Finite Group Databasebeta

← #10799203

Subgroups — #10799203

Order 768.

1562 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#949C211yesmaximal, characteristic
2C2×C3×#413C211yesmaximal, characteristic, complemented
3#84072C211yesmaximal, characteristic, complemented
4C2×#5131C211yesmaximal, characteristic, complemented
5C22×#823C211yesmaximal, characteristic, complemented
6C2×#5124C211yesmaximal, characteristic, complemented
7#83747C211yesmaximal, characteristic, complemented
8#70107C214yesmaximal, complemented
9#70074C214yesmaximal, complemented
10#21249—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×D12C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×#949—24no—
16C2×#949—22no—
17C2×#949—22no—
18C2×#949C418yescomplemented
19C22×#215C2212yes—
20C22×#213C2212yes—
21C2×#926C2211yescharacteristic
22C22×#208C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C3×#413—24no—
25C2×C3×#101C2212yescomplemented
26C2×C3×#100C2212yescomplemented
27#5233—22no—
28#5232—22no—
29#5225C2212yescomplemented
30#5215C2212yescomplemented
31#5131—22no—
32#5131C2214yes—
33C2×#823—24no—
34C2×#823C2214yes—
35#5124—24no—
36C2×#822C2214yescomplemented
37#5122C2214yescomplemented
38Dic3×M16—22no—
39C2×#821—22no—
40#5101—22no—
41C22×AGL(1,ℤ/8)—31no—
42C2×#430—31no—
43#2896—31no—
44C2×#418—31no—
45C2×#413D311yescharacteristic, complemented
46C22×#100—31no—
47#2873—31no—
48#1978—34no—
49#1977—34no—
50C24×D3—21no—
51C2×C3×#41—22no—
52C2×C3×#41—21no—
53C2×C3×#41—21no—
54C2×C3×#41—21no—
55C2×C3×#41—21no—
56C2×C3×#41—22no—
57C2×C3×#41—22no—
58C2×C3×#41C2×C414yes—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4C2311yescharacteristic
64C22×D12—21no—
65C22×D12—21no—
66C22×D12—21no—
67C22×D12—22no—
68C22×D12—21no—
69C22×D12—22no—
70C22×D12—22no—
71C22×D12—21no—
72C22×D12C2×C414yes—
73C22×D12C2312yes—
74C2×C3×M16—21no—
75C2×C3×M16—22no—
76C2×C3×M16—21no—
77C2×C3×M16—22no—
78C2×C3×M16—22no—
79C2×C3×M16D414yes—
80C2×C3×M16C2312yes—
81C2×C3×M16C2312yes—
82C22×C3×C8—21no—
83C22×C3×C8—21no—
84C2×#217—21no—
85#949—216no—
86#949—216no—
87C2×#215—24no—
88C2×#215—24no—
89C2×#215C2×C418yescomplemented
90C2×#213—24no—
91C2×#213—24no—
92C2×#213C2×C418yescomplemented
93#926—21no—
94#926—21no—
95#926D412yes—
96#926C2312yes—
97C2×C4⋊Dic3D412yes—
98C2×C4×Dic3—21no—
99C2×#208—21no—
100C2×#208—22no—
101C2×#208—22no—
102C2×#208C2×C414yes—
103C2×#208D414yes—
104C2×#208D412yes—
105C2×#208D412yes—
106C2×#208C2312yes—
107C22×C3⋊C8—21no—
108C22×C3⋊C8C2312yes—
109C3×#101—24no—
110C3×#100—24no—
111#835—24no—
112#832—24no—
113#823—28no—
114#822—28no—
115#821—28no—
116#815—24no—
117C8×Dic3—24no—
118C22×#41C2×D311yescharacteristic
119C23×D4—31no—
120C2×AGL(1,ℤ/8)—38no—
121C2×AGL(1,ℤ/8)—62no—
122C2×AGL(1,ℤ/8)—62no—
123C2×AGL(1,ℤ/8)—64no—
124C22×SD16—32no—
125C22×D8—32no—
126C22×M16—31no—
127C22×M16—31no—
128C22×M16C2×D311yescharacteristic
129C22×M16C2×D311yescharacteristic
130C2×#117—31no—
131#430—64no—
132C2×#108—34no—
133#419—32no—
134#418—32no—
135#418—34no—
136#418—62no—
137#417—62no—
