Finite Group Databasebeta

← #10844908

Subgroups — #10844908

Order 768.

856 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5165C211yesmaximal, characteristic
2C2×#4679C211yesmaximal, characteristic, complemented
3#76555C211yesmaximal, characteristic, complemented
4#76346C211yesmaximal, characteristic, complemented
5C3×#1968C211yesmaximal, characteristic, complemented
6#72051C211yesmaximal, characteristic, complemented
7#70585C211yesmaximal, characteristic, complemented
8#70563C211yesmaximal, characteristic, complemented
9#70255C211yesmaximal, characteristic, complemented
10#70227C211yesmaximal, characteristic, complemented
11#70191C211yesmaximal, characteristic
12#70152C211yesmaximal, characteristic, complemented
13#69768C211yesmaximal, characteristic, complemented
14#69682C211yesmaximal, characteristic, complemented
15#69650C211yesmaximal, characteristic
16#25309—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C22×#213C2211yescharacteristic
21C2×#878—21no—
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#390C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32#5221C2211yescharacteristic
33C2×#832C2211yescharacteristic
34#5165—24no—
35C2×#821C2211yescharacteristic
36C2×#821C2211yescharacteristic
37C2×#815C2211yescharacteristic
38C2×#815C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41#4988C2211yescharacteristic
42#4984C2211yescharacteristic
43#4975C2211yescharacteristic
44Dic3×C4⋊C4C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#808C2211yescharacteristic
47#4948—21no—
48#4948—21no—
49#4942—21no—
50#4939C2211yescharacteristic
51#4928C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53#4679—24no—
54#4675—21no—
55#4673—21no—
56#4670—21no—
57#4669—21no—
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#463—31no—
63C2×#446—31no—
64#2592—31no—
65#2538—31no—
66#2066—31no—
67#2065—31no—
68#2055—31no—
69#2054—31no—
70#1972—31no—
71#1968D311yescharacteristic, complemented
72#1959—31no—
73#1948—31no—
74#1935—31no—
75#1934—31no—
76#1762—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×Dic3—21no—
80C22×D12C2311yescharacteristic
81C22×Dic6D411yescharacteristic
82C2×C3×SD16—22no—
83C2×C3×SD16—22no—
84C2×C3×SD16—41no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C42C2311yescharacteristic
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95#947—22no—
96#947—22no—
97D4×Dic3—22no—
98D4×Dic3—22no—
99C2×#213—22no—
100C2×#213—22no—
101C2×#213—41no—
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)⋊C4—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3D411yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4D411yescharacteristic
117C2×Dic3⋊C4D411yescharacteristic
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#878—44no—
124C4×Dic6—22no—
125C4×Dic6—22no—
126C3×#106—41no—
127C3×#103—22no—
128C3×#102—22no—
129C3×C4×C8—22no—
130C3×#95—21no—
131C3×#95—21no—
132#835—22no—
133#835—41no—
134#832—22no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#821—22no—
141#821—22no—
142#815—22no—
143#815—22no—
144Dic3⋊C8—22no—
145Dic3⋊C8D414yes—
146#809—41no—
147#808—41no—
148C4⋊(C3⋊C8)—22no—
149C22×SD16C2×D311yescharacteristic
150C22×D8—31no—
151C2×#124—31no—
152C2×C22⋊Q8—61no—
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155#463—64no—
156#446—64no—
157C2×#106—31no—
158C2×#106C2×D311yescharacteristic
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102C2×D311yescharacteristic
170C2×#97—31no—
171C2×#97—31no—
172C2×C4×C8C2×D311yescharacteristic
173#397—61no—
174#396—61no—
175#396—61no—
176#393—61no—
177#391—31no—
178#391C2×D311yescharacteristic
179#390—31no—
180#390—31no—
181#390—31no—
182#390C2×D311yescharacteristic
183#389—31no—
184#389—31no—
185#388—61no—
186#388—61no—
187#384—31no—
188#380—61no—
189C4×C4⋊C4—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194C24×C3—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—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4#4112yes—
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×Dic3—42no—
218C22×Dic3—42no—
219C22×Dic3—42no—
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×Dic6—22no—
231C2×Dic6—22no—
232C2×Dic6—41no—
233C3×SD16—44no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8C2×D412yes—
239C2×C3×C8#4112yes—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C22⋊C4—42no—
245C3×C22⋊C4—81no—
246C3×C42—21no—
247C3×C42—21no—
248#217—42no—
249#217—42no—
250#217—81no—
251#213—44no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—81no—
257(C2×D3)⋊C4—81no—
258C4⋊Dic3—22no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—42no—
266Dic3⋊C4—22no—
267Dic3⋊C4—22no—
268Dic3⋊C4—42no—
269Dic3⋊C4—42no—
270Dic3⋊C4—81no—
271C4×Dic3—21no—
272C4×Dic3—21no—
273C4×Dic3—81no—
274C4×Dic3C2×D412yes—
275C4×Dic3#4112yes—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8C2×D412yes—
282C22×Q8—31no—
283C22×Q8C22×D311yescharacteristic
284C22×D4—31no—
285C22×D4C22×D311yescharacteristic
286C23×C4—61no—
287C2×SD16—22no—
288C2×SD16—22no—
289C2×SD16—41no—
290C2×D8—62no—
291C2×D8—62no—
292C2×D8—121no—
293C22×C8—31no—
294C22×C8—31no—
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297#124—62no—
298#124—62no—
299C22⋊Q8—124no—
300C4×Q8—62no—
301C4×Q8—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—31no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C42—31no—
328C2×C42—31no—
329C2×C42—61no—
330C2×C42C22×D311yescharacteristic
331#106—41no—
332#106—121no—
333C4⋊C8—34no—
