Finite Group Databasebeta

← #10845202

Subgroups — #10845202

Order 768.

916 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2366C211yesmaximal, characteristic
2C2×#5075C211yesmaximal, characteristic
3C2×#5071C211yesmaximal, characteristic
4#73719C211yesmaximal, characteristic, complemented
5#71612C211yesmaximal, characteristic, complemented
6#71612C211yesmaximal, characteristic, complemented
7#71605C211yesmaximal, characteristic, complemented
8#71604C211yesmaximal, characteristic, complemented
9#69651C211yesmaximal, characteristic, complemented
10#69651C211yesmaximal, characteristic, complemented
11#69641C211yesmaximal, characteristic, complemented
12#69640C211yesmaximal, characteristic, complemented
13#69491C211yesmaximal, characteristic, complemented
14#69485C211yesmaximal, characteristic, complemented
15#69475C211yesmaximal, characteristic, complemented
16#25332—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×C3×C4×D4—21no—
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C22×#214C2211yescharacteristic
23C2×#870C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C3×#390—21no—
26C3×#390C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#386—21no—
29C3×#386—21no—
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C4⋊C4C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38#5075—24no—
39#5071—24no—
40#4981—21no—
41#4981—21no—
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44#4936C2211yescharacteristic
45#4935C2211yescharacteristic
46#4929C2211yescharacteristic
47#4929C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C2×#803C2211yescharacteristic
52#4653—21no—
53#4653—21no—
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#490—31no—
63C2×#460—31no—
64#2565—31no—
65#2366D311yescharacteristic
66#2073—31no—
67#2071—31no—
68#2055—31no—
69#2055—31no—
70#1989—31no—
71#1940—31no—
72#1877—31no—
73#1877—31no—
74#1804—31no—
75#1802—31no—
76#1763—31no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4—21no—
82C23×C3×C4—21no—
83C22×Dic6C2311yescharacteristic
84C3×#127—22no—
85C3×#127D414yes—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4—41no—
89C3×C4×D4—41no—
90C3×C4×D4—44no—
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C42—21no—
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C2311yescharacteristic
105C2×#214—22no—
106C2×#214—22no—
107C2×#214—22no—
108C2×#214—22no—
109C2×#214—41no—
110C2×#214—41no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×C4×Dic3—21no—
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#870—22no—
125#870—22no—
126C4×Dic6—22no—
127C4×Dic6—22no—
128C3×#95—21no—
129C3×#95D411yescharacteristic
130C3×#95D411yescharacteristic
131#832—22no—
132#832—22no—
133#832—22no—
134#832—22no—
135#832—22no—
136#832—22no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143#810—22no—
144#810—22no—
145C4⋊(C3⋊C8)—22no—
146C4⋊(C3⋊C8)—41no—
147C4⋊(C3⋊C8)—41no—
148#803—22no—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×#127C2×D311yescharacteristic
152C2×#125—31no—
153C2×C4×Q8—31no—
154C2×C4×D4—21no—
155C2×C4×D4C2×D311yescharacteristic
156C2×C4×D4C2×D311yescharacteristic
157#490—64no—
158#460—64no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#97—31no—
171#398—61no—
172#398—61no—
173#391—31no—
174#391—31no—
175#390—21no—
176#390C2×D311yescharacteristic
177#390C2×D311yescharacteristic
178#387—61no—
179#387—61no—
180#386—21no—
181#386—21no—
182#384—31no—
183#382—31no—
184C4×C4⋊C4C2×D311yescharacteristic
185C4×C4⋊C4C2×D311yescharacteristic
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#336—31no—
193#336—31no—
194C24×C3—21no—
195C24×C3—21no—
196C2×C3×D4—24no—
