Finite Group Databasebeta

← #10824505

Subgroups — #10824505

Order 768.

908 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4710C211yesmaximal, characteristic, complemented
2C2×C4×#204C211yesmaximal, characteristic, complemented
3#73389C211yesmaximal, characteristic
4#73386C211yesmaximal, characteristic, complemented
5C3×#1768C211yesmaximal, characteristic, complemented
6#70484C211yesmaximal, characteristic, complemented
7#70483C211yesmaximal, characteristic, complemented
8#70482C211yesmaximal, characteristic, complemented
9#70482C211yesmaximal, characteristic, complemented
10#69812C211yesmaximal, characteristic, complemented
11C4×#821C211yesmaximal, characteristic, complemented
12#68676C211yesmaximal, characteristic
13#68641C211yesmaximal, characteristic, complemented
14#68638C211yesmaximal, characteristic, complemented
15#68629C211yesmaximal, characteristic
16#23690—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C22×#204C2211yescharacteristic
19C2×#874C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×#376C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#818C2211yescharacteristic
35C2×#817C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38#4939C2211yescharacteristic
39#4939C2211yescharacteristic
40#4936C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42#4929C2211yescharacteristic
43#4929C2211yescharacteristic
44#4928C2211yescharacteristic
45#4928C2211yescharacteristic
46#4710—24no—
47C4×#204—24no—
48#4665—21no—
49#4665—21no—
50#4665—21no—
51#4665—21no—
52#4652—21no—
53#4652—21no—
54#4651—21no—
55#4650—21no—
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#440—31no—
63C2×C4×SD16—31no—
64#2328—31no—
65#2320—31no—
66#2085—31no—
67#2074—31no—
68#2067—31no—
69#2065—31no—
70#2055—31no—
71#2055—31no—
72#2041—31no—
73#2014—31no—
74#1768D311yescharacteristic, complemented
75#1762—31no—
76C4×#102—31no—
77C22×D12—21no—
78C22×D12—21no—
79C22×D12C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×C4×D3—21no—
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×#204—22no—
108C2×#204—22no—
109C2×#204—22no—
110C2×#204—22no—
111C2×#204—22no—
112C2×#204—22no—
113#874—22no—
114#874—22no—
115C4×D12—22no—
116C4×D12—22no—
117C4×D12—22no—
118C4×D12—22no—
119C4×Dic6—22no—
120C4×Dic6D414yescomplemented
121C3×C4⋊C8—22no—
122C3×C4⋊C8—22no—
123C3×#97—22no—
124C3×C4×C8—22no—
125C3×#95D411yescharacteristic
126C3×#95D411yescharacteristic
127C3×#95D411yescharacteristic
128C3×#95D411yescharacteristic
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138#821—22no—
139#821—22no—
140#821—22no—
141#821—22no—
142#821—22no—
143#821—22no—
144#818—22no—
145#817—22no—
146#816—22no—
147#816—22no—
148C22×SD16—31no—
149C22×SD16—31no—
150C2×#127—31no—
151C2×C4×Q8—31no—
152C2×C4×D4—31no—
153C2×C4×D4—31no—
154#440—64no—
155C4×SD16—64no—
156C2×#107—31no—
157C2×#106—31no—
158C2×C4⋊C8C2×D311yescharacteristic
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#102—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97C2×D311yescharacteristic
169C2×C4×C8C2×D311yescharacteristic
170#391—31no—
171#391—31no—
172#390—31no—
173#390—31no—
174#389—31no—
175#389—31no—
176#387—61no—
177#387—61no—
178#386—61no—
179#386—61no—
180#386—61no—
181#386—61no—
182#383—61no—
183#382—31no—
184#382—61no—
185C4×C4⋊C4—31no—
186#376C2×D311yescharacteristic
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336C2×D311yescharacteristic
192#336C2×D311yescharacteristic
193C23×D3—21no—
194C23×D3—21no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2411yescharacteristic
