Finite Group Databasebeta

← #10823585

Subgroups — #10823585

Order 768.

1058 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic
2C2×#5030C211yesmaximal, characteristic, complemented
3C2×C4×#216C211yesmaximal, characteristic, complemented
4#82235C211yesmaximal, characteristic
5#71793C211yesmaximal, characteristic, complemented
6#71781C211yesmaximal, characteristic, complemented
7#71744C211yesmaximal, characteristic, complemented
8#71739C211yesmaximal, characteristic, complemented
9#71700C211yesmaximal, characteristic
10#71700C211yesmaximal, characteristic
11#71695C211yesmaximal, characteristic
12#71694C211yesmaximal, characteristic
13#71631C211yesmaximal, characteristic, complemented
14C4×#835C211yesmaximal, characteristic, complemented
15#71582C211yesmaximal, characteristic
16#23586—31nomaximal, Sylow
17C3×#541—24no—
18C3×#541—24no—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×#126—21no—
21C2×C3×#126—21no—
22C2×C3×#125—21no—
23C2×C3×#125—21no—
24C2×C3×#125C2211yescharacteristic
25C2×C3×C22⋊Q8C2211yescharacteristic
26C2×C3×C22⋊Q8C2211yescharacteristic
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×Q8C2211yescharacteristic
29C2×C3×#117C2211yescharacteristic
30C22×#216C2211yescharacteristic
31C22×#216C2211yescharacteristic
32C2×#924C2211yescharacteristic
33C2×C4×Dic6C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#835C2211yescharacteristic
39C2×#835C2211yescharacteristic
40#5212C2211yescharacteristic
41C2×#830C2211yescharacteristic
42C2×#830C2211yescharacteristic
43#5030—24no—
44C4×#216—24no—
45#4978—21no—
46#4976—21no—
47C2×#810C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#808C2211yescharacteristic
50C2×#807C2211yescharacteristic
51#4945—21no—
52#4944—21no—
53C4×C4⋊Dic3C2211yescharacteristic
54#4929C2211yescharacteristic
55#4928C2211yescharacteristic
56C2×#803C2211yescharacteristic
57C2×C4×C3⋊C8C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4524C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#541D311yescharacteristic
65C2×#441—31no—
66C2×C4×Q16—31no—
67#2323—31no—
68#2059—31no—
69#2043—31no—
70#2036—31no—
71#2035—31no—
72#2026—31no—
73#2016—31no—
74#2001—31no—
75#2001—31no—
76#1763—31no—
77C4×#103—31no—
78#1750—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×Q8C2311yescharacteristic
81C23×C3×C4C2311yescharacteristic
82C22×Dic6C2311yescharacteristic
83C3×C4⋊Q8—22no—
84C3×C4⋊Q8—22no—
85C3×#126—28no—
86C3×#126—28no—
87C3×#125—28no—
88C3×#125—28no—
89C3×#125—22no—
90C3×#125—22no—
91C3×C22⋊Q8—22no—
92C3×C22⋊Q8—22no—
93C3×C22⋊Q8—22no—
94C3×C22⋊Q8—22no—
95C3×C4×Q8—22no—
96C3×C4×Q8—22no—
97C3×C4×Q8D414yes—
98C3×C4×Q8D414yes—
99C3×#117—22no—
100C3×#117—22no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4D411yescharacteristic
110C2×C3×C4⋊C4D411yescharacteristic
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C22⋊C4D411yescharacteristic
116C2×C3×C22⋊C4D411yescharacteristic
117C2×C3×C42—21no—
118C2×C3×C42—21no—
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2311yescharacteristic
121C2×#217—21no—
122C2×#217—21no—
123C2×#216—22no—
124C2×#216—22no—
125C2×#216—22no—
126C2×#216—22no—
127C2×#216—22no—
128C2×#216—22no—
129C2×C4⋊Dic3C2311yescharacteristic
130C2×C4⋊Dic3C2311yescharacteristic
