Finite Group Databasebeta

← #10964097

Subgroups — #10964097

Order 768.

2522 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic, complemented
2C2×#5522C211yesmaximal, characteristic
3#83236C212yesmaximal, complemented
4#83235C212yesmaximal, complemented
5#83030C212yesmaximal, complemented
6#83029C212yesmaximal, complemented
7#82765C214yesmaximal, complemented
8#82764C214yesmaximal, complemented
9#82491C214yesmaximal, complemented
10C4×#946C214yesmaximal, complemented
11#82228C211yesmaximal, characteristic
12#82227C211yesmaximal, characteristic, complemented
13#82226C211yesmaximal, characteristic, complemented
14#82070C211yesmaximal, characteristic
15C2×C4×Dic3⋊C4C211yesmaximal, characteristic, complemented
16#34745—31nomaximal, Sylow
17C22×#235C2211yescharacteristic
18C3×#533—28no—
19C2×C3×C4⋊Q8C2212yes—
20C2×C3×#125C2212yes—
21C2×C3×C22⋊Q8C2214yes—
22C2×C3×C4×Q8C2214yes—
23C2×C3×#117C2211yescharacteristic
24C2×C3×#117C2211yescharacteristic
25C22×C3×C42C2211yescharacteristic
26C2×#946C2214yes—
27C2×#926C2211yescharacteristic
28C22×Dic3⋊C4C2212yes—
29C22×C4×Dic3C2211yescharacteristic
30C22×C4×Dic3C2211yescharacteristic
31#5522—28no—
32C2×#893C2212yes—
33C2×#887C2212yes—
34C2×#67⋊C4C2214yes—
35C2×#872C2212yes—
36C2×C42×D3C2212yes—
37#5231C2214yes—
38#5230C2214yes—
39#5229C2214yes—
40#5213C2214yes—
41C2×C3⋊#95C2212yes—
42#4994C2214yes—
43#4994C2214yes—
44#4993C2214yes—
45#4989C2214yes—
46#4989C2214yes—
47#4988C2212yes—
48#4987C2214yes—
49#4985C2214yes—
50#4984C2212yes—
51#4984C2212yes—
52#4982C2214yes—
53#4982C2214yes—
54#4978C2214yes—
55#4978C2214yes—
56#4973C2212yes—
57#4972C2212yes—
58#4972C2212yes—
59#4972C2214yes—
60#4958C2212yes—
61#4958C2212yes—
62#4950C2212yes—
63#4950C2212yes—
64#4946C2214yes—
65#4939C2211yescharacteristic
66#4939C2211yescharacteristic
67C4×(C2×D3)⋊C4C2214yes—
68C4×(C2×D3)⋊C4C2214yes—
69C4×C4⋊Dic3C2214yes—
70C4×Dic3⋊C4C2212yes—
71C4×Dic3⋊C4C2212yes—
72C4×Dic3⋊C4C2212yes—
73C4×Dic3⋊C4C2212yes—
74C4×Dic3⋊C4C2212yes—
75C4×Dic3⋊C4C2212yes—
76#4929C2211yescharacteristic
77#4929C2211yescharacteristic
78C2×#533D311yescharacteristic, complemented
79C2×C4×#41—31no—
80#2454—32no—
81#2453—32no—
82#2452—32no—
83#2451—32no—
84#2450—34no—
85#2449—34no—
86#2396—34no—
87C4×C22⋊Q8—34no—
88#2297—31no—
89#2297—31no—
90#2296—31no—
91#2296—31no—
92C2×C4×C4⋊C4—31no—
93C22×C3×Q8C2314yes—
94C23×C3×C4C2312yes—
95C23×C3×C4C2311yescharacteristic
96C22×#67C2312yes—
97C23×Dic3C2311yescharacteristic
98C23×Dic3C2312yes—
99C23×Dic3C2311yescharacteristic
100C2×#235—24no—
101C2×#235—24no—
102C2×#235Q818yescomplemented
103C22×D12C2311yescharacteristic
104C22×C4×D3C2314yes—
105C22×C4×D3C2312yes—
106C22×Dic6C2311yescharacteristic
107C3×C4⋊Q8—28no—
108C3×#125—28no—
109C3×C22⋊Q8—216no—
