Finite Group Databasebeta

← #10690582

Subgroups — #10690582

Order 768.

712 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4719C211yesmaximal, characteristic, complemented
2C2×#4670C211yesmaximal, characteristic
3C2×#4468C211yesmaximal, characteristic
4C2×#4468C211yesmaximal, characteristic
5#69910C211yesmaximal, characteristic, complemented
6C3×#1280C211yesmaximal, characteristic, complemented
7#67613C211yesmaximal, characteristic, complemented
8#9084—31nomaximal, Sylow
9C22×(C2×D3)⋊C4C411yescharacteristic
10C22×(C2×D3)⋊C4C411yescharacteristic
11C22×(C2×D3)⋊C4C2211yescharacteristic
12C22×C4⋊Dic3C411yescharacteristic
13C22×C4×Dic3C2211yescharacteristic
14C2×C3×C22⋊C8C2211yescharacteristic
15C2×C3×C22⋊C8C2211yescharacteristic
16C2×C3×#95C2211yescharacteristic
17C2×C3⋊#95C411yescharacteristic
18C2×#830C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#815—21no—
21C2×Dic3⋊C8—21no—
22#4719—22no—
23#4719—22no—
24#4670—21no—
25#4670—21no—
26#4670—21no—
27#4670—21no—
28#4550—21no—
29#4550—21no—
30#4468—22no—
31#4468—22no—
32C2×#432—31no—
33C2×#388—31no—
34#1817—31no—
35C2×#323—31no—
36C2×#323—31no—
37#1280—31no—
38#1280D311yescharacteristic, complemented
39C24×D3C2×C411yescharacteristic
40C23×C3×C4C2×C411yescharacteristic
41C23×C3×C4C2×C411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C23×Dic3C2×C411yescharacteristic
44C23×Dic3C2×C411yescharacteristic
45C23×Dic3C2×C411yescharacteristic
46C22×C3×C8—21no—
47C22×C3×C8—21no—
48C2×(C2×D3)⋊C4—22no—
49C2×(C2×D3)⋊C4—22no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—42no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—42no—
64C2×C4×Dic3—21no—
65C2×C4×Dic3—21no—
66C2×C4×Dic3—21no—
67C2×C4×Dic3—21no—
68C2×C4×Dic3D411yescharacteristic
69C2×C4×Dic3D411yescharacteristic
70C2×C4×Dic3D411yescharacteristic
71C2×C4×Dic3D411yescharacteristic
72C22×C3⋊C8—21no—
73C22×C3⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8D412yes—
78C3×#95—21no—
79C3×#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82#830—21no—
83#830—21no—
84#830—21no—
85#830D412yes—
86#815—24no—
87Dic3⋊C8—24no—
88C22×C4⋊C4—31no—
89C22×C22⋊C4—31no—
90C22×C22⋊C4—31no—
91C22×C22⋊C4—31no—
92C22×C42—31no—
93#432—62no—
94#432—62no—
95C2×C4⋊C8—61no—
96C2×C22⋊C8—31no—
97C2×C22⋊C8—31no—
98C2×C22⋊C8C2×D311yescharacteristic
99C2×C22⋊C8C2×D311yescharacteristic
100C2×#97—61no—
101#388—61no—
102#388—61no—
103#388—61no—
104#388—61no—
105C2×#95—31no—
106C2×#95C2×D311yescharacteristic
107#336—61no—
108#336—61no—
109#323—62no—
110#323—62no—
111C24×C3C22×C411yescharacteristic
112C23×D3—21no—
113C23×D3—21no—
114C23×D3—21no—
115C23×D3—21no—
116C23×D3—21no—
117C23×D3—21no—
118C23×D3—41no—
119C23×D3—41no—
120C23×D3—41no—
121C23×D3—41no—
122C23×D3M1612yes—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4M1612yes—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C22×Dic3—22no—
150C22×Dic3—21no—
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—41no—
154C22×Dic3—41no—
155C22×Dic3—41no—
156C22×Dic3C22⋊C411yescharacteristic
157C22×Dic3C22⋊C411yescharacteristic
158C22×Dic3C22⋊C411yescharacteristic
159C22×Dic3C22⋊C411yescharacteristic
160C2×C3×C8—22no—
161C2×C3×C8—41no—
162C2×C3×C8—41no—
163C2×C3×C8—41no—
164C2×C3×C8—41no—
165C2×C3×C8—41no—
166(C2×D3)⋊C4—24no—
167(C2×D3)⋊C4—24no—
168(C2×D3)⋊C4—42no—
169(C2×D3)⋊C4—41no—
170(C2×D3)⋊C4—41no—
171(C2×D3)⋊C4—41no—
172(C2×D3)⋊C4—41no—
173(C2×D3)⋊C4—42no—
174C4⋊Dic3—42no—
175C4⋊Dic3—42no—
176C4×Dic3—21no—
177C4×Dic3—21no—
