Finite Group Databasebeta

← #10838949

Subgroups — #10838949

Order 768.

988 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4806C211yesmaximal, characteristic
2C2×C4×#204C211yesmaximal, characteristic, complemented
3#76614C211yesmaximal, characteristic, complemented
4#76213C211yesmaximal, characteristic
5C3×#1933C211yesmaximal, characteristic, complemented
6#70280C211yesmaximal, characteristic, complemented
7#70266C211yesmaximal, characteristic, complemented
8#70252C211yesmaximal, characteristic, complemented
9#69785C211yesmaximal, characteristic, complemented
10#69687C211yesmaximal, characteristic, complemented
11#69631C211yesmaximal, characteristic, complemented
12#68909C211yesmaximal, characteristic, complemented
13#68905C211yesmaximal, characteristic, complemented
14#68832C211yesmaximal, characteristic, complemented
15#68822C211yesmaximal, characteristic
16#24894—31nomaximal, Sylow
17C22×D3×Q8C2211yescharacteristic
18C22×#215C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C2×#895C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#389C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×(C2×D3)⋊C8C2211yescharacteristic
33C2×(C2×D3)⋊C8C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37#4992—21no—
38#4991—21no—
39#4988C2211yescharacteristic
40#4987C2211yescharacteristic
41#4987C2211yescharacteristic
42#4977C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C4×(C2×D3)⋊C4C2211yescharacteristic
48#4806—24no—
49C4×#204—24no—
50#4667—21no—
51#4666—21no—
52#4665—21no—
53#4664—21no—
54#4646—21no—
55#4646—21no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#448—31no—
63C2×C4×SD16—31no—
64#2420—31no—
65Q8×C22⋊C4—31no—
66#2097—31no—
67#2077—31no—
68#2055—31no—
69#2033—31no—
70#1933D311yescharacteristic, complemented
71#1923—31no—
72#1880—31no—
73#1877—31no—
74#1863—31no—
75#1805—31no—
76#1787—31no—
77C22×C3×Q8C2311yescharacteristic
78C2×D3×Q8—22no—
79C2×D3×Q8—22no—
80C2×D3×Q8—22no—
81C2×D3×Q8—22no—
82C2×D3×Q8—41no—
83C2×D3×Q8—41no—
84C22×D12—21no—
85C22×D12C2311yescharacteristic
86C22×C4×D3—21no—
87C22×C4×D3—21no—
88C22×C4×D3C2311yescharacteristic
89C22×Dic6—21no—
90C22×Dic6C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×#215—22no—
99C2×#215—22no—
100C2×#215—41no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4D411yescharacteristic
108C2×(C2×D3)⋊C4D411yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C2×#204—22no—
119C2×#204—22no—
120C2×#204D414yes—
121#895—22no—
122#895—41no—
123#887—22no—
124#887—41no—
125C4×D12—22no—
126C4×D12—41no—
127C4×Dic6—22no—
128C4×Dic6—41no—
129C3×#103—22no—
130C3×#103—22no—
131C3×C4×C8—22no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135#835—41no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95D411yescharacteristic
143C3⋊#95D411yescharacteristic
144#821—22no—
145(C2×D3)⋊C8—22no—
146(C2×D3)⋊C8—41no—
147#817—22no—
148#816—22no—
149Dic3⋊C8—41no—
150#810—22no—
151#810—22no—
152#809—41no—
153#807—41no—
154C23×Q8—31no—
155C22×SD16—31no—
156C22×SD16—31no—
157C2×#121—31no—
158C2×C4×Q8—31no—
159C2×C4×Q8—31no—
160C2×C4×D4—31no—
161#448—64no—
162C4×SD16—64no—
163C2×#107—31no—
164C2×#106—31no—
