Finite Group Databasebeta

← #10855236

Subgroups — #10855236

Order 768.

918 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2445C211yesmaximal, characteristic
2C2×#5075C211yesmaximal, characteristic
3C2×C4×#214C211yesmaximal, characteristic, complemented
4C4⋊C4×Dic6C211yesmaximal, characteristic, complemented
5#71762C211yesmaximal, characteristic
6#71607C211yesmaximal, characteristic, complemented
7#69660C211yesmaximal, characteristic, complemented
8#69660C211yesmaximal, characteristic, complemented
9#69660C211yesmaximal, characteristic, complemented
10#69656C211yesmaximal, characteristic, complemented
11#69644C211yesmaximal, characteristic, complemented
12#69640C211yesmaximal, characteristic, complemented
13#69632C211yesmaximal, characteristic, complemented
14#69624C211yesmaximal, characteristic, complemented
15#69482C211yesmaximal, characteristic, complemented
16#26163—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×C3×#123—21no—
19C2×C3×C4×D4C2211yescharacteristic
20C22×#214C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C3×#394—21no—
26C3×#394—21no—
27C3×#390C2211yescharacteristic
28C3×#390C2211yescharacteristic
29C3×#390C2211yescharacteristic
30C3×#389C2211yescharacteristic
31C3×#386—21no—
32C3×C4×C4⋊C4C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38C2×#832C2211yescharacteristic
39#5075—24no—
40C4×#214—24no—
41#4975—21no—
42Dic3×C4⋊C4—21no—
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47#4935C2211yescharacteristic
48#4928C2211yescharacteristic
49#4928C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C2×C4×C3⋊C8C2211yescharacteristic
52#4647—21no—
53#4647—21no—
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#460—31no—
63C2×C4×SD16—31no—
64#2445D311yescharacteristic
65Q8×C4⋊C4—31no—
66#2094—31no—
67#2088—31no—
68#2071—31no—
69#2041—31no—
70#1934—31no—
71#1916—31no—
72#1880—31no—
73#1880—31no—
74#1880—31no—
75#1865—31no—
76#1864—31no—
77C22×C3×D4—21no—
78C22×C3×D4—21no—
79C22×C3×D4C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4—21no—
82C22×Dic6C2311yescharacteristic
83C3×#127—22no—
84C3×#127—41no—
85C3×#123—44no—
86C3×C4×D4—22no—
87C3×C4×D4—41no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
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×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2311yescharacteristic
103C2×#214—22no—
104C2×#214—22no—
105C2×#214—22no—
106C2×#214—22no—
107C2×#214—22no—
108C2×#214D414yes—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3—21no—
121C2×C4×Dic3—21no—
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C4×Dic6—22no—
127C4×Dic6—22no—
128C4×Dic6—22no—
129C4×Dic6—22no—
130C4×Dic6—41no—
131C4×Dic6—41no—
132C3×#95—21no—
133C3×#95D411yescharacteristic
134C3×#95D411yescharacteristic
135#832—22no—
136#832—22no—
137#832—22no—
138#832—22no—
139#832—22no—
140#832—41no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143#810—22no—
144#810—22no—
145#808—22no—
146#808—41no—
147C4⋊(C3⋊C8)—22no—
148C4×C3⋊C8—22no—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×#127C2×D311yescharacteristic
152C2×#123—21no—
153C2×C4×Q8—31no—
154C2×C4×Q8—31no—
155C2×C4×Q8—31no—
156C2×C4×D4C2×D311yescharacteristic
157#460—64no—
158C4×SD16—64no—
159C2×#106—31no—
160C2×#106—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×C4×C8—31no—
171#394—21no—
172#394—21no—
173#390C2×D311yescharacteristic
174#390C2×D311yescharacteristic
175#390C2×D311yescharacteristic
176#389—31no—
177#389—31no—
178#389—61no—
179#389—61no—
180#389C2×D311yescharacteristic
181#386—21no—
182#384—31no—
183#384—61no—
184C4×C4⋊C4—61no—
185C4×C4⋊C4C2×D311yescharacteristic
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#336—31no—
191#336—31no—
192#336—31no—
193#336—31no—
194C24×C3—21no—
