Finite Group Databasebeta

← #10694317

Subgroups — #10694317

Order 768.

742 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
1#82698C211yesmaximal, characteristic
2C2×C3×#327C211yesmaximal, characteristic
3C2×#4537C211yesmaximal, characteristic
4C2×#4467C211yesmaximal, characteristic, complemented
5#69908C211yesmaximal, characteristic, complemented
6Dic3×#95C211yesmaximal, characteristic, complemented
7C2×#4452C211yesmaximal, characteristic, complemented
8#9713—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#936C411yescharacteristic, complemented
11C2×#924C2211yescharacteristic
12C22×C4×Dic3—21no—
13C22×C4×Dic3C2211yescharacteristic
14C2×C3×C22⋊C8C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16#5221C411yescharacteristic, complemented
17#5220—21no—
18#5220—21no—
19C2×C3⋊#95—21no—
20C2×C3⋊#95C2211yescharacteristic
21C2×#830C2211yescharacteristic
22C2×Dic3⋊C8—21no—
23#4977C411yescharacteristic, complemented
24#4973C411yescharacteristic, complemented
25#4945—21no—
26#4945—21no—
27C3×#327—22no—
28#4550—21no—
29#4537—22no—
30#4467—22no—
31#4452—22no—
32#2850—31no—
33#1797—31no—
34C2×#328—31no—
35C2×#328—31no—
36C2×#327—31no—
37C2×#327D311yescharacteristic
38C4×#95—31no—
39C22×C3×D4—21no—
40C22×C3×D4C2×C411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C23×Dic3—21no—
44C23×Dic3D411yescharacteristic
45C23×Dic3D411yescharacteristic
46C22×Dic6C2×C411yescharacteristic
47C22×C3×C8—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C2×C3×C4⋊C4C2×C411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×#217—21no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#217—21no—
58C2×#217—21no—
59#936—42no—
60C2×C4⋊Dic3C2×C411yescharacteristic
61#924—21no—
62#924—21no—
63#924—21no—
64#924—21no—
65C2×Dic3⋊C4—21no—
66C2×C4×Dic3—24no—
67C2×C4×Dic3—42no—
68C2×C4×Dic3—42no—
69C2×C4×Dic3—42no—
70C2×C4×Dic3C2×C411yescharacteristic
71C2×C4×Dic3C2×C411yescharacteristic
72C2×C4×Dic3D411yescharacteristic
73C2×C4×Dic3D411yescharacteristic
74C22×C3⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×#95—21no—
78C3×#95D412yes—
79C3⋊#95—24no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84C3⋊#95—21no—
85C3⋊#95D412yes—
86#830—21no—
87#830—21no—
88Dic3⋊C8—42no—
89C2×#124—31no—
90C2×C22⋊Q8—31no—
91C2×#121C2×D311yescharacteristic
92C22×C42—31no—
93C22×C42—61no—
94C2×C4⋊C8—61no—
95C2×C22⋊C8—31no—
96C2×C22⋊C8C2×D311yescharacteristic
97#391—31no—
98#390—31no—
99#388—61no—
100#388—61no—
101#386—61no—
102#386—61no—
103#382—31no—
104C2×#95—31no—
105C2×#95—61no—
106C2×#95C2×D311yescharacteristic
107#336—61no—
108#328—62no—
109#328—62no—
110#327—22no—
111#327—62no—
112C24×C3—21no—
113C24×C3C2×D411yescharacteristic
114C2×C3×D4—21no—
115C2×C3×D4—21no—
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—
123C2×C3×D4—41no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
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×C4C22×C411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×Dic3—21no—
141C22×Dic3—21no—
142C22×Dic3—22no—
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—22no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C22×Dic3—22no—
150C22×Dic3—21no—
151C22×Dic3—21no—
152C22×Dic3—41no—
153C22×Dic3—41no—
154C22×Dic3—41no—
155C22×Dic3—41no—
156C22×Dic3—41no—
157C22×Dic3C22⋊C411yescharacteristic
158C22×Dic3C22⋊C411yescharacteristic