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—24no—
219C2×D12—24no—
220C2×D12—81no—
221C2×D12—81no—
222C2×D12—81no—
223C2×D12—81no—
224C2×D12#4112yes—
225C2×D12#4112yes—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×C4×D3—81no—
229C2×C4×D3—81no—
230C2×C4×D3—81no—
231C2×C4×D3—81no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6C2×D412yes—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8#4112yes—
249C2×C3×C8#4112yes—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42C2×D412yes—
255C3×C42#4112yes—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—81no—
259(C2×D3)⋊C4—81no—
260(C2×D3)⋊C4—81no—
261(C2×D3)⋊C4—81no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3C2×D412yes—
272Dic3⋊C4—24no—
273Dic3⋊C4—81no—
274Dic3⋊C4—81no—
275C4×Dic3—24no—
276C4×Dic3—81no—
277#204—28no—
278#204—28no—
279C22×Q8—31no—
280C22×D4—31no—
281C22×D4—31no—
282C22×D4—61no—
283C22×D4—61no—
284C23×C4—61no—
285C23×C4—61no—
286C2×SD16—62no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—62no—
290C2×SD16—62no—
291C2×SD16—62no—
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294C22×C8C22×D311yescharacteristic
295C22×C8C22×D311yescharacteristic
296#127—62no—
297#127—62no—
298C4×Q8—34no—
299C4×Q8—62no—
300C4×D4—62no—
301C4×D4—62no—
302C4×D4—62no—
303C4×D4—62no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C42—61no—
319C2×C42—61no—
320C2×C42C22×D311yescharacteristic
321C2×C42C22×D311yescharacteristic
322C2×C42C22×D311yescharacteristic
323#107—62no—
324#106—62no—
325C4⋊C8—22no—
326C4⋊C8—22no—
327#103—62no—
328#103—62no—
329#102—62no—
330#102—62no—
331#102—62no—
332#102—62no—
333#102—62no—
334#102—62no—
335#97—22no—
336C4×C8—22no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95D1211yescharacteristic
347#95D1211yescharacteristic
348#95D1211yescharacteristic
349#95D1211yescharacteristic
350C23×C3C22×D411yescharacteristic
351C22×D3—22no—
352C22×D3—22no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4#12112yes—
382C2×C3×C4#12112yes—
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407D12—22no—
408D12—22no—
409D12—22no—
410D12—22no—
411D12—41no—
412D12—41no—
413D12—41no—
414D12—41no—
415D12—84no—
416D12—84no—
417C4×D3—84no—
418C4×D3—84no—
419Dic6—22no—
420Dic6—22no—
421Dic6—41no—
422Dic6—41no—
423C3×C8—22no—
424C3×C8—22no—
425C3×C8—41no—
426C3×C8—41no—
427C24—61no—
428C24—61no—
429C2×Q8—32no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×D4—32no—
436C2×D4—32no—
437C2×D4—64no—
438C2×D4—64no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—241no—
450C2×D4—241no—
451C2×D4—241no—
452C2×D4—241no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—122no—
460C22×C4—122no—
461C22×C4—241no—
462C22×C4—241no—
463C22×C4—241no—
464C22×C4—241no—
465C22×C4C2×D1211yescharacteristic
466C22×C4C2×D1211yescharacteristic
467C22×C4C2×D1211yescharacteristic
468C22×C4C2×D1211yescharacteristic
469C22×C4C2×D1211yescharacteristic
470C22×C4C2×D1211yescharacteristic
471C22×C4C23×D311yescharacteristic
472SD16—68no—
473SD16—68no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8#23512yes—
485C2×C8#23512yes—
486C4⋊C4—32no—
487C4⋊C4—64no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—241no—
498C4⋊C4—241no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—241no—
502C22⋊C4—241no—
503C22⋊C4—241no—
504C22⋊C4—241no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—64no—
510C42—241no—
511C42D3×D412yes—
512C42#23912yes—
513C22×C3C2×#12111yescharacteristic
514C22×C3#52511yescharacteristic
515C22×C3#53211yescharacteristic
