Finite Group Databasebeta

← #10772017

Subgroups — #10772017

Order 768.

930 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3515C211yesmaximal, characteristic, complemented
2#83347C211yesmaximal, characteristic, complemented
3#83150C211yesmaximal, characteristic, complemented
4#82525C211yesmaximal, characteristic, complemented
5#71937C211yesmaximal, characteristic, complemented
6#71935C211yesmaximal, characteristic, complemented
7#71926C211yesmaximal, characteristic, complemented
8#71925C211yesmaximal, characteristic, complemented
9#71923C211yesmaximal, characteristic, complemented
10#71923C211yesmaximal, characteristic, complemented
11#71912C211yesmaximal, characteristic, complemented
12#71910C211yesmaximal, characteristic, complemented
13#71830C211yesmaximal, characteristic, complemented
14#71751C211yesmaximal, characteristic, complemented
15#71641C211yesmaximal, characteristic, complemented
16#18596—31nomaximal, Sylow
17C3×#557C2211yescharacteristic
18C3×#554C2211yescharacteristic
19C3×#554C2211yescharacteristic
20C3×#553—21no—
21C3×#552—21no—
22C3×#552—22no—
23C3×#552—21no—
24C3×#547—21no—
25C3×#547C2211yescharacteristic
26C3×#541—21no—
27C3×#539—22no—
28C3×#533C2211yescharacteristic
29C3×#529—21no—
30C3×#529C2211yescharacteristic
31C2×C3×#125—21no—
32C3×#520—21no—
33C3×C4×#41C2211yescharacteristic
34C2×#950—21no—
35#5477C2211yescharacteristic
36#5088C2211yescharacteristic
37#5085C2211yescharacteristic
38#5079C2211yescharacteristic
39#5078C2211yescharacteristic
40#5066—21no—
41#5057—21no—
42#5049—22no—
43#5049—21no—
44#5039—21no—
45#5031C2211yescharacteristic
46#5028—21no—
47#5027C2211yescharacteristic
48#5027C2211yescharacteristic
49#5026—22no—
50#5020C2211yescharacteristic
51#5019C2211yescharacteristic
52#5016—21no—
53#5015C2211yescharacteristic
54#5015C2211yescharacteristic
55#5007C2211yescharacteristic
56#5002—21no—
57#4924C2211yescharacteristic
58C2×C4⋊(C3⋊C8)C2211yescharacteristic
59C4×#208C2211yescharacteristic
60#4547C2211yescharacteristic
61#4547C2211yescharacteristic
62#4546C2211yescharacteristic
63#4546C2211yescharacteristic
64#4535C2211yescharacteristic
65#4535C2211yescharacteristic
66#4535C2211yescharacteristic
67#4535C2211yescharacteristic
68D4⋊(C3⋊C8)C2211yescharacteristic
69D4⋊(C3⋊C8)C2211yescharacteristic
70D4⋊(C3⋊C8)C2211yescharacteristic
71D4⋊(C3⋊C8)C2211yescharacteristic
72#3515D311yescharacteristic, complemented
73#3251—31no—
74#3182—31no—
75#3035—31no—
76#1648—31no—
77#1647—31no—
78#1644—31no—
79#1643—31no—
80#1626—31no—
81#1625—31no—
82#1622—31no—
83#1622—31no—
84#1550—31no—
85#1489—31no—
86#1483—31no—
87C2×C3×#41—21no—