195C24×C3—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—
213C2×C3×D4C2×D412yes—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—42no—
217C22×C3×C4—42no—
218C22×C3×C4—81no—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2411yescharacteristic
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6C2×D412yes—
234C2×Dic6#4112yes—
235C2×Dic6#4112yes—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—41no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4SD1614yescomplemented
243C3×C22⋊C4—42no—
244C3×C22⋊C4—42no—
245C3×C22⋊C4—42no—
246C3×C22⋊C4—42no—
247C3×C22⋊C4—81no—
248C3×C22⋊C4—81no—
249C3×C22⋊C4—81no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—21no—
256#214—28no—
257#214—44no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—42no—
264C4⋊Dic3—81no—
265C4⋊Dic3#4112yes—
266Dic3⋊C4—24no—
267Dic3⋊C4—42no—
268Dic3⋊C4—42no—
269Dic3⋊C4—81no—
270Dic3⋊C4—81no—
271C4×Dic3—24no—
272C4×Dic3—42no—
273C4×Dic3—42no—
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—
284C2×C3⋊C8—21no—
285C2×C3⋊C8C2×D412yes—
286C22×Q8—31no—
287C22×D4—21no—
288C22×D4—21no—
289C22×D4C22×D311yescharacteristic
290C22×D4C22×D311yescharacteristic
291C23×C4—21no—
292C2×SD16—34no—
293C2×SD16—62no—
294C2×SD16—62no—
295C2×SD16—62no—
296C2×SD16—62no—
297C2×SD16—62no—
298C22×C8—31no—
299C22×C8—31no—
300C22×C8—31no—
301C22×C8—31no—
302#127—22no—
303#127—41no—
304#123—44no—
305C4×Q8—62no—
306C4×Q8—62no—
307C4×Q8—62no—
308C4×Q8—62no—
309C4×Q8—121no—
310C4×Q8—121no—
311C4×D4—22no—
312C4×D4—41no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4#6711yescharacteristic
325C2×C4⋊C4#6711yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C42—61no—
336C2×C42—61no—
337C2×C42—61no—
338C2×C42C22×D311yescharacteristic
339C2×C42C22×D311yescharacteristic
340C2×C42C22×D311yescharacteristic
341#106—62no—
342#106—121no—
343C4⋊C8—62no—
344#103—62no—
345#103—62no—
346#102—62no—
347#102—62no—
348#102—62no—
349#102—62no—
350#102—62no—
351#102—121no—
352C4×C8—62no—
353#95—21no—
354#95—61no—
355#95—61no—
356#95#6711yescharacteristic
357#95#6711yescharacteristic
358C23×C3—22no—
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×C3—41no—
371C23×C3—41no—
372C23×C3C22×D411yescharacteristic
373C3×D4—22no—
374C3×D4—22no—
375C3×D4—41no—
376C3×D4—41no—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—41no—
381C3×D4—84no—
382C3×D4—84no—
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—21no—
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—21no—
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—84no—
405C2×C3×C4C2×SD1612yes—
406C2×C3×C4C2×SD1612yes—
407C2×C3×C4C2×SD1612yes—
408C2×C3×C4C22×D411yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×C3×C4C2×#4111yescharacteristic
412C2×C3×C4H(5,𝔽2)11yescharacteristic
413C2×C3×C42−1+411yescharacteristic
414C2×Dic3—22no—
415C2×Dic3—22no—
416C2×Dic3—22no—
417C2×Dic3—22no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426Dic6—22no—
427Dic6—22no—
428Dic6—22no—
429Dic6#12314yescomplemented
430C3⋊C8—22no—
431C3⋊C8—41no—
432C3⋊C8—41no—
433C3⋊C8—41no—
434C24—21no—
435C24—21no—
436C2×Q8—32no—
437C2×Q8—32no—
438C2×Q8—32no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×Q8—61no—
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—21no—
452C2×D4—21no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—81no—
458C2×D4—81no—
459C2×D4D3×D412yes—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—81no—
469C22×C4C2×#6711yescharacteristic
470C22×C4C2×#6711yescharacteristic
471C22×C4C2×#6711yescharacteristic
472C22×C4C2×#6711yescharacteristic
473C22×C4C2×#6711yescharacteristic
474C22×C4C2×#6711yescharacteristic
475C22×C4C23×D311yescharacteristic
476SD16—68no—
477SD16—124no—
478C2×C8—32no—
479C2×C8—61no—
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—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—32no—
