Finite Group Databasebeta

← #10800181

Subgroups — #10800181

Order 768.

1426 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#237C211yesmaximal, characteristic
2C22×#815C211yesmaximal, characteristic, complemented
3C2×#4719C211yesmaximal, characteristic, complemented
4C3×#1869C211yesmaximal, characteristic, complemented
5#71975C211yesmaximal, characteristic, complemented
6#71610C211yesmaximal, characteristic, complemented
7#69952C211yesmaximal, characteristic, complemented
8#69917C212yesmaximal, complemented
9#69906C212yesmaximal, complemented
10#68620C212yesmaximal, complemented
11#68600C212yesmaximal, complemented
12#21379—31nomaximal, Sylow
13C22×#237C411yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×D4×Dic3C411yescharacteristic
17C2×C4×#67C2211yescharacteristic
18C2×C4×#67C2211yescharacteristic
19C22×C4×Dic3C2211yescharacteristic
20C22×C4×Dic3C2211yescharacteristic
21C23×C3⋊C8C2211yescharacteristic
22C4×#237—24no—
23C4×#237—24no—
24C2×#892C411yescharacteristic
25C2×#887C411yescharacteristic
26C2×#67⋊C4C411yescharacteristic
27C2×#67⋊C4C411yescharacteristic
28C2×#877C411yescharacteristic
29C2×#877C411yescharacteristic
30C2×C42×D3C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C2×C3×C22⋊C8C2212yes—
33C2×C3×C22⋊C8C2211yescharacteristic
34C2×#830C2212yes—
35C2×#830C2211yescharacteristic
36C2×(C2×D3)⋊C8C2212yes—
37C2×(C2×D3)⋊C8C2212yes—
38C2×#815—24no—
39C2×#815C2212yes—
40C2×#815C2212yes—
41C2×#815C2212yes—
42C2×Dic3⋊C8C2212yes—
43#4719—24no—
44C3×#336C2212yes—
45#4550C2212yes—
46#4550C2212yes—
47#4524C2212yes—
48C2×C4×#41—31no—
49C2×#432—31no—
50C22×#97—31no—
51#1985—32no—
52#1955—32no—
53#1869—31no—
54#1869—31no—
55#1869—31no—
56#1869D311yescharacteristic, complemented
57#1749—32no—
58#1749—32no—
59C22×C3×D4C2×C411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67C2×C411yescharacteristic
63C22×#67C2×C411yescharacteristic
64C23×Dic3C2×C411yescharacteristic
65C23×Dic3C2×C411yescharacteristic
66C23×Dic3C2×C411yescharacteristic
67C23×Dic3C2×C411yescharacteristic
68C2×#237—24no—
69C2×#237—24no—
70C2×#237—24no—
71C22×C4×D3C2×C411yescharacteristic
72C22×C4×D3C2×C411yescharacteristic
73C22×C4×D3C2311yescharacteristic
74C22×Dic6C2×C411yescharacteristic
75C22×C3×C8—22no—
76C22×C3×C8—24no—
77C22×C3×C8C2312yes—
78C22×C3×C8C2312yes—
79C3×C4×D4—22no—
80C3×C4×D4—22no—
81C2×C3×C4⋊C4C2×C411yescharacteristic
82C2×C3×C22⋊C4C2×C411yescharacteristic
83C2×C3×C22⋊C4C2×C411yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×#217C2×C411yescharacteristic
86C2×#217C2×C411yescharacteristic
87D4×Dic3—24no—
88C4×#67—22no—
89C4×#67—22no—
90C4×#67—22no—
91C4×#67—22no—
92C2×(C2×D3)⋊C4C2×C411yescharacteristic
93C2×(C2×D3)⋊C4C2×C411yescharacteristic
94C2×C4⋊Dic3C2×C411yescharacteristic
95C2×Dic3⋊C4C2×C411yescharacteristic
96C2×Dic3⋊C4C2×C411yescharacteristic
97C2×Dic3⋊C4C2×C411yescharacteristic
98C2×Dic3⋊C4C2×C411yescharacteristic
99C2×C4×Dic3—22no—
100C2×C4×Dic3—22no—
101C2×C4×Dic3C2×C411yescharacteristic
102C2×C4×Dic3C2×C411yescharacteristic
103C2×C4×Dic3C2×C411yescharacteristic
104C2×C4×Dic3C2×C411yescharacteristic
105C2×C4×Dic3C2×C411yescharacteristic
106C2×C4×Dic3C2×C411yescharacteristic
107C2×C4×Dic3D414yes—
108C2×C4×Dic3D414yes—
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C22×C3⋊C8—22no—
113C22×C3⋊C8—24no—
114C22×C3⋊C8C2312yes—
