Finite Group Databasebeta

← #10698510

Subgroups — #10698510

Order 768.

760 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82680C211yesmaximal, characteristic
2C2×C3×#327C211yesmaximal, characteristic, complemented
3#71677C211yesmaximal, characteristic, complemented
4#69932C211yesmaximal, characteristic, complemented
5#69419C211yesmaximal, characteristic, complemented
6C2×#4451C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#10402—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#939—21no—
11C2×#939—21no—
12C2×#930C2211yescharacteristic
13C22×C4⋊Dic3—21no—
14C22×C4⋊Dic3C2211yescharacteristic
15C2×C3×C22⋊C8C2211yescharacteristic
16C2×C3×#95C2211yescharacteristic
17#5223—21no—
18C2×#832—21no—
19C2×C3⋊#95—21no—
20C2×C3⋊#95C2211yescharacteristic
21C2×#830C2211yescharacteristic
22C2×#821—21no—
23#4993—21no—
24#4960—21no—
25#4949—21no—
26C3×#327—22no—
27#4551—21no—
28#4529—21no—
29#4451—22no—
30#2656—31no—
31#1874—31no—
32#1874—31no—
33C2×#327—31no—
34C2×#327D311yescharacteristic, complemented
35#1444—31no—
36#1280—31no—
37C22×C3×D4—21no—
38C22×C3×D4D411yescharacteristic
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C22×#67—21no—
43C22×#67—21no—
44C23×Dic3—21no—
45C23×Dic3D411yescharacteristic
46C23×Dic3D411yescharacteristic
47C22×D12D411yescharacteristic
48C22×C3×C8—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C22⋊C4—21no—
55C2×#217—21no—
56C2×#217—21no—
57#939—42no—
58#939—42no—
59#939—42no—
60#939—42no—
61#930—21no—
62#930—21no—
63#930—21no—
64#930—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3—42no—
71C2×C4⋊Dic3—42no—
72C2×C4⋊Dic3—42no—
73C2×C4⋊Dic3—42no—
74C2×C4⋊Dic3D411yescharacteristic
75C2×Dic3⋊C4—21no—
76C2×Dic3⋊C4—21no—
77C22×C3⋊C8—21no—
78C3×C22⋊C8—21no—
79C3×C22⋊C8—21no—
80C3×#95—21no—
81C3×#95D412yescomplemented
82#832—42no—
83C3⋊#95—21no—
84C3⋊#95—21no—
85C3⋊#95—42no—
86#830—21no—
87#830—21no—
88#821—42no—
89C2×#121—31no—
90C2×#121—61no—
91C2×#121—61no—
92C2×#121C2×D311yescharacteristic
93C22×C4⋊C4—31no—
94C22×C4⋊C4—61no—
95C2×#102—61no—
96C2×#102—61no—
97C2×C22⋊C8—31no—
98C2×C22⋊C8C2×D311yescharacteristic
99#399—61no—
100#399—61no—
101#392—61no—
102#392—61no—
103C2×#95—31no—
104C2×#95—61no—
105C2×#95C2×D311yescharacteristic
106#340—61no—
107#340—61no—
108#327—22no—
109#327—62no—
110C24×C3—21no—
111C24×C3C2×D411yescharacteristic
112C23×D3—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C2×#67—42no—
140C2×#67—42no—
141C2×#67—41no—
142C2×#67—41no—
143C2×#67—41no—
144C2×#67—41no—
145C2×#67—41no—
146C2×#67—41no—
147C2×#67—81no—
148C2×#67—81no—
149C2×#67—81no—
150C2×#67—81no—
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—21no—
154C22×Dic3—21no—
155C22×Dic3—22no—
156C22×Dic3—21no—
157C22×Dic3—41no—
158C22×Dic3—41no—
159C22×Dic3—41no—
160C22×Dic3—41no—
161C22×Dic3—41no—
162C22×Dic3—41no—
163C22×Dic3—41no—
164C22×Dic3—41no—
165C22×Dic3—41no—
166C22×Dic3—41no—
167C22×Dic3D812yes—
168C22×Dic3SD1612yes—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—41no—
172C2×D12—41no—
173C2×C3×C8—41no—
174C2×C3×C8—41no—
175C2×C3×C8—41no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C22⋊C4—41no—
179C3×C22⋊C4—41no—
180#217—42no—
181#217—42no—
182(C2×D3)⋊C4—42no—
183(C2×D3)⋊C4—42no—
184(C2×D3)⋊C4—41no—
185(C2×D3)⋊C4—41no—
186C4⋊Dic3—22no—
187C4⋊Dic3—22no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3—41no—
191C4⋊Dic3—41no—
192C4⋊Dic3—48no—
193Dic3⋊C4—42no—
194Dic3⋊C4—42no—
195C2×C3⋊C8—41no—
196C2×C3⋊C8—41no—
