Finite Group Databasebeta

← #10694315

Subgroups — #10694315

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#82809C211yesmaximal, characteristic
2C2×C3×#328C211yesmaximal, characteristic, complemented
3C2×#4538C211yesmaximal, characteristic, complemented
4C2×#4470C211yesmaximal, characteristic
5#69908C211yesmaximal, characteristic, complemented
6Dic3×#95C211yesmaximal, characteristic, complemented
7C2×#4451C211yesmaximal, characteristic
8#9713—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×#947C411yescharacteristic, complemented
11C2×#930C2211yescharacteristic
12C22×C4×Dic3—21no—
13C22×C4×Dic3C2211yescharacteristic
14C2×C3×C22⋊C8C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16#5229C411yescharacteristic, complemented
17C2×C3⋊#95—21no—
18C2×C3⋊#95C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×Dic3⋊C8—21no—
21#4990—21no—
22#4990—21no—
23#4988C411yescharacteristic, complemented
24#4977C411yescharacteristic, complemented
25#4958—21no—
26#4958—21no—
27C3×#328—22no—
28#4550—21no—
29#4538—22no—
30#4470—22no—
31#4451—22no—
32#2850—31no—
33#1797—31no—
34C2×#328—31no—
35C2×#328D311yescharacteristic, complemented
36C2×#327—31no—
37C2×#327—31no—
38C4×#95—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C23×Dic3—21no—
44C23×Dic3D411yescharacteristic
45C23×Dic3D411yescharacteristic
46C22×D12C2×C411yescharacteristic
47C22×C3×C8—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4C2×C411yescharacteristic
54C2×C3×C22⋊C4—21no—
55#947—42no—
56#930—21no—
57#930—21no—
58#930—21no—
59#930—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×C4⋊Dic3C2×C411yescharacteristic
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⋊Q8C2×D311yescharacteristic
91C2×#121—31no—
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—22no—
109#328—62no—
110#327—62no—
111#327—62no—
112C24×C3C2×D411yescharacteristic
113C23×D3—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8—41no—
117C2×C3×Q8—41no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4C22×C411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C2×#67—41no—
136C2×#67—41no—
137C2×#67—41no—
138C2×#67—41no—
139C2×#67—41no—
140C2×#67—41no—
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3—21no—
149C22×Dic3—21no—
150C22×Dic3—22no—
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×D12—21no—
162C2×D12—21no—
163C2×D12—41no—
164C2×D12—41no—
165C2×C3×C8—41no—
166C2×C3×C8—41no—
167C2×C3×C8—41no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—41no—
171C3×C4⋊C4—41no—
172C3×C22⋊C4—41no—
173C3×C22⋊C4—41no—
174(C2×D3)⋊C4—42no—
175(C2×D3)⋊C4—41no—
176(C2×D3)⋊C4—41no—
177(C2×D3)⋊C4—42no—
178(C2×D3)⋊C4—42no—
179(C2×D3)⋊C4—42no—
180C4⋊Dic3—21no—
181C4⋊Dic3—21no—
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×Q8C4×D311yescharacteristic
191C22×D4—31no—
192C22×D4—61no—
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—21no—
202C22⋊Q8—21no—
203C22⋊Q8—21no—
204C22⋊Q8—21no—
205#121—61no—
206#121—61no—
207#121—61no—
208#121—61no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—31no—
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×C3C2×C22⋊C411yescharacteristic
245C23×C3#12111yescharacteristic
246C23×C3#12111yescharacteristic
247C23×C3#13511yescharacteristic
248C23×C3#13511yescharacteristic
249C23×C3AGL(1,ℤ/8)11yescharacteristic
250C23×C3#13711yescharacteristic
251C22×D3—41no—
252C22×D3—41no—
253C22×D3—41no—
254C22×D3—41no—
255C22×D3—41no—
256C22×D3—41no—
257C22×D3—41no—
258C3×Q8—81no—
259C3×Q8—81no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—24no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4C4×D411yescharacteristic
282C2×C3×C4C4×D411yescharacteristic
283C2×C3×C4#12411yescharacteristic
284C2×C3×C4#12711yescharacteristic
285#67—81no—
286#67—81no—
287#67—81no—
288#67—81no—
289C2×Dic3—24no—
290C2×Dic3—24no—
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—21no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—24no—
298C2×Dic3—21no—
299C2×Dic3—21no—
