Finite Group Databasebeta

← #10795106

Subgroups — #10795106

Order 768.

1342 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#235C211yesmaximal, characteristic
2C2×#4840C212yesmaximal, complemented
3C3×#1797C211yesmaximal, characteristic, complemented
4#71530C211yesmaximal, characteristic, complemented
5#70476C212yesmaximal, complemented
6#69956C212yesmaximal
7#69907C212yesmaximal, complemented
8#68664C211yesmaximal, characteristic, complemented
9#68664C211yesmaximal, characteristic, complemented
10#68600C211yesmaximal, characteristic, complemented
11#68599C211yesmaximal, characteristic, complemented
12#20720—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C2×C4×#67C411yescharacteristic
16C2×C4×#67C411yescharacteristic
17C2×C4×#67C411yescharacteristic
18C2×C4×#67C411yescharacteristic
19C2×#926C2211yescharacteristic
20C4×#235—28no—
21C2×C4×D12C411yescharacteristic
22C2×C4×D12C411yescharacteristic
23C2×#872C2211yescharacteristic
24C2×#872C2211yescharacteristic
25C2×C42×D3C2211yescharacteristic
26C2×C42×D3C2211yescharacteristic
27C2×C4×Dic6C411yescharacteristic
28C2×C4×Dic6C411yescharacteristic
29C2×C3×C4⋊C8C2212yes—
30C2×C3×C22⋊C8C2212yes—
31C2×#830C2212yes—
32C2×(C2×D3)⋊C8C2212yes—
33C2×(C2×D3)⋊C8C2212yes—
34C2×(C2×D3)⋊C8C2212yes—
35C2×(C2×D3)⋊C8C2212yes—
36C2×#815C2212yes—
37C2×C8×Dic3C2212yes—
38C2×C4⋊(C3⋊C8)C2212yes—
39#4840—28no—
40C3×#336C2211yescharacteristic
41C3×#336C2211yescharacteristic
42#4550C2212yes—
43#4550C2212yes—
44#4524C2211yescharacteristic
45#4524C2211yescharacteristic
46C2×C4×#41—31no—
47C2×#433—32no—
48#1817—31no—
49#1817—31no—
50#1798—32no—
51#1797—31no—
52#1797D311yescharacteristic, complemented
53#1780—32no—
54#1750—32no—
55#1749—31no—
56C4×C22⋊C8—31no—
57C23×C3×C4C2×C411yescharacteristic
58C23×C3×C4C2×C411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67C2×C411yescharacteristic
61C22×#67C2×C411yescharacteristic
62C2×#235—22no—
63C2×#235—22no—
64C2×#235—24no—
65C2×#235—24no—
66C22×D12C2×C411yescharacteristic
67C22×C4×D3C2×C411yescharacteristic
68C22×C4×D3C2×C411yescharacteristic
69C22×C4×D3C2×C411yescharacteristic
70C22×C4×D3C2×C411yescharacteristic
71C22×C4×D3C2311yescharacteristic
72C22×C4×D3C2311yescharacteristic
73C22×Dic6C2×C411yescharacteristic
74C22×C3×C8C2312yes—
75C22×C3×C8C2312yes—
76C2×C3×C42—24no—
77C2×C3×C42C2×C411yescharacteristic
78C2×C3×C42C2×C411yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×#217C2×C411yescharacteristic
82C2×#217C2×C411yescharacteristic
83C4×#67—24no—
84C4×#67—24no—
85C4×#67—24no—
86C4×#67—24no—
87C2×(C2×D3)⋊C4C2×C411yescharacteristic
88C2×(C2×D3)⋊C4C2×C411yescharacteristic
89C2×(C2×D3)⋊C4C2×C411yescharacteristic
90C2×(C2×D3)⋊C4C2×C411yescharacteristic
91#926—24no—
92C2×C4⋊Dic3C2×C411yescharacteristic
93C2×C4⋊Dic3C2×C411yescharacteristic
94C2×Dic3⋊C4C2×C411yescharacteristic
95C2×Dic3⋊C4C2×C411yescharacteristic
96C2×Dic3⋊C4C2×C411yescharacteristic
97C2×Dic3⋊C4C2×C411yescharacteristic
98C2×C4×Dic3C2×C411yescharacteristic
99C2×C4×Dic3C2×C411yescharacteristic
100C2×C4×Dic3C2×C411yescharacteristic
101C2×C4×Dic3C2×C411yescharacteristic
102C2×C4×Dic3C2311yescharacteristic
103C2×C4×Dic3C2311yescharacteristic