115C22×C3⋊C8C2312yes—
116#892—24no—
117#887—24no—
118#67⋊C4—24no—
119#67⋊C4—24no—
120#877—24no—
121#877—24no—
122C42×D3—22no—
123C42×D3—22no—
124C4×Dic6—22no—
125C4×Dic6—22no—
126C3×C22⋊C8—22no—
127C3×C22⋊C8—24no—
128#830—22no—
129#830—24no—
130(C2×D3)⋊C8—24no—
131(C2×D3)⋊C8—24no—
132#815—24no—
133#815—24no—
134#815—24no—
135Dic3⋊C8—24no—
136C22×#41—31no—
137C23×C8—31no—
138C23×C8C2×D311yescharacteristic
139C4×#41—64no—
140C4×#41—64no—
141C2×C4×Q8—31no—
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×C4×D4—31no—
146C2×C4×D4—31no—
147C2×C4×D4—31no—
148C2×C4×D4C2×D311yescharacteristic
149C2×#117—31no—
150C2×#117—31no—
151C2×#117—31no—
152C22×C42—31no—
153C22×C42—31no—
154C22×C42—31no—
155#432—64no—
156C2×C4⋊C8—32no—
157C2×C22⋊C8—31no—
158C2×C22⋊C8—32no—
159C2×C22⋊C8—32no—
160C2×C22⋊C8—32no—
161C2×C22⋊C8C2×D311yescharacteristic
162C2×C22⋊C8C2×D312yes—
163C2×#97—32no—
164C2×#97—32no—
165C2×#97—32no—
166C2×#97—64no—
167#336—32no—
168#336—32no—
169#336—32no—
170#336C2×D312yes—
171C24×C3C22×C411yescharacteristic
172C24×C3C22×C411yescharacteristic
173C23×D3C22×C411yescharacteristic
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4M1614yes—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4C22×C411yescharacteristic
184C22×C3×C4C22×C411yescharacteristic
185C22×C3×C4C22×C411yescharacteristic
186C22×C3×C4C22×C411yescharacteristic
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2411yescharacteristic
190C2×#67—22no—
191C2×#67—22no—
192C2×#67—22no—
193C2×#67—22no—
194C2×#67—22no—
195C2×#67M1614yes—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—42no—
203C22×Dic3—42no—
204C22×Dic3C22⋊C414yes—
205C22×Dic3C22⋊C414yes—
206C22×Dic3C22×C411yescharacteristic
207C22×Dic3C22×C411yescharacteristic
208C22×Dic3C22×C411yescharacteristic
209C22×Dic3C22×C411yescharacteristic
210C22×Dic3C22×C411yescharacteristic
211C22×Dic3C22×C411yescharacteristic
212C22×Dic3C22×C411yescharacteristic
213#237—232no—
214C2×C4×D3—22no—
215C2×C4×D3—22no—
216C2×C4×D3—22no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—22no—
220C2×C4×D3—22no—
221C2×C4×D3—22no—
222C2×C4×D3—42no—
223C2×Dic6—22no—
224C2×Dic6—22no—
225C2×Dic6—22no—
226C2×C3×C8—28no—
227C2×C3×C8—22no—
228C2×C3×C8—22no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C2×C3×C8—22no—
232C2×C3×C8#4114yescomplemented
233C3×C4⋊C4—22no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C42—21no—
237C3×C42—21no—
238#217—22no—
239#217—22no—
240(C2×D3)⋊C4—22no—
241(C2×D3)⋊C4—22no—
242C4⋊Dic3—22no—
243Dic3⋊C4—22no—
244Dic3⋊C4—22no—
245Dic3⋊C4—22no—
246Dic3⋊C4—22no—
247C4×Dic3—21no—
248C4×Dic3—21no—
249C4×Dic3—22no—
250C4×Dic3—22no—
251C4×Dic3—22no—
252C4×Dic3—22no—
253C4×Dic3C22⋊C414yes—
254C4×Dic3C22⋊C414yes—
255C4×Dic3C2×D412yes—
256C4×Dic3C2×D412yes—
257C4×Dic3C2×D412yes—
258C4×Dic3C2×D412yes—
259C2×C3⋊C8—28no—
260C2×C3⋊C8—22no—
261C2×C3⋊C8—22no—
262C2×C3⋊C8—22no—
263C2×C3⋊C8—22no—
264C2×C3⋊C8—22no—
265C2×C3⋊C8#4114yescomplemented
266C2×#41—64no—
267C2×#41—64no—
268C2×#41—64no—
269C22×Q8—31no—
270C22×D4—31no—
271C22×D4—31no—
272C22×D4C4×D311yescharacteristic
273C23×C4—31no—
274C23×C4—31no—
275C23×C4—31no—
276C23×C4—31no—
277C23×C4—31no—
278C23×C4—31no—
279C23×C4—31no—
280C23×C4C22×D311yescharacteristic
281C23×C4C22×D311yescharacteristic
282C22×C8—22no—
283C22×C8—24no—
284C22×C8—32no—
285C22×C8—32no—