159C22×Dic3C22⋊C411yescharacteristic
160C22×Dic3C22⋊C411yescharacteristic
161C2×Dic6—21no—
162C2×Dic6—21no—
163C2×Dic6—41no—
164C2×Dic6—41no—
165C2×C3×C8—41no—
166C2×C3×C8—41no—
167C2×C3×C8—41no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C22⋊C4—41no—
171C3×C22⋊C4—41no—
172#217—42no—
173#217—41no—
174#217—41no—
175#217—42no—
176#217—42no—
177#217—42no—
178C4⋊Dic3—21no—
179C4⋊Dic3—21no—
180Dic3⋊C4—41no—
181Dic3⋊C4—41no—
182C4×Dic3—22no—
183C4×Dic3—22no—
184C4×Dic3—41no—
185C4×Dic3—41no—
186C4×Dic3—48no—
187C2×C3⋊C8—41no—
188C2×C3⋊C8—41no—
189C2×C3⋊C8—41no—
190C22×Q8—31no—
191C22×D4—21no—
192C22×D4C4×D311yescharacteristic
193C23×C4—21no—
194C23×C4—31no—
195C23×C4—31no—
196C23×C4—61no—
197C23×C4C22×D311yescharacteristic
198C22×C8—21no—
199C22×C8—61no—
200#124—122no—
201C22⋊Q8—61no—
202C22⋊Q8—61no—
203C22⋊Q8—61no—
204C22⋊Q8—61no—
205#121—21no—
206#121—21no—
207#121—21no—
208#121—21no—
209C2×C4⋊C4—31no—
210C2×C4⋊C4—61no—
211C2×C4⋊C4C4×D311yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C22⋊C4—61no—
217C2×C22⋊C4—61no—
218C2×C42—31no—
219C2×C42—31no—
220C2×C42—31no—
221C2×C42—31no—
222C2×C42—64no—
223C2×C42—122no—
224C2×C42—122no—
225C2×C42—122no—
226C4⋊C8—122no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—61no—
230C22⋊C8—61no—
231#95—21no—
232#95—32no—
233#95—61no—
234#95—61no—
235#95—61no—
236#95—61no—
237#95—64no—
238#95—61no—
239#95#6712yes—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3C2×C22⋊C411yescharacteristic
252C23×C3#12111yescharacteristic
253C23×C3#12111yescharacteristic
254C23×C3#13511yescharacteristic
255C23×C3#13511yescharacteristic
256C23×C3AGL(1,ℤ/8)11yescharacteristic
257C23×C3#13711yescharacteristic
258C3×D4—81no—
259C3×D4—81no—
260C3×D4—81no—
261C3×D4—81no—
262C3×D4—81no—
263C3×D4—81no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—24no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4C4×D411yescharacteristic
283C2×C3×C4C4×D411yescharacteristic
284C2×C3×C4#12411yescharacteristic
285C2×C3×C4#12711yescharacteristic
286C2×Dic3—24no—
287C2×Dic3—24no—
288C2×Dic3—21no—
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—21no—
294C2×Dic3—24no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—21no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3#9912yes—
313C2×Dic3#10412yes—
314C2×Dic3#10412yes—
315Dic6—81no—
316Dic6—81no—
317C3×C8—81no—
318C3⋊C8—81no—
319C24—21no—
320C24D3×D411yescharacteristic
321C2×Q8—61no—
322C2×Q8—61no—
323C2×Q8—121no—
324C2×Q8—121no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
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—62no—
351C22×C4—62no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—61no—
361C22×C4—121no—
362C22×C4—121no—
363C22×C4—121no—
364C22×C4—121no—
365C22×C4—121no—
366C22×C4C2×C4×D311yescharacteristic
367C22×C4#23511yescharacteristic
368C22×C4D3×D411yescharacteristic
369C22×C4D3×D411yescharacteristic
370C22×C4#23711yescharacteristic
371C22×C4C2×#6711yescharacteristic
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—121no—
376C2×C8—121no—
377C2×C8—121no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—61no—
381C4⋊C4—61no—
382C4⋊C4—121no—
383C4⋊C4—121no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—122no—
387C22⋊C4—122no—
388C22⋊C4—122no—
389C22⋊C4—122no—
390C22⋊C4—121no—
391C22⋊C4—121no—
392C42—62no—
393C42—62no—
394C42—121no—
395C42—121no—