110C3×C4×Q8—216no—
111C3×#117—24no—
112C3×#117D418yes—
113C2×C3×C4⋊C4C2312yes—
114C2×C3×C4⋊C4C2314yes—
115C2×C3×C4⋊C4C2314yes—
116C2×C3×C4⋊C4C2314yes—
117C2×C3×C4⋊C4C2312yes—
118C2×C3×C22⋊C4C2312yes—
119C2×C3×C22⋊C4C2312yes—
120C2×C3×C42—24no—
121C2×C3×C42C2311yescharacteristic
122C2×C3×C42C2311yescharacteristic
123C2×C3×C42C2312yes—
124C2×C3×C42C2312yes—
125C2×C3×C42C2312yes—
126C2×#217C2312yes—
127#946—216no—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2314yes—
131C2×(C2×D3)⋊C4C2314yes—
132C2×(C2×D3)⋊C4C2314yes—
133#926—24no—
134C2×C4⋊Dic3C2311yescharacteristic
135C2×C4⋊Dic3C2311yescharacteristic
136C2×C4⋊Dic3C2314yes—
137C2×C4⋊Dic3C2314yes—
138C2×Dic3⋊C4—28no—
139C2×Dic3⋊C4C2312yes—
140C2×Dic3⋊C4C2312yes—
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2314yes—
144C2×Dic3⋊C4C2314yes—
145C2×Dic3⋊C4C2312yes—
146C2×C4×Dic3—24no—
147C2×C4×Dic3—24no—
148C2×C4×Dic3C2311yescharacteristic
149C2×C4×Dic3C2311yescharacteristic
150C2×C4×Dic3C2312yes—
151C2×C4×Dic3C2312yes—
152C2×C4×Dic3C2312yes—
153C2×C4×Dic3C2312yes—
154C2×C4×Dic3C2314yes—
155#893—28no—
156#887—28no—
157#67⋊C4—216no—
158#872—28no—
159C42×D3—28no—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162C3⋊#95C2312yes—
163C3⋊#95C2312yes—
164C3⋊#95C2312yes—
165C3⋊#95C2312yes—
166C3⋊#95C2314yes—
167C3⋊#95C2314yes—
168C3⋊#95C2314yes—
169C3⋊#95C2314yes—
170C3⋊#95C2314yes—
171C3⋊#95C2314yes—
172C22×#41—31no—
173#533—28no—
174C2×C4⋊Q8C2×D312yes—
175C2×#125C2×D312yes—
176C2×C22⋊Q8—34no—
177C2×C22⋊Q8C2×D314yes—
178C4×#41—68no—
179C2×C4×Q8—32no—
180C2×C4×Q8C2×D314yes—
181C2×C4×D4—32no—
182C2×C4×D4—34no—
183C2×#117—31no—
184C2×#117—32no—
185C2×#117C2×D311yescharacteristic
186C2×#117C2×D311yescharacteristic
187C22×C4⋊C4—32no—
188C22×C42—31no—
189C22×C42—31no—
190C22×C42—32no—
191C22×C42C2×D311yescharacteristic
192#400—34no—
193#399—34no—
194#398—34no—
195#398—34no—
196#396—34no—
197#396—34no—
198#395—34no—
199#393—34no—
200#391—34no—
201#391—32no—
202#391—31no—
203#391—31no—
204#390—32no—
205#390—31no—
206#390—31no—
207#389—34no—
208#389—32no—
209#389—32no—
210#389—32no—
211#389—32no—
212#387—34no—
213#387—34no—
214#386—34no—
215#386—32no—
216#386—32no—
217#385—34no—
218#385—34no—
219#385—34no—
220#385—32no—
221#385—32no—
222#380—34no—
223C4×C4⋊C4—32no—
224C4×C4⋊C4—32no—
225C4×C4⋊C4—32no—
226C4×C4⋊C4—32no—
227C4×C4⋊C4—32no—
228C4×C4⋊C4—32no—
229C4×C4⋊C4—34no—
230C4×C22⋊C4—34no—
231C4×C22⋊C4—34no—
232C2×#95—32no—
233C24×C3C2411yescharacteristic
234C23×D3C2412yes—
235C2×C3×Q8—28no—
236C2×C3×Q8—28no—
237C2×C3×Q8—44no—
238C22×C3×C4—24no—
239C22×C3×C4—24no—
240C22×C3×C4—42no—
241C22×C3×C4C2×D414yes—
242C22×C3×C4C2×Q814yes—
243C22×C3×C4#4114yes—
244C22×C3×C4C2412yes—
245C22×C3×C4C2414yes—
246C22×C3×C4C2414yes—
247C22×C3×C4C2412yes—
