Finite Group Databasebeta

← #10798417

Subgroups — #10798417

Order 768.

772 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79166C211yesmaximal, characteristic
2#79070C211yesmaximal, characteristic, complemented
3#78549C211yesmaximal, characteristic, complemented
4C3×#1922C211yesmaximal, characteristic, complemented
5#70661C211yesmaximal, characteristic, complemented
6#70545C211yesmaximal, characteristic, complemented
7#69900C211yesmaximal, characteristic, complemented
8#68946C211yesmaximal, characteristic
9#68917C211yesmaximal, characteristic, complemented
10#68774C211yesmaximal, characteristic, complemented
11#68744C211yesmaximal, characteristic, complemented
12#68729C211yesmaximal, characteristic, complemented
13#68729C211yesmaximal, characteristic, complemented
14#68695C211yesmaximal, characteristic
15#68598C211yesmaximal, characteristic, complemented
16#21138—31nomaximal, Sylow
17C2×#907C2211yescharacteristic
18C2×#904C2211yescharacteristic
19C2×#901C2211yescharacteristic
20#5522C2211yescharacteristic
21#5522C2211yescharacteristic
22C3×#427C2211yescharacteristic
23C3×#424C2211yescharacteristic
24C3×#405C2211yescharacteristic
25C2×#825C2211yescharacteristic
26C2×#825C2211yescharacteristic
27#5129C2211yescharacteristic
28C2×#822C2211yescharacteristic
29C2×#822C2211yescharacteristic
30Dic3×M16C2211yescharacteristic
31Dic3×M16C2211yescharacteristic
32#5110C2211yescharacteristic
33C2×#815C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35#5002C2211yescharacteristic
36#5002C2211yescharacteristic
37#4999C2211yescharacteristic
38#4997C2211yescharacteristic
39#4932C2211yescharacteristic
40C2×#805C2211yescharacteristic
41C2×#805C2211yescharacteristic
42#4889C411yescharacteristic, complemented
43#4889C411yescharacteristic, complemented
44#4889C411yescharacteristic, complemented
45#4889C411yescharacteristic, complemented
46#4869C411yescharacteristic, complemented
47#4869C411yescharacteristic, complemented
48#4853C411yescharacteristic, complemented
49#4853C411yescharacteristic, complemented
50#4840—21no—
51#4840—21no—
52#4833—21no—
53#4833—21no—
54#4708—21no—
55#4707—21no—
56#4689—21no—
57C4×D3×C8—21no—
58C3×#341C2211yescharacteristic
59C3×#339C2211yescharacteristic
60C3×#339C2211yescharacteristic
61C3×#336C2211yescharacteristic
62#4552C2211yescharacteristic
63#4550C2211yescharacteristic
64#4550C2211yescharacteristic
65#4532C2211yescharacteristic
66#4532C2211yescharacteristic
67#4524C2211yescharacteristic
68#2943—31no—
69#2922—31no—
70C4×#131—31no—
71#1929—31no—
72#1929—31no—
73#1922—31no—
74#1922D311yescharacteristic, complemented
75#1876—31no—
76#1876—31no—
77#1866—31no—
78#1866—31no—
79#1778—31no—
80#1759—31no—
81#1759—31no—
82#1746—31no—
83C2×#235C2311yescharacteristic
84C2×#235C2311yescharacteristic
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×#117C2311yescharacteristic
90C3×#117C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C4×Dic3C2311yescharacteristic
