Finite Group Databasebeta

← #10844588

Subgroups — #10844588

Order 768.

858 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83178C211yesmaximal, characteristic
2C2×#4765C211yesmaximal, characteristic, complemented
3C2×#4759C211yesmaximal, characteristic, complemented
4#73657C211yesmaximal, characteristic, complemented
5C3×#1982C211yesmaximal, characteristic
6#71762C211yesmaximal, characteristic
7#70489C211yesmaximal, characteristic, complemented
8#70452C211yesmaximal, characteristic, complemented
9#70194C211yesmaximal, characteristic, complemented
10#70178C211yesmaximal, characteristic, complemented
11#70162C211yesmaximal, characteristic, complemented
12#70162C211yesmaximal, characteristic, complemented
13#70155C211yesmaximal, characteristic, complemented
14#70152C211yesmaximal, characteristic, complemented
15#69465C211yesmaximal, characteristic, complemented
16#25285—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×#936C2211yescharacteristic
19C2×#936C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C2×#888C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28#5222—21no—
29#5222—21no—
30#5221C2211yescharacteristic
31#5220—21no—
32#5220—21no—
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#818C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#815C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42C2×Dic3⋊C8C2211yescharacteristic
43#4984C2211yescharacteristic
44Dic3×C4⋊C4C2211yescharacteristic
45Dic3×C4⋊C4C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#807C2211yescharacteristic
49C4×Dic3⋊C4—21no—
50C4×Dic3⋊C4C2211yescharacteristic
51#4929—21no—
52#4929C2211yescharacteristic
53#4928C2211yescharacteristic
54#4765—24no—
55#4759—24no—
56#4647—21no—
57#4646—21no—
58#4551C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#489—31no—
63C2×#474—31no—
64#2544—31no—
65#2369—31no—
66#2068—31no—
67#2067—31no—
68#2042—31no—
69#2041—31no—
70#1983—31no—
71#1983—31no—
72#1982D311yescharacteristic
73#1940—31no—
74#1928—31no—
75#1925—31no—
76#1762—31no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×Dic3—21no—
82C22×Dic6—21no—
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#127—22no—
87C3×#127—41no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99#936—22no—
100#936—22no—
101#936—22no—
102#936—41no—
103D4×Dic3—22no—
104D4×Dic3—22no—
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×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119#888—22no—
120#888D414yescomplemented
121C3×C4⋊C8—41no—
122C3×#102—22no—
123C3×#102—22no—
124C3×#102—22no—
125C3×#102—22no—
126C3×#95—21no—
127#832—22no—
128#832—22no—
129#832—22no—
130#832—41no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139#818—22no—
140#816—22no—
141#816—41no—
142#815—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—41no—
145#810—22no—
146#810—22no—
147#807—22no—
148C2×C4⋊Q8—31no—
149C2×#127C2×D311yescharacteristic
150C2×#124—31no—
151C2×#124—31no—
152C2×C4×D4—31no—
153#489—64no—
154#474—64no—
155C2×#107—31no—
156C2×#107—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102C2×D311yescharacteristic
169C2×#102C2×D311yescharacteristic
170C2×#102C2×D311yescharacteristic
171C2×#102C2×D311yescharacteristic
172C2×#97—31no—
173#394—61no—
174#394—61no—
175#391—31no—
176#391—61no—
177#391—61no—
178#390—31no—
179#389—31no—
180#389—31no—
181#389—61no—
182#386—61no—
183#386—61no—
184C4×C4⋊C4—31no—
185C4×C4⋊C4—31no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—61no—
188C4×C4⋊C4C2×D311yescharacteristic
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C24×C3—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—42no—
207C2×C3×D4—42no—
208C2×C3×D4—42no—
209C2×C3×D4—42no—
210C2×C3×D4—81no—
211C2×C3×D4—81no—
212C2×C3×D4#4112yes—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2411yescharacteristic
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—42no—
223C22×Dic3—81no—
224C22×Dic3—81no—
225C22×Dic3C2×D411yescharacteristic
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—24no—
236C2×Dic6—81no—
237C2×Dic6—81no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4C2×D412yes—
247C3×C4⋊C4#4112yes—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—81no—
251#217—42no—
252#217—42no—
253#217—42no—
254#217—42no—
255#217—42no—
256#217—81no—
257#217—81no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—81no—
264C4⋊Dic3C2×D412yes—
265Dic3⋊C4—22no—
266Dic3⋊C4—24no—
267Dic3⋊C4—41no—
268Dic3⋊C4—81no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—42no—
272C4×Dic3—81no—
273C4×Dic3C2×D412yes—
274C4×Dic3#4112yes—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C22×Q8—31no—
282C22×Q8—61no—
283C22×D4—21no—
284C22×D4—21no—
285C22×D4C22×D311yescharacteristic
286C22×D4C22×D311yescharacteristic
287C23×C4—61no—
288C22×C8—31no—
289C22×C8—31no—
290C22×C8C22×D311yescharacteristic
291C22×C8C22×D311yescharacteristic
292C4⋊Q8—34no—
293C4⋊Q8—62no—
294#127—22no—
295#127—41no—
296#124—62no—
297#124—62no—
298#124—62no—
299#124—121no—
300C4×D4—62no—
301C4×D4—62no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
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×C22⋊C4—61no—
319C2×C42—21no—
320C2×C42—31no—
321C2×C42—31no—
322C2×C42—61no—
323C2×C42—61no—
324C2×C42C22×D311yescharacteristic
325#107—62no—
326#107—62no—
327C4⋊C8—41no—
328C4⋊C8—62no—
329C4⋊C8—121no—
330#103—62no—
331#103—62no—
332#103—62no—
333#103—121no—
