Finite Group Databasebeta

← #10828850

Subgroups — #10828850

Order 768.

1052 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5533C211yesmaximal, characteristic
2C2×#4998C211yesmaximal, characteristic, complemented
3C2×#4723C211yesmaximal, characteristic, complemented
4#74026C211yesmaximal, characteristic
5C3×#1825C211yesmaximal, characteristic, complemented
6#69952C211yesmaximal, characteristic
7#69514C211yesmaximal, characteristic, complemented
8#68907C211yesmaximal, characteristic, complemented
9#68906C211yesmaximal, characteristic, complemented
10#68877C211yesmaximal, characteristic, complemented
11#68876C211yesmaximal, characteristic, complemented
12#68827C211yesmaximal, characteristic, complemented
13#68826C211yesmaximal, characteristic, complemented
14#68825C211yesmaximal, characteristic, complemented
15#68824C211yesmaximal, characteristic, complemented
16#24096—31nomaximal, Sylow
17C2×C3×#117C2211yescharacteristic
18C2×C4×#67C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C23×C3⋊C8C2211yescharacteristic
22#5533—24no—
23#5533—24no—
24C2×#897C2211yescharacteristic
25C2×#892C2211yescharacteristic
26C2×#888C2211yescharacteristic
27C2×#887C2211yescharacteristic
28C2×#885—21no—
29C2×#879—21no—
30C2×#876—21no—
31C2×C4×Dic6—21no—
32C2×C3×C22⋊C8C2211yescharacteristic
33C3×#380C2211yescharacteristic
34C2×(C2×D3)⋊C8C2211yescharacteristic
35C2×(C2×D3)⋊C8C2211yescharacteristic
36C2×#818C2211yescharacteristic
37C2×#817C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40#4998—24no—
41#4987C2211yescharacteristic
42#4987C2211yescharacteristic
43#4984C2211yescharacteristic
44#4984C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#807C2211yescharacteristic
50C2×#807C2211yescharacteristic
51#4961—21no—
52#4953—21no—
53#4723—24no—
54#4670—21no—
55#4653—21no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4529C2211yescharacteristic
64C2×#556—31no—
65C2×#485—31no—
66C2×#427—31no—
67#2485—31no—
68#2058—31no—
69#2057—31no—
70#2036—31no—
71#2035—31no—
72#1892—31no—
73#1869—31no—
74#1865—31no—
75#1863—31no—
76#1854—31no—
77#1854—31no—
78#1825D311yescharacteristic, complemented
79C23×C3×C4C2311yescharacteristic
80C22×#67D411yescharacteristic
81C22×C4×D3C2311yescharacteristic
82C22×Dic6—21no—
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#117—22no—
87C3×#117D414yes—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C42—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217D411yescharacteristic
98C4×#67—22no—
99C4×#67—22no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4D411yescharacteristic
105#926—22no—
106#926—22no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113#924—22no—
114#924D414yes—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4D411yescharacteristic
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3—21no—
122C2×C4×Dic3C2311yescharacteristic
123C22×C3⋊C8—22no—
124C22×C3⋊C8—22no—
125C22×C3⋊C8—22no—
126C22×C3⋊C8C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128#897—22no—
129#897—22no—
130#892—22no—
131#892—22no—
132#888—22no—
133#888—22no—
134#887—22no—
135#887—22no—
136#885—28no—
137#879—28no—
138#876—28no—
139C4×Dic6—28no—
140C3×C22⋊C8—22no—
141C3×#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146(C2×D3)⋊C8—22no—
147(C2×D3)⋊C8—22no—
148#818—22no—
149#817—22no—
150#816—22no—
151#816—22no—
152#810—22no—
153#810—22no—
154#808—22no—
155#808—22no—
156#807—22no—
157#807—22no—
158C23×C8—31no—
159#556—64no—
160#556—64no—
161C2×C4⋊Q8—31no—
162C2×#126—61no—
163C2×#126—61no—
164C2×#124—61no—
165C2×C22⋊Q8—31no—
166C2×C22⋊Q8—31no—
167C2×C4×Q8—31no—
168C2×C4×Q8—61no—
169C2×C4×D4—31no—
170C2×#117—31no—
171C2×#117—31no—
172C2×#117C2×D311yescharacteristic
173#485—64no—
174#427—64no—
175C2×#107—31no—
176C2×#107—31no—
177C2×#107—31no—
178C2×#106—31no—
179C2×#106—31no—
180C2×#106—31no—
181C2×#103—31no—
182C2×#103—31no—
183C2×#103—31no—
184C2×#103—31no—
185C2×C22⋊C8—31no—
186C2×C22⋊C8—31no—
187C2×C22⋊C8C2×D311yescharacteristic
188#397—61no—
