes—
301C2×C3⋊C8—21no—
302C2×C3⋊C8—21no—
303C2×C3⋊C8—21no—
304C2×C3⋊C8—21no—
305C2×C3⋊C8—21no—
306C2×C3⋊C8#4112yes—
307C22×Q8—31no—
308C22×Q8—31no—
309C22×Q8—31no—
310C22×Q8C22×D311yescharacteristic
311C22×D4C22×D311yescharacteristic
312C23×C4—61no—
313C2×SD16—22no—
314C2×SD16—22no—
315C2×SD16—41no—
316C2×SD16—62no—
317C2×SD16—62no—
318C2×SD16—62no—
319C2×SD16—62no—
320C2×SD16—62no—
321C2×SD16—121no—
322C22×C8—31no—
323C22×C8—31no—
324C22×C8C22×D311yescharacteristic
325C22×C8C22×D311yescharacteristic
326C4⋊Q8—34no—
327C4⋊Q8—62no—
328C4⋊Q8—62no—
329C4⋊Q8—62no—
330C4⋊Q8—62no—
331C4⋊Q8—62no—
332#125—68no—
333#125—68no—
334#125—68no—
335C4×Q8—22no—
336C4×Q8—22no—
337C4×Q8—34no—
338C4×Q8—62no—
339C4×Q8—62no—
340C4×Q8—68no—
341C4×Q8—62no—
342C4×Q8—62no—
343C4×Q8—62no—
344C4×D4—62no—
345C4×D4—62no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—31no—
348C2×C4⋊C4—31no—
349C2×C4⋊C4—31no—
350C2×C4⋊C4—31no—
351C2×C4⋊C4—31no—
352C2×C4⋊C4—61no—
353C2×C4⋊C4—61no—
354C2×C4⋊C4—61no—
355C2×C4⋊C4—61no—
356C2×C4⋊C4—61no—
357C2×C4⋊C4—61no—
358C2×C4⋊C4—61no—
359C2×C4⋊C4—61no—
360C2×C4⋊C4—61no—
361C2×C4⋊C4—61no—
362C2×C4⋊C4C22×D311yescharacteristic
363C2×C4⋊C4C22×D311yescharacteristic
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—61no—
367C2×C22⋊C4—61no—
368C2×C22⋊C4—61no—
369C2×C42—21no—
370C2×C42—31no—
371C2×C42—31no—
372C2×C42—61no—
373C2×C42—61no—
374C2×C42C22×D311yescharacteristic
375#107—62no—
376C4⋊C8—62no—
377#103—41no—
378#103—62no—
379#103—62no—
380#103—62no—
381#103—62no—
382#103—62no—
383#103—121no—
384#103—121no—
385#103—121no—
386#102—22no—
387#97—62no—
388#95—61no—
389#95—61no—
390#95—61no—
391#95—61no—
392C23×C3—22no—
393C23×C3—41no—
394C23×C3—41no—
395C23×C3—41no—
396C23×C3—41no—
397C23×C3—41no—
398C23×C3—41no—
399C23×C3C22×D411yescharacteristic
400C3×Q8—22no—
401C3×Q8—22no—
402C3×Q8—41no—
403C3×Q8—41no—
404C3×D4—22no—
405C3×D4—22no—
406C3×D4—41no—
407C3×D4—41no—
408C2×C3×C4—21no—
409C2×C3×C4—21no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—41no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4—41no—
422C2×C3×C4#13712yes—
423C2×C3×C4C22×D411yescharacteristic
424C2×C3×C4C22×D411yescharacteristic
425C2×C3×C4C2×#4111yescharacteristic
426C2×C3×C4C2×#4111yescharacteristic
427C2×C3×C4H(5,𝔽2)11yescharacteristic
428C2×C3×C42−1+411yescharacteristic
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—21no—
434C2×Dic3—21no—
435C2×Dic3—21no—
436C2×Dic3—22no—
437C2×Dic3—22no—
438C2×Dic3—21no—
439C2×Dic3—21no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—41no—
445C2×Dic3—41no—
446C2×Dic3—41no—
447C2×Dic3—41no—
448C2×Dic3—41no—
449C2×Dic3—41no—
450C2×Dic3—84no—
451C2×Dic3#12012yes—
452C2×Dic3#12012yes—
453C2×Dic3#13712yes—
454Dic6—28no—
455Dic6—22no—
456Dic6—22no—
457Dic6—22no—
458Dic6—22no—
459Dic6—41no—
460Dic6—41no—
461Dic6—41no—
462Dic6—41no—
463C3×C8—41no—
464C3×C8—41no—
465C3⋊C8—22no—
466C3⋊C8—41no—
467C24—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—32no—
474C2×Q8—32no—
475C2×Q8—62no—
476C2×Q8—62no—
477C2×Q8—62no—
478C2×Q8—61no—
479C2×Q8—61no—
480C2×Q8—61no—
481C2×Q8—61no—
482C2×Q8—61no—
483C2×Q8—61no—
484C2×Q8—61no—
485C2×Q8—61no—
486C2×Q8—61no—
487C2×Q8—61no—
488C2×Q8D3×D412yes—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4#23512yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—31no—
499C22×C4—31no—
500C22×C4—31no—
501C22×C4—31no—
502C22×C4—61no—
503C22×C4—61no—
504C22×C4—61no—
505C22×C4—61no—
506C22×C4—122no—
507C22×C4—241no—
508C22×C4—241no—
509C22×C4D3×D411yescharacteristic
510C22×C4D3×D411yescharacteristic
511C22×C4C23×D311yescharacteristic
512SD16—44no—
513SD16—68no—
514SD16—124no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—32no—
522C2×C8—61no—
523C2×C8—61no—
524C2×C8—61no—
525C2×C8—61no—
526C2×C8—61no—
527C4⋊C4—24no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—62no—
532C4⋊C4—62no—
533C4⋊C4—62no—
534C4⋊C4—64no—
535C4⋊C4—64no—
