Finite Group Databasebeta

← #10798415

Subgroups — #10798415

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
1Dic3×#131C211yesmaximal, characteristic, complemented
2#79070C211yesmaximal, characteristic, complemented
3#78819C211yesmaximal, characteristic
4C3×#1866C211yesmaximal, characteristic, complemented
5#72010C211yesmaximal, characteristic
6#70699C211yesmaximal, characteristic, complemented
7#70692C211yesmaximal, characteristic, complemented
8#70534C211yesmaximal, characteristic, complemented
9#70377C211yesmaximal, characteristic
10#70354C211yesmaximal, characteristic, complemented
11#70292C211yesmaximal, characteristic, complemented
12#68738C211yesmaximal, characteristic, complemented
13#68729C211yesmaximal, characteristic, complemented
14#68646C211yesmaximal, characteristic, complemented
15#68598C211yesmaximal, characteristic, complemented
16#21138—31nomaximal, Sylow
17C2×C3×#131C2211yescharacteristic
18Dic3×#41C2211yescharacteristic
19C2×#950C2211yescharacteristic
20C2×#948C2211yescharacteristic
21#5522C2211yescharacteristic
22C2×C3×#108C2211yescharacteristic
23C3×#424C2211yescharacteristic
24C2×C3×#104C2211yescharacteristic
25C3×#416C2211yescharacteristic
26C2×#837C2211yescharacteristic
27C2×#825C2211yescharacteristic
28#5129C2211yescharacteristic
29#5122—21no—
30#5122C2211yescharacteristic
31Dic3×M16—21no—
32Dic3×M16C2211yescharacteristic
33Dic3×M16C2211yescharacteristic
34#5113C2211yescharacteristic
35#5110C2211yescharacteristic
36C2×#819C2211yescharacteristic
37#5101—21no—
38#5101C2211yescharacteristic
39C2×#815C2211yescharacteristic
40C2×C8×Dic3—21no—
41C2×C8×Dic3C2211yescharacteristic
42#4999C2211yescharacteristic
43#4998C2211yescharacteristic
44#4932C2211yescharacteristic
45C2×#805C2211yescharacteristic
46#4840—21no—
47#4840—21no—
48#4833—21no—
49#4833—21no—
50#4817C411yescharacteristic, complemented
51#4817C411yescharacteristic, complemented
52#4789C411yescharacteristic, complemented
53#4789C411yescharacteristic, complemented
54#4785C411yescharacteristic, complemented
55#4785C411yescharacteristic, complemented
56#4757C411yescharacteristic, complemented
57#4757C411yescharacteristic, complemented
58C3×#339C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#4558C2211yescharacteristic
61#4558C2211yescharacteristic
62#4555C2211yescharacteristic
63#4550C2211yescharacteristic
64#4550C2211yescharacteristic
65#4532C2211yescharacteristic
66#4524C2211yescharacteristic
67#4523C2211yescharacteristic
68#2943—31no—
69#2922—31no—
70C4×#131—31no—
71#1929—31no—
72#1929—31no—
73#1922—31no—
74#1922—31no—
75#1876—31no—
76#1876—31no—
77#1866—31no—
78#1866D311yescharacteristic, complemented
79#1778—31no—
80#1759—31no—
81#1759—31no—
82#1746—31no—
83C2×C3×#41C2311yescharacteristic
84C2×#235C2311yescharacteristic
85C3×#131—22no—
86C3×#131—22no—
87C2×C3×M16—21no—
88C2×C3×M16C2311yescharacteristic
89C2×C3×M16C2311yescharacteristic
90C22×C3×C8—21no—
91C22×C3×C8C2311yescharacteristic
92C22×C3×C8C2311yescharacteristic
93C3×#117C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95#950—22no—
96#950—22no—
97#948—22no—
98#948—22no—
99Q8×Dic3C2×C411yescharacteristic
100Q8×Dic3C2×C411yescharacteristic
101C2×#216C2×C411yescharacteristic
102C2×#215C2×C411yescharacteristic
103D4×Dic3—21no—
104D4×Dic3—21no—
105D4×Dic3C2×C411yescharacteristic
106D4×Dic3C2×C411yescharacteristic
107C2×#214C2×C411yescharacteristic
108C2×#213C2×C411yescharacteristic
109#926—21no—
110#926C2311yescharacteristic
111C2×C4×Dic3—21no—
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C2×#208—21no—
115C2×#208C2311yescharacteristic
116C2×#208C2311yescharacteristic
117C22×C3⋊C8—21no—
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120#893C2×C411yescharacteristic
