Finite Group Databasebeta

← #10698480

Subgroups — #10698480

Order 768.

694 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2626C211yesmaximal, characteristic
2#71692C211yesmaximal, characteristic, complemented
3C2×#4538C211yesmaximal, characteristic, complemented
4#71546C211yesmaximal, characteristic
5C2×#4525C211yesmaximal, characteristic
6#69505C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#10398—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×#217—21no—
12C22×C4⋊Dic3C2211yescharacteristic
13C3×#399—21no—
14C3×#396—21no—
15C3×#396—21no—
16C3×#393—21no—
17C3×#385—21no—
18C3×#380C2211yescharacteristic
19C3×#379—21no—
20C2×#835—21no—
21C2×C3⋊#95—21no—
22C2×C3⋊#95C2211yescharacteristic
23C2×#830C2211yescharacteristic
24C2×#830C2211yescharacteristic
25C2×#808—21no—
26#4538—22no—
27#4529—21no—
28#4529—21no—
29#4525—22no—
30#2626D311yescharacteristic
31#1872—31no—
32#1853—31no—
33C2×#328—31no—
34C2×#323—31no—
35#1439—31no—
36#1294—31no—
37C25×C3D411yescharacteristic
38C22×C3×Q8D411yescharacteristic
39C23×C3×C4D411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C23×Dic3—21no—
42C23×Dic3—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
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×#217—42no—
66C2×#217—42no—
67C2×#217—42no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3—42no—
71C22×C3⋊C8—21no—
72C22×C3⋊C8—21no—
73C3×#95—21no—
74C3×#95—21no—
75C3×#95—21no—
76#835—42no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—42no—
80#830—21no—
81#830—21no—
82#830—21no—
83#830D412yes—
84#808—42no—
85C2×C22⋊Q8C2×D311yescharacteristic
86C22×C4⋊C4—31no—
87C22×C22⋊C4—61no—
88C22×C22⋊C4C2×D311yescharacteristic
89C2×#106—61no—
90C2×#103—61no—
91C2×C22⋊C8—31no—
92C2×C22⋊C8—31no—
93#399—21no—
94#396—21no—
95#396—21no—
96#393—21no—
97#385—21no—
98#380C2×D311yescharacteristic
99#379—21no—
100C2×#95—31no—
101C2×#95—61no—
102#340—61no—
103#340—61no—
104#328—62no—
105#323—62no—
106C24×C3—21no—
107C24×C3—21no—
108C24×C3—21no—
109C24×C3—21no—
110C24×C3—21no—
111C24×C3—21no—
112C24×C3—41no—
113C24×C3—41no—
114C24×C3—41no—
115C24×C3—41no—
116C24×C3SD1612yes—
117C24×C3C2×D411yescharacteristic
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—41no—
121C2×C3×Q8—41no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
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
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—41no—
142C22×Dic3—41no—
143C22×Dic3—41no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3—41no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C22×Dic3—41no—
150C22×Dic3—41no—
151C22×Dic3—41no—
152C22×Dic3—41no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—41no—
156C3×C4⋊C4—41no—
157C3×C4⋊C4—41no—
158C3×C4⋊C4—41no—
159C3×C4⋊C4—81no—
160C3×C4⋊C4—81no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164C3×C22⋊C4—41no—
165C3×C22⋊C4—41no—
166C3×C22⋊C4—41no—
167C3×C22⋊C4—41no—
168C3×C22⋊C4—41no—
169C3×C22⋊C4—81no—
170C3×C22⋊C4—81no—
171C3×C22⋊C4—81no—
172C3×C22⋊C4—81no—
173#217—48no—
174C4⋊Dic3—22no—
175C4⋊Dic3—22no—
176C4⋊Dic3—41no—
177C4⋊Dic3—41no—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—41no—
180C2×C3⋊C8—41no—
181C2×C3⋊C8—41no—
182C2×C3⋊C8—41no—
183C2×C3⋊C8—41no—
184C25#6711yescharacteristic
185C22×Q8#6711yescharacteristic
186C23×C4—61no—
187C23×C4—61no—
188C23×C4#6711yescharacteristic
189C23×C4C22×D311yescharacteristic
190C22×C8—61no—
191C22×C8—61no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—122no—
202C2×C4⋊C4#6711yescharacteristic
203C2×C4⋊C4#6711yescharacteristic
204C2×C4⋊C4#6711yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—122no—
218C2×C22⋊C4—122no—
219C2×C22⋊C4—122no—
220#106—122no—
221#103—122no—
222C22⋊C8—32no—
223C22⋊C8—61no—
224C22⋊C8—61no—
225C22⋊C8—61no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—61no—
230#95—61no—
231#95—122no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—22no—
237C23×C3—22no—
238C23×C3—22no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3—81no—
254C23×C3—81no—
255C23×C3—81no—
256C23×C3—81no—
257C23×C3—81no—
258C23×C3—81no—
259C23×C3#12011yescharacteristic
260C23×C3#12111yescharacteristic
261C23×C3#12311yescharacteristic
262C23×C3C2×SD1611yescharacteristic
263C23×C3#13511yescharacteristic
264C23×C3#13711yescharacteristic
265C23×C3#13711yescharacteristic
266C3×Q8—42no—
267C3×Q8—81no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—81no—
293C2×C3×C4#12111yescharacteristic
