Finite Group Databasebeta

← #10800182

Subgroups — #10800182

Order 768.

1426 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#235C211yesmaximal, characteristic
2C2×#4705C211yesmaximal, characteristic, complemented
3C2×D3×#97C211yesmaximal, characteristic, complemented
4C3×#1749C211yesmaximal, characteristic, complemented
5#71529C211yesmaximal, characteristic, complemented
6#69957C212yesmaximal, complemented
7#69917C212yesmaximal, complemented
8#69807C211yesmaximal, characteristic, complemented
9#69807C211yesmaximal, characteristic, complemented
10#68668C212yesmaximal, complemented
11#68620C212yesmaximal, complemented
12#21379—31nomaximal, Sylow
13C22×#235C411yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C2×C4×#67C411yescharacteristic
16C2×C4×#67C411yescharacteristic
17C2×C4×#67C2211yescharacteristic
18C2×C4×#67C2211yescharacteristic
19C2×#926C411yescharacteristic
20C22×D3×C8C2212yes—
21C4×#235—24no—
22C4×#235—24no—
23C2×C4×D12C411yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×#872C411yescharacteristic
26C2×#872C411yescharacteristic
27C2×C42×D3C2211yescharacteristic
28C2×C42×D3C2211yescharacteristic
29C2×C4×Dic6C411yescharacteristic
30C2×C4×Dic6C2211yescharacteristic
31C2×C3×C22⋊C8C2212yes—
32C2×C3×#97C2211yescharacteristic
33C2×C3×#97C2211yescharacteristic
34C2×#830C2212yes—
35C2×(C2×D3)⋊C8C2212yes—
36C2×(C2×D3)⋊C8C2212yes—
37C2×(C2×D3)⋊C8C2212yes—
38C2×#815C2212yes—
39C2×Dic3⋊C8C2212yes—
40C2×#803C2211yescharacteristic
41C2×#803C2211yescharacteristic
42#4705—24no—
43D3×#97—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—32no—
54#1869—32no—
55#1749—31no—
56#1749—31no—
57#1749—31no—
58#1749D311yescharacteristic, complemented
59C23×C3×C4C2×C411yescharacteristic
60C23×C3×C4C2×C411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67C2×C411yescharacteristic
63C22×#67C2×C411yescharacteristic
64C2×#235—24no—
65C2×#235—24no—
66C2×#235—24no—
67C22×D12C2×C411yescharacteristic
68C22×C4×D3C2×C411yescharacteristic
69C22×C4×D3C2×C411yescharacteristic
70C22×C4×D3C2×C411yescharacteristic
71C22×C4×D3C2×C411yescharacteristic
72C22×C4×D3C2311yescharacteristic
73C22×C4×D3C2311yescharacteristic
74C22×Dic6C2×C411yescharacteristic
75C22×C3×C8C2312yes—
76C22×C3×C8C2312yes—
77C2×C3×C42—22no—
78C2×C3×C42—22no—
79C2×C3×C42C2×C411yescharacteristic
80C2×C3×C42C2×C411yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×#217C2×C411yescharacteristic
84C2×#217C2×C411yescharacteristic
85C4×#67—24no—
86C4×#67—22no—
87C4×#67—22no—
88C4×#67—24no—
89C4×#67—22no—
90C4×#67—22no—
91C2×(C2×D3)⋊C4C2×C411yescharacteristic
92C2×(C2×D3)⋊C4C2×C411yescharacteristic
93C2×(C2×D3)⋊C4C2×C411yescharacteristic
94C2×(C2×D3)⋊C4C2×C411yescharacteristic
95#926—24no—