197C2×C3⋊C8—41no—
198C22×D4—21no—
199C22×D4—31no—
200C22×D4—61no—
201C22×D4—61no—
202C22×D4—61no—
203C22×D4D1211yescharacteristic
204C23×C4—21no—
205C23×C4—31no—
206C23×C4—31no—
207C23×C4—61no—
208C23×C4C22×D311yescharacteristic
209C22×C8—21no—
210C22×C8—61no—
211#121—21no—
212#121—21no—
213#121—21no—
214#121—21no—
215#121—61no—
216#121—61no—
217#121—61no—
218#121—61no—
219#121—122no—
220#121—122no—
221#121—122no—
222#121—122no—
223C2×C4⋊C4—31no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—61no—
226C2×C4⋊C4—61no—
227C2×C4⋊C4—61no—
228C2×C4⋊C4—122no—
229C2×C4⋊C4—122no—
230C2×C4⋊C4—122no—
231C2×C4⋊C4—122no—
232C2×C4⋊C4D1211yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—61no—
235C2×C22⋊C4—61no—
236C2×C22⋊C4—61no—
237C2×C22⋊C4—61no—
238C2×C22⋊C4—61no—
239#102—122no—
240#102—122no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—61no—
244C22⋊C8—61no—
245#95—21no—
246#95—61no—
247#95—61no—
248#95—122no—
249#95#6712yescomplemented
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—41no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3#12011yescharacteristic
262C23×C3#12111yescharacteristic
263C23×C3#12411yescharacteristic
264C23×C3C2×D811yescharacteristic
265C23×C3C2×SD1611yescharacteristic
266C23×C3AGL(1,ℤ/8)11yescharacteristic
267C23×C3AGL(1,ℤ/8)11yescharacteristic
268C22×D3—41no—
269C22×D3—41no—
270C22×D3—41no—
271C22×D3—41no—
272C22×D3—41no—
273C22×D3—41no—
274C22×D3—41no—
275C3×D4—42no—
276C3×D4—81no—
277C3×D4—81no—
278C3×D4—81no—
279C3×D4—81no—
280C3×D4—81no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—24no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4#12111yescharacteristic
300C2×C3×C4#12111yescharacteristic
301C2×C3×C4#12311yescharacteristic
302C2×C3×C4#12311yescharacteristic
303#67—84no—
304#67—84no—
305#67—81no—
306#67—81no—
307#67—81no—
308#67—81no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—42no—
323C2×Dic3—81no—
324C2×Dic3—81no—
325C2×Dic3—81no—
326C2×Dic3—81no—
327C2×Dic3—81no—
328C2×Dic3—81no—
329D12—42no—
330D12—81no—
331C3×C8—81no—
332C3⋊C8—81no—
333C24—21no—
334C24—61no—
335C24D3×D411yescharacteristic
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—61no—
347C2×D4—61no—
348C2×D4—121no—
349C2×D4—121no—
350C2×D4—122no—
351C2×D4—122no—
352C2×D4—121no—
353C2×D4—121no—
354C2×D4—121no—
355C2×D4—121no—
356C2×D4—121no—
357C2×D4—121no—
358C2×D4—241no—
359C2×D4—241no—
360C2×D4—241no—
361C2×D4—241no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—32no—
367C22×C4—32no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—62no—
379C22×C4—61no—
380C22×C4—121no—
381C22×C4—121no—
382C22×C4—121no—
383C22×C4—121no—
384C22×C4—121no—
385C22×C4—121no—
386C22×C4—121no—
387C22×C4—121no—
388C22×C4—121no—
389C22×C4—121no—
390C22×C4C2×D1211yescharacteristic
391C22×C4#23511yescharacteristic
392C22×C4D3×D411yescharacteristic
393C22×C4#23711yescharacteristic
394C22×C4#23711yescharacteristic
395C22×C4C2×#6711yescharacteristic
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—121no—
400C2×C8—121no—
401C2×C8—121no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—61no—
407C4⋊C4—61no—
408C4⋊C4—121no—
409C4⋊C4—121no—
410C4⋊C4—122no—
411C4⋊C4—122no—
412C4⋊C4—128no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—122no—
416C22⋊C4—122no—
417C22⋊C4—122no—
418C22⋊C4—122no—
419C22⋊C4—121no—
420C22⋊C4—121no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—81no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3—81no—
