Finite Group Databasebeta

← #10855261

Subgroups — #10855261

Order 768.

944 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4819C211yesmaximal, characteristic
2C2×C4×D24C211yesmaximal, characteristic
3#76505C211yesmaximal, characteristic, complemented
4#76202C211yesmaximal, characteristic, complemented
5C3×#1933C211yesmaximal, characteristic, complemented
6#70276C211yesmaximal, characteristic, complemented
7#70249C211yesmaximal, characteristic, complemented
8#70245C211yesmaximal, characteristic, complemented
9#69784C211yesmaximal, characteristic, complemented
10#69687C211yesmaximal, characteristic, complemented
11#69625C211yesmaximal, characteristic, complemented
12#68902C211yesmaximal, characteristic, complemented
13#68897C211yesmaximal, characteristic
14#68894C211yesmaximal, characteristic, complemented
15#68839C211yesmaximal, characteristic, complemented
16#26164—31nomaximal, Sylow
17C2×#947C2211yescharacteristic
18C22×#215C2211yescharacteristic
19C22×D24C2211yescharacteristic
20C2×#893C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×D3×C4⋊C4—21no—
23C2×C4×D12C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C3×#391C2211yescharacteristic
29C3×#389C2211yescharacteristic
30#5229C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#818C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36#4989—21no—
37#4988C2211yescharacteristic
38#4984C2211yescharacteristic
39#4984C2211yescharacteristic
40#4975C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#807C2211yescharacteristic
45C4×Dic3⋊C4C2211yescharacteristic
46#4819—24no—
47C4×D24—24no—
48#4675—21no—
49#4674—21no—
50#4668—21no—
51#4668—21no—
52#4667—21no—
53#4664—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#463—31no—
63C2×C4×D8—31no—
64#2447—31no—
65D4×C4⋊C4—31no—
66#2095—31no—
67#2087—31no—
68#2071—31no—
69#2042—31no—
70#1933D311yescharacteristic, complemented
71#1916—31no—
72#1879—31no—
73#1877—31no—
74#1877—31no—
75#1862—31no—
76#1862—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×C4×D3—21no—
83C22×C4×D3—21no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91#947—22no—
92#947—41no—
93C2×#215—22no—
94C2×#215—22no—
95C2×#215—22no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
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×(C2×D3)⋊C4—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4D411yescharacteristic
111C2×Dic3⋊C4D411yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C2×D24—22no—
117C2×D24—22no—
118C2×D24D414yes—
119#893—22no—
120#893—22no—
121#893—22no—
122#893—41no—
123D3×C4⋊C4—44no—
124C4×D12—22no—
125C4×D12—22no—
126C4×D12—41no—
127C4×D12—41no—
128C3×#103—22no—
129C3×#103—22no—
130C3×C4×C8—22no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134#835—41no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95D411yescharacteristic
140C3⋊#95D411yescharacteristic
141#821—22no—
142#821—22no—
143#818—22no—
144Dic3⋊C8—22no—
145#809—22no—
146#809—22no—
147#809—22no—
148#807—41no—
149C22×SD16—31no—
150C22×D8—31no—
151C2×#124—31no—
152C2×C4×D4—31no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155C2×C4×D4—31no—
156C22×C4⋊C4—61no—
157#463—64no—
158C4×D8—64no—
159C2×#107—31no—
160C2×#107—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103C2×D311yescharacteristic
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×C4×C8C2×D311yescharacteristic
171#397—61no—
172#397—61no—
173#391—31no—
174#391C2×D311yescharacteristic
175#390—31no—
176#389—31no—
177#389—31no—
178#389C2×D311yescharacteristic
179#385—61no—
180#385—61no—
181#385—61no—
182#384—31no—
183#381—61no—
184#380—61no—
185C4×C4⋊C4—31no—
186#340—31no—
187#340—31no—