96C2×C4⋊Dic3C2×C411yescharacteristic
97C2×C4⋊Dic3C2×C411yescharacteristic
98C2×Dic3⋊C4C2×C411yescharacteristic
99C2×Dic3⋊C4C2×C411yescharacteristic
100C2×Dic3⋊C4C2×C411yescharacteristic
101C2×Dic3⋊C4C2×C411yescharacteristic
102C2×C4×Dic3C2×C411yescharacteristic
103C2×C4×Dic3C2×C411yescharacteristic
104C2×C4×Dic3C2×C411yescharacteristic
105C2×C4×Dic3C2×C411yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C22×C3⋊C8C2312yes—
109C22×C3⋊C8C2312yes—
110C2×D3×C8—24no—
111C2×D3×C8—24no—
112C2×D3×C8—24no—
113C4×D12—24no—
114C4×D12—22no—
115C4×D12—22no—
116#872—24no—
117#872—24no—
118C42×D3—22no—
119C42×D3—22no—
120C42×D3D414yes—
121C42×D3D414yes—
122C4×Dic6—24no—
123C4×Dic6—22no—
124C4×Dic6—22no—
125C3×C22⋊C8—24no—
126C3×#97—22no—
127C3×#97—22no—
128#830—24no—
129(C2×D3)⋊C8—24no—
130(C2×D3)⋊C8—24no—
131(C2×D3)⋊C8—24no—
132#815—24no—
133Dic3⋊C8—24no—
134#803—22no—
135#803—22no—
136C22×#41—31no—
137C23×C8—32no—
138C4×#41—64no—
139C4×#41—64no—
140C2×C4×Q8—31no—
141C2×C4×Q8—31no—
142C2×C4×D4—31no—
143C2×C4×D4—31no—
144C2×C4×D4—31no—
145C2×C4×D4—31no—
146C2×C4×D4—31no—
147C2×C4×D4—31no—
148C2×#117—31no—
149C2×#117—31no—
150C2×#117—31no—
151C22×C42—31no—
152C22×C42—31no—
153C22×C42C2×D311yescharacteristic
154#432—64no—
155C2×C4⋊C8—32no—
156C2×C22⋊C8—32no—
157C2×C22⋊C8—32no—
158C2×C22⋊C8—32no—
159C2×C22⋊C8—32no—
160C2×C22⋊C8C2×D312yes—
161C2×#97—32no—
162C2×#97—31no—
163C2×#97—31no—
164C2×#97—64no—
165C2×#97C2×D311yescharacteristic
166C2×#97C2×D311yescharacteristic
167#336—32no—
168#336—32no—
169#336—32no—
170#336C2×D312yes—
171C24×C3C22×C411yescharacteristic
172C23×D3C22×C411yescharacteristic
173C23×D3C22×C411yescharacteristic
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—41no—
183C22×C3×C4—41no—
184C22×C3×C4C22×C411yescharacteristic
185C22×C3×C4C22×C411yescharacteristic
186C22×C3×C4C22×C411yescharacteristic
187C22×C3×C4C22×C411yescharacteristic
188C22×C3×C4C22×C411yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C2×#67—22no—
191C2×#67—22no—
192C2×#67—22no—
193C2×#67—22no—
194C2×#67—22no—
195C2×#67M1614yes—
196C22×Dic3C22×C411yescharacteristic
197C22×Dic3C22×C411yescharacteristic
198C22×Dic3C22×C411yescharacteristic
199C22×Dic3C22×C411yescharacteristic
200C22×Dic3C22×C411yescharacteristic
201C22×Dic3C22×C411yescharacteristic
202#235—232no—
203C2×D12—22no—
204C2×D12—22no—
205C2×D12M1614yes—
206C2×C4×D3—22no—
207C2×C4×D3—22no—
208C2×C4×D3—22no—
209C2×C4×D3—22no—
210C2×C4×D3—22no—
211C2×C4×D3—22no—
212C2×C4×D3—22no—
213C2×C4×D3—22no—
214C2×C4×D3—21no—
215C2×C4×D3—21no—
216C2×C4×D3—22no—
217C2×C4×D3—22no—
218C2×C4×D3—41no—
219C2×C4×D3—41no—
220C2×C4×D3—41no—
221C2×C4×D3—41no—
222C2×C4×D3C22⋊C414yes—
223C2×C4×D3C22⋊C414yes—
224C2×C4×D3C2×D412yes—