121#893C2×C411yescharacteristic
122#887C2×C411yescharacteristic
123#887C2×C411yescharacteristic
124#67⋊C4—21no—
125#67⋊C4—21no—
126#872—21no—
127C42×D3—21no—
128C3×#108—22no—
129C3×C4⋊C8—21no—
130C3×C4⋊C8—21no—
131C3×#104—22no—
132C3×#103C2×C411yescharacteristic
133C3×#103C2×C411yescharacteristic
134C3×#102C2×C411yescharacteristic
135C3×#102C2×C411yescharacteristic
136#837—22no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139#825—22no—
140#821C2×C411yescharacteristic
141#821C2×C411yescharacteristic
142(C2×D3)⋊C8—21no—
143(C2×D3)⋊C8—21no—
144(C2×D3)⋊C8—21no—
145(C2×D3)⋊C8—21no—
146#819—22no—
147#816C2×C411yescharacteristic
148#816C2×C411yescharacteristic
149#815—21no—
150#815—21no—
151#815—21no—
152#815—21no—
153#815C2×C411yescharacteristic
154#815C2×C411yescharacteristic
155#815C2×C411yescharacteristic
156#815C2×C411yescharacteristic
157C8×Dic3—21no—
158C8×Dic3—21no—
159C8×Dic3—21no—
160C8×Dic3—21no—
161C8×Dic3D411yescharacteristic
162C8×Dic3D411yescharacteristic
163C8×Dic3D411yescharacteristic
164C8×Dic3D411yescharacteristic
165#808C2×C411yescharacteristic
166#808C2×C411yescharacteristic
167#807C2×C411yescharacteristic
168#807C2×C411yescharacteristic
169#805—22no—
170C4⋊(C3⋊C8)—21no—
171C4⋊(C3⋊C8)—21no—
172C2×#135—31no—
173C2×#131—31no—
174C2×#131C2×D311yescharacteristic
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×#108C2×D311yescharacteristic
191#427—31no—
192#424—31no—
193#424C2×D311yescharacteristic
194C2×#104—31no—
195C2×#104—31no—
196C2×#104—31no—
197C2×#104C2×D311yescharacteristic
198#416—31no—
199#416C2×D311yescharacteristic
200#408—31no—
201#408—31no—
202#405—31no—
203#405—31no—
204#405—61no—
205#405—61no—
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#341—31no—
215#339—31no—
216#339—31no—
217#339—31no—
218#339C2×D311yescharacteristic
219#336—31no—
220#336—31no—
221#336—31no—
222#336C2×D311yescharacteristic
223C3×#41—21no—
224C3×#41—21no—
225C3×#41#4112yes—
226C3×#41#4112yes—
227C2×C3×Q8C22×C411yescharacteristic
228C2×C3×D4—21no—
229C2×C3×D4C22×C411yescharacteristic
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4C2411yescharacteristic
233C2×#67—21no—
234C22×Dic3—21no—
235C22×Dic3—21no—
236C22×Dic3—21no—
237C22×Dic3#4111yescharacteristic
238C22×Dic3#4111yescharacteristic
239#235—21no—
240#235—21no—
241#235—21no—
242#235—21no—
243C2×D12C22×C411yescharacteristic
244C2×C4×D3—21no—
245C2×C4×D3—21no—
246C2×C4×D3—21no—
247C2×Dic6C22×C411yescharacteristic
248C3×M16—24no—
249C3×M16—21no—
250C3×M16—21no—
251C3×M16—21no—
252C3×M16#4112yes—
253C2×C3×C8—24no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C2×C3×C8C22×C411yescharacteristic
261C2×C3×C8C22×C411yescharacteristic
262C2×C3×C8C2×D411yescharacteristic
263C2×C3×C8C2×D411yescharacteristic
264C3×C4⋊C4C22×C411yescharacteristic
265C3×C4⋊C4C22×C411yescharacteristic
266C3×C22⋊C4—21no—
267C3×C42—21no—
268C3×C42—21no—
269C3×C42—21no—
270#217—21no—
271#217—21no—
272#217—21no—
273#216—22no—
274#215—22no—
275#214—22no—
276#213—22no—
277(C2×D3)⋊C4—21no—
278(C2×D3)⋊C4—21no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3C22×C411yescharacteristic
282C4⋊Dic3C22×C411yescharacteristic
283Dic3⋊C4—21no—
284Dic3⋊C4—21no—
285C4×Dic3—21no—
286C4×Dic3—21no—
287C4×Dic3—21no—
288C4×Dic3—21no—
289C4×Dic3—21no—
290C4×Dic3—21no—
291C4×Dic3—21no—
292C4×Dic3C22×C411yescharacteristic
293C4×Dic3C22×C411yescharacteristic
294C4×Dic3C2×D411yescharacteristic
295C4×Dic3C2×D411yescharacteristic
296#208—24no—
297#208—21no—
298#208—21no—
299#208—21no—
300#208#4112yes—
301C2×C3⋊C8—24no—
302C2×C3⋊C8—21no—
303C2×C3⋊C8—21no—
304C2×C3⋊C8—21no—
305C2×C3⋊C8—21no—
306C2×C3⋊C8—21no—
307C2×C3⋊C8—21no—