93C2×C4×Dic3C2311yescharacteristic
94C2×#208C2311yescharacteristic
95C2×#208C2311yescharacteristic
96C22×C3⋊C8C2311yescharacteristic
97C22×C3⋊C8C2311yescharacteristic
98#907—22no—
99#907—22no—
100C2×Dic12C2×C411yescharacteristic
101#904—22no—
102#904—22no—
103C2×D24C2×C411yescharacteristic
104C2×#204C2×C411yescharacteristic
105C2×#204C2×C411yescharacteristic
106#901—22no—
107#901—22no—
108C2×ΓL(1,𝔽25)—21no—
109C2×ΓL(1,𝔽25)—21no—
110C2×D3×C8—21no—
111C2×D3×C8—21no—
112#893C2×C411yescharacteristic
113#893C2×C411yescharacteristic
114#893C2×C411yescharacteristic
115#893C2×C411yescharacteristic
116#887C2×C411yescharacteristic
117#887C2×C411yescharacteristic
118#887C2×C411yescharacteristic
119#887C2×C411yescharacteristic
120#67⋊C4—21no—
121#67⋊C4—21no—
122#67⋊C4—21no—
123#67⋊C4—21no—
124#872—21no—
125#872—21no—
126C42×D3—21no—
127C42×D3—21no—
128C3×#107C2×C411yescharacteristic
129C3×#107C2×C411yescharacteristic
130C3×#106C2×C411yescharacteristic
131C3×#106C2×C411yescharacteristic
132C3×C4⋊C8—21no—
133C3×C4⋊C8—21no—
134C3×#97—21no—
135C3×C4×C8—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138#825—22no—
139#825—22no—
140#822—22no—
141#822—22no—
142(C2×D3)⋊C8—21no—
143(C2×D3)⋊C8—21no—
144(C2×D3)⋊C8—21no—
145(C2×D3)⋊C8—21no—
146#815—21no—
147#815—21no—
148#815C2×C411yescharacteristic
149#815C2×C411yescharacteristic
150#815C2×C411yescharacteristic
151#815C2×C411yescharacteristic
152C8×Dic3—21no—
153C8×Dic3—21no—
154C8×Dic3D411yescharacteristic
155C8×Dic3D411yescharacteristic
156C8×Dic3D411yescharacteristic
157C8×Dic3D411yescharacteristic
158#810C2×C411yescharacteristic
159#810C2×C411yescharacteristic
160#810C2×C411yescharacteristic
161#810C2×C411yescharacteristic
162#809C2×C411yescharacteristic
163#809C2×C411yescharacteristic
164#809C2×C411yescharacteristic
165#809C2×C411yescharacteristic
166#805—22no—
167#805—22no—
168C4⋊(C3⋊C8)—21no—
169C4⋊(C3⋊C8)—21no—
170#803—21no—
171C4×C3⋊C8—21no—
172C2×#135—31no—
173C2×#131—31no—
174C2×#131—31no—
175C4×#41—31no—
176C4×#41—31no—
177#442—31no—
178#442—31no—
179#441—31no—
180#441—31no—
181#440—31no—
182#440—31no—
183#440—31no—
184#440—31no—
185#433—61no—
186#433—61no—
187#431—61no—
188#431—61no—
189C2×#108—31no—
190C2×#108—31no—
191#427C2×D311yescharacteristic
192#424—31no—
193#424C2×D311yescharacteristic
194C2×#104—31no—
195C2×#104—31no—
196C2×#104—31no—
197C2×#104—31no—
198#416—31no—
199#416—31no—
200#408—31no—
201#408—31no—
202#405—31no—
203#405—61no—
204#405—61no—
205#405C2×D311yescharacteristic
206C4×M16—31no—
207C4×M16—31no—
208C4×M16—61no—
209C2×#97—31no—
210C2×C4×C8—31no—
211C2×C4×C8—61no—
212#376—31no—
213#341—31no—
214#341C2×D311yescharacteristic
215#339—31no—
216#339—31no—
217#339C2×D311yescharacteristic
218#339C2×D311yescharacteristic
219#336—31no—
220#336—31no—
221#336—31no—
222#336C2×D311yescharacteristic
223C22×C3×C4—21no—
224C22×C3×C4C2411yescharacteristic
225C2×#67—21no—
226C2×#67—21no—
227C22×Dic3—21no—
228C22×Dic3#4111yescharacteristic