248C22×C3×C4C2411yescharacteristic
249C22×C3×C4C2411yescharacteristic
250C2×#67—24no—
251C2×#67—24no—
252C2×#67C2×Q818yes—
253C22×Dic3—22no—
254C22×Dic3—24no—
255C22×Dic3—24no—
256C22×Dic3—22no—
257C22×Dic3—24no—
258C22×Dic3—42no—
259C22×Dic3—42no—
260C22×Dic3C2412yes—
261C22×Dic3C2412yes—
262C22×Dic3C2412yes—
263C22×Dic3C2412yes—
264C22×Dic3C2412yes—
265C22×Dic3C2412yes—
266C22×Dic3C2412yes—
267#235—232no—
268C2×D12—22no—
269C2×D12—22no—
270C2×D12C2×Q814yes—
271C2×C4×D3—24no—
272C2×C4×D3—24no—
273C2×C4×D3—28no—
274C2×C4×D3—28no—
275C2×C4×D3—44no—
276C2×C4×D3C2×Q818yes—
277C2×Dic6—22no—
278C2×Dic6—22no—
279C2×Dic6C2×Q814yes—
280C3×C4⋊C4—24no—
281C3×C4⋊C4—28no—
282C3×C4⋊C4—28no—
283C3×C4⋊C4—28no—
284C3×C4⋊C4C2×D418yes—
285C3×C22⋊C4—24no—
286C3×C22⋊C4C2×D418yes—
287C3×C42—24no—
288C3×C42—22no—
289C3×C42—22no—
290C3×C42—24no—
291C3×C42C2×D418yes—
292#217—24no—
293(C2×D3)⋊C4—28no—
294(C2×D3)⋊C4—28no—
295(C2×D3)⋊C4—28no—
296(C2×D3)⋊C4—44no—
297(C2×D3)⋊C4—44no—
298C4⋊Dic3—22no—
299C4⋊Dic3—22no—
300C4⋊Dic3—28no—
301C4⋊Dic3—44no—
302Dic3⋊C4—28no—
303Dic3⋊C4—24no—
304Dic3⋊C4—24no—
305Dic3⋊C4—24no—
306Dic3⋊C4—28no—
307Dic3⋊C4—44no—
308Dic3⋊C4#4118yescomplemented
309C4×Dic3—22no—
310C4×Dic3—28no—
311C4×Dic3—22no—
312C4×Dic3—24no—
313C4×Dic3—24no—
314C4×Dic3—24no—
315C4×Dic3—24no—
316C2×#41—38no—
317C2×#41—64no—
318C2×#41—64no—
319C22×Q8—31no—
320C22×Q8C22×D314yes—
321C22×D4—32no—
322C22×D4—31no—
323C23×C4—32no—
324C23×C4—31no—
325C23×C4—32no—
326C23×C4—31no—
327C23×C4—34no—
328C23×C4C22×D312yes—
329C23×C4C22×D311yescharacteristic
330C4⋊Q8—28no—
331#125—28no—
332C22⋊Q8—216no—
333C22⋊Q8—616no—
334C4×Q8—216no—
335C4×Q8—68no—
336C4×D4—68no—
337C4×D4—616no—
338#117—24no—
339#117—64no—
340#117—68no—
341#117#6718yes—
342C2×C4⋊C4—32no—
343C2×C4⋊C4—32no—
344C2×C4⋊C4—32no—
345C2×C4⋊C4—32no—
346C2×C4⋊C4—34no—
347C2×C4⋊C4—31no—
348C2×C4⋊C4—31no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—34no—
351C2×C4⋊C4—34no—
352C2×C4⋊C4—34no—
353C2×C4⋊C4—68no—
354C2×C4⋊C4C22×D312yes—
355C2×C4⋊C4C22×D314yes—
356C2×C4⋊C4C22×D314yes—
357C2×C4⋊C4C22×D314yes—
358C2×C4⋊C4C22×D312yes—
359C2×C22⋊C4—32no—
360C2×C22⋊C4—34no—
361C2×C22⋊C4—34no—
362C2×C22⋊C4—34no—
363C2×C22⋊C4—34no—
364C2×C22⋊C4—34no—
365C2×C22⋊C4C22×D312yes—
366C2×C22⋊C4C22×D312yes—
367C2×C42—24no—
368C2×C42—32no—
369C2×C42—34no—
370C2×C42—32no—
371C2×C42—31no—
372C2×C42—32no—
373C2×C42—32no—
374C2×C42—31no—
375C2×C42—64no—
376C2×C42—64no—
377C2×C42—68no—
378C2×C42C22×D311yescharacteristic
379C2×C42C22×D311yescharacteristic
380C2×C42C22×D312yes—
381C2×C42C22×D312yes—
382C2×C42C22×D312yes—
383#95—32no—
384#95—32no—
385#95—32no—
386#95—32no—
387#95—34no—
388#95—34no—
389#95—34no—
390#95—34no—
391#95—34no—
392#95—34no—
393#95—34no—
394#95—34no—
395C23×C3—22no—
396C23×C3—22no—
