Finite Group Databasebeta

← #10858305

Subgroups — #10858305

Order 768.

828 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic3⋊C4C211yesmaximal, characteristic, complemented
2C2×C4×#215C211yesmaximal, characteristic
3C2×#4799C211yesmaximal, characteristic, complemented
4#76015C211yesmaximal, characteristic
5C3×#1863C211yesmaximal, characteristic, complemented
6#70641C211yesmaximal, characteristic
7#70618C211yesmaximal, characteristic, complemented
8#70240C211yesmaximal, characteristic
9#70240C211yesmaximal, characteristic
10#70235C211yesmaximal, characteristic, complemented
11#69619C211yesmaximal, characteristic, complemented
12#68901C211yesmaximal, characteristic
13#68835C211yesmaximal, characteristic, complemented
14#68671C211yesmaximal, characteristic
15#68655C211yesmaximal, characteristic, complemented
16#26379—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×#890C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32C4×#215—24no—
33#4994—21no—
34#4993—21no—
35#4988C2211yescharacteristic
36#4988C2211yescharacteristic
37#4975—21no—
38#4975C2211yescharacteristic
39#4972—21no—
40#4972C2211yescharacteristic
41#4972C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#807C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C4×Dic3⋊C4—21no—
47C4×Dic3⋊C4C2211yescharacteristic
48#4928—21no—
49#4928C2211yescharacteristic
50C2×C4×C3⋊C8C2211yescharacteristic
51#4799—24no—
52#4668—21no—
53#4667—21no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#479—31no—
63C2×C4×SD16—31no—
64#2454—31no—
65Q8×C4⋊C4—31no—
66#2093—31no—
67#2084—31no—
68#2067—31no—
69#2059—31no—
70#1942—31no—
71#1916—31no—
72#1880—31no—
73#1880—31no—
74#1864—31no—
75#1863—31no—
76#1863D311yescharacteristic, complemented
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4×Q8—22no—
83C3×C4×Q8—41no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91Q8×Dic3—22no—
92Q8×Dic3—22no—
93Q8×Dic3—22no—
94Q8×Dic3—41no—
95C2×#215—22no—
96C2×#215—22no—
97C2×#215Q814yes—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4D411yescharacteristic
116C2×Dic3⋊C4D411yescharacteristic
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#890—22no—
124#890—41no—
125C4×D12—22no—
126C4×D12—41no—
127C3×#107—41no—
128C3×#103—22no—
129C3×#103—22no—
130C3×#95—21no—
131#835D414yes—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137#821—22no—
138#821—41no—
139#817—22no—
140Dic3⋊C8—22no—
141#809—22no—
142#808—22no—
143#807—22no—
144#807—22no—
145C4×C3⋊C8—22no—
146C22×SD16—31no—
147C2×#125—31no—
148C2×C4×Q8—31no—
149C2×C4×Q8—31no—
150C2×C4×Q8C2×D311yescharacteristic
151C2×C4×D4—31no—
152#479—64no—
153C4×SD16—64no—
154C2×#107—31no—
155C2×#107—31no—
156C2×#107C2×D311yescharacteristic
157C2×#106—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103C2×D311yescharacteristic
162C2×#103C2×D311yescharacteristic
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×C4×C8—31no—
167#399—61no—
168#396—61no—
169#390—31no—
170#390—31no—
171#389—31no—
172#389—31no—
173#389—31no—
174#389—61no—
175#389—61no—
176#389C2×D311yescharacteristic
177#385—61no—
178#384—31no—
179#384—61no—
180#380—61no—
181C4×C4⋊C4—31no—
182C4×C4⋊C4—61no—
183#340—31no—
184#340—31no—
185#340—31no—
186#340C2×D311yescharacteristic
187#336—31no—
188#336—31no—
189#336—31no—
190#336C2×D311yescharacteristic