300C2×Dic3—21no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—41no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3#9912yes—
313C2×Dic3#10412yes—
314C2×Dic3#10412yes—
315D12—81no—
316D12—81no—
317C3×C8—81no—
318C3⋊C8—81no—
319C24—61no—
320C24D3×D411yescharacteristic
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×D4—61no—
326C2×D4—61no—
327C2×D4—121no—
328C2×D4—121no—
329C2×D4—121no—
330C2×D4—121no—
331C2×D4—121no—
332C2×D4—121no—
333C2×D4—121no—
334C2×D4—121no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—31no—
341C22×C4—31no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—62no—
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—41no—
381C4⋊C4—41no—
382C4⋊C4—61no—
383C4⋊C4—61no—
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#39011yescharacteristic
410C22×C3C2×#9911yescharacteristic
411C22×C3#41011yescharacteristic
412C22×C3#41611yescharacteristic
413C22×C3#41711yescharacteristic
414C22×C3C2×#10411yescharacteristic
415C22×C3C2×#10411yescharacteristic
416C22×C3#46311yescharacteristic
417C22×C3#46311yescharacteristic
418C22×C3#46411yescharacteristic
419C22×C3#46411yescharacteristic
420C2×D3—81no—
421C2×D3—81no—
422C2×D3—81no—
423C2×D3—81no—
424C2×D3—81no—
425C2×D3—81no—
426C2×D3—81no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—44no—
432C3×C4—81no—
433C3×C4—81no—
434Dic3—22no—
435Dic3—22no—
436Dic3—22no—
437Dic3—44no—
438Dic3—41no—
439Dic3—81no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—121no—
445C23—121no—
446C23—121no—
447C23—121no—
448C23—121no—
449C23—121no—
450C23—121no—
451C23D3×C22⋊C411yescharacteristic
452C23#88411yescharacteristic
453C23#91411yescharacteristic
454C23#91611yescharacteristic
455C23#91811yescharacteristic
456C23#91911yescharacteristic
457C23#93911yescharacteristic
458Q8—81no—
459Q8—81no—
460D4—241no—
461D4—241no—
462D4—241no—
463D4—241no—
464D4—241no—
465D4—241no—
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—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—64no—
491C2×C4—64no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—61no—
497C2×C4—61no—
498C2×C4—64no—
499C2×C4—61no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—122no—
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#150011yescharacteristic
524C2×C3#150111yescharacteristic
525C2×C3#150411yescharacteristic
526C2×C3#150911yescharacteristic
527C2×C3#196211yescharacteristic
528C2×C3#196411yescharacteristic
529C2×C3#198311yescharacteristic
530D3—161no—
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—241no—
544C22—241no—
545C22—241no—
546C22—241no—
547C22—241no—
548C22—241no—
549C22—241no—
550C22#474011yescharacteristic
551C22D3×#9911yescharacteristic
552C22#479811yescharacteristic
553C22#480311yescharacteristic
554C22#480411yescharacteristic
555C22#481911yescharacteristic
556C22D3×#10411yescharacteristic
557C22D3×#10411yescharacteristic
558C22#495711yescharacteristic
559C22#516511yescharacteristic
560C22#518511yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—44no—
566C4—62no—
567C4—62no—
568C4—62no—
569C4—81no—
570C4—81no—
571C4—124no—
572C4—121no—
573C4—241no—
574C3#971311yescharacteristic, complemented, Sylow
575C2—21no—
576C2—21no—
577C2—21no—
578C2—21no—
579C2—481no—
580C2#6830811yescharacteristic
581C2#6831111yescharacteristic
582C2#6833311yescharacteristic
583C2#6833411yescharacteristic
584C2#7003011yescharacteristic
585C2#7023011yescharacteristic
586C2#7035611yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#328C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenterC2×#328D3
hypocenterC3#9713
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
2. C2×C3×#328C2semidirect
3. C2×#4538C2semidirect
5. #69908C2semidirect
6. Dic3×#95C2semidirect
10. C2×#947C4semidirect
16. #5229C4semidirect
23. #4988C4semidirect
24. #4977C4semidirect
35. C2×#328D3semidirect
574. C3#9713semidirect

Maximal subgroups

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

subgroupindex
#828092
C2×C3×#3282
C2×#45382
C2×#44702
#699082
Dic3×#952
C2×#44512
#97133