197C2×C3×D4—24no—
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—21no—
208C2×C3×D4—42no—
209C2×C3×D4—42no—
210C2×C3×D4—81no—
211C2×C3×D4—81no—
212C2×C3×D4C2×D412yes—
213C2×C3×D4C2×D412yes—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—42no—
217C22×C3×C4—42no—
218C22×C3×C4—42no—
219C22×C3×C4—42no—
220C22×C3×C4—81no—
221C22×C3×C4—81no—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2411yescharacteristic
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6#4112yes—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4#4112yes—
245C3×C22⋊C4—42no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—42no—
248C3×C22⋊C4—42no—
249C3×C22⋊C4—81no—
250C3×C22⋊C4—81no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—42no—
256C3×C42C2×D412yes—
257C3×C42#4112yes—
258#214—44no—
259#214—44no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—42no—
267C4⋊Dic3—42no—
268Dic3⋊C4—42no—
269Dic3⋊C4—42no—
270Dic3⋊C4—42no—
271C4×Dic3—42no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C22×Q8—31no—
285C22×D4—21no—
286C22×D4—21no—
287C22×D4C22×D311yescharacteristic
288C22×D4C22×D311yescharacteristic
289C23×C4—21no—
290C23×C4—21no—
291C2×SD16—62no—
292C2×SD16—62no—
293C2×SD16—62no—
294C2×SD16—62no—
295C2×SD16—121no—
296C2×SD16—121no—
297C22×C8—31no—
298C22×C8—31no—
299C22×C8—31no—
300C22×C8—31no—
301#127—22no—
302#127D1214yes—
303#125—62no—
304#125—62no—
305C4×Q8—62no—
306C4×Q8—62no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—41no—
310C4×D4—41no—
311C4×D4—44no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4#6711yescharacteristic
321C2×C4⋊C4#6711yescharacteristic
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C42—21no—
331C2×C42—61no—
332C2×C42C22×D311yescharacteristic
333C2×C42C22×D311yescharacteristic
334C2×C42C22×D311yescharacteristic
335C4⋊C8—62no—
336C4⋊C8—121no—
337C4⋊C8—121no—
338#103—62no—
339#103—62no—
340#102—62no—
341#102—62no—
342#102—62no—
343#102—62no—
344#102—62no—
345#102—62no—
346#97—62no—
347#95—21no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95#6711yescharacteristic
355#95#6711yescharacteristic
356C23×C3—22no—
357C23×C3—22no—
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×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—41no—
370C23×C3C22×D411yescharacteristic
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—22no—
374C3×D4—22no—
375C3×D4—48no—
376C3×D4—41no—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—84no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—84no—
403C2×C3×C4—84no—
404C2×C3×C4#12112yes—
405C2×C3×C4#12112yes—
406C2×C3×C4AGL(1,ℤ/8)12yes—
407C2×C3×C4C22×D411yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×C3×C4H(5,𝔽2)11yescharacteristic
412C2×C3×C42−1+411yescharacteristic
413C2×Dic3—22no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425Dic6—22no—
426Dic6—22no—
427Dic6—41no—
428Dic6—41no—
429C3⋊C8—41no—
430C3⋊C8—41no—
431C3⋊C8—41no—
432C3⋊C8—41no—
433C24—21no—
434C24—21no—
435C2×Q8—32no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×D4—24no—
442C2×D4—24no—
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—21no—
452C2×D4—21no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—81no—
456C2×D4—81no—
457C2×D4C2×D1212yes—
458C2×D4C2×D1212yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4C2×#6711yescharacteristic
472C22×C4C2×#6711yescharacteristic
473C22×C4C2×#6711yescharacteristic
474C22×C4C2×#6711yescharacteristic
475C22×C4C2×#6711yescharacteristic
476C22×C4C2×#6711yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—124no—
479SD16—124no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—41no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—122no—