229C22×Dic3#4111yescharacteristic
230#235—21no—
231#235—21no—
232#235—21no—
233#235—21no—
234#235—21no—
235#235—21no—
236#235#4112yes—
237#235#4112yes—
238C2×D12C22×C411yescharacteristic
239C2×D12C22×C411yescharacteristic
240C2×C4×D3—21no—
241C2×C4×D3—21no—
242C2×C4×D3—21no—
243C2×C4×D3—21no—
244C2×C4×D3—21no—
245C2×C4×D3—21no—
246C2×Dic6C22×C411yescharacteristic
247C2×Dic6C22×C411yescharacteristic
248C3×M16—21no—
249C3×M16—21no—
250C3×M16—21no—
251C3×M16#4112yes—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8C22×C411yescharacteristic
259C2×C3×C8C22×C411yescharacteristic
260C2×C3×C8C2×D411yescharacteristic
261C2×C3×C8C2×D411yescharacteristic
262C3×C4⋊C4C22×C411yescharacteristic
263C3×C4⋊C4C22×C411yescharacteristic
264C3×C4⋊C4C22×C411yescharacteristic
265C3×C4⋊C4C22×C411yescharacteristic
266C3×C22⋊C4—21no—
267C3×C22⋊C4—21no—
268C3×C42—21no—
269C3×C42—21no—
270C3×C42—21no—
271C3×C42—21no—
272(C2×D3)⋊C4—21no—
273(C2×D3)⋊C4—21no—
274(C2×D3)⋊C4—21no—
275(C2×D3)⋊C4—21no—
276Dic3⋊C4—21no—
277Dic3⋊C4—21no—
278Dic3⋊C4—21no—
279Dic3⋊C4—21no—
280C4×Dic3—21no—
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3—21no—
284C4×Dic3—21no—
285C4×Dic3—21no—
286C4×Dic3C22×C411yescharacteristic
287C4×Dic3C22×C411yescharacteristic
288C4×Dic3C2×D411yescharacteristic
289C4×Dic3C2×D411yescharacteristic
290#208—21no—
291#208—21no—
292#208—21no—
293#208#4112yes—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299C2×C3⋊C8—21no—
300C2×C3⋊C8C22×C411yescharacteristic
301C2×C3⋊C8C22×C411yescharacteristic
302C2×C3⋊C8C2×D411yescharacteristic
303C2×C3⋊C8C2×D411yescharacteristic
304Dic12—22no—
305D24—22no—
306#204—22no—
307#204—22no—
308ΓL(1,𝔽25)—24no—
309ΓL(1,𝔽25)—24no—
310D3×C8—24no—
311D3×C8—24no—
312C2×#41—31no—
313C2×#41—31no—
314#135—62no—
315#135—62no—
316C2×Q16—31no—
317C2×SD16—31no—
318C2×SD16—31no—
319C2×D8—31no—
320#131—62no—
321#131—62no—
322#131—62no—
323#131—62no—
324C2×M16—31no—
325C2×M16—31no—
326C2×M16—61no—
327C2×M16—61no—
328C2×M16C22×D311yescharacteristic
329C2×M16C22×D311yescharacteristic
330C22×C8—31no—
331C22×C8—31no—
332C22×C8—61no—
333C22×C8—61no—
334C22×C8C22×D311yescharacteristic
335C22×C8C22×D311yescharacteristic
336C4×Q8—31no—
337C4×Q8—31no—
338C4×Q8—31no—
339C4×Q8—31no—
340C4×D4—31no—
341C4×D4—31no—
342C4×D4—31no—
343C4×D4—31no—
344C4×D4—61no—
345C4×D4—61no—
346C4×D4—61no—
347C4×D4—61no—
348#117—61no—
349#117—61no—
350#117C22×D311yescharacteristic
351#117C22×D311yescharacteristic
352C2×C42—31no—
353C2×C42—31no—
354C2×C42—61no—
355C2×C42—61no—
356C2×C42C22×D311yescharacteristic
357#108—62no—
358#108—62no—
359#107C4×D311yescharacteristic
360#107C4×D311yescharacteristic
361#106C4×D311yescharacteristic
362#106C4×D311yescharacteristic
363C4⋊C8—21no—
364C4⋊C8—21no—
365C4⋊C8—61no—
366C4⋊C8—61no—
367#104—62no—
368#104—62no—
369#104—62no—
370#104—62no—
371#103—31no—
372#103—31no—
373#103—31no—
374#103—31no—
375#102—31no—
376#102—31no—
377#102—31no—
378#102—31no—
379C22⋊C8—61no—
380C22⋊C8—61no—