286C22×C8—64no—
287C22×C8—62no—
288C22×C8C22×D312yes—
289C22×C8C22×D312yes—
290C4×Q8—62no—
291C4×Q8—62no—
292C4×Q8—64no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—64no—
296C4×D4—62no—
297C4×D4—62no—
298C4×D4—64no—
299C4×D4—64no—
300C4×D4—62no—
301C4×D4—62no—
302#117—64no—
303#117—64no—
304#117—64no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4C4×D311yescharacteristic
311C2×C22⋊C4—31no—
312C2×C22⋊C4—31no—
313C2×C22⋊C4—31no—
314C2×C22⋊C4—31no—
315C2×C22⋊C4C4×D311yescharacteristic
316C2×C22⋊C4C4×D311yescharacteristic
317C2×C42—31no—
318C2×C42—31no—
319C2×C42—34no—
320C2×C42—34no—
321C2×C42—31no—
322C2×C42—31no—
323C2×C42—31no—
324C2×C42—31no—
325C2×C42—31no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—62no—
329C2×C42—62no—
330C2×C42—62no—
331C2×C42—62no—
332C2×C42C22×D311yescharacteristic
333C4⋊C8—64no—
334C22⋊C8—22no—
335C22⋊C8—24no—
336C22⋊C8—64no—
337C22⋊C8—64no—
338C22⋊C8—62no—
339C22⋊C8—64no—
340#97—64no—
341#97—64no—
342#97—64no—
343C23×C3—22no—
344C23×C3—22no—
345C23×C3—21no—
346C23×C3—22no—
347C23×C3—21no—
348C23×C3—22no—
349C23×C3C2×C22⋊C412yes—
350C23×C3C2×M1612yes—
351C23×C3C2×M1612yes—
352C23×C3C2×M1612yes—
353C23×C3C23×C411yescharacteristic
354C22×D3—21no—
355C22×D3—22no—
356C22×D3—21no—
357C22×D3—22no—
358C22×D3C2×M1612yes—
359C3×D4—28no—
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—24no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—42no—
373C2×C3×C4C2×C22⋊C412yes—
374C2×C3×C4C2×M1612yes—
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C2×#4112yes—
378C2×C3×C4C2×#4112yes—
379#67—28no—
380#67—28no—
381C2×Dic3—28no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—28no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—44no—
396C2×Dic3—44no—
397C2×Dic3C2×C22⋊C412yes—
398C2×Dic3C2×C22⋊C412yes—
399C2×Dic3C2×C22⋊C412yes—
400C2×Dic3C2×C22⋊C412yes—
401C2×Dic3C2×M1612yes—
402C4×D3—28no—
403C4×D3—44no—
404C4×D3—41no—
405C4×D3—41no—
406C4×D3—42no—
407Dic6—28no—
408C3×C8—24no—
409C3×C8—42no—
410C3⋊C8—24no—
411C3⋊C8—42no—
412C24—31no—
413C24C2×C4×D311yescharacteristic
414C24C2×C4×D311yescharacteristic
415#41—632no—
416C2×Q8—62no—
417C2×Q8—62no—
418C2×Q8—62no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—34no—
422C2×D4—62no—
423C2×D4—62no—
424C2×D4—62no—
425C2×D4—62no—
426C2×D4—62no—
427C2×D4ΓL(1,𝔽25)14yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—34no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—34no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—122no—
458C22×C4—122no—
459C22×C4—122no—
460C22×C4C2×C4×D311yescharacteristic
461C22×C4C2×C4×D311yescharacteristic
462C22×C4C2×C4×D311yescharacteristic
463C22×C4C2×C4×D311yescharacteristic
464C22×C4D3×D412yes—
465C22×C4D3×D412yes—
466C22×C4C23×D311yescharacteristic
467C2×C8—28no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—34no—
474C2×C8—68no—
475C2×C8—62no—
476C2×C8—62no—
477C2×C8—62no—
478C2×C8—62no—
479C2×C8—62no—
480C2×C8#23714yescomplemented
481C4⋊C4—22no—
482C4⋊C4—62no—
483C4⋊C4—62no—
484C4⋊C4—62no—
485C4⋊C4—62no—
486C4⋊C4—62no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—62no—
490C22⋊C4—62no—
491C22⋊C4—62no—
492C22⋊C4—62no—
493C42—21no—
494C42—21no—
495C42—34no—
496C42—34no—
497C42—32no—
498C42—32no—
499C42—32no—
500C42—32no—
501C42—62no—
502C42—62no—
503C42—62no—
504C42—62no—
505C42—61no—
506C42—61no—