104C22×C3⋊C8C2312yes—
105C22×C3⋊C8C2312yes—
106C4×D12—24no—
107C4×D12—24no—
108#872—24no—
109#872D418yes—
110C42×D3—24no—
111C42×D3—24no—
112C4×Dic6—24no—
113C4×Dic6—24no—
114C3×C4⋊C8—24no—
115C3×C22⋊C8—24no—
116#830—24no—
117(C2×D3)⋊C8—24no—
118(C2×D3)⋊C8—24no—
119(C2×D3)⋊C8—24no—
120(C2×D3)⋊C8—24no—
121#815—24no—
122C8×Dic3—24no—
123C4⋊(C3⋊C8)—24no—
124C22×#41—31no—
125C4×#41—68no—
126C2×C4×Q8—31no—
127C2×C4×Q8—31no—
128C2×C4×D4—31no—
129C2×C4×D4—31no—
130C2×C4×D4—31no—
131C2×C4×D4—31no—
132C2×C4×D4—31no—
133C2×C4×D4—31no—
134C2×#117—31no—
135C2×#117—31no—
136C2×#117—31no—
137C22×C42—31no—
138C22×C42—31no—
139C22×C42C2×D311yescharacteristic
140#433—68no—
141C2×C4⋊C8—32no—
142C2×C4⋊C8C2×D312yes—
143C2×C22⋊C8—32no—
144C2×C22⋊C8—32no—
145C2×C22⋊C8—32no—
146C2×C22⋊C8—32no—
147C2×C22⋊C8—32no—
148C2×C22⋊C8C2×D312yes—
149C2×#97—32no—
150C2×C4×C8—32no—
151#336—32no—
152#336—32no—
153#336—31no—
154#336—31no—
155#336C2×D311yescharacteristic
156#336C2×D311yescharacteristic
157C24×C3C22×C411yescharacteristic
158C23×D3C22×C411yescharacteristic
159C23×D3C22×C411yescharacteristic
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—41no—
167C22×C3×C4—41no—
168C22×C3×C4C22×C411yescharacteristic
169C22×C3×C4C22×C411yescharacteristic
170C22×C3×C4C22×C411yescharacteristic
171C22×C3×C4C22×C411yescharacteristic
172C22×C3×C4C22×C411yescharacteristic
173C22×C3×C4#4112yes—
174C22×C3×C4#4112yes—
175C22×C3×C4C2411yescharacteristic
176C2×#67—22no—
177C2×#67—22no—
178C2×#67—22no—
179C2×#67—22no—
180C2×#67—22no—
181C2×#67—22no—
182C22×Dic3C22×C411yescharacteristic
183C22×Dic3C22×C411yescharacteristic
184C22×Dic3C22×C411yescharacteristic
185C22×Dic3C22×C411yescharacteristic
186C22×Dic3C22×C411yescharacteristic
187C22×Dic3C22×C411yescharacteristic
188#235—216no—
189#235—216no—
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—22no—
193C2×C4×D3—22no—
194C2×C4×D3—21no—
195C2×C4×D3—21no—
196C2×C4×D3—21no—
197C2×C4×D3—21no—
198C2×C4×D3—22no—
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—41no—
208C2×C4×D3—41no—
209C2×C4×D3—41no—
210C2×C4×D3—41no—
211C2×C4×D3C2×D414yes—
212C2×C4×D3#4114yes—
213C2×Dic6—22no—
214C2×Dic6—22no—
215C2×Dic6—22no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C3×C42—22no—
223C3×C42—22no—
224C3×C42—22no—
225C3×C42C2×D414yes—
226#217—22no—
227#217—22no—
228(C2×D3)⋊C4—22no—
229(C2×D3)⋊C4—22no—
230(C2×D3)⋊C4C22⋊C414yes—
231(C2×D3)⋊C4C22⋊C414yes—
232C4⋊Dic3—22no—
233C4⋊Dic3—22no—
234Dic3⋊C4—22no—
235Dic3⋊C4—22no—
236Dic3⋊C4C22⋊C414yes—
237Dic3⋊C4C22⋊C414yes—
238C4×Dic3—22no—
239C4×Dic3—22no—
240C4×Dic3—22no—
241C4×Dic3—22no—
242C4×Dic3—22no—
243C4×Dic3C2×D414yes—
244C2×C3⋊C8—22no—
245C2×C3⋊C8—22no—
246C2×C3⋊C8—22no—
247C2×C3⋊C8—22no—
248C2×C3⋊C8—22no—
249C2×C3⋊C8—22no—
250C2×#41—64no—
251C2×#41—64no—
252C2×#41—62no—
253C2×#41—62no—
254C22×Q8—31no—
255C22×D4—31no—
256C22×D4—31no—
257C22×D4—31no—
258C23×C4—31no—
259C23×C4—31no—
260C23×C4—31no—
261C23×C4—31no—
262C23×C4—31no—
263C23×C4—31no—
264C23×C4C4×D311yescharacteristic
265C23×C4C4×D311yescharacteristic
266C23×C4C22×D311yescharacteristic
267C22×C8—32no—
268C22×C8—32no—
269C22×C8C22×D312yes—
270C22×C8C22×D312yes—
271C4×Q8—64no—
