Finite Group Databasebeta

← #10698459

Subgroups — #10698459

Order 768.

772 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2617C211yesmaximal, characteristic
2#71677C211yesmaximal, characteristic, complemented
3C2×#4537C211yesmaximal, characteristic, complemented
4#71546C211yesmaximal, characteristic, complemented
5C2×#4525C211yesmaximal, characteristic, complemented
6#69505C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#10395—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C3×#120—21no—
11C22×C3×C22⋊C4C2211yescharacteristic
12C22×#217—21no—
13C22×C4⋊Dic3C2211yescharacteristic
14C3×#397—21no—
15C3×#394—21no—
16C3×#394—21no—
17C3×#392—21no—
18C3×#386—21no—
19C3×#380C2211yescharacteristic
20C2×#832—21no—
21C2×C3⋊#95—21no—
22C2×C3⋊#95C2211yescharacteristic
23C2×#830C2211yescharacteristic
24C2×#830C2211yescharacteristic
25C2×#808—21no—
26#4537—22no—
27#4529—21no—
28#4529—21no—
29#4525—22no—
30#2617D311yescharacteristic
31#1874—31no—
32#1853—31no—
33C2×#327—31no—
34C2×#323—31no—
35#1439—31no—
36#1294—31no—
37C25×C3D411yescharacteristic
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×Dic3—21no—
45C23×Dic3—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#120—42no—
51C3×#120—42no—
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—21no—
67C2×#217—42no—
68C2×#217—42no—
69C2×#217—42no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3—42no—
73C22×C3⋊C8—21no—
74C22×C3⋊C8—21no—
75C3×#95—21no—
76C3×#95—21no—
77C3×#95—21no—
78#832—42no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—42no—
82#830—21no—
83#830—21no—
84#830—21no—
85#830D412yescomplemented
86#808—42no—
87C2×#121C2×D311yescharacteristic
88C2×#120—21no—
89C22×C4⋊C4—31no—
90C22×C22⋊C4—61no—
91C22×C22⋊C4C2×D311yescharacteristic
92C2×#106—61no—
93C2×#102—61no—
94C2×C22⋊C8—31no—
95C2×C22⋊C8—31no—
96#397—21no—
97#394—21no—
98#394—21no—
99#392—21no—
100#386—21no—
101#380C2×D311yescharacteristic
102C2×#95—31no—
103C2×#95—61no—
104#340—61no—
105#340—61no—
106#327—62no—
107#323—62no—
108C24×C3—21no—
109C24×C3—21no—
110C24×C3—21no—
111C24×C3—21no—
112C24×C3—21no—
113C24×C3—21no—
114C24×C3—21no—
115C24×C3—41no—
116C24×C3—41no—
117C24×C3—41no—
118C24×C3—41no—
119C24×C3SD1612yes—
120C24×C3C2×D411yescharacteristic
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—42no—
126C2×C3×D4—42no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C2×C3×D4—41no—
132C2×C3×D4—41no—
133C2×C3×D4—81no—
134C2×C3×D4—81no—
135C2×C3×D4—81no—
136C2×C3×D4—81no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—41no—
144C22×C3×C4—41no—
145C22×C3×C4—41no—
146C22×C3×C4—41no—
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4#4111yescharacteristic
151C22×C3×C4#4111yescharacteristic
152C22×C3×C4#4111yescharacteristic
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—41no—
156C22×Dic3—41no—
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—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—41no—
170C3×C4⋊C4—41no—
171C3×C22⋊C4—42no—
172C3×C22⋊C4—42no—
173C3×C22⋊C4—41no—
174C3×C22⋊C4—41no—
175C3×C22⋊C4—41no—
176C3×C22⋊C4—41no—
177C3×C22⋊C4—41no—
178C3×C22⋊C4—41no—
179C3×C22⋊C4—41no—
180C3×C22⋊C4—41no—
181C3×C22⋊C4—81no—
182C3×C22⋊C4—81no—
183#217—48no—
184C4⋊Dic3—22no—
185C4⋊Dic3—22no—
186C4⋊Dic3—41no—
187C4⋊Dic3—41no—
188C2×C3⋊C8—22no—
189C2×C3⋊C8—41no—
190C2×C3⋊C8—41no—
191C2×C3⋊C8—41no—
192C2×C3⋊C8—41no—
193C2×C3⋊C8—41no—
194C25#6711yescharacteristic
195C22×D4—21no—
196C22×D4—21no—
197C22×D4—21no—
198C22×D4#6711yescharacteristic
199C23×C4—61no—
200C23×C4—61no—
201C23×C4#6711yescharacteristic
202C23×C4C22×D311yescharacteristic
203C22×C8—61no—
204C22×C8—61no—
205#121—21no—
206#121—21no—
207#121—21no—
208#121—21no—
209#120—42no—
210#120—42no—
211C2×C4⋊C4—61no—
212C2×C4⋊C4—61no—
213C2×C4⋊C4—122no—
214C2×C4⋊C4#6711yescharacteristic
215C2×C4⋊C4#6711yescharacteristic
216C2×C4⋊C4#6711yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—122no—
230C2×C22⋊C4—122no—
231C2×C22⋊C4—122no—
232#106—122no—
233#102—122no—
234C22⋊C8—32no—
235C22⋊C8—61no—
236C22⋊C8—61no—
237C22⋊C8—61no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—61no—
242#95—61no—
243#95—122no—
244C23×C3—21no—
245C23×C3—21no—
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—22no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—81no—
273C23×C3—81no—
274C23×C3—81no—
275C23×C3—81no—
276C23×C3—81no—
277C23×C3—81no—
278C23×C3#12011yescharacteristic
279C23×C3#12111yescharacteristic
280C23×C3#12311yescharacteristic
281C23×C3C2×SD1611yescharacteristic
282C23×C3#13511yescharacteristic
283C23×C3AGL(1,ℤ/8)11yescharacteristic
284C23×C3AGL(1,ℤ/8)11yescharacteristic
285C3×D4—42no—
286C3×D4—81no—
287C3×D4—84no—
288C3×D4—84no—
289C3×D4—81no—
290C3×D4—81no—
