Finite Group Databasebeta

← #11659009

Subgroups — #11659009

Order 768.

2038 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85834C211yesmaximal, characteristic
2C3×#2906C211yesmaximal, characteristic, complemented
3#83793C211yesmaximal, characteristic
4#83793C211yesmaximal, characteristic
5#83782C211yesmaximal, characteristic, complemented
6Dic3×#131C211yesmaximal, characteristic, complemented
7#82512C211yesmaximal, characteristic, complemented
8#79072C213yesmaximal, complemented
9#79072C213yesmaximal, complemented
10#79071C213yesmaximal, complemented
11#79070C213yesmaximal, complemented
12#79040C213yesmaximal, complemented
13#79039C213yesmaximal, complemented
14#79000C213yesmaximal, complemented
15C2×#4840C213yesmaximal, complemented
16#60129—31nomaximal, Sylow
17C2×D3×#41C2211yescharacteristic
18C2×C3×#131C2211yescharacteristic
19C2×C3×#131C2211yescharacteristic
20C3×C4×#41C2211yescharacteristic
21Dic3×#41C2211yescharacteristic
22C2×#948C2211yescharacteristic
23C2×#948C2211yescharacteristic
24#5574C411yescharacteristic
25#5574C411yescharacteristic
26#5572C411yescharacteristic
27#5572C411yescharacteristic
28#5546C413yes—
29#5546C413yes—
30#5545C413yes—
31#5545C413yes—
32#5522C2213yes—
33D3×#117C2213yes—
34C4×#235C2213yes—
35C2×#872C2213yes—
36C3×#424C2213yescomplemented
37C3×#424C2213yescomplemented
38C3×#423C2213yescomplemented
39C2×C3×C4⋊C8C2213yescomplemented
40#5129C2213yes—
41#5129C2213yes—
42#5126C2213yes—
43#5126C2213yes—
44#5122C2213yescomplemented
45#5122C2213yescomplemented
46Dic3×M16C2213yescomplemented
47Dic3×M16C2213yescomplemented
48#5110C2213yes—
49#5110C2213yes—
50C2×(C2×D3)⋊C8C2213yes—
51C2×(C2×D3)⋊C8C2213yes—
52#5101C2213yescomplemented
53#5101C2213yescomplemented
54C2×#815C2213yescomplemented
55C2×C8×Dic3C2213yescomplemented
56#4999C2213yescomplemented
57#4999C2213yescomplemented
58#4924C2213yescomplemented
59C2×C4⋊(C3⋊C8)C2213yescomplemented
60#4840C2211yescharacteristic
61#4840C2213yes—
62#4840C2213yes—
63#4840C2213yes—
64#4840C2216yes—
65#4840C2213yescomplemented
66#4840C2213yescomplemented
67#4840C2213yescomplemented
68#4840C2213yescomplemented
69#4840C2216yescomplemented
70#4840C2216yescomplemented
71#4835C2216yescomplemented
72#4835C2216yescomplemented
73#4833C2216yescomplemented
74#4833C2216yescomplemented
75#3428—31no—
76#2925—33no—
77#2925—33no—
78#2924—33no—
79#2924—33no—
80#2923—33no—
81#2922—33no—
82#2921—33no—
83C2×#433—33no—
84#2906—31no—
85#2906D311yescharacteristic, complemented