165C2×C4⋊C8—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103C2×D311yescharacteristic
171C2×#103C2×D311yescharacteristic
172C2×#102—31no—
173C2×#102—31no—
174C2×C22⋊C8—31no—
175C2×C22⋊C8—31no—
176C2×C4×C8C2×D311yescharacteristic
177#397—61no—
178#394—61no—
179#391—31no—
180#391—61no—
181#391—61no—
182#391C2×D311yescharacteristic
183#390—31no—
184#389C2×D311yescharacteristic
185#386—61no—
186#382—31no—
187#381—61no—
188#380—31no—
189#380—31no—
190#380—61no—
191C4×C22⋊C4—31no—
192C4×C22⋊C4—61no—
193#340—31no—
194#340—31no—
195#340C2×D311yescharacteristic
196#340C2×D311yescharacteristic
197#336—31no—
198#336—31no—
199C23×D3—21no—
200C23×D3C2×D411yescharacteristic
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8#4112yes—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220D3×Q8—44no—
221D3×Q8—44no—
222D3×Q8—44no—
223D3×Q8—44no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—42no—
230C2×D12—42no—
231C2×D12—81no—
232C2×D12C2×D412yes—
233C2×C4×D3—24no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—21no—
237C2×C4×D3—21no—
238C2×C4×D3—21no—
239C2×C4×D3—21no—
240C2×C4×D3—42no—
241C2×C4×D3—42no—
242C2×C4×D3—42no—
243C2×C4×D3—42no—
244C2×C4×D3—81no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—24no—
249C2×Dic6—42no—
250C2×Dic6—42no—
251C2×Dic6C2×D412yes—
252C2×Dic6#4112yes—
253C2×Dic6#4112yes—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8C2×D412yes—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—21no—
265C3×C4⋊C4—21no—
266C3×C4⋊C4—81no—
267C3×C42—21no—
268C3×C42—21no—
269#215—44no—
270(C2×D3)⋊C4—22no—
271(C2×D3)⋊C4—22no—
272(C2×D3)⋊C4—42no—
273(C2×D3)⋊C4—42no—
274(C2×D3)⋊C4—42no—
275(C2×D3)⋊C4—81no—
276(C2×D3)⋊C4—81no—
277(C2×D3)⋊C4—81no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282Dic3⋊C4—42no—
283Dic3⋊C4—42no—
284C4×Dic3—21no—
285C4×Dic3—21no—
286C4×Dic3—42no—
287C4×Dic3—42no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294#204—28no—
295C22×Q8—31no—
296C22×Q8—61no—
297C22×Q8—62no—
298C22×Q8—62no—
299C22×Q8—62no—
300C22×Q8—62no—
301C22×Q8—121no—
302C22×Q8—121no—
303C22×Q8C22×D311yescharacteristic
304C22×D4—31no—
305C22×D4—61no—
306C23×C4—31no—
307C23×C4—61no—
308C23×C4—61no—
309C2×SD16—34no—
310C2×SD16—62no—
311C2×SD16—62no—
312C2×SD16—62no—
313C2×SD16—62no—
314C2×SD16—121no—
315C22×C8—31no—
316C22×C8—31no—
317C22×C8C22×D311yescharacteristic
318C22×C8C22×D311yescharacteristic
319#121—62no—
320#121—121no—
321C4×Q8—62no—
322C4×Q8—62no—
323C4×Q8—121no—
324C4×Q8—121no—
325C4×D4—62no—
326C4×D4—121no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4C22×D311yescharacteristic
333C2×C4⋊C4C22×D311yescharacteristic
334C2×C4⋊C4C22×D311yescharacteristic
335C2×C22⋊C4—31no—
336C2×C22⋊C4—31no—
337C2×C22⋊C4—61no—
338C2×C22⋊C4—61no—
339C2×C22⋊C4—61no—
340C2×C22⋊C4—61no—
341C2×C22⋊C4—61no—
342C2×C22⋊C4—61no—
343C2×C42—31no—
344C2×C42—61no—
345C2×C42—61no—
346C2×C42C22×D311yescharacteristic
347#107—121no—
348#106—62no—
349C4⋊C8—121no—
350#103—22no—
351#103—22no—
352#103—62no—
353#103—62no—
354#103—62no—
355#103—121no—
356#102—62no—
357#102—121no—
358C22⋊C8—62no—
359C22⋊C8—121no—
360C4×C8—22no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—31no—