272C4×Q8—64no—
273C4×D4—64no—
274C4×D4—64no—
275C4×D4—64no—
276C4×D4—64no—
277C4×D4—64no—
278C4×D4—64no—
279#117—38no—
280#117—64no—
281#117—64no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C22⋊C4—31no—
289C2×C22⋊C4—31no—
290C2×C22⋊C4—31no—
291C2×C22⋊C4—31no—
292C2×C22⋊C4—31no—
293C2×C22⋊C4—31no—
294C2×C42—24no—
295C2×C42—31no—
296C2×C42—31no—
297C2×C42—31no—
298C2×C42—31no—
299C2×C42—31no—
300C2×C42—31no—
301C2×C42—64no—
302C2×C42—64no—
303C2×C42C4×D311yescharacteristic
304C2×C42C4×D311yescharacteristic
305C2×C42C22×D311yescharacteristic
306C2×C42C22×D311yescharacteristic
307C4⋊C8—24no—
308C4⋊C8—64no—
309C22⋊C8—24no—
310C22⋊C8—64no—
311C22⋊C8—64no—
312C22⋊C8—64no—
313C22⋊C8—64no—
314C22⋊C8—64no—
315#97—64no—
316C4×C8—64no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3#11712yes—
324C23×C3C23×C411yescharacteristic
325C22×D3—21no—
326C22×D3—21no—
327C22×D3—21no—
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3—21no—
333C22×D3—21no—
334C22×D3—21no—
335C22×D3—21no—
336C22×D3—21no—
337C22×D3C2×C22⋊C412yes—
338C22×D3#11712yes—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—44no—
352C2×C3×C4—44no—
353C2×C3×C4C2×C22⋊C412yes—
354C2×C3×C4C2×C22⋊C412yes—
355C2×C3×C4#11712yes—
356C2×C3×C4#13112yes—
357C2×C3×C4#13112yes—
358C2×C3×C4C22×D412yes—
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4112yes—
362#67—28no—
363#67—28no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3C2×C22⋊C412yes—
379C2×Dic3C2×C22⋊C412yes—
380C2×Dic3C2×C22⋊C412yes—
381C2×Dic3#11712yes—
382D12—28no—
383C4×D3—24no—
384C4×D3—24no—
385C4×D3—24no—
386C4×D3—24no—
387C4×D3—44no—
388C4×D3—44no—
389C4×D3—44no—
390C4×D3—44no—
391Dic6—28no—
392C3×C8—42no—
393C3×C8—42no—
394C3⋊C8—42no—
395C3⋊C8—42no—
396C24—31no—
397C24—31no—
398C24C2×C4×D311yescharacteristic
399#41—616no—
400#41—616no—
401C2×Q8—62no—
402C2×Q8—62no—
403C2×Q8—62no—
404C2×D4—62no—
405C2×D4—62no—
406C2×D4—62no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4—62no—
412C2×D4—62no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—34no—
424C22×C4—34no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—121no—
444C22×C4—121no—
445C22×C4—121no—
446C22×C4—121no—
447C22×C4C2×C4×D311yescharacteristic
448C22×C4C2×C4×D311yescharacteristic
449C22×C4C2×C4×D311yescharacteristic
450C22×C4C2×C4×D311yescharacteristic
451C22×C4C2×C4×D311yescharacteristic
452C22×C4#23712yes—
453C22×C4#23912yes—
454C22×C4C23×D311yescharacteristic
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—62no—
462C2×C8—62no—
463C2×C8—62no—
464C2×C8—62no—
465C2×C8—62no—
466C2×C8—62no—
467C4⋊C4—34no—
468C4⋊C4—34no—
469C4⋊C4—62no—
470C4⋊C4—62no—
471C4⋊C4—62no—
472C4⋊C4—62no—
473C22⋊C4—34no—
474C22⋊C4—34no—
475C22⋊C4—62no—
476C22⋊C4—62no—
477C22⋊C4—62no—
478C22⋊C4—62no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—34no—
483C42—62no—
484C42—62no—
485C42—62no—
486C42—62no—
487C42—62no—
488C42D3×D414yes—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—41no—
496C22×C3C22×C22⋊C411yescharacteristic
497C22×C3C2×#11711yescharacteristic
498C22×C3C2×#11711yescharacteristic
499C22×C3C2×#13111yescharacteristic
500C22×C3C2×#13111yescharacteristic
501C22×C3#56811yescharacteristic