188#340C2×D311yescharacteristic
189#340C2×D311yescharacteristic
190#336—31no—
191#336—31no—
192#336—31no—
193#336—31no—
194C23×D3—21no—
195C23×D3—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12C2×D412yes—
224C2×D12C2×D412yes—
225C2×D12#4112yes—
226C2×D12#4112yes—
227C2×C4×D3—24no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×C4×D3—42no—
231C2×C4×D3—42no—
232C2×C4×D3—42no—
233C2×C4×D3—42no—
234C2×C4×D3—42no—
235C2×C4×D3—42no—
236C2×C4×D3—81no—
237C2×C4×D3—81no—
238C2×C4×D3—81no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8C2×D412yes—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—42no—
251C3×C4⋊C4#4112yes—
252C3×C42—21no—
253C3×C42—21no—
254#215—44no—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—81no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—42no—
268C4⋊Dic3—81no—
269Dic3⋊C4—22no—
270Dic3⋊C4—22no—
271Dic3⋊C4—42no—
272Dic3⋊C4—81no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283D24—28no—
284C22×Q8C22×D311yescharacteristic
285C22×D4—31no—
286C22×D4—31no—
287C23×C4—61no—
288C23×C4—61no—
289C23×C4—61no—
290C23×C4—61no—
291C2×SD16—62no—
292C2×SD16—62no—
293C2×SD16—62no—
294C2×D8—34no—
295C2×D8—62no—
296C2×D8—62no—
297C22×C8—31no—
298C22×C8—31no—
299C22×C8C22×D311yescharacteristic
300C22×C8C22×D311yescharacteristic
301#124—62no—
302#124—121no—
303C4×D4—62no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—62no—
307C4×D4—62no—
308C4×D4—121no—
309C4×D4—121no—
310C4×D4—121no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—124no—
321C2×C4⋊C4C22×D311yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C42—31no—
333C2×C42—31no—
334C2×C42C22×D311yescharacteristic
335#107—62no—
336#107—121no—
337C4⋊C8—62no—
338#103—22no—
339#103—22no—
340#103—121no—
341#102—62no—
342#102—62no—
343#102—62no—
344#102—62no—
345#102—62no—
346C4×C8—22no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—31no—
351#95—31no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3C22×D411yescharacteristic
357C22×D3—22no—
358C22×D3—22no—
359C22×D3—22no—
360C22×D3—22no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C3×Q8—41no—
372C3×Q8—41no—
373C3×Q8—41no—
374C3×Q8—41no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4AGL(1,ℤ/8)12yes—
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4H(5,𝔽2)11yescharacteristic
398C2×C3×C42−1+411yescharacteristic
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3#13512yes—
417D12—22no—
418D12—22no—
419D12—22no—
420D12—22no—
421D12—22no—
422D12—41no—
423D12—41no—
424D12#12314yes—
425C4×D3—48no—
426C4×D3—84no—
427C4×D3—84no—
428C4×D3—84no—
429C3×C8—22no—
430C3×C8—41no—
431C3⋊C8—41no—
432C3⋊C8—41no—
433C24—61no—
434C24—61no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×D4—32no—
442C2×D4—32no—
443C2×D4—32no—
444C2×D4—32no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—31no—
461C22×C4—64no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—122no—
465C22×C4—122no—
466C22×C4—122no—
467C22×C4—122no—
468C22×C4—122no—
469C22×C4—122no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4—241no—
473C22×C4—241no—
474C22×C4—241no—
475C22×C4D3×D411yescharacteristic
476C22×C4D3×D411yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—124no—
479D8—68no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8D3×D412yes—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—62no—
499C4⋊C4—62no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—122no—
505C4⋊C4—122no—
506C4⋊C4—241no—
507C4⋊C4—241no—
508C4⋊C4#23512yes—
509C22⋊C4—64no—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C22⋊C4—122no—
515C22⋊C4—122no—
516C22⋊C4—241no—
517C42—21no—
518C42—21no—
519C42—61no—