397C23×C3C22×D412yes—
398C23×C3C22×Q812yes—
399C23×C3C2×#4112yes—
400C23×C3C2511yescharacteristic
401C22×D3—22no—
402C22×D3—24no—
403C22×D3—22no—
404C22×D3—24no—
405C22×D3C22×Q814yes—
406C3×Q8—416no—
407C2×C3×C4—22no—
408C2×C3×C4—24no—
409C2×C3×C4—24no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—24no—
413C2×C3×C4—22no—
414C2×C3×C4—24no—
415C2×C3×C4—24no—
416C2×C3×C4—22no—
417C2×C3×C4—48no—
418C2×C3×C4C22⋊Q8116yescomplemented
419C2×C3×C4C22×D418yes—
420C2×C3×C4C22×D412yes—
421C2×C3×C4C22×D412yes—
422C2×C3×C4C22×Q812yes—
423C2×C3×C4C22×Q812yes—
424C2×C3×C4C2×#4112yes—
425C2×C3×C4C2×#4112yes—
426C2×C3×C4C2×#4114yes—
427#67—216no—
428C2×Dic3—28no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—22no—
436C2×Dic3—22no—
437C2×Dic3—22no—
438C2×Dic3—22no—
439C2×Dic3—22no—
440C2×Dic3—24no—
441C2×Dic3—22no—
442C2×Dic3—24no—
443C2×Dic3—22no—
444C2×Dic3—22no—
445C2×Dic3—48no—
446C2×Dic3—44no—
447C2×Dic3C22×Q814yes—
448C2×Dic3C2×#4114yes—
449C2×Dic3C2×#4114yes—
450D12—28no—
451C4×D3—216no—
452C4×D3—416no—
453Dic6—28no—
454C24—32no—
455C24C23×D311yescharacteristic
456#41—632no—
457C2×Q8—28no—
458C2×Q8—28no—
459C2×Q8—34no—
460C2×Q8—44no—
461C2×Q8—62no—
462C2×Q8—62no—
463C2×D4—38no—
464C2×D4—34no—
465C2×D4—64no—
466C2×D4—64no—
467C2×D4—62no—
468C2×D4—62no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—32no—
472C22×C4—32no—
473C22×C4—32no—
474C22×C4—38no—
475C22×C4—32no—
476C22×C4—32no—
477C22×C4—32no—
478C22×C4—32no—
479C22×C4—42no—
480C22×C4—68no—
481C22×C4—64no—
482C22×C4—64no—
483C22×C4—62no—
484C22×C4—64no—
485C22×C4—64no—
486C22×C4—62no—
487C22×C4—64no—
488C22×C4—68no—
489C22×C4—124no—
490C22×C4—122no—
491C22×C4—122no—
492C22×C4D3×Q814yes—
493C22×C4#23914yes—
494C22×C4C2×#6714yes—
495C22×C4C23×D312yes—
496C22×C4C23×D314yes—
497C22×C4C23×D312yes—
498C22×C4C23×D314yes—
499C22×C4C23×D311yescharacteristic
500C22×C4C23×D311yescharacteristic
501C4⋊C4—24no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—38no—
506C4⋊C4—64no—
507C4⋊C4—64no—
508C4⋊C4—68no—
509C4⋊C4—68no—
510C4⋊C4—62no—
511C4⋊C4—62no—
512C4⋊C4—64no—
513C4⋊C4—68no—
514C4⋊C4—124no—
515C4⋊C4—124no—
516C4⋊C4C2×#6718yes—
517C22⋊C4—24no—
518C22⋊C4—68no—
519C22⋊C4—64no—
520C22⋊C4—68no—
521C22⋊C4—68no—
522C22⋊C4—124no—
523C22⋊C4—124no—
524C22⋊C4C2×#6718yes—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—24no—
529C42—64no—
530C42—64no—
531C42—64no—
532C42—62no—
533C42—62no—
534C42—68no—
535C42—64no—
536C42C2×#6718yes—
537C22×C3—22no—
538C22×C3—22no—
539C22×C3—24no—
540C22×C3—24no—
541C22×C3C2×C22⋊Q814yes—
542C22×C3C23×D411yescharacteristic
543C22×C3C23×Q811yescharacteristic
544C22×C3C22×#4111yescharacteristic
545C22×C3C22×#4112yes—
546C22×C3#58512yes—
547C2×D3—24no—
548C2×D3—28no—
549C2×D3—42no—
550C2×D3—42no—
551C2×D3—44no—