381C22⋊C8—61no—
382C22⋊C8—61no—
383#97—21no—
384#97—31no—
385#97—31no—
386#97—31no—
387#97—31no—
388#97—61no—
389#97—61no—
390#97—61no—
391C4×C8—21no—
392C4×C8—31no—
393C4×C8—31no—
394C4×C8—31no—
395C4×C8—31no—
396C4×C8—61no—
397C4×C8—61no—
398C4×C8—61no—
399#95—61no—
400#95—61no—
401C23×C3C2×#4111yescharacteristic
402C22×D3—21no—
403C22×D3—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—21no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4C23×C411yescharacteristic
412C2×C3×C4C22×D411yescharacteristic
413C2×C3×C4C2×#4111yescharacteristic
414C2×C3×C4C2×#4111yescharacteristic
415C2×C3×C4C2×#4111yescharacteristic
416C2×C3×C4C2×#4111yescharacteristic
417#67—22no—
418#67—22no—
419#67—41no—
420#67—41no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3C4×D411yescharacteristic
429C2×Dic3C4×D411yescharacteristic
430C2×Dic3C4×D411yescharacteristic
431C2×Dic3C4×D411yescharacteristic
432D12—21no—
433D12—21no—
434D12—21no—
435D12#11712yes—
436C4×D3—22no—
437C4×D3—22no—
438C4×D3—42no—
439C4×D3—42no—
440C4×D3—42no—
441C4×D3—42no—
442C4×D3—41no—
443C4×D3—41no—
444Dic6—21no—
445Dic6—21no—
446Dic6—21no—
447Dic6#11712yes—
448C3×C8—21no—
449C3×C8—21no—
450C3×C8—21no—
451C3×C8—22no—
452C3×C8—21no—
453C3×C8—41no—
454C3⋊C8—21no—
455C3⋊C8—21no—
456C3⋊C8—21no—
457C3⋊C8—22no—
458C3⋊C8—21no—
459C3⋊C8—41no—
460#41—32no—
461#41—32no—
462#41—61no—
463#41—61no—
464#41—61no—
465#41—61no—
466#41—61no—
467#41—61no—
468C2×Q8—31no—
469C2×Q8—31no—
470C2×D4—31no—
471C2×D4—31no—
472C2×D4—61no—
473C2×D4—61no—
474C22×C4—21no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4C23×D311yescharacteristic
485Q16—62no—
486SD16—62no—
487SD16—62no—
488D8—62no—
489M16—21no—
490M16—21no—
491M16—21no—
492M16—32no—
493M16—64no—
494M16—64no—
495M16—61no—
496M16—61no—
497M16—61no—
498M16#23912yes—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—31no—
506C2×C8—31no—
507C2×C8—31no—
508C2×C8—31no—
509C2×C8—64no—
510C2×C8—64no—
511C2×C8—61no—
512C2×C8—61no—
513C2×C8—61no—
514C2×C8—61no—
515C2×C8—61no—
516C2×C8—61no—
517C2×C8C2×C4×D311yescharacteristic
518C2×C8C2×C4×D311yescharacteristic
519C2×C8D3×D411yescharacteristic
520C2×C8D3×D411yescharacteristic
521C4⋊C4—61no—
522C4⋊C4—61no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4C2×C4×D311yescharacteristic
526C4⋊C4C2×C4×D311yescharacteristic
527C4⋊C4C2×C4×D311yescharacteristic
528C4⋊C4C2×C4×D311yescharacteristic
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—61no—
532C22⋊C4—61no—
533C22⋊C4—61no—
534C22⋊C4—61no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—31no—
540C42—31no—
541C42—31no—
542C42—31no—
543C42—61no—
544C42—61no—
545C42—61no—
546C42—61no—
547C42—61no—
548C42—61no—
549C22×C3—21no—
550C22×C3—21no—
551C22×C3C2×C4×D411yescharacteristic
552C22×C3#52911yescharacteristic
553C22×C3#52911yescharacteristic
554C2×D3—22no—
555C2×D3—41no—
556C2×D3—41no—
557C2×D3—41no—
558C2×D3—41no—
559C2×D3—41no—
560C3×C4—41no—
561C3×C4—41no—
562C3×C4—41no—
563C3×C4—41no—
564C3×C4—81no—