86#2880—31no—
87#2880—31no—
88#2875—31no—
89C4×#131—31no—
90C2×C3×#41C2311yescharacteristic
91D3×#41—24no—
92D3×#41—24no—
93C2×#239C2×C411yescharacteristic
94C2×D3×Q8C2×C411yescharacteristic
95C2×#237C2×C413yes—
96C2×D3×D4C2×C413yes—
97C2×#235C2313yes—
98C22×C4×D3C2313yes—
99C3×#131—24no—
100C3×#131—24no—
101C2×C3×M16C2313yes—
102C2×C3×M16C2313yes—
103C22×C3×C8C2313yes—
104C22×C3×C8C2313yes—
105C3×C4×Q8C2×C411yescharacteristic
106C3×C4×Q8C2×C411yescharacteristic
107C3×C4×D4C2×C413yes—
108C3×C4×D4C2×C413yes—
109C3×#117C2313yes—
110C2×C3×C42C2313yes—
111#948—24no—
112#948—24no—
113Q8×Dic3C2×C411yescharacteristic
114Q8×Dic3C2×C411yescharacteristic
115D4×Dic3C2×C413yes—
116D4×Dic3C2×C413yes—
117C4×#67C2×C416yes—
118C4×#67C2×C416yes—
119C2×(C2×D3)⋊C4C2×C413yes—
120C2×(C2×D3)⋊C4C2×C413yes—
121#926C2313yes—
122C2×Dic3⋊C4C2×C413yes—
123C2×Dic3⋊C4C2×C413yes—
124C2×C4×Dic3C2313yes—
125C2×#208C2313yes—
126C2×#208C2313yes—
127C22×C3⋊C8C2313yes—
128C22×C3⋊C8C2313yes—
129#893C2×C413yes—
130#893C2×C413yes—
131#892C2×C413yes—
132#892C2×C413yes—
133D3×C4⋊C4C2×C413yes—
134D3×C4⋊C4C2×C413yes—
135#887C2×C413yes—
136#887C2×C413yes—
137#67⋊C4C2×C416yes—
138#67⋊C4C2×C416yes—
139D3×C22⋊C4C2×C416yes—
140#877C2×C416yes—
141C4×D12C2×C413yes—
142C4×D12C2×C413yes—
143#872C2×C413yes—
144#872C2×C413yes—
145#872C2×C413yes—
146#872C2×C413yes—
147#872C2311yescharacteristic
148#872C2313yes—
149#872C2313yes—
150#872C2313yes—
151C42×D3C2316yes—
152C4×Dic6C2×C413yes—
153C4×Dic6C2×C413yes—
154C3×C4⋊C8C2311yescharacteristic
155C3×C4⋊C8C2313yes—
156C3×C4⋊C8C2313yes—
157C3×C4⋊C8C2313yes—
158C3×C4⋊C8C2316yes—
159(C2×D3)⋊C8C2313yes—
160(C2×D3)⋊C8C2313yes—
161(C2×D3)⋊C8C2316yes—
162(C2×D3)⋊C8C2316yes—
163(C2×D3)⋊C8C2313yes—
164(C2×D3)⋊C8C2313yes—
165(C2×D3)⋊C8C2313yes—
166(C2×D3)⋊C8C2313yes—
167(C2×D3)⋊C8C2311yescharacteristic
168(C2×D3)⋊C8C2311yescharacteristic
169#815C2313yes—
170#815C2316yes—
171#815C2313yes—
172#815C2311yescharacteristic
173#815C2313yes—
174C8×Dic3C2313yes—
175C8×Dic3C2316yes—
176C8×Dic3C2313yes—
177C8×Dic3C2311yescharacteristic
178C8×Dic3C2313yes—
179C4⋊(C3⋊C8)C2311yescharacteristic
180C4⋊(C3⋊C8)C2313yes—
181C4⋊(C3⋊C8)C2313yes—
182C4⋊(C3⋊C8)C2313yes—
183C4⋊(C3⋊C8)C2316yes—
184C22×#41—31no—
185C2×#131—31no—
186C2×#131—31no—
187C2×#131C2×D311yescharacteristic
188C2×#131C2×D311yescharacteristic
189#520—31no—
190#520—31no—
191#520—33no—
192#520—33no—
193#519—31no—
194#519—31no—
195#518—33no—
196#518—33no—
197C4×#41—31no—
