Finite Group Databasebeta

← #10794571

Subgroups — #10794571

Order 768.

1498 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#235C211yesmaximal, characteristic
2C2×#4839C212yesmaximal, complemented
3C2×#4727C212yesmaximal
4C3×#1797C211yesmaximal, characteristic, complemented
5#71530C211yesmaximal, characteristic, complemented
6#70476C212yesmaximal, complemented
7#69909C212yesmaximal, complemented
8#68662C211yesmaximal, characteristic, complemented
9#68662C211yesmaximal, characteristic, complemented
10#68660C211yesmaximal, characteristic, complemented
11D3×#336C211yesmaximal, characteristic, complemented
12#20630—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C2×C4×#67C411yescharacteristic
16C2×C4×#67C411yescharacteristic
17C2×C4×#67C411yescharacteristic
18C2×C4×#67C411yescharacteristic
19C2×#926C2211yescharacteristic
20C22×ΓL(1,𝔽25)C2212yes—
21C22×D3×C8C2212yes—
22C4×#235—28no—
23C2×C4×D12C411yescharacteristic
24C2×C4×D12C411yescharacteristic
25C2×#872C2211yescharacteristic
26C2×#872C2211yescharacteristic
27C2×C42×D3C2211yescharacteristic
28C2×C42×D3C2211yescharacteristic
29C2×C4×Dic6C411yescharacteristic
30C2×C4×Dic6C411yescharacteristic
31C2×C3×C4⋊C8C2212yes—
32C2×C3×C22⋊C8C2212yes—
33C2×#830C2212yes—
34C2×(C2×D3)⋊C8C2212yes—
35C2×(C2×D3)⋊C8C2212yes—
36C2×Dic3⋊C8C2212yes—
37C2×Dic3⋊C8C2212yes—
38C2×C4⋊(C3⋊C8)C2212yes—
39#4839—28no—
40#4727—28no—
41C3×#336C2211yescharacteristic
42C3×#336C2211yescharacteristic
43#4550C2212yes—
44#4550C2212yes—
45#4524C2211yescharacteristic
46#4524C2211yescharacteristic
47C2×C4×#41—31no—
48C2×#424—32no—
49C2×#408—32no—
50#1798—32no—
51#1798—32no—
52#1797—31no—
53#1797—31no—
54#1797—31no—
55#1797D311yescharacteristic, complemented
56#1728—31no—
57C2×#336—31no—
58C23×C3×C4C2×C411yescharacteristic
59C23×C3×C4C2×C411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C22×#67C2×C411yescharacteristic
62C22×#67C2×C411yescharacteristic
63C2×#235—24no—
64C2×#235—24no—
65C2×#235D414yes—
66C2×#235Q814yes—
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—24no—
78C2×C3×C42C2×C411yescharacteristic
79C2×C3×C42C2×C411yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×#217C2×C411yescharacteristic
83C2×#217C2×C411yescharacteristic
84C4×#67—24no—
85C4×#67—24no—
86C4×#67—24no—
87C4×#67—24no—
88C2×(C2×D3)⋊C4C2×C411yescharacteristic
89C2×(C2×D3)⋊C4C2×C411yescharacteristic
90C2×(C2×D3)⋊C4C2×C411yescharacteristic
91C2×(C2×D3)⋊C4C2×C411yescharacteristic
92#926—24no—
93C2×C4⋊Dic3C2×C411yescharacteristic
94C2×C4⋊Dic3C2×C411yescharacteristic
95C2×Dic3⋊C4C2×C411yescharacteristic
96C2×Dic3⋊C4C2×C411yescharacteristic
97C2×Dic3⋊C4C2×C411yescharacteristic
98C2×Dic3⋊C4C2×C411yescharacteristic
99C2×C4×Dic3C2×C411yescharacteristic
100C2×C4×Dic3C2×C411yescharacteristic
101C2×C4×Dic3C2×C411yescharacteristic
102C2×C4×Dic3C2×C411yescharacteristic
103C2×C4×Dic3C2311yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2312yes—
106C22×C3⋊C8C2312yes—
107C2×ΓL(1,𝔽25)—24no—
108C2×ΓL(1,𝔽25)—24no—
109C2×ΓL(1,𝔽25)—24no—
110C2×D3×C8—24no—
111C2×D3×C8—24no—
112C2×D3×C8—24no—
113C4×D12—24no—
114C4×D12—24no—
115#872—24no—
116#872D418yes—
117C42×D3—24no—
118C42×D3—24no—
119C4×Dic6—24no—
120C4×Dic6—24no—
121C3×C4⋊C8—24no—
122C3×C22⋊C8—24no—
123#830—24no—
124(C2×D3)⋊C8—24no—
125(C2×D3)⋊C8—24no—
126Dic3⋊C8—24no—
127Dic3⋊C8—24no—