495C4⋊C4—41no—
496C4⋊C4—42no—
497C4⋊C4—64no—
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—241no—
507C4⋊C4—241no—
508C4⋊C4—241no—
509C4⋊C4#21414yescomplemented
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—81no—
515C22⋊C4—81no—
516C22⋊C4—81no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—64no—
524C42—122no—
525C42—122no—
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×C3—81no—
538C22×C3—81no—
539C22×C3—81no—
540C22×C3#52511yescharacteristic
541C22×C3D4211yescharacteristic
542C22×C3#54811yescharacteristic
543C22×C3C22×SD1611yescharacteristic
544C22×C3#57511yescharacteristic
545C22×C3#57611yescharacteristic
546C22×C3#57711yescharacteristic
547C3×C4—22no—
548C3×C4—22no—
549C3×C4—22no—
550C3×C4—41no—
551C3×C4—41no—
552C3×C4—41no—
553C3×C4—81no—
554C3×C4—81no—
555C3×C4C2×#12311yescharacteristic
556C3×C4#53911yescharacteristic
557C3×C4#54711yescharacteristic
558C3×C4#55311yescharacteristic
559Dic3—24no—
560Dic3—42no—
561Dic3—81no—
562Dic3—81no—
563C23—22no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23C22×#6711yescharacteristic
578Q8—34no—
579Q8—62no—
580Q8—62no—
581Q8—62no—
582D4—22no—
583D4—22no—
584D4—41no—
585D4—41no—
586D4—41no—
587D4—41no—
588D4—41no—
589D4—41no—
590D4—84no—
591D4—84no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—62no—
616C2×C4—62no—
617C2×C4—84no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4C2×#21412yes—
627C2×C4C2×#21412yes—
628C2×C4C2×#21412yes—
629C2×C4C2×D3×D411yescharacteristic
630C2×C4#100411yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4D3×#4111yescharacteristic
633C2×C4D3×#4111yescharacteristic
634C2×C4#101011yescharacteristic
635C8—62no—
636C8—121no—
637C8—121no—
638C8—121no—
639C2×C3—82no—
640C2×C3—161no—
641C2×C3#244311yescharacteristic
642C2×C3#310311yescharacteristic
643C2×C3#311911yescharacteristic
644C2×C3#326811yescharacteristic
645C2×C3D4×SD1611yescharacteristic
646C2×C3#336211yescharacteristic
647C2×C3#337111yescharacteristic
648C22—42no—
649C22—42no—
650C22—42no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22C22×#21411yescharacteristic
663C22D4×#6711yescharacteristic
664C22#582211yescharacteristic
665C22#582811yescharacteristic
666C22#583111yescharacteristic
667C22#583511yescharacteristic
668C22#583611yescharacteristic
669C4—22no—
670C4—22no—
671C4—22no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—64no—
676C4—81no—
677C4—81no—
678C4—122no—
679C4—241no—
680C4—241no—
681C4D3×#12311yescharacteristic
682C4#565511yescharacteristic
683C4#566011yescharacteristic
684C4#566111yescharacteristic
685C3#2616311yescharacteristic, complemented, Sylow
686C2—82no—
687C2—161no—
688C2#8289711yescharacteristic
689C2#8331511yescharacteristic
690C2#8343411yescharacteristic
691C2D4×#21411yescharacteristic
692C2#8350711yescharacteristic
693C2#8358611yescharacteristic
694C2#8371811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2445C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenter#2445D3
hypocenterC3#26163
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2445

Complements

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

NH ≅ G/Ntype
3. C2×C4×#214C2semidirect
4. C4⋊C4×Dic6C2semidirect
6. #71607C2semidirect
7. #69660C2semidirect
8. #69660C2semidirect
9. #69660C2semidirect
10. #69656C2semidirect
11. #69644C2semidirect
12. #69640C2semidirect
13. #69632C2semidirect
14. #69624C2semidirect
15. #69482C2semidirect
242. C3×C4⋊C4SD16semidirect
429. Dic6#123semidirect
509. C4⋊C4#214semidirect
685. C3#26163semidirect

Maximal subgroups

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

subgroupindex
C3×#24452
C2×#50752
C2×C4×#2142
C4⋊C4×Dic62
#717622
#716072
#696602
#696562
#696442
#696402
#696322
#696242
#694822
#261633