Finite Group Databasebeta

← #10844674

Subgroups — #10844674

Order 768.

828 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82974C211yesmaximal, characteristic
2C2×#4796C211yesmaximal, characteristic, complemented
3C2×#4764C211yesmaximal, characteristic
4#75988C211yesmaximal, characteristic, complemented
5C3×#2042C211yesmaximal, characteristic, complemented
6#71721C211yesmaximal, characteristic, complemented
7#70279C211yesmaximal, characteristic, complemented
8#70230C211yesmaximal, characteristic, complemented
9#70220C211yesmaximal, characteristic, complemented
10#70180C211yesmaximal, characteristic, complemented
11#70174C211yesmaximal, characteristic, complemented
12#70161C211yesmaximal, characteristic
13#69629C211yesmaximal, characteristic, complemented
14#68625C211yesmaximal, characteristic, complemented
15#68618C211yesmaximal, characteristic, complemented
16#25291—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×#940C2211yescharacteristic
21C2×#890C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29#5221C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#818C2211yescharacteristic
35C2×#815C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38#4976—21no—
39Dic3×C4⋊C4—21no—
40Dic3×C4⋊C4C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46#4961—21no—
47#4961—21no—
48#4957—21no—
49#4957—21no—
50#4939C2211yescharacteristic
51C4×C4⋊Dic3C2211yescharacteristic
52C4×Dic3⋊C4C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54#4796—24no—
55#4764—24no—
56#4653—21no—
57#4652—21no—
58C3×#336C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#490—31no—
63C2×#479—31no—
64#2548—31no—
65#2370—31no—
66#2067—31no—
67#2065—31no—
68#2042—31no—
69#2042D311yescharacteristic, complemented
70#1983—31no—
71#1983—31no—
72#1982—31no—
73#1940—31no—
74#1926—31no—
75#1925—31no—
76#1763—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C22×#67—21no—
80C22×#67—21no—
81C22×D12C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×#124—22no—
85C3×#124—41no—
86C2×C3×C4⋊C4—21no—
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—
94#947—22no—
95#947—41no—
96Q8×Dic3—22no—
97Q8×Dic3—22no—
98#940—22no—
99#940—22no—
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×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4D411yescharacteristic
112C2×Dic3⋊C4D411yescharacteristic
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3C2311yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#890—22no—
122#890D414yes—
123C3×#103—22no—
124C3×#103—22no—
125C3×#102—22no—
126C3×#102—41no—
127C3×#95—21no—
128#835—22no—
129#835—22no—
130#832—22no—
131#832—22no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95D411yescharacteristic
137C3⋊#95D411yescharacteristic
138#821—22no—
139#821—22no—
140#818—22no—
141#815—22no—
142Dic3⋊C8—22no—
143Dic3⋊C8—41no—
144#809—22no—
145#809—41no—
146#808—22no—
147C4⋊(C3⋊C8)—41no—
148C2×#127—31no—
149C2×#125—31no—
150C2×#124—31no—
151C2×#124C2×D311yescharacteristic
152C2×C4×Q8—31no—
153#490—64no—
154#479—64no—
155C2×#107—31no—
156C2×#106—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103C2×D311yescharacteristic
163C2×#103C2×D311yescharacteristic
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#102C2×D311yescharacteristic
171C2×#102C2×D311yescharacteristic
172C2×#97—31no—
173#394—61no—
174#394—61no—
175#390—31no—
176#390—31no—
177#390—61no—
178#390—61no—
179#389—31no—
180#389—61no—
181#389C2×D311yescharacteristic
182#387—61no—
183#387—61no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—31no—
188C4×C4⋊C4—61no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336C2×D311yescharacteristic
193C24×C3—21no—
194C23×D3—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8#4112yes—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C2×#67—42no—
214C2×#67—42no—
215C2×#67—42no—
216C2×#67—42no—
217C2×#67—81no—
218C2×#67—81no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4C2×D412yes—
243C3×C22⋊C4—42no—
244C3×C22⋊C4—42no—
245C3×C42—21no—
246C3×C42—21no—
247#217—81no—
248#217—81no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—81no—
252(C2×D3)⋊C4—81no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—24no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—42no—
259C4⋊Dic3C2×D412yes—
260C4⋊Dic3#4112yes—
261Dic3⋊C4—24no—
262Dic3⋊C4—41no—
263Dic3⋊C4—41no—
264Dic3⋊C4—81no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—42no—
268C4×Dic3—42no—
269C4×Dic3—81no—
270C4×Dic3—81no—
271C4×Dic3C2×D412yes—
272C4×Dic3#4112yes—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C22×Q8C22×D311yescharacteristic
280C22×D4—31no—
281C22×D4—61no—
282C22×D4—61no—
283C22×D4C22×D311yescharacteristic
284C22×C8—31no—
285C22×C8—31no—
286C22×C8C22×D311yescharacteristic
287C22×C8C22×D311yescharacteristic
288#127—62no—
289#127—62no—
290#125—34no—
291#125—62no—
292#124—22no—
293#124—41no—
294#124—62no—
295#124—121no—
296C4×Q8—62no—
297C4×Q8—62no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C42—31no—
319C2×C42—31no—
320C2×C42—61no—
321C2×C42—61no—
322C2×C42—61no—
323C2×C42—61no—
324C2×C42C22×D311yescharacteristic
325#107—62no—
326#106—62no—
327C4⋊C8—62no—
328C4⋊C8—121no—
329C4⋊C8—121no—
330#103—22no—
331#103—22no—
332#103—62no—
333#103—62no—
334#102—22no—
335#102—41no—
336#102—62no—
337#102—62no—
338#102—62no—
339#102—62no—
340#102—62no—
