Finite Group Databasebeta

← #10830098

Subgroups — #10830098

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×#5721C211yesmaximal, characteristic, complemented
2C2×#5162C211yesmaximal, characteristic
3C2×#5164C211yesmaximal, characteristic
4#76364C211yesmaximal, characteristic
5C3×#2013C211yesmaximal, characteristic
6#72057C211yesmaximal, characteristic
7#72048C211yesmaximal, characteristic
8#71781C211yesmaximal, characteristic, complemented
9#70246C211yesmaximal, characteristic, complemented
10#70236C211yesmaximal, characteristic, complemented
11#70228C211yesmaximal, characteristic, complemented
12#70222C211yesmaximal, characteristic, complemented
13#69660C211yesmaximal, characteristic
14#68683C211yesmaximal, characteristic, complemented
15#68620C211yesmaximal, characteristic, complemented
16#24199—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C22×C3×SD16C2211yescharacteristic
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×#944C2211yescharacteristic
23C2×#944C2211yescharacteristic
24C2×D4×Dic3C2211yescharacteristic
25C22×#214C2211yescharacteristic
26#5721—24no—
27#5721—24no—
28C2×#890—21no—
29C2×#889—21no—
30C2×#889—21no—
31C2×#887—21no—
32C2×C4×Dic6C2211yescharacteristic
33C2×C4×Dic6C2211yescharacteristic
34C2×C3×#103C2211yescharacteristic
35C2×C3×#103C2211yescharacteristic
36C3×#390C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#835C2211yescharacteristic
39C2×#835C2211yescharacteristic
40C2×#835C2211yescharacteristic
41#5221C2211yescharacteristic
42C2×#832C2211yescharacteristic
43#5164—24no—
44#5162—24no—
45C2×#818C2211yescharacteristic
46C2×#816C2211yescharacteristic
47C2×#816C2211yescharacteristic
48C2×#815C2211yescharacteristic
49C2×Dic3⋊C8C2211yescharacteristic
50#4984C2211yescharacteristic
51#4977C2211yescharacteristic
52C2×#810C2211yescharacteristic
53#4950—21no—
54#4949—21no—
55#4947—21no—
56#4946—21no—
57#4928C2211yescharacteristic
58C3×#336C2211yescharacteristic
59#4551C2211yescharacteristic
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#2013D311yescharacteristic
72#1985—31no—
73#1960—31no—
74#1880—31no—
75#1880—31no—
76#1863—31no—
77#1863—31no—
78#1805—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×Q8C2311yescharacteristic
81C22×C3×D4C2311yescharacteristic
82C23×Dic3—21no—
83C22×Dic6D411yescharacteristic
84C22×Dic6C2311yescharacteristic
85C2×C3×SD16—22no—
86C2×C3×SD16—22no—
87C2×C3×SD16—22no—
88C2×C3×SD16—22no—
89C2×C3×SD16—22no—
90C2×C3×SD16—41no—
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C3×C4⋊Q8—22no—
94C3×C4⋊Q8—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C42C2311yescharacteristic
101C2×#217—21no—
102C2×#217—21no—
103C2×#217—21no—
104Q8×Dic3—22no—
105Q8×Dic3—22no—
106Q8×Dic3—22no—
107Q8×Dic3D414yes—
108#944—22no—
109#944—22no—
110#944—22no—
111#944D414yes—
112D4×Dic3—22no—
113D4×Dic3—22no—
114C2×#214—22no—
115C2×#214—22no—
116C2×#214—41no—
117C2×C4⋊Dic3—21no—
118C2×C4⋊Dic3—21no—