178C4×Dic3—21no—
179C4×Dic3—21no—
180C4×Dic3—21no—
181C4×Dic3—21no—
182C4×Dic3—21no—
183C4×Dic3—21no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—41no—
186C2×C3⋊C8—41no—
187C2×C3⋊C8—41no—
188C2×C3⋊C8—41no—
189C2×C3⋊C8—41no—
190C25—31no—
191C23×C4—31no—
192C23×C4—31no—
193C23×C4—31no—
194C23×C4C4×D311yescharacteristic
195C23×C4C4×D311yescharacteristic
196C23×C4C22×D311yescharacteristic
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—61no—
200C22×C8—61no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—122no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—61no—
206C2×C22⋊C4—62no—
207C2×C22⋊C4—62no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—122no—
217C2×C42—31no—
218C2×C42—31no—
219C2×C42—31no—
220C2×C42—31no—
221C2×C42—61no—
222C2×C42—61no—
223C2×C42—61no—
224C2×C42—61no—
225C4⋊C8—64no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—32no—
230C22⋊C8—61no—
231C22⋊C8—61no—
232C22⋊C8—61no—
233C22⋊C8D1212yes—
234#97—64no—
235#95—21no—
236#95—21no—
237#95—61no—
238#95—61no—
239C23×C3—21no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3C2×C22⋊C411yescharacteristic
244C23×C3C4×D411yescharacteristic
245C23×C3C4×D411yescharacteristic
246C23×C3C2×M1611yescharacteristic
247C23×C3C2×M1611yescharacteristic
248C23×C3#13111yescharacteristic
249C23×C3#13111yescharacteristic
250C22×D3—22no—
251C22×D3—22no—
252C22×D3—22no—
253C22×D3—42no—
254C22×D3—42no—
255C22×D3—41no—
256C22×D3—41no—
257C22×D3—41no—
258C22×D3—41no—
259C22×D3—41no—
260C22×D3—41no—
261C22×D3—41no—
262C22×D3—41no—
263C22×D3—42no—
264C22×D3—42no—
265C22×D3—81no—
266C22×D3—81no—
267C22×D3—81no—
268C22×D3—81no—
269C22×D3—81no—
270C22×D3—81no—
271C22×D3—81no—
272C22×D3—81no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—22no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—81no—
290C2×C3×C4—81no—
291C2×C3×C4#12011yescharacteristic
292C2×C3×C4#12111yescharacteristic
293C2×C3×C4#12111yescharacteristic
294C2×C3×C4#12311yescharacteristic
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—21no—
298C2×Dic3—21no—
299C2×Dic3—21no—
300C2×Dic3—21no—
301C2×Dic3—21no—
302C2×Dic3—21no—
303C2×Dic3—21no—
304C2×Dic3—21no—
305C2×Dic3—21no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—81no—
316C2×Dic3—81no—
317C2×Dic3#10012yes—
318C3×C8—42no—
319C3×C8—81no—
320C3⋊C8—42no—
321C3⋊C8—81no—
322C24—32no—
323C24—61no—
324C24—61no—
325C24—61no—
326C24—61no—
327C24—61no—
328C24—61no—
329C24—121no—
330C24—121no—
331C24—121no—
332C24—121no—
333C24C2×C4×D311yescharacteristic
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—31no—
340C22×C4—31no—
341C22×C4—31no—
342C22×C4—31no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—62no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—61no—
361C22×C4—121no—
362C22×C4—121no—
363C22×C4—121no—
364C22×C4ΓL(1,𝔽25)12yes—
365C22×C4C2×D1211yescharacteristic
366C22×C4#23511yescharacteristic
367C22×C4D3×D411yescharacteristic
368C22×C4D3×D411yescharacteristic
369C22×C4D3×D411yescharacteristic
370C22×C4#23911yescharacteristic
371C2×C8—22no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—62no—
378C2×C8—121no—
379C2×C8—121no—
380C2×C8—121no—
381C2×C8—121no—
382C2×C8—121no—
383C4⋊C4—122no—
384C4⋊C4—122no—
385C22⋊C4—64no—
386C22⋊C4—64no—
387C22⋊C4—122no—
388C22⋊C4—122no—
389C22⋊C4—121no—
390C22⋊C4—121no—
391C22⋊C4—121no—
392C22⋊C4—121no—
393C42—61no—
394C42—61no—
395C42—61no—
396C42—61no—
397C42—61no—
398C42—61no—
399C42—61no—