438C22×C3—81no—
439C22×C3—81no—
440C22×C3#39411yescharacteristic
441C22×C3#44711yescharacteristic
442C22×C3#45011yescharacteristic
443C22×C3#45311yescharacteristic
444C22×C3#45611yescharacteristic
445C22×C3B(4,𝔽2)11yescharacteristic
446C22×C3#45811yescharacteristic
447C22×C3#45911yescharacteristic
448C22×C3#46011yescharacteristic
449C22×C3#48611yescharacteristic
450C22×C3#49011yescharacteristic
451C2×D3—81no—
452C2×D3—81no—
453C2×D3—81no—
454C2×D3—81no—
455C2×D3—81no—
456C2×D3—81no—
457C2×D3—81no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—44no—
463C3×C4—81no—
464Dic3—42no—
465Dic3—42no—
466Dic3—82no—
467Dic3—81no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—121no—
480C23—121no—
481C23—121no—
482C23—121no—
483C23—121no—
484C23—121no—
485C23—121no—
486C23#88211yescharacteristic
487C23#88511yescharacteristic
488C23D3×D811yescharacteristic
489C23#91111yescharacteristic
490C23D3×SD1611yescharacteristic
491C23#91411yescharacteristic
492C23#93911yescharacteristic
493D4—42no—
494D4—81no—
495D4—81no—
496D4—81no—
497D4—81no—
498D4—81no—
499D4—122no—
500D4—241no—
501D4—244no—
502D4—244no—
503D4—241no—
504D4—241no—
505D4—241no—
506D4—241no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—24no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—121no—
536C2×C4—122no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—241no—
540C2×C4—241no—
541C2×C4—241no—
542C2×C4—241no—
543C2×C4—241no—
544C2×C4—241no—
545C2×C4#88311yescharacteristic
546C2×C4#88611yescharacteristic
547C2×C4#93011yescharacteristic
548C2×C4#93811yescharacteristic
549C8—81no—
550C8—241no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—161no—
556C2×C3#159511yescharacteristic
557C2×C3#160411yescharacteristic
558C2×C3#161011yescharacteristic
559C2×C3#161311yescharacteristic
560C2×C3#203911yescharacteristic
561C2×C3#204111yescharacteristic
562C2×C3#204411yescharacteristic
563D3—161no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—241no—
584C22—241no—
585C22—241no—
586C22—241no—
587C22—241no—
588C22—241no—
589C22—241no—
590C22#474111yescharacteristic
591C22#474211yescharacteristic
592C22#476011yescharacteristic
593C22#476411yescharacteristic
594C22#477311yescharacteristic
595C22#477611yescharacteristic
596C22#482211yescharacteristic
597C22#482611yescharacteristic
598C22#496111yescharacteristic
599C22#515011yescharacteristic
600C22#516311yescharacteristic
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—44no—
606C4—81no—
607C4—122no—
608C4—122no—
609C4—241no—
610C4—242no—
611C3#1040211yescharacteristic, complemented, Sylow
612C2—21no—
613C2—21no—
614C2—21no—
615C2—21no—
616C2—161no—
617C2—481no—
618C2#6828411yescharacteristic
619C2#6829011yescharacteristic
620C2#6829511yescharacteristic
621C2#6829711yescharacteristic
622C2#7003211yescharacteristic
623C2#7017711yescharacteristic
624C2#7036111yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#327C2
socleC23×C3#120
prime radical#82680C2
hypercenterC2×#327D3
hypocenterC3#10402
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. C2×C3×#327C2semidirect
3. #71677C2semidirect
4. #69932C2semidirect
5. #69419C2semidirect
6. C2×#4451C2semidirect
7. #67674C2semidirect
34. C2×#327D3semidirect
81. C3×#95D4semidirect
249. #95#67semidirect
611. C3#10402semidirect

Maximal subgroups

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

subgroupindex
#826802
C2×C3×#3272
#716772
#699322
#694192
C2×#44512
#676742
#104023