119C2×C4⋊Dic3—21no—
120C2×C4⋊Dic3D411yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122C2×C4⋊Dic3C2311yescharacteristic
123C2×C4⋊Dic3C2311yescharacteristic
124C2×Dic3⋊C4—21no—
125C2×Dic3⋊C4—21no—
126C2×Dic3⋊C4—21no—
127C2×Dic3⋊C4—21no—
128C2×Dic3⋊C4—21no—
129C2×Dic3⋊C4—21no—
130C2×Dic3⋊C4D411yescharacteristic
131C2×Dic3⋊C4D411yescharacteristic
132C2×C4×Dic3—21no—
133C2×C4×Dic3—21no—
134C2×C4×Dic3—21no—
135C2×C4×Dic3C2311yescharacteristic
136C2×C4×Dic3C2311yescharacteristic
137C22×C3⋊C8C2311yescharacteristic
138C22×C3⋊C8C2311yescharacteristic
139#890—28no—
140#889—28no—
141#889—28no—
142#887—28no—
143C4×Dic6—22no—
144C4×Dic6—22no—
145C4×Dic6—22no—
146C4×Dic6—22no—
147C3×#103—22no—
148C3×#103—41no—
149#835—22no—
150#835—22no—
151#835—41no—
152#835—41no—
153#832—22no—
154C3⋊#95—21no—
155C3⋊#95—21no—
156C3⋊#95—21no—
157C3⋊#95—21no—
158#818—22no—
159#816—22no—
160#816—22no—
161#815—22no—
162Dic3⋊C8—22no—
163#810—41no—
164C22×SD16—31no—
165C22×SD16C2×D311yescharacteristic
166C22×SD16C2×D311yescharacteristic
167#557—64no—
168#557—64no—
169C2×C4⋊Q8—31no—
170C2×C4⋊Q8—31no—
171C2×C4⋊Q8C2×D311yescharacteristic
172C2×#125—61no—
173C2×#125—61no—
174C2×#125—61no—
175C2×C4×Q8—31no—
176C2×C4×Q8—31no—
177C2×C4×Q8—31no—
178C2×C4×Q8—31no—
179C2×C4×Q8—61no—
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×#103C2×D311yescharacteristic
193C2×#103C2×D311yescharacteristic
194C2×#102—31no—
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#340—31no—
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×Q8—21no—
218C2×C3×Q8—21no—
219C2×C3×Q8—21no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8C2×D412yes—
223C2×C3×Q8C2×D412yes—
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—21no—
227C2×C3×D4—21no—
228C2×C3×D4—21no—
229C2×C3×D4#4112yes—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4C2×D411yescharacteristic
234C22×C3×C4C2×D411yescharacteristic
235C22×C3×C4C2411yescharacteristic
236C22×Dic3—21no—
237C22×Dic3—21no—
238C22×Dic3—21no—
239C22×Dic3—21no—
240C22×Dic3—42no—
241C22×Dic3—81no—
242C22×Dic3—81no—
243C22×Dic3C2×D411yescharacteristic
244C22×Dic3C2×D411yescharacteristic
245C22×Dic3C2×D411yescharacteristic
246C22×Dic3C2×D411yescharacteristic
247C2×Dic6—22no—
248C2×Dic6—22no—
249C2×Dic6—22no—
250C2×Dic6—21no—
251C2×Dic6—21no—
252C2×Dic6—21no—
253C2×Dic6—21no—
254C2×Dic6—21no—
255C2×Dic6C2×D412yes—
256C3×SD16—28no—
257C3×SD16—44no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C2×C3×C8—21no—
263C2×C3×C8#4112yes—
264C3×C4⋊C4—24no—
265C3×C4⋊C4—24no—
266C3×C4⋊C4—21no—
267C3×C4⋊C4—21no—
268C3×C22⋊C4—42no—
269C3×C22⋊C4—81no—
270C3×C42—21no—
271C3×C42—21no—
272#217—42no—
273#217—42no—
274#217—81no—
275#214—44no—
276C4⋊Dic3—24no—
277C4⋊Dic3—21no—
278C4⋊Dic3—22no—
279C4⋊Dic3—24no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283C4⋊Dic3—21no—
284C4⋊Dic3—81no—
285C4⋊Dic3C2×D412yes—
286Dic3⋊C4—24no—
287Dic3⋊C4—24no—
288Dic3⋊C4—24no—
289Dic3⋊C4—22no—