138#416—62no—
139C2×#102—62no—
140#413—24no—
141C2×#101C2×D312yescomplemented
142C2×#100—34no—
143C2×#100—64no—
144C2×#100C2×D312yescomplemented
145#405—34no—
146#405—62no—
147C4×M16—62no—
148C24×C3—21no—
149C24×C3#4111yescharacteristic
150C23×D3—24no—
151C23×D3—22no—
152C23×D3—41no—
153C23×D3—41no—
154C23×D3—42no—
155C23×D3—84no—
156C3×#41—24no—
157C3×#41—24no—
158C3×#41—24no—
159C3×#41—24no—
160C3×#41—42no—
161C3×#41—42no—
162C3×#41—42no—
163C3×#41—42no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8C22×C412yes—
170C2×C3×D4—24no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4—41no—
177C2×C3×D4—41no—
178C2×C3×D4—42no—
179C2×C3×D4C22×C412yes—
180C22×C3×C4—21no—
181C22×C3×C4—24no—
182C22×C3×C4—41no—
183C22×C3×C4—41no—
184C22×C3×C4—42no—
185C22×C3×C4C22×C412yes—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D412yes—
192C22×C3×C4#4111yescharacteristic
193C22×C3×C4#4112yes—
194C22×C3×C4C2411yescharacteristic
195C22×Dic3—22no—
196C2×D12—24no—
197C2×D12—24no—
198C2×D12—24no—
199C2×D12—24no—
200C2×D12—22no—
201C2×D12—22no—
202C2×D12—22no—
203C2×D12—22no—
204C2×D12—24no—
205C2×D12—22no—
206C2×D12—41no—
207C2×D12—41no—
208C2×D12—42no—
209C2×D12—42no—
210C2×D12—42no—
211C2×D12—42no—
212C2×D12—42no—
213C2×D12C22×C414yes—
214C3×M16—24no—
215C3×M16—22no—
216C3×M16—22no—
217C3×M16—22no—
218C3×M16—42no—
219C3×M16C2×D414yes—
220C2×C3×C8—24no—
221C2×C3×C8—22no—
222C2×C3×C8—22no—
223C2×C3×C8—42no—
224#217—22no—
225#217—22no—
226#215—216no—
227#213—216no—
228C4⋊Dic3—22no—
229C4⋊Dic3C2×D414yes—
230C4×Dic3—22no—
231C4×Dic3—22no—
232#208—22no—
233#208—22no—
234#208—42no—
235#208C22⋊C418yes—
236#208C22⋊C418yes—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—42no—
239C2×C3⋊C8C22×C414yes—
240C2×C3⋊C8C2×D414yes—
241C2×C3⋊C8C2×D414yes—
242C25—61no—
243C2×#41—22no—
244C2×#41—21no—
245C2×#41—21no—
246C2×#41—22no—
247C2×#41—21no—
248C2×#41—21no—
249C2×#41—22no—
250C2×#41C4×D314yes—
251C22×Q8C22×D311yescharacteristic
252C22×D4—21no—
253C22×D4—34no—
254C22×D4—32no—
255C22×D4—61no—
256C22×D4—61no—
257C22×D4—61no—
258C22×D4—62no—
259C22×D4—61no—
260C22×D4—61no—
261C22×D4—62no—
262C22×D4—62no—
263C22×D4C22×D311yescharacteristic
264C23×C4—21no—
265C23×C4C22×D311yescharacteristic
266AGL(1,ℤ/8)—616no—
267AGL(1,ℤ/8)—616no—
268C2×SD16—38no—
269C2×SD16—64no—
270C2×SD16—64no—
271C2×D8—38no—
272C2×D8—64no—
273C2×D8—64no—
274C2×M16—21no—
275C2×M16—22no—
276C2×M16—21no—
277C2×M16—22no—
278C2×M16—22no—
279C2×M16—34no—
280C2×M16—32no—
281C2×M16—32no—
282C2×M16—34no—
283C2×M16—32no—
284C2×M16—61no—
285C2×M16—62no—
286C2×M16—62no—
287C2×M16#6714yes—
288C2×M16C22×D312yes—
289C2×M16C22×D312yes—
290C22×C8—21no—
291C22×C8—21no—
292C22×C8—32no—
293C22×C8—61no—
294#117—32no—
295#117—32no—
296#117—61no—
297#117—61no—
298C2×C4⋊C4—32no—
299C2×C22⋊C4—61no—
300C2×C42—61no—
301#108—68no—
302#103—64no—
303#102—68no—
304#102—64no—
305#101—24no—
306#100—24no—
307#100—68no—
308#97—64no—
309C4×C8—64no—
310C23×C3—22no—
311C23×C3—42no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—82no—