128C4⋊(C3⋊C8)—24no—
129C22×#41—31no—
130C22×M16—32no—
131C23×C8—32no—
132C4×#41—68no—
133C2×C4×Q8—31no—
134C2×C4×Q8—31no—
135C2×C4×D4—31no—
136C2×C4×D4—31no—
137C2×C4×D4—31no—
138C2×C4×D4—31no—
139C2×C4×D4—31no—
140C2×C4×D4—31no—
141C2×#117—31no—
142C2×#117—31no—
143C2×#117—31no—
144C22×C42—31no—
145C22×C42—31no—
146C22×C42C2×D311yescharacteristic
147#424—68no—
148C2×C4⋊C8—32no—
149C2×C4⋊C8—32no—
150C2×C4⋊C8—32no—
151C2×C4⋊C8C2×D312yes—
152#408—68no—
153C2×C22⋊C8—32no—
154C2×C22⋊C8—32no—
155C2×C22⋊C8—32no—
156C2×C22⋊C8C2×D312yes—
157#336—32no—
158#336—32no—
159#336—31no—
160#336—31no—
161#336C2×D311yescharacteristic
162#336C2×D311yescharacteristic
163C24×C3C22×C411yescharacteristic
164C23×D3C22×C411yescharacteristic
165C23×D3C22×C411yescharacteristic
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—41no—
173C22×C3×C4—41no—
174C22×C3×C4C22×C411yescharacteristic
175C22×C3×C4C22×C411yescharacteristic
176C22×C3×C4C22×C411yescharacteristic
177C22×C3×C4C22×C411yescharacteristic
178C22×C3×C4C22×C411yescharacteristic
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×Q812yes—
181C22×C3×C4C2411yescharacteristic
182C2×#67—22no—
183C2×#67—22no—
184C2×#67—22no—
185C2×#67—22no—
186C2×#67C4⋊C414yes—
187C2×#67C4⋊C414yes—
188C22×Dic3C22×C411yescharacteristic
189C22×Dic3C22×C411yescharacteristic
190C22×Dic3C22×C411yescharacteristic
191C22×Dic3C22×C411yescharacteristic
192C22×Dic3C22×C411yescharacteristic
193C22×Dic3C22×C411yescharacteristic
194#235—216no—
195#235—216no—
196C2×D12—22no—
197C2×D12—22no—
198C2×D12C4⋊C414yes—
199C2×C4×D3—22no—
200C2×C4×D3—22no—
201C2×C4×D3—22no—
202C2×C4×D3—22no—
203C2×C4×D3—22no—
204C2×C4×D3—22no—
205C2×C4×D3—22no—
206C2×C4×D3—22no—
207C2×C4×D3—22no—
208C2×C4×D3—22no—
209C2×C4×D3—41no—
210C2×C4×D3—41no—
211C2×C4×D3—41no—
212C2×C4×D3—41no—
213C2×C4×D3C2×D412yes—
214C2×C4×D3C2×D412yes—
215C2×C4×D3C2×D414yes—
216C2×C4×D3C2×Q812yes—
217C2×C4×D3C2×Q812yes—
218C2×C4×D3#4114yes—
219C2×Dic6—22no—
220C2×Dic6—22no—
221C2×Dic6C4⋊C414yes—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C3×C42—22no—
229C3×C42—22no—
230C3×C42—22no—
231C3×C42C2×D414yes—
232#217—22no—
233#217—22no—
234(C2×D3)⋊C4—22no—
235(C2×D3)⋊C4—22no—
236(C2×D3)⋊C4C22⋊C414yes—
237(C2×D3)⋊C4C22⋊C414yes—
238C4⋊Dic3—22no—
239C4⋊Dic3—22no—
240Dic3⋊C4—22no—
241Dic3⋊C4—22no—
242Dic3⋊C4C22⋊C414yes—
243Dic3⋊C4C22⋊C414yes—
244C4×Dic3—22no—
245C4×Dic3—22no—
246C4×Dic3—22no—
247C4×Dic3—22no—
248C4×Dic3—22no—
249C4×Dic3C2×D414yes—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—22no—
254C2×C3⋊C8—22no—
255C2×C3⋊C8—22no—
256ΓL(1,𝔽25)—48no—
257D3×C8—48no—
258C2×#41—34no—
259C2×#41—34no—
260C2×#41—64no—
261C2×#41—64no—
262C22×Q8—31no—
263C22×D4—31no—
264C22×D4—31no—
265C22×D4—31no—
266C23×C4—31no—
267C23×C4—31no—
268C23×C4—31no—
269C23×C4—31no—
270C23×C4—31no—
271C23×C4—31no—
272C23×C4C4×D311yescharacteristic
273C23×C4C4×D311yescharacteristic
274C23×C4C22×D311yescharacteristic
275C2×M16—64no—
276C2×M16—64no—
277C2×M16—64no—
278C22×C8—32no—
279C22×C8—32no—
280C22×C8—64no—
281C22×C8—64no—
282C22×C8—64no—
283C22×C8C22×D312yes—
284C22×C8C22×D312yes—
285C4×Q8—64no—
286C4×Q8—64no—
287C4×D4—64no—
288C4×D4—64no—
289C4×D4—64no—
290C4×D4—64no—
291C4×D4—64no—
292C4×D4—64no—
293#117—38no—
294#117—64no—
295#117—64no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C22⋊C4—31no—