131C2×C4⋊Dic3C2311yescharacteristic
132#924—22no—
133#924—22no—
134C2×Dic3⋊C4—21no—
135C2×Dic3⋊C4—21no—
136C2×Dic3⋊C4—21no—
137C2×C4×Dic3—21no—
138C2×C4×Dic3—21no—
139C22×C3⋊C8C2311yescharacteristic
140C22×C3⋊C8C2311yescharacteristic
141C22×C3⋊C8C2311yescharacteristic
142C22×C3⋊C8C2311yescharacteristic
143C4×Dic6—22no—
144C4×Dic6—22no—
145#835—22no—
146#835—22no—
147#835—22no—
148#835—22no—
149#835—22no—
150#835—22no—
151C3⋊#95—21no—
152C3⋊#95—21no—
153C3⋊#95—21no—
154C3⋊#95—21no—
155C3⋊#95—21no—
156#830—22no—
157#830—22no—
158#810—22no—
159#810—22no—
160#808—22no—
161#807—22no—
162#803—22no—
163C4×C3⋊C8—22no—
164C22×Q16—31no—
165C22×Q16—31no—
166#541—24no—
167#541—24no—
168C2×C4⋊Q8C2×D311yescharacteristic
169C2×#126—21no—
170C2×#126—21no—
171C2×#125—21no—
172C2×#125—21no—
173C2×#125C2×D311yescharacteristic
174C2×C22⋊Q8—31no—
175C2×C22⋊Q8C2×D311yescharacteristic
176C2×C22⋊Q8C2×D311yescharacteristic
177C2×C4×Q8—31no—
178C2×C4×Q8C2×D311yescharacteristic
179C2×C4×Q8C2×D311yescharacteristic
180C2×#117C2×D311yescharacteristic
181#441—64no—
182C4×Q16—64no—
183C2×#107—31no—
184C2×#106—31no—
185C2×#103—31no—
186C2×#103—31no—
187C2×#103—31no—
188C2×#103—31no—
189C2×#103—31no—
190C2×#103—31no—
191C2×#103—31no—
192C2×#103—31no—
193C2×C22⋊C8—31no—
194C2×C22⋊C8—31no—
195C2×#97—31no—
196C2×C4×C8—31no—
197#391—31no—
198#389—31no—
199#389—61no—
200#388—61no—
201#388—61no—
202#387—61no—
203#380—31no—
204C4×C4⋊C4—31no—
205#340—31no—
206#340—31no—
207#340—31no—
208#340—31no—
209#336—31no—
210#336—31no—
211C24×C3C2×D411yescharacteristic
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—21no—
218C2×C3×Q8—21no—
219C2×C3×Q8—21no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8C2×D412yes—
223C2×C3×Q8C2×D412yes—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—21no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4—21no—
234C22×C3×C4C2×D411yescharacteristic
235C22×C3×C4C2×D411yescharacteristic
236C22×C3×C4C2×D411yescharacteristic
237C22×C3×C4C2×D411yescharacteristic
238C22×C3×C4C2×D411yescharacteristic
239C22×C3×C4C2411yescharacteristic
240C22×Dic3—21no—
241C22×Dic3—21no—
242C22×Dic3—21no—
243C22×Dic3—21no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6—21no—
249C2×Dic6#4112yes—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—24no—
252C3×C4⋊C4—24no—
253C3×C4⋊C4—24no—
254C3×C4⋊C4—24no—
255C3×C4⋊C4—24no—
256C3×C4⋊C4—24no—
257C3×C4⋊C4—24no—
258C3×C4⋊C4—22no—
259C3×C4⋊C4—22no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4C2×D412yes—
263C3×C4⋊C4C2×D412yes—
264C3×C22⋊C4—24no—
265C3×C22⋊C4—24no—
266C3×C22⋊C4—22no—
267C3×C22⋊C4—22no—
268C3×C42—24no—
269C3×C42—24no—
270C3×C42—21no—
271C3×C42—21no—
272C3×C42C2×D412yes—
273C3×C42C2×D412yes—
274#217—24no—
275#217—81no—
276#216—28no—
277#216—28no—
278C4⋊Dic3—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283C4⋊Dic3#4112yes—
284Dic3⋊C4—24no—
285Dic3⋊C4—42no—
286Dic3⋊C4—81no—
287C4×Dic3—42no—
288C4×Dic3—81no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299C2×C3⋊C8#4112yes—
300C2×C3⋊C8#4112yes—
301C22×Q8—31no—
302C22×Q8C22×D311yescharacteristic
303C22×Q8C22×D311yescharacteristic
304C23×C4C22×D311yescharacteristic
305C2×Q16—62no—
306C2×Q16—62no—
307C2×Q16—62no—