290Dic3⋊C4—22no—
291Dic3⋊C4—24no—
292Dic3⋊C4—24no—
293Dic3⋊C4—81no—
294C4×Dic3—24no—
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—21no—
307C22×Q8—31no—
308C22×Q8—31no—
309C22×Q8C22×D311yescharacteristic
310C22×Q8C22×D311yescharacteristic
311C22×D4C22×D311yescharacteristic
312C23×C4—61no—
313C2×SD16—22no—
314C2×SD16—22no—
315C2×SD16—22no—
316C2×SD16—22no—
317C2×SD16—22no—
318C2×SD16—41no—
319C2×SD16—62no—
320C2×SD16—62no—
321C2×SD16—121no—
322C22×C8—31no—
323C22×C8—31no—
324C22×C8C22×D311yescharacteristic
325C22×C8C22×D311yescharacteristic
326C4⋊Q8—22no—
327C4⋊Q8—22no—
328C4⋊Q8—34no—
329C4⋊Q8—62no—
330C4⋊Q8—62no—
331C4⋊Q8—62no—
332#125—68no—
333#125—68no—
334#125—68no—
335C4×Q8—34no—
336C4×Q8—62no—
337C4×Q8—62no—
338C4×Q8—62no—
339C4×Q8—62no—
340C4×Q8—62no—
341C4×Q8—68no—
342C4×Q8—62no—
343C4×Q8—62no—
344C4×D4—62no—
345C4×D4—62no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—31no—
349C2×C4⋊C4—31no—
350C2×C4⋊C4—31no—
351C2×C4⋊C4—31no—
352C2×C4⋊C4—31no—
353C2×C4⋊C4—31no—
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⋊C4—61no—
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—31no—
370C2×C42—31no—
371C2×C42—61no—
372C2×C42—61no—
373C2×C42—61no—
374C2×C42C22×D311yescharacteristic
375#107—62no—
376C4⋊C8—62no—
377#103—22no—
378#103—41no—
379#103—62no—
380#103—62no—
381#103—62no—
382#103—62no—
383#103—121no—
384#103—121no—
385#103—121no—
386#102—62no—
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—22no—
403C3×Q8—22no—
404C3×Q8—41no—
405C3×Q8—41no—
406C3×Q8—41no—
407C3×Q8—41no—
408C3×D4—22no—
409C3×D4—22no—
410C3×D4—41no—
411C3×D4—41no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—21no—
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4—41no—
422C2×C3×C4—41no—
423C2×C3×C4—41no—
424C2×C3×C4—41no—
425C2×C3×C4—41no—
426C2×C3×C4#13712yes—
427C2×C3×C4C22×D411yescharacteristic
428C2×C3×C4C22×D411yescharacteristic
429C2×C3×C4C2×#4111yescharacteristic
430C2×C3×C4C2×#4111yescharacteristic
431C2×C3×C4H(5,𝔽2)11yescharacteristic
432C2×C3×C42−1+411yescharacteristic
433C2×Dic3—21no—
434C2×Dic3—21no—
435C2×Dic3—21no—
436C2×Dic3—21no—
437C2×Dic3—21no—
438C2×Dic3—21no—
439C2×Dic3—22no—
440C2×Dic3—22no—
441C2×Dic3—21no—
442C2×Dic3—21no—
443C2×Dic3—21no—
444C2×Dic3—41no—
445C2×Dic3—41no—
446C2×Dic3—41no—
447C2×Dic3—41no—
448C2×Dic3—41no—
449C2×Dic3—41no—
450C2×Dic3—41no—
451C2×Dic3—41no—
452C2×Dic3—41no—
453C2×Dic3—41no—
454C2×Dic3—84no—
455C2×Dic3#12012yes—
456C2×Dic3#12012yes—
457C2×Dic3#13712yes—
458Dic6—28no—
459Dic6—22no—
460Dic6—22no—
461Dic6—41no—
462Dic6—41no—
463C3×C8—22no—
464C3×C8—41no—
465C3⋊C8—41no—
466C3⋊C8—41no—
467C24—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—32no—
479C2×Q8—62no—
480C2×Q8—62no—
481C2×Q8—62no—
482C2×Q8—61no—
483C2×Q8—61no—
484C2×Q8—61no—
485C2×Q8—61no—
486C2×Q8—61no—
487C2×Q8D3×D412yes—
488C2×Q8D3×D412yes—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4#23712yes—