291C3×D4—81no—
292C3×D4—81no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—81no—
315C2×C3×C4#12111yescharacteristic
316C2×C3×C4#12111yescharacteristic
317C2×C3×C4#12311yescharacteristic
318C2×C3×C4#12411yescharacteristic
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—81no—
326C2×Dic3—81no—
327C2×Dic3—81no—
328C2×Dic3—81no—
329C2×Dic3—81no—
330C2×Dic3—81no—
331C2×Dic3—81no—
332C2×Dic3—81no—
333C3⋊C8—42no—
334C3⋊C8—81no—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—21no—
342C24—41no—
343C24—41no—
344C24—41no—
345C24—41no—
346C24#21412yes—
347C24C2×#6711yescharacteristic
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—81no—
361C2×D4—81no—
362C2×D4—81no—
363C2×D4—81no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—121no—
377C22×C4—121no—
378C22×C4—121no—
379C22×C4—121no—
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×C4D3×D411yescharacteristic
389C22×C4#23711yescharacteristic
390C22×C4#23711yescharacteristic
391C22×C4#23711yescharacteristic
392C22×C4C2×#6711yescharacteristic
393C22×C4C2×#6711yescharacteristic
394C2×C8—62no—
395C2×C8—121no—
396C2×C8—121no—
397C2×C8—121no—
398C2×C8—121no—
399C2×C8—121no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—121no—
407C4⋊C4—121no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—81no—
419C22⋊C4—81no—
420C22⋊C4—128no—
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—44no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—82no—
438C22×C3—82no—
439C22×C3—82no—
440C22×C3—81no—
441C22×C3—81no—
442C22×C3—81no—
443C22×C3—81no—
444C22×C3—81no—
445C22×C3—81no—
446C22×C3—81no—
447C22×C3—81no—
448C22×C3—81no—
449C22×C3—81no—
450C22×C3—81no—
451C22×C3#39411yescharacteristic
452C22×C3#44911yescharacteristic
453C22×C3#45011yescharacteristic
454C22×C3#45311yescharacteristic
455C22×C3#45511yescharacteristic
456C22×C3B(4,𝔽2)11yescharacteristic
457C22×C3#45811yescharacteristic
458C22×C3#46511yescharacteristic
459C22×C3#46911yescharacteristic
460C22×C3#48111yescharacteristic
461C22×C3#48211yescharacteristic
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—81no—
467C3×C4—81no—
468C3×C4—81no—
469C3×C4—81no—
470Dic3—82no—
471Dic3—82no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—81no—
501C23—81no—
502C23—81no—
503C23—81no—
504C23—81no—
505C23—81no—
506C23#93211yescharacteristic
507C23C2×#21411yescharacteristic
508C23#93511yescharacteristic
509C23#93911yescharacteristic
510C23#94911yescharacteristic
511C23#95011yescharacteristic
512C23#95311yescharacteristic
513D4—42no—
514D4—81no—
515D4—84no—
516D4—84no—
517D4—81no—
518D4—81no—
519D4—81no—
520D4—81no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—81no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—241no—
550C2×C4—241no—
551C2×C4—241no—
552C2×C4—241no—
553C2×C4—241no—
554C2×C4—241no—
555C2×C4—241no—
556C2×C4—241no—
557C2×C4#93511yescharacteristic
558C2×C4#93611yescharacteristic
559C2×C4#93811yescharacteristic
560C2×C4#93911yescharacteristic
561C8—122no—
562C8—241no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—82no—
568C2×C3—161no—
569C2×C3#160011yescharacteristic
570C2×C3#160111yescharacteristic
571C2×C3#161211yescharacteristic
572C2×C3#161311yescharacteristic
573C2×C3#203411yescharacteristic
574C2×C3#203711yescharacteristic
575C2×C3#204411yescharacteristic
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22#503411yescharacteristic
607C22#503511yescharacteristic
608C22#503911yescharacteristic
609C22#504111yescharacteristic
610C22#521811yescharacteristic
611C22#522211yescharacteristic
612C22#523811yescharacteristic
613C22#524111yescharacteristic
614C22#524211yescharacteristic
615C22#524311yescharacteristic
616C22#524411yescharacteristic
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—242no—
626C4—242no—
627C3#1039511yescharacteristic, complemented, Sylow
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—82no—
633C2—161no—
634C2#7168011yescharacteristic
635C2#7184411yescharacteristic
636C2#7184611yescharacteristic
637C2#7185011yescharacteristic
638C2#7185211yescharacteristic
639C2#7188211yescharacteristic
640C2#7188411yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2617C2
socleC23×C3#120
hypercenter#2617D3
hypocenterC3#10395
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2617

Complements

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

NH ≅ G/Ntype
2. #71677C2semidirect
3. C2×#4537C2semidirect
4. #71546C2semidirect
5. C2×#4525C2semidirect
6. #69505C2semidirect
7. #67624C2semidirect
85. #830D4semidirect
627. C3#10395semidirect

Maximal subgroups

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

subgroupindex
C3×#26172
#716772
C2×#45372
#715462
C2×#45252
#695052
#676242
#103953