88C2×C3×#41D411yescharacteristic
89C2×#235—21no—
90C3×C4⋊Q8—21no—
91C3×C4⋊Q8C2311yescharacteristic
92C3×C4⋊Q8C2311yescharacteristic
93C3×#126—22no—
94C3×#126—22no—
95C3×#126—22no—
96C3×#126—21no—
97C3×#125—22no—
98C3×#125—22no—
99C3×#125—22no—
100C3×#125—21no—
101C3×#125—21no—
102C3×#125—22no—
103C3×#125—21no—
104C3×#125—21no—
105C3×#125D411yescharacteristic
106C3×#124D411yescharacteristic
107C3×#123—22no—
108C3×#123—22no—
109C3×#123—22no—
110C3×#123—21no—
111C3×#123—22no—
112C3×C22⋊Q8—21no—
113C3×C22⋊Q8—22no—
114C3×C22⋊Q8—22no—
115C3×C22⋊Q8—21no—
116C3×C22⋊Q8—21no—
117C3×C22⋊Q8—21no—
118C3×C22⋊Q8D411yescharacteristic
119C3×C22⋊Q8D412yes—
120C3×#121—22no—
121C3×#121—21no—
122C3×#121D411yescharacteristic
123C3×C4×Q8—21no—
124C3×C4×Q8—21no—
125C3×C4×Q8—21no—
126C3×C4×Q8D412yes—
127C3×C4×Q8D411yescharacteristic
128C3×C4×Q8C2311yescharacteristic
129C3×C4×Q8C2311yescharacteristic
130C3×C4×D4—22no—
131C3×C4×D4—22no—
132C3×C4×D4—21no—
133C3×C4×D4—21no—
134C3×C4×D4—21no—
135C3×C4×D4—21no—
136C3×C4×D4—21no—
137C3×C4×D4D411yescharacteristic
138C3×C4×D4C2311yescharacteristic
139C3×C4×D4C2311yescharacteristic
140C3×#117—22no—
141C3×#117—22no—
142C3×#117—21no—
143C3×#117—21no—
144C3×#117—21no—
145C3×#117D411yescharacteristic
146C2×C3×C4⋊C4—22no—
147C2×C3×C4⋊C4—22no—
148C2×C3×C4⋊C4—21no—
149C2×C3×C4⋊C4—21no—
150C2×C3×C4⋊C4—21no—
151C2×C3×C42—21no—
152C2×C3×C42C2311yescharacteristic
153#950—44no—
154C2×#216—21no—
155C2×#215—22no—
156C2×#215—21no—
157C2×#215—21no—
158C2×#214—21no—
159C2×#213—21no—
160C2×#213—21no—
161#930—21no—
162#930—21no—
163C2×#208—21no—
164C2×#208—21no—
165C22×C3⋊C8—21no—
166#875—21no—
167#874C2311yescharacteristic
168C4×D12—21no—
169C4×D12—21no—
170#869C2311yescharacteristic
171#835—21no—
172#835—22no—
173#832—21no—
174#810—21no—
175#810—21no—
176#809—21no—
177#809—21no—
178#809—21no—
179#809—22no—
180#809—21no—
181#808—21no—
182#807—21no—
183#807—22no—
184C4⋊(C3⋊C8)—21no—
185C4⋊(C3⋊C8)—21no—
186C4⋊(C3⋊C8)C2311yescharacteristic
187C4⋊(C3⋊C8)C2311yescharacteristic
188C4⋊(C3⋊C8)C2311yescharacteristic
189C4⋊(C3⋊C8)C2311yescharacteristic
190#803—21no—
191C4×C3⋊C8C2311yescharacteristic
192C4×C3⋊C8C2311yescharacteristic
193C2×#135—61no—
194#557C2×D311yescharacteristic
195#554C2×D311yescharacteristic
196#554C2×D311yescharacteristic
197#553—21no—
198#552—21no—
199#552—22no—
200#552—21no—
201#547—21no—
202#547C2×D311yescharacteristic
203#541—21no—
204#539—22no—
205#533C2×D311yescharacteristic
206#532—31no—
207#529—21no—
208#529C2×D311yescharacteristic
209C2×#125—21no—
210#520—21no—
211C4×#41C2×D311yescharacteristic
212#500—31no—
213#492—31no—