502C22×C3#56811yescharacteristic
503C2×D3—22no—
504C2×D3—22no—
505C2×D3—22no—
506C2×D3—22no—
507C2×D3—22no—
508C2×D3—22no—
509C2×D3—41no—
510C2×D3—41no—
511C2×D3—41no—
512C2×D3—41no—
513C2×D3—41no—
514C2×D3—41no—
515C2×D3—41no—
516C2×D3—41no—
517C3×C4—22no—
518C3×C4—22no—
519C3×C4—22no—
520C3×C4—22no—
521C3×C4—22no—
522C3×C4#52512yes—
523C3×C4#52512yes—
524Dic3—22no—
525Dic3—22no—
526Dic3—22no—
527Dic3—22no—
528Dic3—41no—
529Dic3—41no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—32no—
537C23—32no—
538C23—61no—
539C23—61no—
540C23—61no—
541C23—61no—
542C23—61no—
543C23—61no—
544C23—61no—
545C23—61no—
546C23—61no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23#89212yes—
551C23C22×C4×D311yescharacteristic
552Q8—68no—
553D4—68no—
554D4—68no—
555D4—68no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—32no—
569C2×C4—32no—
570C2×C4—32no—
571C2×C4—32no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—64no—
583C2×C4—64no—
584C2×C4—64no—
585C2×C4—64no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—124no—
593C2×C4—124no—
594C2×C4—124no—
595C2×C4—124no—
596C2×C4D3×C22⋊C412yes—
597C2×C4D3×C22⋊C412yes—
598C2×C4#89212yes—
599C2×C4#90112yes—
600C2×C4#90712yes—
601C2×C4C2×D3×D412yes—
602C2×C4C2×#23711yescharacteristic
603C2×C4C2×#23911yescharacteristic
604C2×C4D3×#4112yes—
605C8—42no—
606C8—42no—
607C8—122no—
608C8—122no—
609C2×C3—42no—
610C2×C3#228011yescharacteristic
611C2×C3#288011yescharacteristic
612C2×C3#288011yescharacteristic
613C2×C3#292211yescharacteristic
614C2×C3#292311yescharacteristic
615C2×C3#292511yescharacteristic
616C2×C3#292511yescharacteristic
617D3—42no—
618D3—42no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—62no—
627C22—62no—
628C22—62no—
629C22—62no—
630C22—62no—
631C22—62no—
632C22—121no—
633C22—121no—
634C22—121no—
635C22—121no—
636C22—121no—
637C22—121no—
638C22—121no—
639C22—121no—
640C22C2×D3×C22⋊C411yescharacteristic
641C22C2×#89211yescharacteristic
642C22D3×#11711yescharacteristic
643C22C2×#90111yescharacteristic
644C22C2×#90711yescharacteristic
645C22#574511yescharacteristic
646C22#574511yescharacteristic
647C4—22no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—62no—
653C4—62no—
654C4—62no—
655C4—62no—
656C4—121no—
657C4—121no—
658C4#560612yes—
659C4#562712yes—
660C3#2072011yescharacteristic, complemented, Sylow
661C2—42no—
662C2—122no—
663C2—122no—
664C2#7386211yescharacteristic
665C2#7860011yescharacteristic
666C2#7863111yescharacteristic
667C2#7900011yescharacteristic
668C2#7904011yescharacteristic
669C2#7928311yescharacteristic
670C2#7933211yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1797C2
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenter#1797D3
hypocenterC3#20720
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. C2×#4840C2semidirect
3. C3×#1797C2semidirect
4. #71530C2semidirect
5. #70476C2semidirect
7. #69907C2semidirect
8. #68664C2semidirect
9. #68664C2semidirect
10. #68600C2semidirect
11. #68599C2semidirect
52. #1797D3semidirect
660. C3#20720semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2352
C2×#48402
C3×#17972
#715302
#704762
#699562
#699072
#686642
#686002
#685992
#207203