308C2×Q16—62no—
309C2×Q16—62no—
310C2×Q16—62no—
311C22×C8—31no—
312C22×C8—31no—
313C22×C8—31no—
314C22×C8—31no—
315C4⋊Q8—22no—
316C4⋊Q8—22no—
317#126—28no—
318#126—28no—
319#125—28no—
320#125—28no—
321#125—22no—
322#125—22no—
323C22⋊Q8—22no—
324C22⋊Q8—22no—
325C22⋊Q8—22no—
326C22⋊Q8—22no—
327C22⋊Q8—62no—
328C22⋊Q8—62no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8—62no—
332C4×Q8—62no—
333C4×Q8#6714yes—
334C4×Q8#6714yes—
335#117—22no—
336#117—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—31no—
346C2×C4⋊C4—31no—
347C2×C4⋊C4—31no—
348C2×C4⋊C4—61no—
349C2×C4⋊C4—61no—
350C2×C4⋊C4—61no—
351C2×C4⋊C4#6711yescharacteristic
352C2×C4⋊C4#6711yescharacteristic
353C2×C4⋊C4C22×D311yescharacteristic
354C2×C4⋊C4C22×D311yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—61no—
358C2×C22⋊C4—61no—
359C2×C22⋊C4#6711yescharacteristic
360C2×C22⋊C4#6711yescharacteristic
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—61no—
364C2×C42—61no—
365C2×C42C22×D311yescharacteristic
366C2×C42C22×D311yescharacteristic
367#107—62no—
368#106—62no—
369#103—62no—
370#103—62no—
371#103—62no—
372#103—62no—
373#103—62no—
374#103—62no—
375#103—62no—
376#103—62no—
377C22⋊C8—62no—
378C22⋊C8—62no—
379#97—62no—
380C4×C8—62no—
381#95—61no—
382#95—61no—
383#95—61no—
384#95—61no—
385#95—61no—
386C23×C3—21no—
387C23×C3—21no—
388C23×C3—21no—
389C23×C3—21no—
390C23×C3—21no—
391C23×C3—21no—
392C23×C3—21no—
393C23×C3C22×D411yescharacteristic
394C3×Q8—22no—
395C3×Q8—22no—
396C3×Q8—22no—
397C3×Q8—22no—
398C3×Q8—41no—
399C3×Q8—41no—
400C3×Q8—41no—
401C3×Q8—41no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—21no—
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4—41no—
422C2×C3×C4—41no—
423C2×C3×C4—41no—
424C2×C3×C4—41no—
425C2×C3×C4—41no—
426C2×C3×C4—41no—
427C2×C3×C4—41no—
428C2×C3×C4—41no—
429C2×C3×C4—41no—
430C2×C3×C4—41no—
431C2×C3×C4#12012yes—
432C2×C3×C4#12012yes—
433C2×C3×C4C22×D411yescharacteristic
434C2×C3×C4C22×D411yescharacteristic
435C2×C3×C4C2×#4111yescharacteristic
436C2×C3×C4C2×#4111yescharacteristic
437C2×C3×C4C2×#4111yescharacteristic
438C2×C3×C4C2×#4111yescharacteristic
439C2×Dic3—22no—
440C2×Dic3—22no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—41no—
445C2×Dic3—41no—
446C2×Dic3—41no—
447C2×Dic3—41no—
448C2×Dic3—41no—
449C2×Dic3—41no—
450C2×Dic3—41no—
451Dic6—22no—
452Dic6—22no—
453Dic6—41no—
454Dic6—41no—
455C3⋊C8—22no—
456C3⋊C8—22no—
457C3⋊C8—41no—
458C3⋊C8—41no—
459C24C2×#6711yescharacteristic
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—32no—
471C2×Q8—61no—
472C2×Q8—61no—
473C2×Q8—61no—
474C2×Q8—61no—
475C2×Q8—61no—
476C2×Q8C2×#6712yes—
477C2×Q8C2×#6712yes—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—61no—
489C22×C4—61no—
490C22×C4—61no—
491C22×C4—61no—
492C22×C4C2×#6711yescharacteristic
493C22×C4C2×#6711yescharacteristic
494C22×C4C2×#6711yescharacteristic
495C22×C4C2×#6711yescharacteristic
496C22×C4C2×#6711yescharacteristic
497C22×C4C23×D311yescharacteristic
498Q16—68no—
499Q16—68no—
500C2×C8—32no—
501C2×C8—32no—
502C2×C8—61no—
503C2×C8—61no—
504C2×C8—61no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8—61no—
510C2×C8—61no—
511C2×C8—61no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—32no—