536C4⋊C4—64no—
537C4⋊C4—64no—
538C4⋊C4—64no—
539C4⋊C4—64no—
540C4⋊C4—64no—
541C4⋊C4—64no—
542C4⋊C4—61no—
543C4⋊C4—61no—
544C4⋊C4—61no—
545C4⋊C4—61no—
546C4⋊C4—241no—
547C4⋊C4—241no—
548C4⋊C4D3×D412yes—
549C22⋊C4—42no—
550C22⋊C4—81no—
551C22⋊C4—122no—
552C22⋊C4—122no—
553C22⋊C4—241no—
554C42—24no—
555C42—21no—
556C42—21no—
557C42—32no—
558C42—32no—
559C42—64no—
560C42—64no—
561C42—61no—
562C42—61no—
563C22×C3—42no—
564C22×C3—42no—
565C22×C3—42no—
566C22×C3—81no—
567C22×C3—81no—
568C22×C3—81no—
569C22×C3—81no—
570C22×C3C2×#12011yescharacteristic
571C22×C3#54811yescharacteristic
572C22×C3#55011yescharacteristic
573C22×C3C2×#13711yescharacteristic
574C22×C3C2×#13711yescharacteristic
575C22×C3#57711yescharacteristic
576C22×C3#57711yescharacteristic
577C3×C4—22no—
578C3×C4—42no—
579C3×C4—41no—
580C3×C4—42no—
581C3×C4—81no—
582C3×C4#52511yescharacteristic
583C3×C4#53711yescharacteristic
584C3×C4#54611yescharacteristic
585C3×C4#54711yescharacteristic
586Dic3—22no—
587Dic3—22no—
588Dic3—22no—
589Dic3—42no—
590Dic3—42no—
591Dic3—41no—
592Dic3—81no—
593Dic3—81no—
594C23—22no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—41no—
599C23—41no—
600C23—41no—
601C23C2×D3×D411yescharacteristic
602Q8—22no—
603Q8—22no—
604Q8—41no—
605Q8—41no—
606Q8—68no—
607Q8—62no—
608Q8—62no—
609Q8—62no—
610Q8—62no—
611Q8—121no—
612Q8—121no—
613Q8—121no—
614Q8—121no—
615D4—22no—
616D4—22no—
617D4—41no—
618D4—41no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—22no—
626C2×C4—32no—
627C2×C4—32no—
628C2×C4—32no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—61no—
639C2×C4—61no—
640C2×C4—61no—
641C2×C4—61no—
642C2×C4—61no—
643C2×C4—61no—
644C2×C4—61no—
645C2×C4—62no—
646C2×C4—62no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4—121no—
651C2×C4—121no—
652C2×C4—121no—
653C2×C4—121no—
654C2×C4—121no—
655C2×C4—121no—
656C2×C4—121no—
657C2×C4—244no—
658C2×C4#91512yes—
659C2×C4C2×#23511yescharacteristic
660C2×C4C2×D3×D411yescharacteristic
661C2×C4C2×D3×D411yescharacteristic
662C2×C4#100411yescharacteristic
663C2×C4D3×#4111yescharacteristic
664C2×C4#101011yescharacteristic
665C8—41no—
666C8—41no—
667C8—62no—
668C8—121no—
669C2×C3—82no—
670C2×C3#240711yescharacteristic
671C2×C3#300411yescharacteristic
672C2×C3#301011yescharacteristic
673C2×C3#337211yescharacteristic
674C2×C3#337211yescharacteristic
675C2×C3#340011yescharacteristic
676C2×C3#340011yescharacteristic
677C22—42no—
678C22—42no—
679C22—42no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22#555211yescharacteristic
685C22D3×#12011yescharacteristic
686C22#559911yescharacteristic
687C22C2×#91511yescharacteristic
688C22#577011yescharacteristic
689C22#577511yescharacteristic
690C22D3×#13711yescharacteristic
691C4—22no—
692C4—42no—
693C4—41no—
694C4—42no—
695C4—62no—
696C4—62no—
697C4—62no—
698C4—81no—
699C4—122no—
700C4—122no—
701C4—121no—
702C4—241no—
703C4—241no—
704C4#549011yescharacteristic
705C4#554711yescharacteristic
706C4#560811yescharacteristic
707C4#565511yescharacteristic
708C3#2419911yescharacteristic, complemented, Sylow
709C2—82no—
710C2#7468811yescharacteristic
711C2#7962811yescharacteristic
712C2#7975511yescharacteristic
713C2#8003911yescharacteristic
714C2#8008611yescharacteristic
715C2#8038011yescharacteristic
716C2#8059111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1880C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1880D3
hypocenterC3#24199
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
1. C2×#5567C2semidirect
7. #71628C2semidirect
8. #70246C2semidirect
9. #70228C2semidirect
12. #68906C2semidirect
13. #68905C2semidirect
14. #68631C2semidirect
15. #68620C2semidirect
708. C3#24199semidirect

Maximal subgroups

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

subgroupindex
C2×#55672
C2×#51622
C2×#47572
#763642
C3×#18802
#720572
#716282
#702462
#702282
#702132
#696612
#689062
#689052
#686312
#686202
#241993