Finite Group Databasebeta

← #10844679

Subgroups — #10844679

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
1C2×C3×#490C211yesmaximal, characteristic
2#83179C211yesmaximal, characteristic
3#83025C211yesmaximal, characteristic, complemented
4C2×#5063C211yesmaximal, characteristic, complemented
5#71761C211yesmaximal, characteristic
6#70194C211yesmaximal, characteristic, complemented
7#70193C211yesmaximal, characteristic, complemented
8#70178C211yesmaximal, characteristic, complemented
9#70178C211yesmaximal, characteristic, complemented
10#70174C211yesmaximal, characteristic, complemented
11#70173C211yesmaximal, characteristic, complemented
12#70162C211yesmaximal, characteristic, complemented
13#70162C211yesmaximal, characteristic, complemented
14#69836C211yesmaximal, characteristic, complemented
15#69813C211yesmaximal, characteristic, complemented
16#25291—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×#936C2211yescharacteristic
20C2×#936C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C3×#490—24no—
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28#5222—21no—
29#5222—21no—
30#5221—21no—
31#5221—21no—
32#5221C2211yescharacteristic
33#5221C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42#5063—24no—
43#4984—21no—
44#4984C2211yescharacteristic
45#4984C2211yescharacteristic
46#4980—21no—
47#4980—21no—
48Dic3×C4⋊C4—21no—
49Dic3×C4⋊C4C2211yescharacteristic
50Dic3×C4⋊C4C2211yescharacteristic
51C2×#810C2211yescharacteristic
52C2×#810C2211yescharacteristic
53C2×#808C2211yescharacteristic
54C2×#807C2211yescharacteristic
55C4×C4⋊Dic3C2211yescharacteristic
56C4×C4⋊Dic3C2211yescharacteristic
57C2×C4⋊(C3⋊C8)C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62C2×#490D311yescharacteristic
63C2×#479—31no—
64#2548—31no—
65#2370—31no—
66#2067—31no—
67#2065—31no—
68#2042—31no—
69#2042—31no—
70#1983—31no—
71#1983—31no—
72#1982—31no—
73#1940—31no—
74#1926—31no—
75#1925—31no—
76#1763—31no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×#127—22no—
85C3×#127—22no—
86C3×#125—22no—
87C3×#125D414yes—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101#936—22no—
102#936—22no—
103#936—41no—
104#936—41no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3D411yescharacteristic
108C2×C4⋊Dic3D411yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
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
121C4×Dic6—22no—
122C4×Dic6—22no—
123C3×#102—22no—
124C3×#102—22no—
125C3×#102—22no—
126C3×#102—22no—
127C3×#97—22no—
128#832—22no—
129#832—22no—
130#832—41no—
131#832—41no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139#816—22no—
140#816—22no—
141Dic3⋊C8—41no—
142Dic3⋊C8—41no—
143#810—22no—
144#810—22no—
145#808—22no—
146#807—22no—
147C4⋊(C3⋊C8)—22no—
148C2×#127C2×D311yescharacteristic
149C2×#125C2×D311yescharacteristic
150C2×#124—31no—
151C2×#124—31no—
152C2×C4×Q8—31no—
153#490—24no—
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×#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×#97C2×D311yescharacteristic
173#394—61no—
174#394—61no—
175#390—31no—
176#390—31no—
177#390—61no—
178#390—61no—
179#389—31no—
180#389—31no—
181#389—61no—
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#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—21no—
207C2×C3×D4—42no—
208C2×C3×D4—42no—
209C2×C3×D4—42no—
210C2×C3×D4—42no—
211C2×C3×D4—81no—
212C2×C3×D4—81no—
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×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C22×Dic3C2×D411yescharacteristic
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6#4112yes—
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—24no—
239C3×C4⋊C4—24no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4C2×D412yes—
243C3×C4⋊C4C2×D412yes—
244C3×C42C2×D412yes—
245C3×C42#4112yes—
246#217—42no—
247#217—42no—
248#217—42no—
249#217—42no—
250#217—81no—
251#217—81no—
252#217—81no—
253#217—81no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—41no—
260C4⋊Dic3—41no—
261C4⋊Dic3—42no—
262C4⋊Dic3—81no—
263C4⋊Dic3#4112yes—
264Dic3⋊C4—42no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—42no—
270C4×Dic3—42no—
271C4×Dic3—81no—
272C4×Dic3—81no—
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×Q8—31no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C22×D311yescharacteristic
283C22×D4C22×D311yescharacteristic
284C22×C8—31no—
285C22×C8—31no—
286C22×C8C22×D311yescharacteristic
287C22×C8C22×D311yescharacteristic
288#127—22no—
289#127—22no—
290#125—22no—
291#125#6714yes—
292#124—62no—
293#124—62no—
294#124—121no—
295#124—121no—
296C4×Q8—62no—
297C4×Q8—62no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4C22×D311yescharacteristic
309C2×C4⋊C4C22×D311yescharacteristic
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
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—62no—
331#103—62no—
332#103—62no—
333#103—62no—
334#102—22no—