525C4⋊C4—64no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C4⋊C4—61no—
530C4⋊C4—61no—
531C4⋊C4—122no—
532C4⋊C4—241no—
533C4⋊C4C2×#6712yes—
534C4⋊C4C2×#6712yes—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—64no—
540C22⋊C4—241no—
541C42—24no—
542C42—24no—
543C42—21no—
544C42—21no—
545C42—122no—
546C42—241no—
547C42C2×#6712yes—
548C42C2×#6712yes—
549C22×C3—41no—
550C22×C3—41no—
551C22×C3—41no—
552C22×C3—41no—
553C22×C3—41no—
554C22×C3—41no—
555C22×C3—41no—
556C22×C3C2×#12011yescharacteristic
557C22×C3#52511yescharacteristic
558C22×C3#53211yescharacteristic
559C22×C3#57711yescharacteristic
560C22×C3#57811yescharacteristic
561C22×C3#57811yescharacteristic
562C22×C3#57811yescharacteristic
563C3×C4—22no—
564C3×C4—22no—
565C3×C4—41no—
566C3×C4—42no—
567C3×C4—42no—
568C3×C4—41no—
569C3×C4—42no—
570C3×C4—41no—
571C3×C4—42no—
572C3×C4#52511yescharacteristic
573C3×C4#52511yescharacteristic
574C3×C4#53211yescharacteristic
575C3×C4#53211yescharacteristic
576Dic3—42no—
577Dic3—42no—
578Dic3—81no—
579Dic3—81no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23C22×#6711yescharacteristic
588Q8—22no—
589Q8—22no—
590Q8—22no—
591Q8—22no—
592Q8—41no—
593Q8—41no—
594Q8—41no—
595Q8—41no—
596Q8—62no—
597Q8—62no—
598Q8—121no—
599Q8—121no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4#95312yes—
642C2×C4#95312yes—
643C2×C4D3×#4111yescharacteristic
644C2×C4D3×#4111yescharacteristic
645C2×C4D3×#4111yescharacteristic
646C2×C4D3×#4111yescharacteristic
647C2×C4C22×#6711yescharacteristic
648C2×C4C22×#6711yescharacteristic
649C8—62no—
650C8—62no—
651C8—121no—
652C8—121no—
653C2×C3—81no—
654C2×C3#238811yescharacteristic
655C2×C3#301111yescharacteristic
656C2×C3#301311yescharacteristic
657C2×C3#312011yescharacteristic
658C2×C3#312711yescharacteristic
659C2×C3#317711yescharacteristic
660C2×C3#317811yescharacteristic
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22#582811yescharacteristic
669C22#583211yescharacteristic
670C22#583711yescharacteristic
671C22#583811yescharacteristic
672C22#583811yescharacteristic
673C22#583811yescharacteristic
674C22C2×#95311yescharacteristic
675C4—22no—
676C4—22no—
677C4—41no—
678C4—42no—
679C4—42no—
680C4—41no—
681C4—42no—
682C4—41no—
683C4—42no—
684C4—122no—
685C4—122no—
686C4—241no—
687C4—241no—
688C4#582811yescharacteristic
689C4#582811yescharacteristic
690C4#583211yescharacteristic
691C4#583211yescharacteristic
692C3#2358611yescharacteristic, complemented, Sylow
693C2—81no—
694C2#8344411yescharacteristic
695C2#8344511yescharacteristic
696C2#8344611yescharacteristic
697C2#8345111yescharacteristic
698C2#8409511yescharacteristic
699C2#8411411yescharacteristic
700C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#541C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#541D3
hypocenterC3#23586
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
2. C2×#5030C2semidirect
3. C2×C4×#216C2semidirect
5. #71793C2semidirect
6. #71781C2semidirect
7. #71744C2semidirect
8. #71739C2semidirect
13. #71631C2semidirect
14. C4×#835C2semidirect
692. C3#23586semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×#50302
C2×C4×#2162
#822352
#717932
#717812
#717442
#717392
#717002
#716952
#716942
#716312
C4×#8352
#715822
#235863