516C22×C3#57611yescharacteristic
517C22×C3#57711yescharacteristic
518C22×C3#57711yescharacteristic
519C22×C3#57711yescharacteristic
520C2×D3—42no—
521C2×D3—42no—
522C2×D3—42no—
523C2×D3—42no—
524C2×D3—42no—
525C2×D3—42no—
526C2×D3—81no—
527C2×D3—81no—
528C2×D3—81no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—81no—
532C2×D3—81no—
533C2×D3—81no—
534C3×C4—22no—
535C3×C4—22no—
536C3×C4—41no—
537C3×C4—41no—
538C3×C4—41no—
539C3×C4—41no—
540C3×C4#52511yescharacteristic
541C3×C4#52511yescharacteristic
542C3×C4#52911yescharacteristic
543C3×C4#52911yescharacteristic
544Dic3—42no—
545Dic3—42no—
546Dic3—81no—
547Dic3—81no—
548Dic3—81no—
549Dic3—81no—
550C23—62no—
551C23—62no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23—121no—
563C23—121no—
564C23C22×D1211yescharacteristic
565Q8—62no—
566Q8—62no—
567Q8—121no—
568Q8—121no—
569D4—62no—
570D4—62no—
571D4—62no—
572D4—62no—
573D4—121no—
574D4—121no—
575D4—121no—
576D4—121no—
577D4—244no—
578D4—244no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—62no—
596C2×C4—62no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—244no—
614C2×C4—244no—
615C2×C4#89512yes—
616C2×C4#89512yes—
617C2×C4C2×#23511yescharacteristic
618C2×C4C2×#23511yescharacteristic
619C2×C4C2×D3×D411yescharacteristic
620C2×C4C2×#23911yescharacteristic
621C2×C4D3×#4111yescharacteristic
622C2×C4D3×#4111yescharacteristic
623C8—22no—
624C8—22no—
625C8—41no—
626C8—41no—
627C2×C3#239211yescharacteristic
628C2×C3#304311yescharacteristic
629C2×C3#304511yescharacteristic
630C2×C3#311811yescharacteristic
631C2×C3#312311yescharacteristic
632C2×C3#317411yescharacteristic
633C2×C3#317911yescharacteristic
634D3—82no—
635D3—82no—
636C22—122no—
637C22—122no—
638C22—122no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22—241no—
643C22—241no—
644C22—241no—
645C22—241no—
646C22—241no—
647C22—241no—
648C22—241no—
649C22—241no—
650C22#547711yescharacteristic
651C22C2×#89511yescharacteristic
652C22#552411yescharacteristic
653C22#575111yescharacteristic
654C22#575111yescharacteristic
655C22#575111yescharacteristic
656C22#575211yescharacteristic
657C4—22no—
658C4—22no—
659C4—41no—
660C4—41no—
661C4—41no—
662C4—41no—
663C4—122no—
664C4—122no—
665C4—241no—
666C4—241no—
667C4—241no—
668C4—241no—
669C4#554711yescharacteristic
670C4#554711yescharacteristic
671C4#558011yescharacteristic
672C4#558011yescharacteristic
673C3#2369011yescharacteristic, complemented, Sylow
674C2—242no—
675C2—242no—
676C2#7377611yescharacteristic
677C2#7856711yescharacteristic
678C2#7857811yescharacteristic
679C2#7908811yescharacteristic
680C2#7909911yescharacteristic
681C2#7938311yescharacteristic
682C2#7938711yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1768C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#1768D3
hypocenterC3#23690
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #1768

Complements

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

NH ≅ G/Ntype
1. C2×#4710C2semidirect
2. C2×C4×#204C2semidirect
4. #73386C2semidirect
5. C3×#1768C2semidirect
6. #70484C2semidirect
7. #70483C2semidirect
8. #70482C2semidirect
9. #70482C2semidirect
10. #69812C2semidirect
11. C4×#821C2semidirect
13. #68641C2semidirect
14. #68638C2semidirect
74. #1768D3semidirect
120. C4×Dic6D4semidirect
673. C3#23690semidirect

Maximal subgroups

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

subgroupindex
C2×#47102
C2×C4×#2042
#733892
#733862
C3×#17682
#704842
#704832
#704822
#698122
C4×#8212
#686762
#686412
#686382
#686292
#236903