335#102—22no—
336#102—22no—
337#102—22no—
338#102—62no—
339#102—62no—
340#102—121no—
341#102—121no—
342#97—22no—
343#95—31no—
344#95—31no—
345#95—61no—
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×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3C22×D411yescharacteristic
365C3×D4—41no—
366C3×D4—41no—
367C3×D4—41no—
368C3×D4—41no—
369C3×D4—41no—
370C3×D4—41no—
371C3×D4—41no—
372C3×D4—41no—
373C3×D4—84no—
374C3×D4—84no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4#12112yes—
386C2×C3×C4#12112yes—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4H(5,𝔽2)11yescharacteristic
392C2×C3×C42−1+411yescharacteristic
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—22no—
398C2×Dic3—22no—
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—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—
417Dic6—22no—
418Dic6—22no—
419Dic6—41no—
420Dic6—41no—
421C3×C8—41no—
422C3×C8—41no—
423C3⋊C8—41no—
424C3⋊C8—41no—
425C24—21no—
426C24—21no—
427C2×Q8—32no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—81no—
450C2×D4—81no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—61no—
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—24no—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—32no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—121no—
487C4⋊C4—121no—
488C4⋊C4—122no—
489C4⋊C4—122no—
490C4⋊C4—241no—
491C4⋊C4C2×#6712yes—
492C4⋊C4C2×#6712yes—
493C22⋊C4—122no—
494C22⋊C4—122no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—241no—
498C22⋊C4—241no—
499C22⋊C4—241no—
500C22⋊C4—241no—
501C42—61no—
502C42—61no—
503C42—61no—
504C42—61no—
505C42—122no—
506C42—122no—
507C42—241no—
508C42—241no—
509C42#23712yes—
510C42C2×#6712yes—
511C22×C3—81no—
512C22×C3—81no—
513C22×C3—81no—
514C22×C3—81no—
515C22×C3—81no—
516C22×C3—81no—
517C22×C3—81no—
518C22×C3—81no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—81no—
525C22×C3C2×#12111yescharacteristic
526C22×C3#54611yescharacteristic
527C22×C3#54811yescharacteristic
528C22×C3#57511yescharacteristic
529C22×C3#57511yescharacteristic
530C22×C3#57611yescharacteristic
531C22×C3#57711yescharacteristic
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—42no—
535C3×C4—42no—
536C3×C4#52911yescharacteristic
537C3×C4#54211yescharacteristic
538C3×C4#54811yescharacteristic
539C3×C4#55011yescharacteristic
540Dic3—42no—
541Dic3—42no—
542Dic3—41no—
543Dic3—41no—
544Dic3—41no—
545Dic3—41no—
546Dic3—81no—
547Dic3—81no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23C2×D3×D411yescharacteristic
563Q8—62no—
564Q8—62no—
565Q8—121no—
566Q8—121no—
567D4—41no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572D4—41no—
573D4—41no—
574D4—41no—
575D4—84no—
576D4—84no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4#93912yes—
612C2×C4#93912yes—
613C2×C4C2×#23711yescharacteristic
614C2×C4D3×#4111yescharacteristic
615C2×C4D3×#4111yescharacteristic
616C2×C4#100911yescharacteristic
617C2×C4#101011yescharacteristic
618C2×C4C22×#6711yescharacteristic
619C8—41no—
620C8—41no—
621C8—121no—
622C8—121no—
623C2×C3—161no—
624C2×C3—161no—
625C2×C3#242611yescharacteristic
626C2×C3#304011yescharacteristic
627C2×C3#304511yescharacteristic
628C2×C3#330911yescharacteristic
629C2×C3#331711yescharacteristic
630C2×C3#337011yescharacteristic
631C2×C3#337111yescharacteristic
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22#567611yescharacteristic
647C22#567711yescharacteristic
648C22#577311yescharacteristic
649C22#577311yescharacteristic
650C22#577411yescharacteristic
651C22#577511yescharacteristic
652C22C2×#93911yescharacteristic
653C4—22no—
654C4—22no—
655C4—42no—
656C4—42no—
657C4—122no—
658C4—122no—
659C4—121no—
660C4—121no—
661C4—121no—
662C4—121no—
663C4—241no—
664C4—241no—
665C4#568011yescharacteristic
666C4#568211yescharacteristic
667C4#583111yescharacteristic
668C4#583411yescharacteristic
669C3#2529111yescharacteristic, complemented, Sylow
670C2—161no—
671C2—161no—
672C2#8171011yescharacteristic
673C2#8171111yescharacteristic
674C2#8172411yescharacteristic
675C2#8172611yescharacteristic
676C2#8299511yescharacteristic
677C2#8393311yescharacteristic
678C2#8393611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#490C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenterC2×#490D3
hypocenterC3#25291
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#490

Complements

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

NH ≅ G/Ntype
3. #83025C2semidirect
4. C2×#5063C2semidirect
6. #70194C2semidirect
7. #70193C2semidirect
8. #70178C2semidirect
9. #70178C2semidirect
10. #70174C2semidirect
11. #70173C2semidirect
12. #70162C2semidirect
13. #70162C2semidirect
14. #69836C2semidirect
15. #69813C2semidirect
669. C3#25291semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4902
#831792
#830252
C2×#50632
#717612
#701942
#701932
#701782
#701742
#701732
#701622
#698362
#698132
#252913