334C4⋊C8—62no—
335C4⋊C8—62no—
336#103—22no—
337#103—62no—
338#103—121no—
339#102—22no—
340#102—62no—
341#102—62no—
342#102—62no—
343#102—121no—
344#97—62no—
345#97—62no—
346C4×C8—22no—
347#95—21no—
348#95—21no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354C23×C3—22no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3C22×D411yescharacteristic
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C3×Q8—41no—
370C3×Q8—41no—
371C3×Q8—41no—
372C3×Q8—41no—
373C3×D4—22no—
374C3×D4—22no—
375C3×D4—41no—
376C3×D4—41no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
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×C4AGL(1,ℤ/8)12yes—
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4H(5,𝔽2)11yescharacteristic
397C2×C3×C42−1+411yescharacteristic
398C2×Dic3—21no—
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—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—84no—
418C2×Dic3#12112yes—
419C2×Dic3#12112yes—
420D12—41no—
421D12—41no—
422D12—41no—
423D12—41no—
424Dic6—44no—
425C3×C8—22no—
426C3×C8—22no—
427C3⋊C8—22no—
428C3⋊C8—41no—
429C24—21no—
430C24—61no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—62no—
438C2×Q8—62no—
439C2×Q8—121no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4#23512yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—122no—
463C22×C4—122no—
464C22×C4—122no—
465C22×C4D3×D411yescharacteristic
466C22×C4D3×D411yescharacteristic
467C22×C4C23×D311yescharacteristic
468SD16—44no—
469D8—124no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—32no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8D3×D412yes—
481C2×C8#23712yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—62no—
487C4⋊C4—62no—
488C4⋊C4—62no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—122no—
496C4⋊C4—122no—
497C4⋊C4—122no—
498C4⋊C4—241no—
499C22⋊C4—42no—
500C22⋊C4—81no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C22⋊C4—241no—
508C22⋊C4—241no—
509C22⋊C4—241no—
510C42—21no—
511C42—21no—
512C42—32no—
513C42—32no—
514C42—61no—
515C42—61no—
516C42—241no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3C2×#12111yescharacteristic
525C22×C3#54811yescharacteristic
526C22×C3#55011yescharacteristic
527C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
528C22×C3#57511yescharacteristic
529C22×C3#57711yescharacteristic
530C22×C3#57711yescharacteristic
531C2×D3—81no—
532C2×D3—81no—
533C2×D3—81no—
534C2×D3—81no—
535C2×D3—81no—
536C2×D3—81no—
537C2×D3—81no—
538C3×C4—22no—
539C3×C4—41no—
540C3×C4—81no—
541C3×C4—81no—
542C3×C4—81no—
543C3×C4#52911yescharacteristic
544C3×C4#54211yescharacteristic
545C3×C4#54611yescharacteristic
546C3×C4#54811yescharacteristic
547Dic3—22no—
548Dic3—22no—
549Dic3—41no—
550Dic3—41no—
551Dic3—81no—
552Dic3—81no—
553Dic3—81no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—121no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23C2×D3×D411yescharacteristic
569Q8—41no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573Q8—124no—
574D4—22no—
575D4—22no—
576D4—41no—
577D4—41no—
578D4—121no—
579D4—121no—
580D4—121no—
581D4—121no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—32no—
588C2×C4—32no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—244no—
618C2×C4#91412yes—
619C2×C4C2×#23511yescharacteristic
620C2×C4C2×D3×D411yescharacteristic
621C2×C4C2×#23711yescharacteristic
622C2×C4#100411yescharacteristic
623C2×C4D3×#4111yescharacteristic
624C2×C4#101011yescharacteristic
625C8—22no—
626C8—22no—
627C8—62no—
628C8—121no—
629C2×C3—82no—
630C2×C3#242611yescharacteristic
631C2×C3#305511yescharacteristic
632C2×C3#306011yescharacteristic
633C2×C3#337311yescharacteristic
634C2×C3#337611yescharacteristic
635C2×C3#339711yescharacteristic
636C2×C3#340111yescharacteristic
637D3—161no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—241no—
646C22—241no—
647C22—241no—
648C22—241no—
649C22—241no—
650C22—241no—
651C22—241no—
652C22#555211yescharacteristic
653C22D3×#12111yescharacteristic
654C22#571811yescharacteristic
655C22C2×#91411yescharacteristic
656C22#576211yescharacteristic
657C22#576811yescharacteristic
658C22#577011yescharacteristic
659C4—22no—
660C4—41no—
661C4—62no—
662C4—62no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4—121no—
667C4—121no—
668C4—241no—
669C4—241no—
670C4—241no—
671C4#549011yescharacteristic
672C4#553811yescharacteristic
673C4#559811yescharacteristic
674C4#566911yescharacteristic
675C3#2530911yescharacteristic, complemented, Sylow
676C2—82no—
677C2—481no—
678C2#7631211yescharacteristic
679C2#7962911yescharacteristic
680C2#7963011yescharacteristic
681C2#8065711yescharacteristic
682C2#8066011yescharacteristic
683C2#8175711yescharacteristic
684C2#8181611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1968C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#1968D3
hypocenterC3#25309
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
2. C2×#4679C2semidirect
3. #76555C2semidirect
4. #76346C2semidirect
5. C3×#1968C2semidirect
6. #72051C2semidirect
7. #70585C2semidirect
8. #70563C2semidirect
9. #70255C2semidirect
10. #70227C2semidirect
12. #70152C2semidirect
13. #69768C2semidirect
14. #69682C2semidirect
71. #1968D3semidirect
675. C3#25309semidirect

Maximal subgroups

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

subgroupindex
C2×#51652
C2×#46792
#765552
#763462
C3×#19682
#720512
#705852
#705632
#702552
#702272
#701912
#701522
#697682
#696822
#696502
#253093