198C4×#41—33no—
199C4×#41—33no—
200C4×#41C2×D311yescharacteristic
201C2×#117—33no—
202C2×#117—33no—
203#433—31no—
204#433—33no—
205#433—33no—
206#433—33no—
207#433—33no—
208#433—33no—
209#433—33no—
210#433—33no—
211#433—36no—
212#433—36no—
213#433—36no—
214#432—36no—
215#432—36no—
216#431—36no—
217#431—36no—
218#424—33no—
219#424—33no—
220#424C2×D313yescomplemented
221#424C2×D313yescomplemented
222#423—33no—
223#423C2×D313yescomplemented
224C2×C4⋊C8—33no—
225C2×C4⋊C8C2×D313yescomplemented
226#408—33no—
227#408—33no—
228#408—33no—
229#408—33no—
230#407—33no—
231#407—33no—
232C2×C22⋊C8—33no—
233C2×C22⋊C8—33no—
234#405—33no—
235#405—33no—
236#405—33no—
237#405—33no—
238C4×M16—33no—
239C4×M16—33no—
240C2×#97—33no—
241C2×C4×C8—33no—
242C23×D3C22×C413yes—
243C3×#41#4114yes—
244C3×#41#4114yes—
245C2×C3×Q8C22×C411yescharacteristic
246C2×C3×D4C22×C413yes—
247C22×C3×C4C22×C413yes—
248C22×C3×C4C22×C413yes—
249C22×C3×C4C2413yes—
250C2×#67C22×C416yes—
251C22×Dic3C22×C413yes—
252C22×Dic3C22×C413yes—
253C22×Dic3C22×C413yes—
254#239—24no—
255D3×Q8—24no—
256#237—212no—
257D3×D4—212no—
258#235—26no—
259#235—26no—
260C2×D12C22×C413yes—
261C2×C4×D3—26no—
262C2×C4×D3—26no—
263C2×C4×D3C22×C413yes—
264C2×C4×D3C22×C413yes—
265C2×C4×D3C22×C416yes—
266C2×C4×D3C22×C416yes—
267C2×C4×D3C2413yes—
268C2×C4×D3C2411yescharacteristic
269C2×Dic6C22×C413yes—
270C3×M16—26no—
271C3×M16—26no—
272C2×C3×C8—26no—
273C2×C3×C8—26no—
274C2×C3×C8C2411yescharacteristic
275C2×C3×C8C2413yes—
276C2×C3×C8C2413yes—
277C2×C3×C8C2411yescharacteristic
278C3×C4⋊C4C22×C413yes—
279C3×C4⋊C4C22×C413yes—
280C3×C22⋊C4C22×C416yes—
281C3×C42C22×C413yes—
282C3×C42C22×C413yes—
283C3×C42C2413yes—
284C3×C42C2411yescharacteristic
285#217C22×C416yes—
286(C2×D3)⋊C4C22×C411yescharacteristic
287(C2×D3)⋊C4C22×C411yescharacteristic
288(C2×D3)⋊C4C22×C413yes—
289(C2×D3)⋊C4C22×C413yes—
290(C2×D3)⋊C4C22×C413yes—
291(C2×D3)⋊C4C22×C413yes—
292(C2×D3)⋊C4C22×C413yes—
293(C2×D3)⋊C4C22×C413yes—
294C4⋊Dic3C22×C413yes—
295C4⋊Dic3C22×C413yes—
296Dic3⋊C4C22×C413yes—
297Dic3⋊C4C22×C413yes—
298Dic3⋊C4C22×C411yescharacteristic
299Dic3⋊C4C22×C411yescharacteristic
300Dic3⋊C4C22×C413yes—
301Dic3⋊C4C22×C413yes—
302Dic3⋊C4C22×C413yes—
303Dic3⋊C4C22×C413yes—
304C4×Dic3C22×C416yes—
305C4×Dic3C22×C416yes—
306C4×Dic3C22×C413yes—
307C4×Dic3C22×C413yes—
308C4×Dic3C2413yes—
309C4×Dic3C2411yescharacteristic
310#208—26no—
311#208—26no—
312C2×C3⋊C8—26no—
313C2×C3⋊C8—26no—
314C2×C3⋊C8C2411yescharacteristic
315C2×C3⋊C8C2413yes—