315C23×C3C4×D412yes—
316C23×C3#12111yescharacteristic
317C23×C3#12111yescharacteristic
318C23×C3#12412yes—
319C23×C3C2×#4111yescharacteristic
320C22×D3—24no—
321C22×D3—42no—
322C22×D3—42no—
323C22×D3—44no—
324C22×D3—44no—
325C22×D3—84no—
326C22×D3—84no—
327C22×D3—88no—
328C22×D3—84no—
329C22×D3—88no—
330C3×Q8—22no—
331C3×Q8—22no—
332C3×Q8—41no—
333C3×Q8—41no—
334C3×D4—22no—
335C3×D4—22no—
336C3×D4—44no—
337C3×D4—44no—
338C3×D4—41no—
339C3×D4—41no—
340C2×C3×C4—22no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—44no—
344C2×C3×C4—44no—
345C2×C3×C4C2×C22⋊C412yes—
346C2×C3×C4C2×C22⋊C412yes—
347C2×C3×C4C4×D412yes—
348C2×C3×C4C4×D412yes—
349C2×C3×C4C4×D412yes—
350C2×C3×C4#12111yescharacteristic
351C2×C3×C4#12111yescharacteristic
352C2×C3×C4#12112yes—
353C2×C3×C4#12112yes—
354C2×C3×C4#12412yes—
355C2×C3×C4#12712yes—
356C2×C3×C4#12712yes—
357C2×C3×C4C23×C411yescharacteristic
358C2×C3×C4C22×D411yescharacteristic
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×Dic3—24no—
364C2×Dic3—82no—
365D12—24no—
366D12—24no—
367D12—42no—
368D12—42no—
369D12—44no—
370D12—44no—
371C3×C8—24no—
372C3×C8—42no—
373C3⋊C8—42no—
374C3⋊C8C2×C22⋊C418yes—
375C24—21no—
376C24—64no—
377C24—62no—
378C24—121no—
379C24—121no—
380C24—122no—
381C24—244no—
382C24#23511yescharacteristic
383#41—24no—
384#41—24no—
385#41—24no—
386#41—24no—
387#41—42no—
388#41—42no—
389#41—42no—
390#41—42no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8C2×C4×D312yes—
397C2×D4—24no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—34no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—42no—
407C2×D4—64no—
408C2×D4—64no—
409C2×D4—64no—
410C2×D4—64no—
411C2×D4—64no—
412C2×D4—62no—
413C2×D4—62no—
414C2×D4—62no—
415C2×D4—62no—
416C2×D4—62no—
417C2×D4—121no—
418C2×D4—121no—
419C2×D4—122no—
420C2×D4—122no—
421C2×D4—122no—
422C2×D4—122no—
423C2×D4—122no—
424C2×D4C2×C4×D312yes—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—62no—
431C22×C4C2×C4×D312yes—
432C22×C4#23511yescharacteristic
433C22×C4D3×D411yescharacteristic
434C22×C4D3×D411yescharacteristic
435C22×C4D3×D412yes—
436C22×C4D3×D411yescharacteristic
437C22×C4D3×D411yescharacteristic
438C22×C4#23712yes—
439C22×C4C2×#6712yes—
440C22×C4C23×D311yescharacteristic
441SD16—616no—
442D8—616no—
443M16—24no—
444M16—22no—
445M16—22no—
446M16—22no—
447M16—38no—
448M16—38no—
449M16—42no—
450M16—62no—
451M16—62no—
452M16—122no—
453M16C2×#6714yes—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—34no—
458C2×C8—34no—
459C2×C8—34no—
460C2×C8—42no—
461C2×C8—62no—
462C2×C8—122no—
463C4⋊C4—34no—
464C4⋊C4—62no—
465C22⋊C4—62no—
466C22⋊C4—62no—
467C42—62no—
468C42—62no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—82no—
476C22×C3—82no—
477C22×C3—82no—
478C22×C3#39012yes—
479C22×C3#39012yes—
480C22×C3C2×C4×D411yescharacteristic
481C22×C3C2×#12111yescharacteristic
482C22×C3C2×#12411yescharacteristic
483C2×D3—44no—
484C2×D3—88no—
485C2×D3—84no—
486C2×D3—88no—
487C2×D3—84no—
488C2×D3—84no—
489C3×C4—42no—
490C3×C4#39012yes—
491C3×C4#39012yes—