303C2×C22⋊C4—31no—
304C2×C22⋊C4—31no—
305C2×C22⋊C4—31no—
306C2×C22⋊C4—31no—
307C2×C22⋊C4—31no—
308C2×C42—24no—
309C2×C42—31no—
310C2×C42—31no—
311C2×C42—31no—
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—31no—
315C2×C42—64no—
316C2×C42—64no—
317C2×C42C4×D311yescharacteristic
318C2×C42C4×D311yescharacteristic
319C2×C42C22×D311yescharacteristic
320C2×C42C22×D311yescharacteristic
321C4⋊C8—24no—
322C4⋊C8—64no—
323C4⋊C8—64no—
324C4⋊C8—64no—
325C22⋊C8—24no—
326C22⋊C8—64no—
327C22⋊C8—64no—
328C22⋊C8—64no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3C2×C4⋊C412yes—
336C23×C3C23×C411yescharacteristic
337C22×D3—21no—
338C22×D3—21no—
339C22×D3—21no—
340C22×D3—21no—
341C22×D3—21no—
342C22×D3—21no—
343C22×D3—21no—
344C22×D3—21no—
345C22×D3—21no—
346C22×D3—21no—
347C22×D3C2×C22⋊C412yes—
348C22×D3C2×C4⋊C412yes—
349C22×D3C2×C4⋊C412yes—
350C22×D3#11712yes—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—44no—
364C2×C3×C4—44no—
365C2×C3×C4C2×C22⋊C412yes—
366C2×C3×C4C2×C22⋊C412yes—
367C2×C3×C4C2×C4⋊C412yes—
368C2×C3×C4#13112yes—
369C2×C3×C4#13112yes—
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×D412yes—
372C2×C3×C4C22×Q811yescharacteristic
373C2×C3×C4C2×#4112yes—
374#67—28no—
375#67—28no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3C2×C22⋊C412yes—
389C2×Dic3C2×C22⋊C412yes—
390C2×Dic3C2×C22⋊C412yes—
391C2×Dic3C2×C4⋊C412yes—
392C2×Dic3C2×C4⋊C412yes—
393C2×Dic3#11712yes—
394D12—28no—
395C4×D3—24no—
396C4×D3—24no—
397C4×D3—44no—
398C4×D3—44no—
399C4×D3—44no—
400C4×D3—44no—
401C4×D3#12118yes—
402C4×D3C22⋊Q818yes—
403Dic6—28no—
404C3×C8—42no—
405C3×C8—42no—
406C3⋊C8—42no—
407C3⋊C8—42no—
408C24—31no—
409C24—31no—
410C24C2×C4×D311yescharacteristic
411#41—616no—
412#41—616no—
413C2×Q8—34no—
414C2×Q8—62no—
415C2×Q8—62no—
416C2×D4—34no—
417C2×D4—34no—
418C2×D4—34no—
419C2×D4—62no—
420C2×D4—62no—
421C2×D4—62no—
422C2×D4—62no—
423C2×D4—62no—
424C2×D4—62no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—31no—
432C22×C4—34no—
433C22×C4—32no—
434C22×C4—32no—
435C22×C4—34no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—32no—
440C22×C4—32no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—41no—
444C22×C4—41no—
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—62no—
454C22×C4—62no—
455C22×C4—121no—
456C22×C4—121no—
457C22×C4—121no—
458C22×C4—121no—
459C22×C4C2×C4×D311yescharacteristic
460C22×C4C2×C4×D311yescharacteristic
461C22×C4C2×C4×D311yescharacteristic
462C22×C4C2×C4×D311yescharacteristic
463C22×C4C2×C4×D311yescharacteristic
464C22×C4D3×D412yes—
465C22×C4D3×Q812yes—
466C22×C4C23×D311yescharacteristic
467M16—128no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—62no—
475C2×C8—62no—
476C2×C8—62no—
477C2×C8—62no—
478C2×C8—62no—
479C2×C8—62no—
480C2×C8—128no—
481C4⋊C4—34no—
482C4⋊C4—34no—
483C4⋊C4—62no—
484C4⋊C4—62no—
485C4⋊C4—62no—
486C4⋊C4—62no—
487C22⋊C4—34no—
488C22⋊C4—34no—
489C22⋊C4—62no—
490C22⋊C4—62no—
491C22⋊C4—62no—
492C22⋊C4—62no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—34no—
497C42—62no—
498C42—62no—
499C42—62no—
500C42—62no—
501C42—62no—
502C42D3×D414yes—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3C22×C22⋊C411yescharacteristic
511C22×C3C22×C4⋊C411yescharacteristic
512C22×C3C2×#11711yescharacteristic
513C22×C3C2×#13111yescharacteristic
514C22×C3C2×#13111yescharacteristic