400C42—61no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3#38611yescharacteristic
414C22×C3#40711yescharacteristic
415C22×C3#40811yescharacteristic
416C22×C3#41011yescharacteristic
417C22×C3#41011yescharacteristic
418C22×C3C2×#10011yescharacteristic
419C22×C3#41311yescharacteristic
420C22×C3#43511yescharacteristic
421C22×C3#43511yescharacteristic
422C22×C3#43611yescharacteristic
423C22×C3#43611yescharacteristic
424C2×D3—42no—
425C2×D3—42no—
426C2×D3—42no—
427C2×D3—82no—
428C2×D3—82no—
429C2×D3—81no—
430C2×D3—81no—
431C2×D3—81no—
432C2×D3—81no—
433C2×D3—82no—
434C2×D3—82no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—42no—
440C3×C4—81no—
441Dic3—22no—
442Dic3—41no—
443Dic3—41no—
444Dic3—41no—
445Dic3—81no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—62no—
451C23—62no—
452C23—62no—
453C23—122no—
454C23—122no—
455C23—122no—
456C23—122no—
457C23—121no—
458C23—121no—
459C23—121no—
460C23—121no—
461C23—121no—
462C23—121no—
463C23—121no—
464C23—121no—
465C23—241no—
466C23—241no—
467C23—241no—
468C23—241no—
469C23—241no—
470C23—241no—
471C23—241no—
472C23—241no—
473C23C4×D1211yescharacteristic
474C23D3×C22⋊C411yescharacteristic
475C23#89311yescharacteristic
476C23C2×ΓL(1,𝔽25)11yescharacteristic
477C23#90111yescharacteristic
478C23D3×M1611yescharacteristic
479C23#90711yescharacteristic
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—32no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—61no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—241no—
520C2×C4—241no—
521C2×C4#88211yescharacteristic
522C2×C4#88411yescharacteristic
523C2×C4#89411yescharacteristic
524C2×C4#89511yescharacteristic
525C8—42no—
526C8—81no—
527C8—122no—
528C8—241no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3#146411yescharacteristic
534C2×C3#146611yescharacteristic
535C2×C3#146911yescharacteristic
536C2×C3#147311yescharacteristic
537C2×C3#187011yescharacteristic
538C2×C3#189911yescharacteristic
539C2×C3#190011yescharacteristic
540D3—82no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—122no—
554C22—122no—
555C22—122no—
556C22—242no—
557C22—242no—
558C22—242no—
559C22—242no—
560C22—241no—
561C22—241no—
562C22—241no—
563C22—241no—
564C22#466511yescharacteristic
565C22#468811yescharacteristic
566C22#470611yescharacteristic
567C22#472611yescharacteristic
568C22#472711yescharacteristic
569C22#474011yescharacteristic
570C22#474011yescharacteristic
571C22D3×#10011yescharacteristic
572C22#475111yescharacteristic
573C22#483611yescharacteristic
574C22#483811yescharacteristic
575C4—21no—
576C4—21no—
577C4—21no—
578C4—21no—
579C4—42no—
580C4—62no—
581C4—81no—
582C4—121no—
583C4—121no—
584C4—121no—
585C4—241no—
586C3#908411yescharacteristic, complemented, Sylow
587C2—21no—
588C2—21no—
589C2—21no—
590C2—21no—
591C2—242no—
592C2#6810311yescharacteristic
593C2#6810411yescharacteristic
594C2#6812611yescharacteristic
595C2#6812911yescharacteristic
596C2#6866911yescharacteristic
597C2#6895411yescharacteristic
598C2#6899211yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1280C2
socleC23×C3C2×C22⋊C4
prime radicalC24×D3C2×C4
cosocleC23×C3×C4C23
hypercenter#1280D3
hypocenterC3#9084
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
1. C2×#4719C2semidirect
5. #69910C2semidirect
6. C3×#1280C2semidirect
7. #67613C2semidirect
38. #1280D3semidirect
586. C3#9084semidirect

Maximal subgroups

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

subgroupindex
C2×#47192
C2×#46702
C2×#44682
#699102
C3×#12802
#676132
#90843