507C22×C3—22no—
508C22×C3—24no—
509C22×C3—24no—
510C22×C3—22no—
511C22×C3—22no—
512C22×C3—24no—
513C22×C3—41no—
514C22×C3—42no—
515C22×C3—41no—
516C22×C3#40714yes—
517C22×C3C22×C22⋊C411yescharacteristic
518C22×C3C4×#4112yes—
519C22×C3C22×M1611yescharacteristic
520C22×C3C22×M1611yescharacteristic
521C22×C3#56812yes—
522C2×D3—22no—
523C2×D3—24no—
524C2×D3—41no—
525C2×D3—42no—
526C2×D3—41no—
527C3×C4—22no—
528C3×C4—22no—
529C3×C4—41no—
530C3×C4—41no—
531C3×C4#52511yescharacteristic
532C3×C4#52511yescharacteristic
533C3×C4#53212yes—
534Dic3—22no—
535Dic3—22no—
536Dic3—22no—
537Dic3—22no—
538Dic3—22no—
539Dic3—41no—
540Dic3—41no—
541C23—22no—
542C23—22no—
543C23—21no—
544C23—21no—
545C23—22no—
546C23—22no—
547C23—32no—
548C23—61no—
549C23—61no—
550C23—62no—
551C23—62no—
552C23D3×C22⋊C412yes—
553C23C2×ΓL(1,𝔽25)12yes—
554C23C2×ΓL(1,𝔽25)12yes—
555C23D3×M1612yes—
556C23C22×C4×D311yescharacteristic
557Q8—68no—
558D4—28no—
559D4—68no—
560D4—68no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—32no—
572C2×C4—32no—
573C2×C4—32no—
574C2×C4—32no—
575C2×C4—32no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—68no—
580C2×C4—68no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—68no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—124no—
595C2×C4—124no—
596C2×C4—124no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—122no—
600C2×C4D3×C22⋊C412yes—
601C2×C4C2×ΓL(1,𝔽25)12yes—
602C2×C4C2×D3×D411yescharacteristic
603C2×C4C2×D3×D411yescharacteristic
604C2×C4C2×#23712yes—
605C2×C4D3×#4112yes—
606C8—24no—
607C8—42no—
608C8—64no—
609C8—122no—
610C2×C3—24no—
611C2×C3—42no—
612C2×C3#229711yescharacteristic
613C2×C3C2×#40711yescharacteristic
614C2×C3#287911yescharacteristic
615C2×C3#297012yes—
616C2×C3#297012yes—
617D3—42no—
618C22—24no—
619C22—22no—
620C22—24no—
621C22—22no—
622C22—22no—
623C22—24no—
624C22—41no—
625C22—42no—
626C22—41no—
627C22—62no—
628C22—64no—
629C22—122no—
630C22—121no—
631C22—121no—
632C22#472614yes—
633C22C2×D3×C22⋊C411yescharacteristic
634C22C4×#23712yes—
635C22C22×ΓL(1,𝔽25)11yescharacteristic
636C22C2×D3×M1611yescharacteristic
637C22#575012yes—
638C4—22no—
639C4—22no—
640C4—41no—
641C4—41no—
642C4—62no—
643C4—62no—
644C4—62no—
645C4—62no—
646C4—62no—
647C4—121no—
648C4—121no—
649C4#558611yescharacteristic
650C4#559612yes—
651C4#565611yescharacteristic
652C3#2137911yescharacteristic, complemented, Sylow
653C2—24no—
654C2—42no—
655C2—122no—
656C2#7465811yescharacteristic
657C2C2×#472611yescharacteristic
658C2#7866211yescharacteristic
659C2#7940412yes—
660C2#7946012yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1869C2
prime radicalC22×#237C4
cosocleC22×C3×C4C24
hypercenter#1869D3
hypocenterC3#21379
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1869

Complements

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

NH ≅ G/Ntype
2. C22×#815C2semidirect
3. C2×#4719C2semidirect
4. C3×#1869C2semidirect
5. #71975C2semidirect
6. #71610C2semidirect
7. #69952C2semidirect
8. #69917C2semidirect
9. #69906C2semidirect
10. #68620C2semidirect
11. #68600C2semidirect
56. #1869D3semidirect
232. C2×C3×C8#41semidirect
265. C2×C3⋊C8#41semidirect
480. C2×C8#237semidirect
652. C3#21379semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2372
C22×#8152
C2×#47192
C3×#18692
#719752
#716102
#699522
#699172
#699062
#686202
#686002
#213793