191C23×D3—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8C2×D412yes—
196C2×C3×Q8C2×Q812yes—
197C2×C3×Q8#4112yes—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12C2×Q812yes—
218C2×C4×D3—42no—
219C2×C4×D3—42no—
220C2×C4×D3—81no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—42no—
234C3×C4⋊C4—81no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—42no—
238#215—28no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—42no—
241(C2×D3)⋊C4—81no—
242(C2×D3)⋊C4—81no—
243(C2×D3)⋊C4—81no—
244C4⋊Dic3—24no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—42no—
250C4⋊Dic3—42no—
251C4⋊Dic3C2×D412yes—
252C4⋊Dic3#4112yes—
253Dic3⋊C4—22no—
254Dic3⋊C4—22no—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257Dic3⋊C4—81no—
258Dic3⋊C4—81no—
259Dic3⋊C4—81no—
260Dic3⋊C4—81no—
261C4×Dic3—24no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—42no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8C2×D412yes—
272C2×C3⋊C8C2×Q812yes—
273C22×Q8C22×D311yescharacteristic
274C22×D4—31no—
275C23×C4—61no—
276C2×SD16—34no—
277C2×SD16—62no—
278C2×SD16—62no—
279C22×C8—31no—
280C22×C8—31no—
281C22×C8C22×D311yescharacteristic
282C22×C8C22×D311yescharacteristic
283#125—62no—
284#125—121no—
285C4×Q8—22no—
286C4×Q8—41no—
287C4×Q8—62no—
288C4×Q8—62no—
289C4×Q8—62no—
290C4×Q8—121no—
291C4×D4—62no—
292C4×D4—121no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C42—21no—
318C2×C42—31no—
319C2×C42—31no—
320C2×C42—61no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323#107—41no—
324#107—62no—
325#107—62no—
326#106—62no—
327#106—62no—
328C4⋊C8—62no—
329#103—22no—
330#103—22no—
331#103—34no—
332#102—62no—
333#102—62no—
334#102—121no—
335C4×C8—62no—
336#95—21no—
337#95—31no—
338#95—31no—
339#95—61no—
340#95—61no—
341#95—61no—
342C23×C3C22×D411yescharacteristic
343C22×D3—22no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C3×Q8—22no—
351C3×Q8—22no—
352C3×Q8—22no—
353C3×Q8C22⋊Q814yes—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4#13712yes—
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C22×Q811yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4H(5,𝔽2)11yescharacteristic
377C2×C3×C42−1+411yescharacteristic
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—22no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—22no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3#13512yes—
399D12—22no—
400D12—22no—
401D12—41no—
402D12—41no—
403C4×D3—84no—
404C3×C8—41no—
405C3×C8—41no—
406C3⋊C8—22no—
407C3⋊C8—22no—
408C24—61no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8#23512yes—
413C2×Q8D3×D412yes—
414C2×Q8D3×Q812yes—
415C2×D4—32no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—122no—
433C22×C4—122no—
434C22×C4—241no—
435C22×C4D3×D411yescharacteristic
436C22×C4D3×D411yescharacteristic
437C22×C4C23×D311yescharacteristic
438SD16—68no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—32no—
446C2×C8—32no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—32no—
458C4⋊C4—32no—
459C4⋊C4—42no—
460C4⋊C4—62no—
461C4⋊C4—64no—
462C4⋊C4—62no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—61no—
467C4⋊C4—81no—
468C4⋊C4—122no—
469C4⋊C4—122no—
470C4⋊C4—122no—
471C4⋊C4—122no—
472C4⋊C4—241no—
473C4⋊C4—241no—
474C4⋊C4—241no—
475C4⋊C4—241no—
476C22⋊C4—122no—
477C22⋊C4—122no—
478C22⋊C4—241no—