552C3×C4—28no—
553C3×C4—24no—
554C3×C4—42no—
555C3×C4—44no—
556C3×C4C2×C22⋊Q814yes—
557C3×C4C2×C22⋊Q814yes—
558Dic3—24no—
559Dic3—24no—
560Dic3—24no—
561Dic3—42no—
562Dic3—42no—
563Dic3—42no—
564Dic3—42no—
565C23—22no—
566C23—22no—
567C23—34no—
568C23—62no—
569C23—64no—
570C23—64no—
571C23—62no—
572C23C2×D3×Q812yes—
573C23C2×#23912yes—
574C23C22×#6712yes—
575C23C24×D311yescharacteristic
576Q8—416no—
577Q8—68no—
578D4—616no—
579D4—68no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—34no—
591C2×C4—34no—
592C2×C4—34no—
593C2×C4—48no—
594C2×C4—68no—
595C2×C4—62no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—616no—
602C2×C4—62no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—64no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—64no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—1216no—
613C2×C4—128no—
614C2×C4—124no—
615C2×C4#946116yescomplemented
616C2×C4C2×D3×Q812yes—
617C2×C4C2×D3×Q812yes—
618C2×C4C2×#23912yes—
619C2×C4C2×#23912yes—
620C2×C4D3×#4114yes—
621C2×C4C22×#6718yes—
622C2×C4C22×#6712yes—
623C2×C4C22×#6712yes—
624C2×C3—24no—
625C2×C3C22×C22⋊Q811yescharacteristic
626C2×C3#344411yescharacteristic
627C2×C3#345511yescharacteristic
628C2×C3D4×#4112yes—
629C2×C3Q8×#4112yes—
630D3—44no—
631C22—22no—
632C22—24no—
633C22—24no—
634C22—22no—
635C22—68no—
636C22—64no—
637C22—122no—
638C22—124no—
639C22—122no—
640C22C2×#94614yes—
641C22C22×D3×Q811yescharacteristic
642C22C22×#23911yescharacteristic
643C22C2×D3×#4112yes—
644C22#596412yes—
645C22C23×#6711yescharacteristic
646C4—24no—
647C4—28no—
648C4—42no—
649C4—44no—
650C4—64no—
651C4—64no—
652C4—64no—
653C4—122no—
654C4—122no—
655C4—122no—
656C4—122no—
657C4C2×#94614yes—
658C4C2×#94614yes—
659C3#3474511yescharacteristic, complemented, Sylow
660C2—24no—
661C2—124no—
662C2#8625711yescharacteristic
663C2#8627311yescharacteristic
664C2#8627712yes—
665C2C22×#94611yescharacteristic
666C2#41×#6712yes—
subgroupG/H
centerC22×C4C23×D3
commutator, socle, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#533C2
prime radicalC22×#235C22
hypercenterC2×#533D3
hypocenterC3#34745
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×C3×#533C2semidirect
3. #83236C2semidirect
4. #83235C2semidirect
5. #83030C2semidirect
6. #83029C2semidirect
7. #82765C2semidirect
8. #82764C2semidirect
9. #82491C2semidirect
10. C4×#946C2semidirect
12. #82227C2semidirect
13. #82226C2semidirect
15. C2×C4×Dic3⋊C4C2semidirect
78. C2×#533D3semidirect
102. C2×#235Q8semidirect
308. Dic3⋊C4#41semidirect
418. C2×C3×C4C22⋊Q8semidirect
615. C2×C4#946semidirect
659. C3#34745semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×#55222
#832362
#832352
#830302
#830292
#827652
#827642
#824912
C4×#9462
#822282
#822272
#822262
#820702
C2×C4×Dic3⋊C42
#347453