214#487—31no—
215#486—31no—
216#478—61no—
217#475—31no—
218#474—31no—
219#469—61no—
220#468—62no—
221#468—61no—
222#464—61no—
223#463—61no—
224#463—61no—
225#462—31no—
226#461—31no—
227#460—31no—
228#460—31no—
229#459—31no—
230#459—31no—
231#440—62no—
232#423—31no—
233C2×C4⋊C8—31no—
234#416—61no—
235C4×M16—31no—
236#333—31no—
237#333—31no—
238#332—31no—
239#332—31no—
240Q8⋊C8—31no—
241Q8⋊C8—31no—
242Q8⋊C8—31no—
243Q8⋊C8—31no—
244D4⋊C8—31no—
245D4⋊C8—31no—
246D4⋊C8—31no—
247D4⋊C8—31no—
248C3×#41—22no—
249C3×#41—41no—
250C3×#41—42no—
251C3×#41—42no—
252C2×C3×Q8—21no—
253C2×C3×Q8C2×D411yescharacteristic
254C2×C3×Q8C2×D411yescharacteristic
255C2×C3×Q8C2×D412yes—
256C2×C3×D4—22no—
257C2×C3×D4—21no—
258C2×C3×D4—21no—
259C2×C3×D4C2×D411yescharacteristic
260C2×C3×D4C2×D411yescharacteristic
261C22×C3×C4—22no—
262C22×C3×C4—22no—
263C22×C3×C4—22no—
264C22×C3×C4—21no—
265C22×C3×C4—21no—
266C22×C3×C4—21no—
267C22×C3×C4—21no—
268C22×C3×C4C2×D411yescharacteristic
269C22×C3×C4C2×D411yescharacteristic
270C22×C3×C4C2×D411yescharacteristic
271C2×#67—41no—
272#235—42no—
273#235—42no—
274C2×D12—21no—
275C2×D12—21no—
276C2×D12—21no—
277C2×C4×D3—41no—
278C2×Dic6—21no—
279C3×C4⋊C4—22no—
280C3×C4⋊C4—22no—
281C3×C4⋊C4—22no—
282C3×C4⋊C4—22no—
283C3×C4⋊C4—22no—
284C3×C4⋊C4—22no—
285C3×C4⋊C4—22no—
286C3×C4⋊C4—22no—
287C3×C4⋊C4—21no—
288C3×C4⋊C4—21no—
289C3×C4⋊C4—22no—
290C3×C4⋊C4—22no—
291C3×C4⋊C4—21no—
292C3×C4⋊C4—21no—
293C3×C4⋊C4—21no—
294C3×C4⋊C4—21no—
295C3×C4⋊C4—21no—
296C3×C4⋊C4—21no—
297C3×C4⋊C4—21no—
298C3×C4⋊C4C2×D411yescharacteristic
299C3×C4⋊C4C2×D412yes—
300C3×C4⋊C4C2×D411yescharacteristic
301C3×C4⋊C4C2×D411yescharacteristic
302C3×C4⋊C4C2×D411yescharacteristic
303C3×C22⋊C4—22no—
304C3×C22⋊C4—22no—
305C3×C22⋊C4—22no—
306C3×C22⋊C4—22no—
307C3×C22⋊C4—21no—
308C3×C22⋊C4—21no—
309C3×C22⋊C4—21no—
310C3×C22⋊C4—21no—
311C3×C22⋊C4—21no—
312C3×C22⋊C4—21no—
313C3×C22⋊C4—21no—
314C3×C22⋊C4—21no—
315C3×C42—22no—
316C3×C42—21no—
317C3×C42—21no—
318C3×C42—22no—
319C3×C42—21no—
320C3×C42—21no—
321C3×C42C2×D411yescharacteristic
322C3×C42C2×D411yescharacteristic
323C3×C42C2×D411yescharacteristic
324C3×C42C2411yescharacteristic
325#216—42no—
326#215—44no—
327#215—42no—
328#215—81no—
329#214—42no—
330#213—42no—
331#213—81no—
332(C2×D3)⋊C4—41no—
333(C2×D3)⋊C4—41no—
334C4⋊Dic3—21no—
335C4⋊Dic3—21no—
336#208—42no—
337#208—81no—
338C2×C3⋊C8—21no—
339C2×C3⋊C8—21no—
340C2×C3⋊C8—21no—
341C2×C3⋊C8—21no—
342C2×C3⋊C8—21no—
343C2×C3⋊C8—21no—
344C2×C3⋊C8—42no—
345C2×#41—21no—
346C2×#41—61no—
347C2×#41#6711yescharacteristic
348#135—124no—
349C2×Q16—61no—
350C2×SD16—61no—
351C2×SD16—62no—