479C22⋊C4—241no—
480C22⋊C4—241no—
481C42—21no—
482C42—21no—
483C42—42no—
484C42—64no—
485C42—61no—
486C42—61no—
487C42—61no—
488C42—61no—
489C42—122no—
490C22×C3#52511yescharacteristic
491C22×C3D4211yescharacteristic
492C22×C3D4×Q811yescharacteristic
493C22×C3C2×#13511yescharacteristic
494C22×C3C2×#13711yescharacteristic
495C22×C3#57611yescharacteristic
496C22×C3#57711yescharacteristic
497C2×D3—42no—
498C2×D3—42no—
499C2×D3—42no—
500C2×D3—81no—
501C2×D3—81no—
502C2×D3—81no—
503C2×D3—81no—
504C3×C4—24no—
505C3×C4—22no—
506C3×C4—41no—
507C3×C4—81no—
508C3×C4—81no—
509C3×C4—81no—
510C3×C4C2×C22⋊Q811yescharacteristic
511C3×C4#53911yescharacteristic
512C3×C4#54711yescharacteristic
513C3×C4#55411yescharacteristic
514Dic3—22no—
515Dic3—41no—
516Dic3—42no—
517Dic3—42no—
518Dic3—41no—
519Dic3—41no—
520Dic3—81no—
521C23—62no—
522C23—121no—
523C23—121no—
524C23—121no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23C2×D3×D411yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—22no—
532Q8#89614yes—
533D4—62no—
534D4—62no—
535D4—121no—
536D4—121no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—32no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—62no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—62no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—244no—
576C2×C4#91812yes—
577C2×C4C2×#23511yescharacteristic
578C2×C4C2×D3×D411yescharacteristic
579C2×C4#100411yescharacteristic
580C2×C4C2×D3×Q811yescharacteristic
581C2×C4D3×#4111yescharacteristic
582C2×C4#101011yescharacteristic
583C8—41no—
584C8—41no—
585C8—62no—
586C8—62no—
587C2×C3#244911yescharacteristic
588C2×C3#310811yescharacteristic
589C2×C3#311911yescharacteristic
590C2×C3#326611yescharacteristic
591C2×C3#328711yescharacteristic
592C2×C3#338111yescharacteristic
593C2×C3#338511yescharacteristic
594D3—82no—
595C22—122no—
596C22—122no—
597C22—122no—
598C22—241no—
599C22—241no—
600C22—241no—
601C22—241no—
602C22#554711yescharacteristic
603C22#559011yescharacteristic
604C22#563211yescharacteristic
605C22C2×#91811yescharacteristic
606C22D3×#13511yescharacteristic
607C22#577411yescharacteristic
608C22#577811yescharacteristic
609C4—22no—
610C4—24no—
611C4—41no—
612C4—62no—
613C4—81no—
614C4—81no—
615C4—81no—
616C4—121no—
617C4—121no—
618C4—122no—
619C4—121no—
620C4—122no—
621C4—241no—
622C4C2×#89611yescharacteristic
623C4#556411yescharacteristic
624C4#563611yescharacteristic
625C4#565511yescharacteristic
626C3#2637911yescharacteristic, complemented, Sylow
627C2—242no—
628C2#7442111yescharacteristic
629C2#7972311yescharacteristic
630C2#7981211yescharacteristic
631C2#8013111yescharacteristic
632C2#8020411yescharacteristic
633C2#8113711yescharacteristic
634C2#8122111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1863C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1863D3
hypocenterC3#26379
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
1. Q8×Dic3⋊C4C2semidirect
3. C2×#4799C2semidirect
5. C3×#1863C2semidirect
7. #70618C2semidirect
10. #70235C2semidirect
11. #69619C2semidirect
13. #68835C2semidirect
15. #68655C2semidirect
76. #1863D3semidirect
626. C3#26379semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic3⋊C42
C2×C4×#2152
C2×#47992
#760152
C3×#18632
#706412
#706182
#702402
#702352
#696192
#689012
#688352
#686712
#686552
#263793