515C22×C3#56811yescharacteristic
516C22×C3#56811yescharacteristic
517C2×D3—22no—
518C2×D3—22no—
519C2×D3—22no—
520C2×D3—22no—
521C2×D3—22no—
522C2×D3—22no—
523C2×D3—22no—
524C2×D3—22no—
525C2×D3—22no—
526C2×D3—41no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—41no—
530C2×D3#38014yes—
531C3×C4—22no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—22no—
535C3×C4—22no—
536C3×C4C2×#12112yes—
537C3×C4C2×C22⋊Q812yes—
538Dic3—22no—
539Dic3—22no—
540Dic3—22no—
541Dic3—41no—
542Dic3—41no—
543Dic3#38014yes—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—32no—
551C23—32no—
552C23—32no—
553C23—32no—
554C23—61no—
555C23—61no—
556C23—61no—
557C23—61no—
558C23—61no—
559C23—61no—
560C23—61no—
561C23—61no—
562C23—61no—
563C23—61no—
564C23D3×C4⋊C412yes—
565C23C22×C4×D311yescharacteristic
566Q8—68no—
567D4—68no—
568D4—68no—
569D4—68no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—32no—
583C2×C4—32no—
584C2×C4—32no—
585C2×C4—38no—
586C2×C4—38no—
587C2×C4—32no—
588C2×C4—32no—
589C2×C4—32no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—64no—
599C2×C4—64no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—124no—
607C2×C4—124no—
608C2×C4—124no—
609C2×C4—124no—
610C2×C4D3×C22⋊C412yes—
611C2×C4D3×C22⋊C412yes—
612C2×C4D3×C4⋊C412yes—
613C2×C4#90112yes—
614C2×C4#90712yes—
615C2×C4C2×D3×D411yescharacteristic
616C2×C4C2×D3×D412yes—
617C2×C4C2×D3×Q811yescharacteristic
618C2×C4D3×#4112yes—
619C8—42no—
620C8—42no—
621C8—122no—
622C8—122no—
623C2×C3—42no—
624C2×C3C2×#38011yescharacteristic
625C2×C3#288011yescharacteristic
626C2×C3#288011yescharacteristic
627C2×C3#290611yescharacteristic
628C2×C3#290611yescharacteristic
629C2×C3#292211yescharacteristic
630C2×C3#292311yescharacteristic
631D3—24no—
632D3—42no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—34no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—62no—
642C22—62no—
643C22—62no—
644C22—62no—
645C22—62no—
646C22—62no—
647C22—62no—
648C22—62no—
649C22—62no—
650C22—121no—
651C22—121no—
652C22—121no—
653C22—121no—
654C22C2×D3×C22⋊C411yescharacteristic
655C22C2×D3×C4⋊C411yescharacteristic
656C22D3×#11711yescharacteristic
657C22C2×#90111yescharacteristic
658C22C2×#90711yescharacteristic
659C22#574511yescharacteristic
660C22#574511yescharacteristic
661C4—22no—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—34no—
667C4—62no—
668C4—62no—
669C4—62no—
670C4—121no—
671C4—121no—
672C4D3×#12112yes—
673C4D3×C22⋊Q812yes—
674C3#2063011yescharacteristic, complemented, Sylow
675C2—42no—
676C2—64no—
677C2—122no—
678C2D3×#38011yescharacteristic
679C2#7860011yescharacteristic
680C2#7863111yescharacteristic
681C2#7901211yescharacteristic
682C2#7904911yescharacteristic
683C2#7928311yescharacteristic
684C2#7933211yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1797C2
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenter#1797D3
hypocenterC3#20630
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. C2×#4839C2semidirect
4. C3×#1797C2semidirect
5. #71530C2semidirect
6. #70476C2semidirect
7. #69909C2semidirect
8. #68662C2semidirect
9. #68662C2semidirect
10. #68660C2semidirect
11. D3×#336C2semidirect
55. #1797D3semidirect
674. C3#20630semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2352
C2×#48392
C2×#47272
C3×#17972
#715302
#704762
#699092
#686622
#686602
D3×#3362
#206303