352C2×SD16—61no—
353C2×SD16—61no—
354C2×D8—61no—
355C2×D8—61no—
356C2×M16—61no—
357C2×M16—61no—
358C22×C8—61no—
359C4⋊Q8—21no—
360C4⋊Q8—31no—
361C4⋊Q8C22×D311yescharacteristic
362C4⋊Q8C22×D311yescharacteristic
363#127—31no—
364#126—22no—
365#126—22no—
366#126—22no—
367#126—21no—
368#125—22no—
369#125—22no—
370#125—22no—
371#125—21no—
372#125—21no—
373#125—22no—
374#125—21no—
375#125—21no—
376#125#6711yescharacteristic
377#124—61no—
378#124#6711yescharacteristic
379#123—22no—
380#123—22no—
381#123—22no—
382#123—21no—
383#123—22no—
384C22⋊Q8—21no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387C22⋊Q8—21no—
388C22⋊Q8—21no—
389C22⋊Q8—21no—
390C22⋊Q8#6711yescharacteristic
391C22⋊Q8#6712yes—
392#121—22no—
393#121—21no—
394#121—61no—
395#121—61no—
396#121#6711yescharacteristic
397C4×Q8—21no—
398C4×Q8—21no—
399C4×Q8—21no—
400C4×Q8#6711yescharacteristic
401C4×Q8#6712yes—
402C4×Q8C22×D311yescharacteristic
403C4×Q8C22×D311yescharacteristic
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4—21no—
409C4×D4—21no—
410C4×D4—21no—
411C4×D4—61no—
412C4×D4—61no—
413C4×D4#6711yescharacteristic
414C4×D4C22×D311yescharacteristic
415C4×D4C22×D311yescharacteristic
416#117—22no—
417#117—22no—
418#117—21no—
419#117—21no—
420#117—21no—
421#117#6711yescharacteristic
422C2×C4⋊C4—22no—
423C2×C4⋊C4—22no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C42—21no—
428C2×C42C22×D311yescharacteristic
429#107—62no—
430#107—61no—
431#106—61no—
432C4⋊C8—31no—
433C4⋊C8—31no—
434C4⋊C8—31no—
435C4⋊C8—31no—
436C4⋊C8—61no—
437C4⋊C8—61no—
438#103—62no—
439#103—61no—
440#103—61no—
441#103—61no—
442#102—61no—
443#102—61no—
444#102—61no—
445#102—62no—
446#102—61no—
447#102—61no—
448#97—61no—
449C4×C8—31no—
450C4×C8—31no—
451C23×C3—21no—
452C23×C3—21no—
453C23×C3#12011yescharacteristic
454C22×D3—41no—
455C3×Q8—21no—
456C3×Q8—21no—
457C3×Q8—21no—
458C3×Q8—21no—
459C3×Q8—42no—
460C3×Q8—41no—
461C3×Q8—41no—
462C3×D4—21no—
463C3×D4—21no—
464C3×D4—21no—
465C3×D4—42no—
466C3×D4—44no—
467C3×D4—81no—
468C3×D4AGL(1,ℤ/8)12yes—
469C2×C3×C4—21no—
470C2×C3×C4—21no—
471C2×C3×C4—21no—
472C2×C3×C4—21no—
473C2×C3×C4—21no—
474C2×C3×C4—21no—
475C2×C3×C4—21no—
476C2×C3×C4—21no—
477C2×C3×C4—22no—
478C2×C3×C4—22no—
479C2×C3×C4—21no—
480C2×C3×C4—21no—
481C2×C3×C4—41no—
482C2×C3×C4—42no—
483C2×C3×C4—44no—
484C2×C3×C4—41no—
485C2×C3×C4—41no—
486C2×C3×C4—41no—
487C2×C3×C4—41no—
488C2×C3×C4—81no—
489C2×C3×C4—82no—
490C2×C3×C4—82no—
491C2×C3×C4#12011yescharacteristic
492C2×C3×C4#12011yescharacteristic
493C2×C3×C4#12011yescharacteristic
494C2×C3×C4C22×D411yescharacteristic
495C2×C3×C4C22×D411yescharacteristic