225C2×C4×D3C2×D412yes—
226C2×Dic6—22no—
227C2×Dic6—22no—
228C2×Dic6—22no—
229C2×C3×C8—22no—
230C2×C3×C8—22no—
231C2×C3×C8—22no—
232C2×C3×C8—22no—
233C2×C3×C8—22no—
234C2×C3×C8#4114yescomplemented
235C3×C42—22no—
236C3×C42—22no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42C2×D412yes—
240C3×C42C2×D412yes—
241#217—22no—
242#217—22no—
243(C2×D3)⋊C4—22no—
244(C2×D3)⋊C4—22no—
245(C2×D3)⋊C4—22no—
246(C2×D3)⋊C4—22no—
247C4⋊Dic3—22no—
248C4⋊Dic3—22no—
249Dic3⋊C4—22no—
250Dic3⋊C4—22no—
251Dic3⋊C4—22no—
252Dic3⋊C4—22no—
253C4×Dic3—22no—
254C4×Dic3—22no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3C22⋊C414yes—
258C4×Dic3C22⋊C414yes—
259C4×Dic3C2×D412yes—
260C4×Dic3C2×D412yes—
261C2×C3⋊C8—22no—
262C2×C3⋊C8—22no—
263C2×C3⋊C8—22no—
264C2×C3⋊C8—22no—
265C2×C3⋊C8—22no—
266C2×C3⋊C8#4114yescomplemented
267D3×C8—216no—
268C2×#41—64no—
269C2×#41—64no—
270C2×#41—64no—
271C22×Q8—31no—
272C22×D4—31no—
273C22×D4—31no—
274C22×D4—31no—
275C23×C4—31no—
276C23×C4—31no—
277C23×C4—31no—
278C23×C4—31no—
279C23×C4—31no—
280C23×C4—31no—
281C23×C4C4×D311yescharacteristic
282C23×C4C4×D311yescharacteristic
283C23×C4C22×D311yescharacteristic
284C22×C8—32no—
285C22×C8—32no—
286C22×C8—64no—
287C22×C8—64no—
288C22×C8—64no—
289C22×C8C22×D312yes—
290C22×C8C22×D312yes—
291C4×Q8—64no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×D4—64no—
295C4×D4—62no—
296C4×D4—62no—
297C4×D4—64no—
298C4×D4—62no—
299C4×D4—62no—
300C4×D4—64no—
301C4×D4—62no—
302C4×D4—62no—
303#117—64no—
304#117—64no—
305#117—64no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C22⋊C4—31no—
313C2×C22⋊C4—31no—
314C2×C22⋊C4—31no—
315C2×C22⋊C4—31no—
316C2×C22⋊C4—31no—
317C2×C22⋊C4—31no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—31no—
321C2×C42—31no—
322C2×C42—31no—
323C2×C42—34no—
324C2×C42—34no—
325C2×C42—31no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—62no—
329C2×C42—62no—
330C2×C42C4×D311yescharacteristic
331C2×C42C4×D311yescharacteristic
332C2×C42C22×D311yescharacteristic
333C2×C42C22×D311yescharacteristic
334C4⋊C8—64no—
335C22⋊C8—24no—
336C22⋊C8—64no—
337C22⋊C8—64no—
338C22⋊C8—64no—
339C22⋊C8—64no—
340#97—22no—
341#97—22no—
342#97—64no—
343#97—62no—
344#97—62no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3C2×M1612yes—
352C23×C3C23×C411yescharacteristic
353C22×D3—21no—
354C22×D3—21no—
355C22×D3—21no—
356C22×D3—21no—
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—21no—
363C22×D3C2×C22⋊C412yes—
364C22×D3C2×M1612yes—
365C22×D3C2×M1612yes—
366C22×D3C2×M1612yes—
367C2×C3×C4—28no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—44no—
381C2×C3×C4—42no—
382C2×C3×C4—42no—
383C2×C3×C4C2×C22⋊C412yes—
384C2×C3×C4C2×C22⋊C412yes—
385C2×C3×C4C2×M1612yes—