365#95—31no—
366#95—61no—
367#95—61no—
368#95—61no—
369#95—61no—
370#95—61no—
371#95—61no—
372C23×C3C22×D411yescharacteristic
373C22×D3—21no—
374C22×D3—21no—
375C22×D3—21no—
376C22×D3—21no—
377C22×D3—21no—
378C22×D3—21no—
379C22×D3—22no—
380C22×D3—41no—
381C22×D3—41no—
382C22×D3—41no—
383C22×D3—41no—
384C22×D3—41no—
385C22×D3—41no—
386C22×D3#13512yes—
387C3×Q8—22no—
388C3×Q8—22no—
389C3×Q8—41no—
390C3×Q8—41no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—22no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4#13712yes—
409C2×C3×C4C22×D411yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×C3×C4C2×#4111yescharacteristic
412C2×C3×C4C2×#4111yescharacteristic
413C2×C3×C4H(5,𝔽2)11yescharacteristic
414C2×C3×C4H(5,𝔽2)11yescharacteristic
415C2×Dic3—22no—
416C2×Dic3—22no—
417C2×Dic3—22no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433D12—22no—
434D12—22no—
435D12—41no—
436D12—41no—
437D12—84no—
438C4×D3—48no—
439C4×D3—41no—
440C4×D3—41no—
441C4×D3—41no—
442C4×D3—41no—
443C4×D3—84no—
444Dic6—22no—
445Dic6—22no—
446Dic6—22no—
447Dic6—48no—
448Dic6#12314yes—
449C3×C8—22no—
450C3×C8—41no—
451C3⋊C8—41no—
452C3⋊C8—41no—
453C24—31no—
454C24—61no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—32no—
461C2×Q8—32no—
462C2×Q8—32no—
463C2×Q8—64no—
464C2×Q8—61no—
465C2×Q8—61no—
466C2×Q8—61no—
467C2×Q8—122no—
468C2×Q8—122no—
469C2×Q8—124no—
470C2×Q8—124no—
471C2×Q8—124no—
472C2×Q8—124no—
473C2×Q8#23512yes—
474C2×D4—32no—
475C2×D4—61no—
476C2×D4—61no—
477C2×D4—61no—
478C2×D4—61no—
479C2×D4—61no—
480C2×D4—122no—
481C2×D4—122no—
482C2×D4—241no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—31no—
488C22×C4—31no—
489C22×C4—31no—
490C22×C4—64no—
491C22×C4—61no—
492C22×C4—61no—
493C22×C4—61no—
494C22×C4—61no—
495C22×C4—61no—
496C22×C4—61no—
497C22×C4—61no—
498C22×C4—61no—
499C22×C4—61no—
500C22×C4—122no—
501C22×C4—122no—
502C22×C4—122no—
503C22×C4—122no—
504C22×C4—241no—
505C22×C4D3×D411yescharacteristic
506C22×C4D3×D411yescharacteristic
507C22×C4C23×D311yescharacteristic
508SD16—68no—
509SD16—124no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—61no—
516C2×C8—61no—
517C2×C8—61no—
518C2×C8—61no—
519C2×C8—61no—
520C2×C8—61no—
521C2×C8D3×D412yes—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—61no—
529C4⋊C4—61no—
530C4⋊C4—61no—
531C4⋊C4—61no—
532C4⋊C4—81no—
533C4⋊C4—122no—
534C4⋊C4—122no—
535C22⋊C4—62no—
536C22⋊C4—62no—
537C22⋊C4—122no—
538C22⋊C4—122no—
539C22⋊C4—122no—
540C22⋊C4—241no—
541C22⋊C4—241no—
542C22⋊C4—241no—
543C42—21no—
544C42—21no—
545C42—61no—
546C42—61no—
547C42—122no—
548C42—122no—
549C22×C3#52511yescharacteristic
550C22×C3D4211yescharacteristic
551C22×C3#54611yescharacteristic
552C22×C3C2×#13511yescharacteristic
553C22×C3C2×#13711yescharacteristic
554C22×C3#57711yescharacteristic
555C22×C3#57711yescharacteristic
556C2×D3—22no—
557C2×D3—22no—
558C2×D3—22no—
559C2×D3—41no—
560C2×D3—41no—
561C2×D3—41no—
562C2×D3—41no—
563C2×D3—42no—
564C2×D3—42no—
565C2×D3—42no—
566C2×D3—81no—
567C2×D3—81no—
568C2×D3—81no—
569C2×D3—81no—
570C3×C4—22no—
571C3×C4—42no—
572C3×C4—41no—
573C3×C4—81no—
574C3×C4—81no—
575C3×C4—81no—
576C3×C4C2×#12311yescharacteristic