396C42—128no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
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—81no—
410C22×C3—81no—
411C22×C3—81no—
412C22×C3—81no—
413C22×C3—81no—
414C22×C3—81no—
415C22×C3—81no—
416C22×C3#39011yescharacteristic
417C22×C3C2×#9911yescharacteristic
418C22×C3#41011yescharacteristic
419C22×C3#41611yescharacteristic
420C22×C3#41711yescharacteristic
421C22×C3C2×#10411yescharacteristic
422C22×C3C2×#10411yescharacteristic
423C22×C3#46311yescharacteristic
424C22×C3#46311yescharacteristic
425C22×C3#46411yescharacteristic
426C22×C3#46411yescharacteristic
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—44no—
432C3×C4—81no—
433Dic3—22no—
434Dic3—22no—
435Dic3—22no—
436Dic3—44no—
437Dic3—41no—
438Dic3—81no—
439Dic3—81no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23D3×C22⋊C411yescharacteristic
452C23#88411yescharacteristic
453C23#91111yescharacteristic
454C23#91211yescharacteristic
455C23#91511yescharacteristic
456C23#91611yescharacteristic
457C23#93911yescharacteristic
458Q8—241no—
459Q8—241no—
460D4—81no—
461D4—81no—
462D4—81no—
463D4—81no—
464D4—81no—
465D4—81no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—24no—
471C2×C4—32no—
472C2×C4—32no—
473C2×C4—32no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—64no—
488C2×C4—64no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—64no—
496C2×C4—61no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4#67⋊C411yescharacteristic
514C2×C4#88511yescharacteristic
515C2×C4C4×#6711yescharacteristic
516C2×C4#94011yescharacteristic
517C8—81no—
518C8—241no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—161no—
524C2×C3#150011yescharacteristic
525C2×C3#150111yescharacteristic
526C2×C3#150411yescharacteristic
527C2×C3#150911yescharacteristic
528C2×C3#196211yescharacteristic
529C2×C3#196411yescharacteristic
530C2×C3#198311yescharacteristic
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22#474011yescharacteristic
551C22D3×#9911yescharacteristic
552C22#476711yescharacteristic
553C22#477111yescharacteristic
554C22#477211yescharacteristic
555C22#478711yescharacteristic
556C22D3×#10411yescharacteristic
557C22D3×#10411yescharacteristic
558C22#495711yescharacteristic
559C22#515311yescharacteristic
560C22#516611yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—44no—
566C4—62no—
567C4—62no—
568C4—62no—
569C4—81no—
570C4—124no—
571C4—121no—
572C4—241no—
573C4—241no—
574C3#971311yescharacteristic, complemented, Sylow
575C2—21no—
576C2—21no—
577C2—21no—
578C2—21no—
579C2—161no—
580C2#6827611yescharacteristic
581C2#6827911yescharacteristic
582C2#6830111yescharacteristic
583C2#6830211yescharacteristic
584C2#7003011yescharacteristic
585C2#7016211yescharacteristic
586C2#7035611yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#327C2
socleC23×C3C2×C22⋊C4
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenterC2×#327D3
hypocenterC3#9713
LCSG ▷ C22×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
4. C2×#4467C2semidirect
5. #69908C2semidirect
6. Dic3×#95C2semidirect
7. C2×#4452C2semidirect
10. C2×#936C4semidirect
16. #5221C4semidirect
23. #4977C4semidirect
24. #4973C4semidirect
574. C3#9713semidirect

Maximal subgroups

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

subgroupindex
#826982
C2×C3×#3272
C2×#45372
C2×#44672
#699082
Dic3×#952
C2×#44522
#97133