504C4⋊C4—122no—
505C4⋊C4—122no—
506C4⋊C4—122no—
507C4⋊C4—122no—
508C4⋊C4#23712yes—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—81no—
514C22⋊C4—81no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—122no—
521C42C2×D1212yes—
522C42#23512yes—
523C22×C3—42no—
524C22×C3—42no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—81no—
530C22×C3—81no—
531C22×C3—81no—
532C22×C3—81no—
533C22×C3—81no—
534C22×C3—81no—
535C22×C3—81no—
536C22×C3—81no—
537C22×C3C2×#12111yescharacteristic
538C22×C3#54611yescharacteristic
539C22×C3#54811yescharacteristic
540C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
541C22×C3#57511yescharacteristic
542C22×C3#57711yescharacteristic
543C22×C3#57711yescharacteristic
544C3×C4—22no—
545C3×C4—22no—
546C3×C4—22no—
547C3×C4—41no—
548C3×C4—42no—
549C3×C4—41no—
550C3×C4—41no—
551C3×C4—81no—
552C3×C4#52911yescharacteristic
553C3×C4#54211yescharacteristic
554C3×C4#54611yescharacteristic
555C3×C4#54711yescharacteristic
556Dic3—42no—
557Dic3—81no—
558Dic3—81no—
559Dic3—81no—
560C23—22no—
561C23—22no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23C22×#6711yescharacteristic
575Q8—62no—
576Q8—62no—
577Q8—121no—
578Q8—121no—
579D4—22no—
580D4—22no—
581D4—22no—
582D4—22no—
583D4—48no—
584D4—41no—
585D4—41no—
586D4—41no—
587D4—41no—
588D4—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—62no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4#93012yes—
625C2×C4#93012yes—
626C2×C4#93212yes—
627C2×C4C22×D1211yescharacteristic
628C2×C4C2×#23511yescharacteristic
629C2×C4C2×#23711yescharacteristic
630C2×C4#100411yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4#101011yescharacteristic
633C8—121no—
634C8—121no—
635C8—121no—
636C8—121no—
637C2×C3—82no—
638C2×C3—82no—
639C2×C3#242711yescharacteristic
640C2×C3#303611yescharacteristic
641C2×C3#304211yescharacteristic
642C2×C3#331011yescharacteristic
643C2×C3#331711yescharacteristic
644C2×C3#336811yescharacteristic
645C2×C3#337111yescharacteristic
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22C2×#93011yescharacteristic
661C22C2×#93211yescharacteristic
662C22#580611yescharacteristic
663C22#582211yescharacteristic
664C22#583111yescharacteristic
665C22#583611yescharacteristic
666C22#583611yescharacteristic
667C4—22no—
668C4—22no—
669C4—22no—
670C4—41no—
671C4—42no—
672C4—41no—
673C4—41no—
674C4—81no—
675C4—122no—
676C4—241no—
677C4—241no—
678C4—241no—
679C4#555411yescharacteristic
680C4#555511yescharacteristic
681C4#555711yescharacteristic
682C4#556111yescharacteristic
683C3#2533211yescharacteristic, complemented, Sylow
684C2—82no—
685C2—82no—
686C2#8230711yescharacteristic
687C2#8240811yescharacteristic
688C2#8254411yescharacteristic
689C2#8348411yescharacteristic
690C2#8357311yescharacteristic
691C2#8358711yescharacteristic
692C2#8371811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2366C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenter#2366D3
hypocenterC3#25332
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2366

Complements

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

NH ≅ G/Ntype
4. #73719C2semidirect
5. #71612C2semidirect
6. #71612C2semidirect
7. #71605C2semidirect
8. #71604C2semidirect
9. #69651C2semidirect
10. #69651C2semidirect
11. #69641C2semidirect
12. #69640C2semidirect
13. #69491C2semidirect
14. #69485C2semidirect
15. #69475C2semidirect
683. C3#25332semidirect

Maximal subgroups

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

subgroupindex
C3×#23662
C2×#50752
C2×#50712
#737192
#716122
#716052
#716042
#696512
#696412
#696402
#694912
#694852
#694752
#253323