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C2×#4112yes—
389C2×C3×C4C2×#4112yes—
390#67—28no—
391#67—28no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3C2×C22⋊C412yes—
407C2×Dic3C2×C22⋊C412yes—
408C2×Dic3C2×C22⋊C412yes—
409C2×Dic3C2×M1612yes—
410D12—28no—
411C4×D3—28no—
412C4×D3—28no—
413C4×D3—24no—
414C4×D3—24no—
415C4×D3—44no—
416C4×D3—44no—
417C4×D3—42no—
418C4×D3—42no—
419Dic6—28no—
420C3×C8—24no—
421C3×C8—42no—
422C3⋊C8—24no—
423C3⋊C8—42no—
424C24—31no—
425C24—31no—
426C24C2×C4×D311yescharacteristic
427#41—632no—
428C2×Q8—62no—
429C2×Q8—62no—
430C2×Q8—62no—
431C2×D4—34no—
432C2×D4—34no—
433C2×D4—62no—
434C2×D4—62no—
435C2×D4—62no—
436C2×D4—62no—
437C2×D4—62no—
438C2×D4—62no—
439C2×D4—62no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—31no—
452C22×C4—34no—
453C22×C4—32no—
454C22×C4—32no—
455C22×C4—34no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—62no—
472C22×C4—121no—
473C22×C4—121no—
474C22×C4—121no—
475C22×C4—121no—
476C22×C4C2×C4×D311yescharacteristic
477C22×C4C2×C4×D311yescharacteristic
478C22×C4C2×C4×D311yescharacteristic
479C22×C4C2×C4×D311yescharacteristic
480C22×C4C2×C4×D311yescharacteristic
481C22×C4C23×D311yescharacteristic
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—34no—
488C2×C8—616no—
489C2×C8—62no—
490C2×C8—62no—
491C2×C8—62no—
492C2×C8—62no—
493C2×C8—62no—
494C2×C8#23514yescomplemented
495C4⋊C4—62no—
496C4⋊C4—62no—
497C4⋊C4—62no—
498C4⋊C4—62no—
499C4⋊C4—62no—
500C4⋊C4—62no—
501C22⋊C4—62no—
502C22⋊C4—62no—
503C22⋊C4—62no—
504C22⋊C4—62no—
505C22⋊C4—62no—
506C22⋊C4—62no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—21no—
511C42—34no—
512C42—34no—
513C42—32no—
514C42—32no—
515C42—62no—
516C42—62no—
517C42—61no—
518C42—61no—
519C42D3×D412yes—
520C42D3×D412yes—
521C22×C3—22no—
522C22×C3—22no—
523C22×C3—22no—
524C22×C3—41no—
525C22×C3—41no—
526C22×C3—41no—
527C22×C3—41no—
528C22×C3C22×C22⋊C411yescharacteristic
529C22×C3C4×#4111yescharacteristic
530C22×C3C4×#4111yescharacteristic
531C22×C3C22×M1611yescharacteristic
532C22×C3C22×M1611yescharacteristic
533C22×C3#56811yescharacteristic
534C22×C3#56811yescharacteristic
535C2×D3—22no—
536C2×D3—22no—
537C2×D3—22no—
538C2×D3—22no—
539C2×D3—22no—
540C2×D3—22no—
541C2×D3—22no—
542C2×D3—22no—
543C2×D3—22no—
544C2×D3—41no—
545C2×D3—41no—
546C2×D3—41no—
547C2×D3—41no—
548C2×D3#40714yes—
549C3×C4—22no—
550C3×C4—22no—
551C3×C4—22no—
552C3×C4—41no—
553C3×C4—41no—
554C3×C4#52512yes—
555C3×C4#53212yes—
556Dic3—22no—
557Dic3—22no—
558Dic3—22no—
559Dic3—22no—
560Dic3—41no—
561Dic3—41no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—32no—
569C23—32no—
570C23—32no—
571C23—32no—
572C23—61no—
573C23—61no—
574C23—61no—
575C23—61no—
576C23—61no—
577C23—61no—
578C23—61no—
579C23—61no—