316C2×C3⋊C8C2413yes—
317C2×C3⋊C8C2411yescharacteristic
318C2×#41—31no—
319C2×#41—33no—
320C2×#41—33no—
321C2×#41—64no—
322C2×#41—64no—
323C2×#41C22×D311yescharacteristic
324C22×Q8—31no—
325C22×D4—33no—
326C23×C4—33no—
327#131—24no—
328#131—24no—
329#131—64no—
330#131—64no—
331C2×M16—33no—
332C2×M16—33no—
333C2×M16C22×D313yes—
334C2×M16C22×D313yes—
335C22×C8—33no—
336C22×C8—33no—
337C22×C8C22×D313yes—
338C22×C8C22×D313yes—
339C4×Q8—31no—
340C4×Q8—31no—
341C4×Q8—33no—
342C4×Q8—33no—
343C4×Q8—33no—
344C4×Q8—33no—
345C4×Q8C4×D311yescharacteristic
346C4×Q8C4×D311yescharacteristic
347C4×D4—33no—
348C4×D4—33no—
349C4×D4—36no—
350C4×D4—36no—
351C4×D4—33no—
352C4×D4—33no—
353C4×D4—36no—
354C4×D4—36no—
355C4×D4—33no—
356C4×D4—33no—
357C4×D4C4×D313yes—
358C4×D4C4×D313yes—
359#117—33no—
360#117—33no—
361#117—33no—
362#117—36no—
363#117—33no—
364#117—33no—
365#117—33no—
366#117—33no—
367#117—33no—
368#117—31no—
369#117—33no—
370#117—33no—
371#117C22×D313yes—
372C2×C4⋊C4—33no—
373C2×C4⋊C4—33no—
374C2×C4⋊C4—33no—
375C2×C4⋊C4—33no—
376C2×C22⋊C4—33no—
377C2×C22⋊C4—33no—
378C2×C22⋊C4—36no—
379C2×C42—33no—
380C2×C42—36no—
381C2×C42C22×D313yes—
382C4⋊C8—31no—
383C4⋊C8—33no—
384C4⋊C8—33no—
385C4⋊C8—33no—
386C4⋊C8—36no—
387C4⋊C8C22×D311yescharacteristic
388C4⋊C8C22×D313yes—
389C4⋊C8C22×D313yes—
390C4⋊C8C22×D313yes—
391C4⋊C8C22×D316yes—
392C22⋊C8—31no—
393C22⋊C8—31no—
394C22⋊C8—33no—
395C22⋊C8—33no—
396C22⋊C8—33no—
397C22⋊C8—33no—
398C22⋊C8—33no—
399C22⋊C8—33no—
400C22⋊C8—36no—
401C22⋊C8—36no—
402#97—31no—
403#97—33no—
404#97—33no—
405#97—33no—
406#97—36no—
407C4×C8—31no—
408C4×C8—33no—
409C4×C8—33no—
410C4×C8—33no—
411C4×C8—36no—
412C23×C3C23×C413yes—
413C22×D3—26no—
414C22×D3—43no—
415C22×D3—43no—
416C22×D3C23×C411yescharacteristic
417C22×D3C23×C413yes—
418C3×Q8#11714yes—
419C3×D4#117112yes—
420C2×C3×C4—43no—
421C2×C3×C4—43no—
422C2×C3×C4C23×C413yes—
423C2×C3×C4C23×C413yes—
424C2×C3×C4C23×C411yescharacteristic
425C2×C3×C4C23×C411yescharacteristic
426C2×C3×C4C23×C413yes—
427C2×C3×C4C2×#4116yes—
428C2×C3×C4C2×#4116yes—
429C2×C3×C4C2511yescharacteristic
430#67—212no—
431C2×Dic3—26no—
432C2×Dic3—43no—
433C2×Dic3—43no—
434C2×Dic3C23×C413yes—
435C2×Dic3C23×C411yescharacteristic
436C2×Dic3C23×C411yescharacteristic
437C2×Dic3C23×C413yes—
438C2×Dic3C23×C413yes—
439C2×Dic3C23×C411yescharacteristic
440D12—26no—
441C4×D3—26no—
442C4×D3—26no—
443C4×D3—21no—
444C4×D3—21no—
445C4×D3—23no—
446C4×D3—23no—
447C4×D3—46no—
448C4×D3—46no—