341#102—121no—
342#97—62no—
343#95—21no—
344#95—31no—
345#95—31no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3C22×D411yescharacteristic
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C3×Q8—22no—
366C3×Q8—22no—
367C3×Q8—41no—
368C3×Q8—41no—
369C3×D4—41no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4—41no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
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×C4C22×D411yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4H(5,𝔽2)11yescharacteristic
393C2×C3×C42−1+411yescharacteristic
394#67—84no—
395#67—84no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—22no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—22no—
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#12112yes—
416C2×Dic3#12112yes—
417D12—41no—
418D12—41no—
419D12—41no—
420D12—41no—
421C3×C8—41no—
422C3×C8—41no—
423C3⋊C8—41no—
424C3⋊C8—41no—
425C24—21no—
426C24—61no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8#23712yes—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—122no—
446C2×D4—122no—
447C2×D4—122no—
448C2×D4—122no—
449C2×D4—241no—
450C2×D4—241no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4D3×D411yescharacteristic
462C22×C4D3×D411yescharacteristic
463C22×C4C23×D311yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—32no—
480C4⋊C4—32no—
481C4⋊C4—42no—
482C4⋊C4—64no—
483C4⋊C4—64no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—121no—
489C4⋊C4—121no—
490C4⋊C4—122no—
491C4⋊C4—241no—
492C4⋊C4D3×D412yes—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—241no—
498C22⋊C4—241no—
499C22⋊C4—241no—
500C22⋊C4—241no—
501C42—21no—
502C42—21no—
503C42—32no—
504C42—32no—
505C42—61no—
506C42—61no—
507C42—122no—
508C42—122no—
509C42—241no—
510C42—241no—
511C22×C3—81no—
512C22×C3—81no—
513C22×C3—81no—
514C22×C3—81no—
515C22×C3—81no—
516C22×C3—81no—
517C22×C3—81no—
518C22×C3C2×#12111yescharacteristic
519C22×C3#54611yescharacteristic
520C22×C3#54811yescharacteristic
521C22×C3#57511yescharacteristic
522C22×C3#57511yescharacteristic
523C22×C3#57611yescharacteristic
524C22×C3#57711yescharacteristic
525C2×D3—81no—
526C2×D3—81no—
527C2×D3—81no—
528C2×D3—81no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—81no—
532C3×C4—42no—
533C3×C4—42no—
534C3×C4—41no—
535C3×C4—41no—
536C3×C4—81no—
537C3×C4#52911yescharacteristic
538C3×C4#54211yescharacteristic
539C3×C4#54811yescharacteristic
540C3×C4#55011yescharacteristic
541Dic3—22no—
542Dic3—22no—
543Dic3—42no—
544Dic3—42no—
545Dic3—41no—
546Dic3—41no—
547Dic3—81no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23C2×D3×D411yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567D4—41no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—121no—
572D4—121no—
573D4—121no—
574D4—121no—
575D4—244no—
576D4—244no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—32no—
586C2×C4—32no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—62no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—62no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4C2×D3×D411yescharacteristic
614C2×C4C2×#23711yescharacteristic
615C2×C4#100411yescharacteristic
616C2×C4D3×#4111yescharacteristic
617C2×C4D3×#4111yescharacteristic
618C2×C4#101011yescharacteristic
619C8—41no—
620C8—41no—
621C8—121no—
622C8—121no—
623C2×C3—161no—
624C2×C3#242611yescharacteristic
625C2×C3#304011yescharacteristic
626C2×C3#304511yescharacteristic
627C2×C3#330911yescharacteristic
628C2×C3#331711yescharacteristic
629C2×C3#337011yescharacteristic
630C2×C3#337111yescharacteristic
631D3—161no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—241no—
640C22—241no—
641C22—241no—
642C22—241no—
643C22—241no—
644C22—241no—
645C22—241no—
646C22#559111yescharacteristic
647C22D3×#12111yescharacteristic
648C22#567711yescharacteristic
649C22#576811yescharacteristic
650C22#576911yescharacteristic
651C22#577311yescharacteristic
652C22#577811yescharacteristic
653C4—42no—
654C4—41no—
655C4—42no—
656C4—41no—
657C4—62no—
658C4—62no—
659C4—81no—
660C4—121no—
661C4—122no—
662C4—121no—
663C4—122no—
664C4—241no—
665C4#559811yescharacteristic
666C4#559911yescharacteristic
667C4#564911yescharacteristic
668C4#566211yescharacteristic
669C3#2529111yescharacteristic, complemented, Sylow
670C2—161no—
671C2—481no—
672C2#7690111yescharacteristic
673C2#8011111yescharacteristic
674C2#8025611yescharacteristic
675C2#8053511yescharacteristic
676C2#8060211yescharacteristic
677C2#8166011yescharacteristic
678C2#8167311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2042C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenter#2042D3
hypocenterC3#25291
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2042

Complements

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

NH ≅ G/Ntype
2. C2×#4796C2semidirect
4. #75988C2semidirect
5. C3×#2042C2semidirect
6. #71721C2semidirect
7. #70279C2semidirect
8. #70230C2semidirect
9. #70220C2semidirect
10. #70180C2semidirect
11. #70174C2semidirect
13. #69629C2semidirect
14. #68625C2semidirect
15. #68618C2semidirect
69. #2042D3semidirect
669. C3#25291semidirect

Maximal subgroups

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

subgroupindex
#829742
C2×#47962
C2×#47642
#759882
C3×#20422
#717212
#702792
#702302
#702202
#701802
#701742
#701612
#696292
#686252
#686182
#252913