565C3×C4C2×#11711yescharacteristic
566C3×C4C4×#4111yescharacteristic
567C3×C4#52511yescharacteristic
568C3×C4#52511yescharacteristic
569Dic3—21no—
570Dic3—21no—
571Dic3—21no—
572Dic3—21no—
573Dic3—22no—
574Dic3—41no—
575Dic3—81no—
576C23—61no—
577C23—61no—
578C23D3×#4111yescharacteristic
579Q8—32no—
580Q8—61no—
581Q8—61no—
582Q8—61no—
583D4—32no—
584D4—62no—
585D4—62no—
586D4—61no—
587D4—61no—
588D4—61no—
589D4—121no—
590D4—121no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—31no—
596C2×C4—31no—
597C2×C4—31no—
598C2×C4—31no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—61no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—61no—
606C2×C4—122no—
607C2×C4—122no—
608C2×C4—122no—
609C2×C4—122no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4C22×C4×D311yescharacteristic
618C2×C4C2×D3×D411yescharacteristic
619C2×C4C2×#23911yescharacteristic
620C2×C4D3×#4111yescharacteristic
621C2×C4D3×#4111yescharacteristic
622C2×C4D3×#4111yescharacteristic
623C8—21no—
624C8—21no—
625C8—21no—
626C8—21no—
627C8—22no—
628C8—41no—
629C8—62no—
630C8—61no—
631C8—61no—
632C8—61no—
633C8—61no—
634C8—121no—
635C2×C3—21no—
636C2×C3—21no—
637C2×C3#229011yescharacteristic
638C2×C3#295511yescharacteristic
639C2×C3#295511yescharacteristic
640D3—42no—
641D3—81no—
642C22—21no—
643C22—21no—
644C22—62no—
645C22—121no—
646C22—121no—
647C22—121no—
648C22—121no—
649C22—121no—
650C22C4×D3×D411yescharacteristic
651C22#569111yescharacteristic
652C22#570611yescharacteristic
653C4—41no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—61no—
658C4—61no—
659C4—61no—
660C4—61no—
661C4—62no—
662C4—81no—
663C4—121no—
664C4—241no—
665C4D3×#11711yescharacteristic
666C4C4×#23911yescharacteristic
667C4#562711yescharacteristic
668C4#565611yescharacteristic
669C3#2113811yescharacteristic, complemented, Sylow
670C2—21no—
671C2—21no—
672C2—122no—
673C2—241no—
674C2#7493711yescharacteristic
675C2#7988811yescharacteristic
676C2#7989711yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1922C2
socleC22×C3C2×C4×D4
prime radicalC2×#904C22
cosocleC22×C3×C4C24
hypercenter#1922D3
hypocenterC3#21138
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1922

Complements

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

NH ≅ G/Ntype
2. #79070C2semidirect
3. #78549C2semidirect
4. C3×#1922C2semidirect
5. #70661C2semidirect
6. #70545C2semidirect
7. #69900C2semidirect
9. #68917C2semidirect
10. #68774C2semidirect
11. #68744C2semidirect
12. #68729C2semidirect
13. #68729C2semidirect
15. #68598C2semidirect
42. #4889C4semidirect
43. #4889C4semidirect
44. #4889C4semidirect
45. #4889C4semidirect
46. #4869C4semidirect
47. #4869C4semidirect
48. #4853C4semidirect
49. #4853C4semidirect
74. #1922D3semidirect
669. C3#21138semidirect

Maximal subgroups

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

subgroupindex
#791662
#790702
#785492
C3×#19222
#706612
#705452
#699002
#689462
#689172
#687742
#687442
#687292
#686952
#685982
#211383