449Dic6—26no—
450C3×C8—21no—
451C3×C8—21no—
452C3×C8—23no—
453C3×C8—23no—
454C3⋊C8—21no—
455C3⋊C8—21no—
456C3⋊C8—23no—
457C3⋊C8—23no—
458C24—33no—
459#41—64no—
460#41—612no—
461#41—66no—
462#41—66no—
463#41#23714yes—
464#41#23914yes—
465C2×Q8—33no—
466C2×Q8—64no—
467C2×Q8C2×C4×D311yescharacteristic
468C2×D4—36no—
469C2×D4—33no—
470C2×D4—612no—
471C2×D4C2×C4×D313yes—
472C22×C4—33no—
473C22×C4—33no—
474C22×C4—33no—
475C22×C4—33no—
476C22×C4—33no—
477C22×C4—33no—
478C22×C4—31no—
479C22×C4—36no—
480C22×C4—36no—
481C22×C4—66no—
482C22×C4—66no—
483C22×C4C2×C4×D313yes—
484C22×C4C2×C4×D313yes—
485C22×C4C23×D313yes—
486M16—26no—
487M16—26no—
488M16—66no—
489M16—66no—
490C2×C8—26no—
491C2×C8—26no—
492C2×C8—31no—
493C2×C8—33no—
494C2×C8—33no—
495C2×C8—31no—
496C2×C8—66no—
497C2×C8—66no—
498C2×C8C23×D311yescharacteristic
499C2×C8C23×D313yes—
500C2×C8C23×D313yes—
501C2×C8C23×D311yescharacteristic
502C4⋊C4—33no—
503C4⋊C4—33no—
504C4⋊C4—31no—
505C4⋊C4—31no—
506C4⋊C4—33no—
507C4⋊C4—33no—
508C4⋊C4—33no—
509C4⋊C4—33no—
510C4⋊C4—33no—
511C4⋊C4—33no—
512C4⋊C4C2×C4×D313yes—
513C4⋊C4C2×C4×D313yes—
514C22⋊C4—36no—
515C22⋊C4—33no—
516C22⋊C4—33no—
517C22⋊C4—31no—
518C22⋊C4—31no—
519C22⋊C4—33no—
520C22⋊C4—33no—
521C22⋊C4—33no—
522C22⋊C4—33no—
523C22⋊C4C2×C4×D316yes—
524C42—33no—
525C42—33no—
526C42—36no—
527C42—36no—
528C42—31no—
529C42—33no—
530C42C2×C4×D313yes—
531C42C2×C4×D313yes—
532C42C23×D313yes—
533C42C23×D311yescharacteristic
534C22×C3—23no—
535C22×C3—23no—
536C22×C3C2×#11716yes—
537C22×C3C24×C411yescharacteristic
538C2×D3—21no—
539C2×D3—23no—
540C2×D3—23no—
541C2×D3—23no—
542C2×D3—21no—
543C2×D3—21no—
544C2×D3—86no—
545C3×C4—23no—
546C3×C4—23no—
547C3×C4—21no—
548C3×C4—21no—
549C3×C4C2×#11716yes—
550C3×C4C22×#4111yescharacteristic
551C3×C4C22×#4111yescharacteristic
552Dic3—23no—
553Dic3—21no—
554Dic3—23no—
555Dic3—23no—
556Dic3—21no—
557Dic3—21no—
558C23—31no—
559C23—33no—
560C23—66no—
561C23—123no—
562C23—123no—
563C23C22×C4×D313yes—
564Q8—66no—
565Q8#89214yes—
566D4—612no—
567D4—66no—
568D4#892112yes—
569C2×C4—33no—
570C2×C4—31no—
571C2×C4—33no—
572C2×C4—31no—
573C2×C4—33no—
574C2×C4—31no—
575C2×C4—43no—
576C2×C4—43no—
577C2×C4—66no—
578C2×C4—61no—
579C2×C4—63no—
580C2×C4—61no—
581C2×C4—63no—
582C2×C4—66no—
583C2×C4—66no—
584C2×C4—126no—
585C2×C4—126no—
586C2×C4—123no—
587C2×C4—123no—
588C2×C4C22×C4×D313yes—
589C2×C4C22×C4×D313yes—
590C2×C4C22×C4×D311yescharacteristic
591C2×C4C22×C4×D311yescharacteristic
592C2×C4C22×C4×D313yes—