492C3×C4C22×C22⋊C411yescharacteristic
493C3×C4C2×C4×D411yescharacteristic
494C3×C4C2×#12111yescharacteristic
495C3×C4C2×#12711yescharacteristic
496Dic3—44no—
497C23—22no—
498C23—42no—
499C23—41no—
500C23—41no—
501C23—64no—
502C23—82no—
503C23—124no—
504C23—124no—
505C23—122no—
506C23—122no—
507C23—244no—
508C23—244no—
509C23—244no—
510C23—248no—
511C23—248no—
512C23#88411yescharacteristic
513C23#88411yescharacteristic
514C23#88512yes—
515C23C4×#6712yes—
516C23C2×#23511yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521D4—22no—
522D4—22no—
523D4—44no—
524D4—44no—
525D4—41no—
526D4—41no—
527D4—64no—
528D4—64no—
529D4—124no—
530D4—124no—
531D4—122no—
532D4—122no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—64no—
539C2×C4—242no—
540C2×C4D3×C22⋊C412yes—
541C2×C4D3×C22⋊C412yes—
542C2×C4#67⋊C412yes—
543C2×C4#67⋊C412yes—
544C2×C4#88411yescharacteristic
545C2×C4#88411yescharacteristic
546C2×C4#88512yes—
547C2×C4C4×#6712yes—
548C2×C4#93912yes—
549C2×C4#93912yes—
550C2×C4#94012yes—
551C2×C4#94012yes—
552C2×C4C22×C4×D311yescharacteristic
553C2×C4C2×D3×D411yescharacteristic
554C2×C4C2×D3×D411yescharacteristic
555C2×C4C2×D3×D411yescharacteristic
556C2×C4C2×#23711yescharacteristic
557C2×C4C22×#6711yescharacteristic
558C8—24no—
559C8—38no—
560C8—42no—
561C8—122no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—22no—
565C2×C3—22no—
566C2×C3—82no—
567C2×C3C2×#39011yescharacteristic
568C2×C3#306812yes—
569D3—84no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—124no—
580C22—244no—
581C22—244no—
582C22—248no—
583C22—248no—
584C22—244no—
585C22#495712yes—
586C22#495712yes—
587C22C2×#88411yescharacteristic
588C22C2×#88511yescharacteristic
589C22C2×C4×#6711yescharacteristic
590C4—42no—
591C4—124no—
592C4#495712yes—
593C4#495712yes—
594C4C2×D3×C22⋊C411yescharacteristic
595C4C2×#67⋊C411yescharacteristic
596C4C2×#93911yescharacteristic
597C4C2×#94011yescharacteristic
598C3#2124911yescharacteristic, complemented, Sylow
599C2—21no—
600C2—21no—
601C2—22no—
602C2—22no—
603C2—82no—
604C2—244no—
605C2#7872012yes—
606C2C2×#495711yescharacteristic
subgroupG/H
centerC22C2×#884
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#413C2
socleC22×C3C2×#121
prime radicalC22×#949C2
cosocleC22×C3×C4C24
hypercenterC2×#413D3
hypocenterC3#21249
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
2. C2×C3×#413C2semidirect
3. #84072C2semidirect
4. C2×#5131C2semidirect
5. C22×#823C2semidirect
6. C2×#5124C2semidirect
7. #83747C2semidirect
8. #70107C2semidirect
9. #70074C2semidirect
18. C2×#949C4semidirect
25. C2×C3×#101C22semidirect
26. C2×C3×#100C22semidirect
29. #5225C22semidirect
30. #5215C22semidirect
36. C2×#822C22semidirect
37. #5122C22semidirect
45. C2×#413D3semidirect
89. C2×#215C2×C4semidirect
92. C2×#213C2×C4semidirect
141. C2×#101C2×D3semidirect
144. C2×#100C2×D3semidirect
598. C3#21249semidirect

Maximal subgroups

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

subgroupindex
C22×#9492
C2×C3×#4132
#840722
C2×#51312
C22×#8232
C2×#51242
#837472
#701072
#700742
#212493