520C42—61no—
521C42—61no—
522C42—61no—
523C22×C3#52511yescharacteristic
524C22×C3D4211yescharacteristic
525C22×C3#54811yescharacteristic
526C22×C3C2×#13511yescharacteristic
527C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
528C22×C3#57611yescharacteristic
529C22×C3#57811yescharacteristic
530C2×D3—24no—
531C2×D3—42no—
532C2×D3—42no—
533C2×D3—42no—
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—42no—
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—42no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C3×C4—22no—
545C3×C4—42no—
546C3×C4—41no—
547C3×C4—81no—
548C3×C4—81no—
549C3×C4—81no—
550C3×C4C2×#12311yescharacteristic
551C3×C4#53911yescharacteristic
552C3×C4#54711yescharacteristic
553C3×C4#55311yescharacteristic
554Dic3—22no—
555Dic3—41no—
556Dic3—41no—
557Dic3—41no—
558Dic3—81no—
559Dic3—81no—
560C23—62no—
561C23—62no—
562C23—62no—
563C23—62no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23—121no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23C2×D3×D411yescharacteristic
575Q8—41no—
576Q8—41no—
577Q8—41no—
578Q8—41no—
579D4—34no—
580D4—62no—
581D4—62no—
582D4—62no—
583D4—62no—
584D4—62no—
585D4—121no—
586D4—121no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—32no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—128no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—244no—
624C2×C4—244no—
625C2×C4—244no—
626C2×C4#91412yes—
627C2×C4C2×#23511yescharacteristic
628C2×C4C2×D3×D411yescharacteristic
629C2×C4D3×#4111yescharacteristic
630C2×C4D3×#4111yescharacteristic
631C2×C4#100911yescharacteristic
632C2×C4#101011yescharacteristic
633C8—22no—
634C8—41no—
635C8—121no—
636C8—121no—
637C2×C3#244311yescharacteristic
638C2×C3#310511yescharacteristic
639C2×C3#312411yescharacteristic
640C2×C3#326411yescharacteristic
641C2×C3#328311yescharacteristic
642C2×C3#336111yescharacteristic
643C2×C3#336711yescharacteristic
644D3—44no—
645D3—82no—
646C22—64no—
647C22—122no—
648C22—122no—
649C22—122no—
650C22—122no—
651C22—122no—
652C22—122no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—241no—
657C22—241no—
658C22—241no—
659C22—241no—
660C22#554711yescharacteristic
661C22#560911yescharacteristic
662C22#567711yescharacteristic
663C22C2×#91411yescharacteristic
664C22#576611yescharacteristic
665C22D3×#13511yescharacteristic
666C22#576911yescharacteristic
667C4—22no—
668C4—42no—
669C4—41no—
670C4—62no—
671C4—81no—
672C4—81no—
673C4—81no—
674C4—121no—
675C4—121no—
676C4—121no—
677C4—241no—
678C4—241no—
679C4#553011yescharacteristic
680C4#555111yescharacteristic
681C4D3×#12311yescharacteristic
682C4#570711yescharacteristic
683C3#2616411yescharacteristic, complemented, Sylow
684C2—124no—
685C2—242no—
686C2#7584811yescharacteristic
687C2#7965411yescharacteristic
688C2#7974911yescharacteristic
689C2#8067411yescharacteristic
690C2#8077411yescharacteristic
691C2#8162211yescharacteristic
692C2#8163311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1933C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1933D3
hypocenterC3#26164
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
3. #76505C2semidirect
4. #76202C2semidirect
5. C3×#1933C2semidirect
6. #70276C2semidirect
7. #70249C2semidirect
8. #70245C2semidirect
9. #69784C2semidirect
10. #69687C2semidirect
11. #69625C2semidirect
12. #68902C2semidirect
14. #68894C2semidirect
15. #68839C2semidirect
70. #1933D3semidirect
683. C3#26164semidirect

Maximal subgroups

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

subgroupindex
C2×#48192
C2×C4×D242
#765052
#762022
C3×#19332
#702762
#702492
#702452
#697842
#696872
#696252
#689022
#688972
#688942
#688392
#261643