593C2×C4C2×#23716yes—
594C2×C4C2×#23916yes—
595C2×C4C24×D311yescharacteristic
596C8—21no—
597C8—21no—
598C8—23no—
599C8—23no—
600C8—61no—
601C8—61no—
602C8—63no—
603C8—63no—
604C2×C3—26no—
605C2×C3C22×#11711yescharacteristic
606C2×C3#357311yescharacteristic
607C2×C3#357311yescharacteristic
608D3—43no—
609D3—41no—
610C22—23no—
611C22—23no—
612C22—63no—
613C22—61no—
614C22—63no—
615C22—61no—
616C22—61no—
617C22—63no—
618C22—246no—
619C22C2×#89216yes—
620C22C23×C4×D311yescharacteristic
621C4—21no—
622C4—23no—
623C4—21no—
624C4—23no—
625C4—63no—
626C4—61no—
627C4—63no—
628C4—61no—
629C4—61no—
630C4—63no—
631C4C2×#89216yes—
632C4C22×#23711yescharacteristic
633C4C22×#23911yescharacteristic
634C3#6012911yescharacteristic, complemented, Sylow
635C2—26no—
636C2—121no—
637C2—123no—
638C2C22×#89211yescharacteristic
639C2#8716011yescharacteristic
640C2#8716611yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D3
commutator, socleC22×C3C24×C4
FittingC3×#2906C2
prime radicalC2×D3×#41C22
cosocleC2×C3×C4C25
hypercenter#2906D3
hypocenterC3#60129
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #2906

Complements

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

NH ≅ G/Ntype
2. C3×#2906C2semidirect
5. #83782C2semidirect
6. Dic3×#131C2semidirect
7. #82512C2semidirect
8. #79072C2semidirect
9. #79072C2semidirect
10. #79071C2semidirect
11. #79070C2semidirect
12. #79040C2semidirect
13. #79039C2semidirect
14. #79000C2semidirect
15. C2×#4840C2semidirect
36. C3×#424C22semidirect
37. C3×#424C22semidirect
38. C3×#423C22semidirect
39. C2×C3×C4⋊C8C22semidirect
44. #5122C22semidirect
45. #5122C22semidirect
46. Dic3×M16C22semidirect
47. Dic3×M16C22semidirect
52. #5101C22semidirect
53. #5101C22semidirect
54. C2×#815C22semidirect
55. C2×C8×Dic3C22semidirect
56. #4999C22semidirect
57. #4999C22semidirect
58. #4924C22semidirect
59. C2×C4⋊(C3⋊C8)C22semidirect
65. #4840C22semidirect
66. #4840C22semidirect
67. #4840C22semidirect
68. #4840C22semidirect
69. #4840C22semidirect
70. #4840C22semidirect
71. #4835C22semidirect
72. #4835C22semidirect
73. #4833C22semidirect
74. #4833C22semidirect
85. #2906D3semidirect
220. #424C2×D3semidirect
221. #424C2×D3semidirect
223. #423C2×D3semidirect
225. C2×C4⋊C8C2×D3semidirect
634. C3#60129semidirect

Maximal subgroups

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

subgroupindex
#858342
C3×#29062
#837932
#837822
Dic3×#1312
#825122
#790722
#790712
#790702
#790402
#790392
#790002
C2×#48402
#601293