496C2×C3×C4C22×D411yescharacteristic
497#67—82no—
498C2×Dic3—41no—
499D12—42no—
500D12—41no—
501D12—41no—
502D12—41no—
503D12—41no—
504C4×D3—82no—
505Dic6—41no—
506Dic6—41no—
507C3⋊C8—22no—
508C3⋊C8—41no—
509C3⋊C8—41no—
510C3⋊C8—41no—
511C3⋊C8—41no—
512C3⋊C8—41no—
513#41—22no—
514#41—41no—
515#41—42no—
516#41—42no—
517#41—122no—
518#41—122no—
519C2×Q8—21no—
520C2×Q8—61no—
521C2×Q8C2×#6711yescharacteristic
522C2×Q8C2×#6711yescharacteristic
523C2×Q8C2×#6712yes—
524C2×D4—22no—
525C2×D4—21no—
526C2×D4—21no—
527C2×D4—61no—
528C2×D4—61no—
529C2×D4—61no—
530C2×D4—121no—
531C2×D4C2×#6711yescharacteristic
532C2×D4C2×#6711yescharacteristic
533C22×C4—22no—
534C22×C4—22no—
535C22×C4—22no—
536C22×C4—21no—
537C22×C4—21no—
538C22×C4—21no—
539C22×C4—21no—
540C22×C4—121no—
541C22×C4C2×#6711yescharacteristic
542C22×C4C2×#6711yescharacteristic
543C22×C4C2×#6711yescharacteristic
544Q16—122no—
545SD16—124no—
546SD16—122no—
547SD16—122no—
548SD16—241no—
549D8—122no—
550D8—241no—
551M16—122no—
552M16—241no—
553C2×C8—61no—
554C2×C8—61no—
555C2×C8—61no—
556C2×C8—61no—
557C2×C8—61no—
558C2×C8—61no—
559C2×C8—122no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—22no—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—22no—
568C4⋊C4—21no—
569C4⋊C4—21no—
570C4⋊C4—22no—
571C4⋊C4—22no—
572C4⋊C4—21no—
573C4⋊C4—21no—
574C4⋊C4—21no—
575C4⋊C4—21no—
576C4⋊C4—21no—
577C4⋊C4—21no—
578C4⋊C4—21no—
579C4⋊C4—61no—
580C4⋊C4—61no—
581C4⋊C4C2×#6711yescharacteristic
582C4⋊C4C2×#6711yescharacteristic
583C4⋊C4C2×#6711yescharacteristic
584C4⋊C4C2×#6711yescharacteristic
585C4⋊C4C2×#6712yes—
586C22⋊C4—22no—
587C22⋊C4—22no—
588C22⋊C4—22no—
589C22⋊C4—22no—
590C22⋊C4—21no—
591C22⋊C4—21no—
592C22⋊C4—21no—
593C22⋊C4—21no—
594C22⋊C4—21no—
595C22⋊C4—21no—
596C22⋊C4—21no—
597C22⋊C4—21no—
598C22⋊C4—121no—
599C22⋊C4—121no—
600C42—22no—
601C42—21no—
602C42—21no—
603C42—22no—
604C42—21no—
605C42—21no—
606C42C2×#6711yescharacteristic
607C42C2×#6711yescharacteristic
608C42C2×#6711yescharacteristic
609C42C23×D311yescharacteristic
610C22×C3—21no—
611C22×C3—22no—
612C22×C3—21no—
613C22×C3—21no—
614C22×C3—41no—
615C22×C3—41no—
616C22×C3—41no—
617C22×C3—41no—
618C22×C3—41no—
619C22×C3C2×#12011yescharacteristic
620C2×D3—81no—
621C2×D3—81no—
622C2×D3—81no—
623C3×C4—21no—
624C3×C4—21no—
625C3×C4—21no—
626C3×C4—21no—
627C3×C4—41no—
628C3×C4—41no—
629C3×C4—42no—
630C3×C4—41no—
631C3×C4—41no—
632C3×C4—41no—
633C3×C4—41no—
634C3×C4—42no—
635C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
636C3×C4#57511yescharacteristic
637C3×C4#57711yescharacteristic
638C3×C4#57811yescharacteristic
639Dic3—81no—
640C23—21no—