189#394—61no—
190#389—31no—
191#389—31no—
192#388—61no—
193#387—61no—
194#380—31no—
195#380—31no—
196#380C2×D311yescharacteristic
197#340—31no—
198#340—31no—
199#340C2×D311yescharacteristic
200#340C2×D311yescharacteristic
201#340C2×D311yescharacteristic
202#340C2×D311yescharacteristic
203#336—31no—
204#336—31no—
205C24×C3C2×D411yescharacteristic
206C23×D3C2×D411yescharacteristic
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2411yescharacteristic
219C2×#67—22no—
220C2×#67—22no—
221C2×#67—41no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C2×C4×D3—21no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—24no—
240C2×Dic6—21no—
241C2×Dic6—42no—
242C2×Dic6—42no—
243C2×Dic6C2×D412yes—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4C2×D412yes—
257C3×C4⋊C4C2×D412yes—
258C3×C22⋊C4—24no—
259C3×C22⋊C4—81no—
260C3×C42—24no—
261#217—22no—
262#217—24no—
263#217—42no—
264(C2×D3)⋊C4—24no—
265(C2×D3)⋊C4—24no—
266(C2×D3)⋊C4—22no—
267(C2×D3)⋊C4—42no—
268(C2×D3)⋊C4—81no—
269C4⋊Dic3—24no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—42no—
276C4⋊Dic3—81no—
277C4⋊Dic3C2×D412yes—
278Dic3⋊C4—24no—
279Dic3⋊C4—24no—
280Dic3⋊C4—22no—
281Dic3⋊C4—24no—
282C4×Dic3—24no—
283C4×Dic3—24no—
284C4×Dic3—21no—
285C4×Dic3—21no—
286C4×Dic3—42no—
287C2×C3⋊C8—28no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8#4112yes—
293C2×C3⋊C8#4112yes—
294C22×Q8—31no—
295C22×Q8—61no—
296C22×D4—31no—
297C23×C4—31no—
298C23×C4C22×D311yescharacteristic
299C22×C8—31no—
300C22×C8—31no—
301C22×C8—62no—
302C22×C8—62no—
303C22×C8—62no—
304C22×C8C22×D311yescharacteristic
305C22×C8C22×D311yescharacteristic
306C4⋊Q8—62no—
307C4⋊Q8—62no—
308#126—68no—
309#126—68no—
310#124—68no—
311C22⋊Q8—34no—
312C22⋊Q8—62no—
313C22⋊Q8—62no—
314C22⋊Q8—62no—
315C4×Q8—68no—
316C4×Q8—62no—
317C4×Q8—62no—
318C4×D4—62no—
319C4×D4—62no—
320#117—22no—
321#117—62no—
322#117—62no—
323#117—62no—
324#117—62no—
325#117D1214yes—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4C22×D311yescharacteristic
337C2×C4⋊C4C22×D311yescharacteristic
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C4⋊C4C22×D311yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—31no—
343C2×C22⋊C4—31no—
344C2×C22⋊C4—61no—
345C2×C22⋊C4—61no—
346C2×C22⋊C4—61no—
347C2×C22⋊C4—61no—
348C2×C22⋊C4—61no—
349C2×C22⋊C4—61no—
350C2×C42—21no—
351C2×C42—31no—
352C2×C42—61no—
353C2×C42—61no—
354C2×C42—61no—
355#107—62no—
356#107—62no—
357#107—62no—
358#106—62no—
359#106—62no—
360#106—62no—
361#103—62no—
362#103—62no—
363#103—62no—
364#103—62no—
365C22⋊C8—22no—
366C22⋊C8—62no—
367C22⋊C8—62no—
368#95—21no—
369#95—61no—
370#95—61no—
371#95—61no—
372#95—61no—
373C23×C3—21no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—21no—
379C23×C3#12012yes—
380C23×C3C22×D411yescharacteristic
381C22×D3—21no—
382C22×D3—21no—
383C22×D3—21no—
384C22×D3—21no—
385C22×D3—21no—
386C22×D3—21no—
387C22×D3—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—41no—
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#12012yes—
407C2×C3×C4C22×D411yescharacteristic
408C2×C3×C4C22×D411yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×C3×C42−1+411yescharacteristic
412C2×C3×C42−1+411yescharacteristic
413#67—44no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—22no—
419C2×Dic3—22no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3#13512yes—
435C4×D3—41no—
436C4×D3—41no—
437C4×D3—41no—
438C4×D3—41no—
439Dic6—22no—
440Dic6—22no—
441Dic6—48no—
442Dic6—41no—
443Dic6—41no—
444C3×C8—41no—
445C3×C8—41no—
446C3⋊C8—22no—
447C3⋊C8—22no—
448C24—31no—
449C24D3×D411yescharacteristic
450C2×Q8—32no—
451C2×Q8—64no—
452C2×Q8—61no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—122no—
458C2×Q8—122no—
459C2×D4—62no—
460C2×D4—62no—
461C2×D4—121no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—31no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4C2×D1212yes—
484C22×C4D3×D411yescharacteristic
485C22×C4D3×D412yes—
486C22×C4C23×D311yescharacteristic