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—28no—
513SD16—44no—
514SD16—124no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—61no—
521C2×C8—61no—
522C2×C8—61no—
523C2×C8—61no—
524C2×C8—61no—
525C2×C8—61no—
526C2×C8#23712yes—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—32no—
532C4⋊C4—62no—
533C4⋊C4—62no—
534C4⋊C4—62no—
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—61no—
547C4⋊C4—241no—
548C4⋊C4—241no—
549C22⋊C4—42no—
550C22⋊C4—81no—
551C22⋊C4—122no—
552C22⋊C4—122no—
553C22⋊C4—241no—
554C42—21no—
555C42—21no—
556C42—32no—
557C42—32no—
558C42—64no—
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—42no—
580C3×C4—41no—
581C3×C4—81no—
582C3×C4#52511yescharacteristic
583C3×C4#53711yescharacteristic
584C3×C4#54611yescharacteristic
585C3×C4#54711yescharacteristic
586Dic3—22no—
587Dic3—22no—
588Dic3—22no—
589Dic3—42no—
590Dic3—41no—
591Dic3—42no—
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—22no—
605Q8—22no—
606Q8—41no—
607Q8—41no—
608Q8—41no—
609Q8—41no—
610Q8—68no—
611Q8—62no—
612Q8—62no—
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—21no—
624C2×C4—22no—
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—62no—
645C2×C4—61no—
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×D3×D411yescharacteristic
660C2×C4C2×D3×D411yescharacteristic
661C2×C4C2×#23711yescharacteristic
662C2×C4C2×#23711yescharacteristic
663C2×C4#100411yescharacteristic
664C2×C4#101011yescharacteristic
665C8—22no—
666C8—41no—
667C8—121no—
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—
684C22D3×#12011yescharacteristic
685C22#559811yescharacteristic
686C22#571811yescharacteristic
687C22C2×#91511yescharacteristic
688C22#576211yescharacteristic
689C22D3×#13711yescharacteristic
690C22#577811yescharacteristic
691C4—22no—
692C4—42no—
693C4—42no—
694C4—41no—
695C4—62no—
696C4—62no—
697C4—62no—
698C4—81no—
699C4—122no—
700C4—121no—
701C4—122no—
702C4—241no—
703C4—241no—
704C4#558611yescharacteristic
705C4#559111yescharacteristic
706C4#559711yescharacteristic
707C4#561111yescharacteristic
708C3#2419911yescharacteristic, complemented, Sylow
709C2—82no—
710C2#7666811yescharacteristic
711C2#8003911yescharacteristic
712C2#8004111yescharacteristic
713C2#8041711yescharacteristic
714C2#8041811yescharacteristic
715C2#8175811yescharacteristic
716C2#8186511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2013C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2013D3
hypocenterC3#24199
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2013

Complements

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

NH ≅ G/Ntype
1. C2×#5721C2semidirect
8. #71781C2semidirect
9. #70246C2semidirect
10. #70236C2semidirect
11. #70228C2semidirect
12. #70222C2semidirect
14. #68683C2semidirect
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×#57212
C2×#51622
C2×#51642
#763642
C3×#20132
#720572
#720482
#717812
#702462
#702362
#702282
#702222
#696602
#686832
#686202
#241993