577C3×C4#53811yescharacteristic
578C3×C4#54611yescharacteristic
579C3×C4#55111yescharacteristic
580Dic3—24no—
581Dic3—41no—
582Dic3—41no—
583Dic3—41no—
584Dic3—81no—
585Dic3—81no—
586C23—32no—
587C23—61no—
588C23—61no—
589C23—61no—
590C23—61no—
591C23—61no—
592C23—61no—
593C23—62no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23—121no—
600C23C2×D3×D411yescharacteristic
601Q8—22no—
602Q8—22no—
603Q8—34no—
604Q8—41no—
605Q8—41no—
606Q8—62no—
607Q8—62no—
608Q8—62no—
609Q8—128no—
610D4—62no—
611D4—62no—
612D4—121no—
613D4—121no—
614D4—244no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—62no—
633C2×C4—62no—
634C2×C4—62no—
635C2×C4—61no—
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—128no—
645C2×C4—121no—
646C2×C4—121no—
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—244no—
656C2×C4#91812yes—
657C2×C4C2×#23511yescharacteristic
658C2×C4C2×D3×D411yescharacteristic
659C2×C4D3×#4111yescharacteristic
660C2×C4D3×#4111yescharacteristic
661C2×C4#100911yescharacteristic
662C2×C4#100911yescharacteristic
663C8—22no—
664C8—41no—
665C8—121no—
666C8—121no—
667C2×C3#241811yescharacteristic
668C2×C3#310811yescharacteristic
669C2×C3#312611yescharacteristic
670C2×C3#326611yescharacteristic
671C2×C3#328211yescharacteristic
672C2×C3#330111yescharacteristic
673C2×C3#332911yescharacteristic
674D3—42no—
675D3—82no—
676C22—62no—
677C22—62no—
678C22—62no—
679C22—121no—
680C22—121no—
681C22—121no—
682C22—121no—
683C22—122no—
684C22—122no—
685C22—122no—
686C22—241no—
687C22—241no—
688C22—241no—
689C22—241no—
690C22#554711yescharacteristic
691C22#560911yescharacteristic
692C22#567611yescharacteristic
693C22#576211yescharacteristic
694C22C2×#91811yescharacteristic
695C22D3×#13511yescharacteristic
696C22#577011yescharacteristic
697C4—22no—
698C4—41no—
699C4—42no—
700C4—64no—
701C4—81no—
702C4—81no—
703C4—81no—
704C4—121no—
705C4—121no—
706C4—121no—
707C4—241no—
708C4—241no—
709C4#552711yescharacteristic
710C4#555011yescharacteristic
711C4D3×#12311yescharacteristic
712C4#570411yescharacteristic
713C3#2489411yescharacteristic, complemented, Sylow
714C2—122no—
715C2—242no—
716C2#7584511yescharacteristic
717C2#7966111yescharacteristic
718C2#7972311yescharacteristic
719C2#8067811yescharacteristic
720C2#8077111yescharacteristic
721C2#8161711yescharacteristic
722C2#8165511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1933C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#1933D3
hypocenterC3#24894
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1933

Complements

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

NH ≅ G/Ntype
2. C2×C4×#204C2semidirect
3. #76614C2semidirect
5. C3×#1933C2semidirect
6. #70280C2semidirect
7. #70266C2semidirect
8. #70252C2semidirect
9. #69785C2semidirect
10. #69687C2semidirect
11. #69631C2semidirect
12. #68909C2semidirect
13. #68905C2semidirect
14. #68832C2semidirect
70. #1933D3semidirect
713. C3#24894semidirect

Maximal subgroups

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

subgroupindex
C2×#48062
C2×C4×#2042
#766142
#762132
C3×#19332
#702802
#702662
#702522
#697852
#696872
#696312
#689092
#689052
#688322
#688222
#248943