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—32no—
494C2×C8—32no—
495C2×C8—68no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—32no—
507C4⋊C4—64no—
508C4⋊C4—64no—
509C4⋊C4—64no—
510C4⋊C4—62no—
511C4⋊C4—64no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C4⋊C4—122no—
518C4⋊C4—241no—
519C4⋊C4C2×D1212yes—
520C4⋊C4C2×D1212yes—
521C22⋊C4—24no—
522C22⋊C4—64no—
523C22⋊C4—64no—
524C22⋊C4—62no—
525C22⋊C4—64no—
526C22⋊C4—62no—
527C22⋊C4—81no—
528C22⋊C4—122no—
529C22⋊C4—122no—
530C22⋊C4—241no—
531C42—24no—
532C42—64no—
533C42—64no—
534C42—61no—
535C42—61no—
536C42—122no—
537C22×C3—22no—
538C22×C3—22no—
539C22×C3—22no—
540C22×C3—41no—
541C22×C3—41no—
542C22×C3—41no—
543C22×C3—41no—
544C22×C3C2×#12011yescharacteristic
545C22×C3#54711yescharacteristic
546C22×C3#54711yescharacteristic
547C22×C3C2×#13511yescharacteristic
548C22×C3#57511yescharacteristic
549C22×C3#57711yescharacteristic
550C22×C3#57811yescharacteristic
551C2×D3—41no—
552C2×D3—41no—
553C2×D3—41no—
554C2×D3—41no—
555C2×D3—41no—
556C2×D3—41no—
557C2×D3—41no—
558C3×C4—22no—
559C3×C4—42no—
560C3×C4—42no—
561C3×C4—81no—
562C3×C4—81no—
563C3×C4#52511yescharacteristic
564C3×C4#53611yescharacteristic
565C3×C4#54711yescharacteristic
566C3×C4#54711yescharacteristic
567Dic3—22no—
568Dic3—42no—
569Dic3—41no—
570Dic3—42no—
571Dic3—41no—
572Dic3—81no—
573Dic3—81no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—61no—
581C23—61no—
582C23—61no—
583C23—61no—
584C23—61no—
585C23—61no—
586C23—61no—
587C23#88212yes—
588C23C2×D3×D411yescharacteristic
589Q8—62no—
590Q8—62no—
591Q8—128no—
592Q8—121no—
593Q8—121no—
594D4—124no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—32no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—62no—
620C2×C4—61no—
621C2×C4—62no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4#88212yes—
639C2×C4C22×D1211yescharacteristic
640C2×C4C2×D3×D411yescharacteristic
641C2×C4D3×#4111yescharacteristic
642C2×C4D3×#4111yescharacteristic
643C2×C4#101011yescharacteristic
644C2×C4#101011yescharacteristic
645C8—41no—
646C8—41no—
647C8—62no—
648C8—62no—
649C2×C3—42no—
650C2×C3#240611yescharacteristic
651C2×C3#300211yescharacteristic
652C2×C3#301311yescharacteristic
653C2×C3#334011yescharacteristic
654C2×C3#334211yescharacteristic
655C2×C3#334911yescharacteristic
656C2×C3#335011yescharacteristic
657D3—81no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—121no—
666C22—121no—
667C22—121no—
668C22—121no—
669C22—121no—
670C22—121no—
671C22—121no—
672C22C2×#88211yescharacteristic
673C22#565511yescharacteristic
674C22#565511yescharacteristic
675C22D3×#13511yescharacteristic
676C22#576811yescharacteristic
677C22#577511yescharacteristic
678C22#577911yescharacteristic
679C4—22no—
680C4—42no—
681C4—42no—
682C4—62no—
683C4—81no—
684C4—81no—
685C4—121no—
686C4—121no—
687C4—122no—
688C4—122no—
689C4—241no—
690C4—241no—
691C4#552411yescharacteristic
692C4#552611yescharacteristic
693C4#565511yescharacteristic
694C4#565511yescharacteristic
695C3#2409611yescharacteristic, complemented, Sylow
696C2—42no—
697C2—241no—
698C2#7406311yescharacteristic
699C2#7883311yescharacteristic
700C2#7888711yescharacteristic
701C2#8136411yescharacteristic
702C2#8139411yescharacteristic
703C2#8150811yescharacteristic
704C2#8153311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1825C2
socleC23×C3C22×D4
prime radicalC22×#67D4
hypercenter#1825D3
hypocenterC3#24096
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
2. C2×#4998C2semidirect
3. C2×#4723C2semidirect
5. C3×#1825C2semidirect
7. #69514C2semidirect
8. #68907C2semidirect
9. #68906C2semidirect
10. #68877C2semidirect
11. #68876C2semidirect
12. #68827C2semidirect
13. #68826C2semidirect
14. #68825C2semidirect
15. #68824C2semidirect
78. #1825D3semidirect
695. C3#24096semidirect

Maximal subgroups

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

subgroupindex
C2×#55332
C2×#49982
C2×#47232
#740262
C3×#18252
#699522
#695142
#689072
#689062
#688772
#688762
#688272
#688262
#688252
#688242
#240963