641C23—21no—
642C23—121no—
643C23#95311yescharacteristic
644Q8—21no—
645Q8—21no—
646Q8—21no—
647Q8—21no—
648Q8—42no—
649Q8—41no—
650Q8—41no—
651Q8—121no—
652Q8—121no—
653D4—21no—
654D4—21no—
655D4—21no—
656D4—42no—
657D4—44no—
658D4—81no—
659D4—122no—
660D4—121no—
661D4—121no—
662D4—121no—
663D4—121no—
664D4—242no—
665D4#94912yes—
666C2×C4—21no—
667C2×C4—21no—
668C2×C4—21no—
669C2×C4—21no—
670C2×C4—21no—
671C2×C4—21no—
672C2×C4—21no—
673C2×C4—21no—
674C2×C4—22no—
675C2×C4—22no—
676C2×C4—21no—
677C2×C4—21no—
678C2×C4—41no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—41no—
685C2×C4—81no—
686C2×C4—82no—
687C2×C4—82no—
688C2×C4—121no—
689C2×C4—242no—
690C2×C4#95311yescharacteristic
691C2×C4#95311yescharacteristic
692C2×C4#95311yescharacteristic
693C2×C4C22×#6711yescharacteristic
694C2×C4C22×#6711yescharacteristic
695C2×C4C22×#6711yescharacteristic
696C8—62no—
697C8—121no—
698C8—121no—
699C8—121no—
700C8—121no—
701C8—121no—
702C2×C3—41no—
703C2×C3—42no—
704C2×C3—81no—
705C2×C3#300611yescharacteristic
706C2×C3#301311yescharacteristic
707C2×C3#301811yescharacteristic
708D3—161no—
709C22—21no—
710C22—22no—
711C22—21no—
712C22—21no—
713C22—41no—
714C22—41no—
715C22—41no—
716C22—41no—
717C22—41no—
718C22—241no—
719C22—241no—
720C22—241no—
721C22C2×#95311yescharacteristic
722C4—21no—
723C4—21no—
724C4—21no—
725C4—21no—
726C4—41no—
727C4—41no—
728C4—42no—
729C4—41no—
730C4—41no—
731C4—41no—
732C4—41no—
733C4—42no—
734C4—241no—
735C4C2×#94911yescharacteristic
736C4#582211yescharacteristic
737C4#583711yescharacteristic
738C4#583811yescharacteristic
739C3#1859611yescharacteristic, complemented, Sylow
740C2—41no—
741C2—42no—
742C2—81no—
743C2—481no—
744C2#8408211yescharacteristic
745C2#8411411yescharacteristic
746C2#8412111yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3515C2
socleC22×C3C2×#120
hypercenter#3515D3
hypocenterC3#18596
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3515

Complements

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

NH ≅ G/Ntype
1. C3×#3515C2semidirect
2. #83347C2semidirect
3. #83150C2semidirect
4. #82525C2semidirect
5. #71937C2semidirect
6. #71935C2semidirect
7. #71926C2semidirect
8. #71925C2semidirect
9. #71923C2semidirect
10. #71923C2semidirect
11. #71912C2semidirect
12. #71910C2semidirect
13. #71830C2semidirect
14. #71751C2semidirect
15. #71641C2semidirect
72. #3515D3semidirect
739. C3#18596semidirect

Maximal subgroups

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

subgroupindex
C3×#35152
#833472
#831502
#825252
#719372
#719352
#719262
#719252
#719232
#719122
#719102
#718302
#717512
#716412
#185963