580C23—61no—
581C23—61no—
582C23D3×M1612yes—
583C23C22×C4×D311yescharacteristic
584Q8—68no—
585D4—68no—
586D4—68no—
587D4—68no—
588C2×C4—28no—
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—21no—
600C2×C4—21no—
601C2×C4—32no—
602C2×C4—32no—
603C2×C4—32no—
604C2×C4—32no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—68no—
609C2×C4—68no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—64no—
619C2×C4—64no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—124no—
627C2×C4—124no—
628C2×C4—122no—
629C2×C4—122no—
630C2×C4D3×C22⋊C412yes—
631C2×C4D3×C22⋊C412yes—
632C2×C4D3×M1612yes—
633C2×C4C2×#23512yes—
634C2×C4C2×D3×D411yescharacteristic
635C2×C4C2×D3×D411yescharacteristic
636C2×C4D3×#4112yes—
637C8—24no—
638C8—42no—
639C8—64no—
640C8—122no—
641C2×C3—42no—
642C2×C3#229711yescharacteristic
643C2×C3C2×#40711yescharacteristic
644C2×C3#287911yescharacteristic
645C2×C3#297011yescharacteristic
646C2×C3#297011yescharacteristic
647C2×C3#297011yescharacteristic
648C2×C3#297011yescharacteristic
649D3—24no—
650D3—42no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—34no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—62no—
660C22—62no—
661C22—62no—
662C22—62no—
663C22—62no—
664C22—62no—
665C22—62no—
666C22—62no—
667C22—62no—
668C22—121no—
669C22—121no—
670C22—121no—
671C22—121no—
672C22C4×#23511yescharacteristic
673C22C2×D3×C22⋊C411yescharacteristic
674C22#552211yescharacteristic
675C22C2×D3×M1611yescharacteristic
676C22C2×D3×M1611yescharacteristic
677C22#574511yescharacteristic
678C22#574511yescharacteristic
679C4—22no—
680C4—22no—
681C4—22no—
682C4—41no—
683C4—41no—
684C4—62no—
685C4—62no—
686C4—62no—
687C4—62no—
688C4—121no—
689C4—121no—
690C4#554712yes—
691C4#554812yes—
692C3#2137911yescharacteristic, complemented, Sylow
693C2—42no—
694C2—64no—
695C2—122no—
696C2#7356311yescharacteristic
697C2#7848211yescharacteristic
698C2#7852411yescharacteristic
699C2D3×#40711yescharacteristic
700C2#7863011yescharacteristic
701C2#7932711yescharacteristic
702C2#7932811yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1749C2
prime radicalC22×#235C4
cosocleC22×C3×C4C24
hypercenter#1749D3
hypocenterC3#21379
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. C2×#4705C2semidirect
3. C2×D3×#97C2semidirect
4. C3×#1749C2semidirect
5. #71529C2semidirect
6. #69957C2semidirect
7. #69917C2semidirect
8. #69807C2semidirect
9. #69807C2semidirect
10. #68668C2semidirect
11. #68620C2semidirect
58. #1749D3semidirect
234. C2×C3×C8#41semidirect
266. C2×C3⋊C8#41semidirect
494. C2×C8#235semidirect
692. C3#21379semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2352
C2×#47052
C2×D3×#972
C3×#17492
#715292
#699572
#699172
#698072
#686682
#686202
#213793