334#102—22no—
335#102—22no—
336#102—22no—
337#102—22no—
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—
350#95—61no—
351#95—61no—
352C23×C3—22no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3C22×D411yescharacteristic
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—41no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4—41no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—84no—
376C3×D4—84no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—21no—
381C2×C3×C4—22no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4#13712yes—
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4H(5,𝔽2)11yescharacteristic
394C2×C3×C42−1+411yescharacteristic
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—22no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—41no—
407C2×Dic3—41no—
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—
420Dic6—41no—
421Dic6—41no—
422Dic6—41no—
423Dic6—41no—
424Dic6—84no—
425C3×C8—41no—
426C3×C8—41no—
427C3⋊C8—41no—
428C3⋊C8—41no—
429C24—21no—
430C24—21no—
431C2×Q8—64no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—241no—
439C2×Q8—241no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—81no—
456C2×D4—81no—
457C2×D4#23512yes—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—31no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—122no—
469C22×C4—241no—
470C22×C4—241no—
471C22×C4D3×D411yescharacteristic
472C22×C4D3×D411yescharacteristic
473C22×C4C23×D311yescharacteristic
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—32no—
489C4⋊C4—64no—
490C4⋊C4—62no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—121no—
497C4⋊C4—241no—
498C4⋊C4—241no—
499C4⋊C4D3×D412yes—
500C4⋊C4#23712yes—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—122no—
506C22⋊C4—241no—
507C22⋊C4—241no—
508C42—21no—
509C42—21no—
510C42—32no—
511C42—32no—
512C42—61no—
513C42—61no—
514C42—81no—
515C42—122no—
516C42—241no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—81no—
525C22×C3—81no—
526C22×C3—81no—
527C22×C3—81no—
528C22×C3—81no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3C2×#12111yescharacteristic
532C22×C3#54611yescharacteristic
533C22×C3#54811yescharacteristic
534C22×C3C2×#13711yescharacteristic
535C22×C3#57511yescharacteristic
536C22×C3#57611yescharacteristic
537C22×C3#57611yescharacteristic
538C3×C4—22no—
539C3×C4—42no—
540C3×C4—42no—
541C3×C4—41no—
542C3×C4#52911yescharacteristic
543C3×C4#54211yescharacteristic
544C3×C4#54811yescharacteristic
545C3×C4#55011yescharacteristic
546Dic3—22no—
547Dic3—22no—
548Dic3—42no—
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—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23C2×D3×D411yescharacteristic
569Q8—121no—
570Q8—121no—
571Q8—121no—
572Q8—121no—
573Q8—244no—
574D4—22no—
575D4—22no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—84no—
583D4—84no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—32no—
592C2×C4—32no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—61no—
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—62no—
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—121no—
618C2×C4—121no—
619C2×C4—244no—
620C2×C4#91512yes—
621C2×C4C2×#23511yescharacteristic
622C2×C4C2×D3×D411yescharacteristic
623C2×C4C2×#23711yescharacteristic
624C2×C4#100411yescharacteristic
625C2×C4D3×#4111yescharacteristic
626C2×C4#101011yescharacteristic
627C8—41no—
628C8—41no—
629C8—121no—
630C8—121no—
631C2×C3—82no—
632C2×C3—161no—
633C2×C3#242611yescharacteristic
634C2×C3#303711yescharacteristic
635C2×C3#304111yescharacteristic
636C2×C3#331411yescharacteristic
637C2×C3#331911yescharacteristic
638C2×C3#337011yescharacteristic
639C2×C3#337511yescharacteristic
640C22—42no—
641C22—42no—
642C22—42no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22#549011yescharacteristic
655C22#559811yescharacteristic
656C22D3×#12111yescharacteristic
657C22#575711yescharacteristic
658C22C2×#91511yescharacteristic
659C22#577311yescharacteristic
660C22#577411yescharacteristic
661C4—22no—
662C4—41no—
663C4—42no—
664C4—42no—
665C4—62no—
666C4—62no—
667C4—121no—
668C4—121no—
669C4—122no—
670C4—241no—
671C4—241no—
672C4—241no—
673C4#553811yescharacteristic
674C4#555211yescharacteristic
675C4#566911yescharacteristic
676C4#571811yescharacteristic
677C3#2528511yescharacteristic, complemented, Sylow
678C2—82no—
679C2—161no—
680C2#7631211yescharacteristic
681C2#7876311yescharacteristic
682C2#7893711yescharacteristic
683C2#8032211yescharacteristic
684C2#8033411yescharacteristic
685C2#8037411yescharacteristic
686C2#8038311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1982C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenter#1982D3
hypocenterC3#25285
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1982

Complements

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

NH ≅ G/Ntype
2. C2×#4765C2semidirect
3. C2×#4759C2semidirect
4. #73657C2semidirect
7. #70489C2semidirect
8. #70452C2semidirect
9. #70194C2semidirect
10. #70178C2semidirect
11. #70162C2semidirect
12. #70162C2semidirect
13. #70155C2semidirect
14. #70152C2semidirect
15. #69465C2semidirect
120. #888D4semidirect
677. C3#25285semidirect

Maximal subgroups

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

subgroupindex
#831782
C2×#47652
C2×#47592
#736572
C3×#19822
#717622
#704892
#704522
#701942
#701782
#701622
#701552
#701522
#694652
#252853