294C2×C3×C4#12111yescharacteristic
295C2×C3×C4#12311yescharacteristic
296C2×C3×C4#12411yescharacteristic
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—81no—
304C2×Dic3—81no—
305C2×Dic3—81no—
306C2×Dic3—81no—
307C2×Dic3—81no—
308C2×Dic3—81no—
309C2×Dic3—81no—
310C2×Dic3—81no—
311C3⋊C8—42no—
312C3⋊C8—81no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—41no—
320C24—41no—
321C24—41no—
322C24—41no—
323C24#21512yes—
324C24C2×#6711yescharacteristic
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—41no—
328C2×Q8—41no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—62no—
341C22×C4—62no—
342C22×C4—121no—
343C22×C4—121no—
344C22×C4—121no—
345C22×C4—121no—
346C22×C4—121no—
347C22×C4—121no—
348C22×C4—121no—
349C22×C4—121no—
350C22×C4—121no—
351C22×C4—121no—
352C22×C4—121no—
353C22×C4—121no—
354C22×C4D3×D411yescharacteristic
355C22×C4#23711yescharacteristic
356C22×C4#23711yescharacteristic
357C22×C4#23711yescharacteristic
358C22×C4C2×#6711yescharacteristic
359C22×C4C2×#6711yescharacteristic
360C2×C8—62no—
361C2×C8—121no—
362C2×C8—121no—
363C2×C8—121no—
364C2×C8—121no—
365C2×C8—121no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—62no—
373C4⋊C4—62no—
374C4⋊C4—81no—
375C4⋊C4—81no—
376C4⋊C4—121no—
377C4⋊C4—121no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—81no—
387C22⋊C4—81no—
388C22⋊C4—81no—
389C22⋊C4—81no—
390C22⋊C4—128no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—44no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—82no—
408C22×C3—82no—
409C22×C3—82no—
410C22×C3—81no—
411C22×C3—81no—
412C22×C3—81no—
413C22×C3—81no—
414C22×C3#39411yescharacteristic
415C22×C3#44811yescharacteristic
416C22×C3#45211yescharacteristic
417C22×C3#45311yescharacteristic
418C22×C3#45511yescharacteristic
419C22×C3B(4,𝔽2)11yescharacteristic
420C22×C3#45811yescharacteristic
421C22×C3#46511yescharacteristic
422C22×C3#47011yescharacteristic
423C22×C3#48311yescharacteristic
424C22×C3#48511yescharacteristic
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—81no—
430C3×C4—81no—
431C3×C4—81no—
432C3×C4—81no—
433C3×C4—81no—
434Dic3—82no—
435Dic3—82no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—81no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23#93511yescharacteristic
464C23#93911yescharacteristic
465C23C2×#21511yescharacteristic
466C23#94211yescharacteristic
467C23#95011yescharacteristic
468C23#95111yescharacteristic
469C23#95311yescharacteristic
470Q8—42no—
471Q8—81no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—81no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—241no—
504C2×C4—241no—
505C2×C4—241no—
506C2×C4—241no—
507C2×C4—241no—
508C2×C4—241no—
509C2×C4—241no—
510C2×C4—241no—
511C2×C4#93511yescharacteristic
512C2×C4#93611yescharacteristic
513C2×C4#93811yescharacteristic
514C2×C4#93911yescharacteristic
515C8—122no—
516C8—241no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—82no—
522C2×C3#159611yescharacteristic
523C2×C3#160611yescharacteristic
524C2×C3#160811yescharacteristic
525C2×C3#161811yescharacteristic
526C2×C3#203511yescharacteristic
527C2×C3#203711yescharacteristic
528C2×C3#204411yescharacteristic
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—44no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22#504311yescharacteristic
553C22#504511yescharacteristic
554C22#504811yescharacteristic
555C22#505211yescharacteristic
556C22#522211yescharacteristic
557C22#522711yescharacteristic
558C22#523911yescharacteristic
559C22#524111yescharacteristic
560C22#524211yescharacteristic
561C22#524311yescharacteristic
562C22#524411yescharacteristic
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—242no—
573C4—242no—
574C3#1039811yescharacteristic, complemented, Sylow
575C2—21no—
576C2—21no—
577C2—21no—
578C2—21no—
579C2—82no—
580C2#7169411yescharacteristic
581C2#7185011yescharacteristic
582C2#7185211yescharacteristic
583C2#7188311yescharacteristic
584C2#7188511yescharacteristic
585C2#7189011yescharacteristic
586C2#7189211yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2626C2
socleC23×C3#120
prime radicalC25×C3D4
hypercenter#2626D3
hypocenterC3#10398
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
2. #71692C2semidirect
3. C2×#4538C2semidirect
6. #69505C2semidirect
7. #67624C2semidirect
574. C3#10398semidirect

Maximal subgroups

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

subgroupindex
C3×#26262
#716922
C2×#45382
#715462
C2×#45252
#695052
#676242
#103983