308C2×C3⋊C8C22×C411yescharacteristic
309C2×C3⋊C8C22×C411yescharacteristic
310C2×C3⋊C8C2×D411yescharacteristic
311C2×C3⋊C8C2×D411yescharacteristic
312C2×#41—31no—
313C2×#41C22×D311yescharacteristic
314#135—62no—
315#135—62no—
316C2×Q16—31no—
317C2×SD16—31no—
318C2×SD16—31no—
319C2×D8—31no—
320#131—22no—
321#131—22no—
322#131—62no—
323#131—62no—
324C2×M16—21no—
325C2×M16—31no—
326C2×M16—31no—
327C2×M16—61no—
328C2×M16C22×D311yescharacteristic
329C2×M16C22×D311yescharacteristic
330C22×C8—21no—
331C22×C8—31no—
332C22×C8—31no—
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—31no—
349#117—61no—
350#117—61no—
351#117C22×D311yescharacteristic
352C2×C42—31no—
353C2×C42—31no—
354C2×C42—61no—
355C2×C42—61no—
356C2×C42C22×D311yescharacteristic
357#108—22no—
358#108—62no—
359#107—31no—
360#107—31no—
361#106—31no—
362#106—31no—
363C4⋊C8—21no—
364C4⋊C8—21no—
365C4⋊C8—61no—
366C4⋊C8—61no—
367#104—22no—
368#104—62no—
369#104—62no—
370#104—62no—
371#103—31no—
372#103—31no—
373#103C4×D311yescharacteristic
374#103C4×D311yescharacteristic
375#102—31no—
376#102—31no—
377#102C4×D311yescharacteristic
378#102C4×D311yescharacteristic
379C22⋊C8—61no—
380C22⋊C8—61no—
381C22⋊C8—61no—
382C22⋊C8—61no—
383#97—31no—
384#97—31no—
385#97—31no—
386#97—31no—
387#97—61no—
388#97—61no—
389#97—61no—
390#97—61no—
391C4×C8—31no—
392C4×C8—31no—
393C4×C8—31no—
394C4×C8—31no—
395C4×C8—61no—
396C4×C8—61no—
397C4×C8—61no—
398C4×C8—61no—
399#95—61no—
400#95—61no—
401C23×C3—21no—
402C23×C3C2×#4111yescharacteristic
403C22×D3—21no—
404C3×Q8—21no—
405C3×Q8#11712yes—
406C3×D4—22no—
407C3×D4—22no—
408C3×D4—21no—
409C3×D4#11712yes—
410C2×C3×C4—21no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4C23×C411yescharacteristic
419C2×C3×C4C22×D411yescharacteristic
420C2×C3×C4C2×#4111yescharacteristic
421C2×C3×C4C2×#4111yescharacteristic
422C2×C3×C4C2×#4111yescharacteristic
423C2×C3×C4C2×#4111yescharacteristic
424#67—41no—
425#67—41no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—42no—
432C2×Dic3—42no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3C4×D411yescharacteristic
437C2×Dic3C4×D411yescharacteristic
438C2×Dic3C4×D411yescharacteristic
439C2×Dic3C4×D411yescharacteristic
440D12—21no—
441D12—21no—
442C4×D3—42no—
443C4×D3—42no—
444C4×D3—41no—
445C4×D3—41no—
446Dic6—21no—
447Dic6—21no—
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—21no—
461#41—21no—
462#41—61no—
463#41—61no—
464#41—61no—
465#41—61no—
466#41#23712yes—
467#41#23912yes—
468C2×Q8—31no—
469C2×Q8C2×C4×D311yescharacteristic
470C2×D4—21no—
471C2×D4—31no—
472C2×D4—61no—
473C2×D4C2×C4×D311yescharacteristic
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—31no—
477C22×C4—31no—
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—24no—
490M16—21no—
491M16—21no—
492M16—21no—
493M16—32no—
494M16—64no—
495M16—61no—
496M16—61no—
497M16—61no—
498M16#23712yes—
499C2×C8—24no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—31no—
507C2×C8—31no—
508C2×C8—31no—
509C2×C8—31no—
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—31no—
522C4⋊C4—31no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4C2×C4×D311yescharacteristic
528C4⋊C4C2×C4×D311yescharacteristic
529C22⋊C4—21no—
530C22⋊C4—61no—
531C22⋊C4—61no—
532C22⋊C4—61no—
533C22⋊C4—61no—
534C22⋊C4—61no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—31no—
539C42—31no—
540C42—31no—
541C42—31no—
542C42—61no—
543C42—61no—
544C42—61no—
545C42—61no—
546C42—61no—
547C42—61no—
548C42—61no—
549C22×C3—21no—
550C22×C3—21no—
551C22×C3—22no—
552C22×C3—41no—
553C22×C3—41no—
554C22×C3C2×C4×D411yescharacteristic
555C22×C3#52911yescharacteristic
556C22×C3#52911yescharacteristic
557C2×D3—41no—
558C2×D3—41no—
559C2×D3—41no—
560C3×C4—22no—
561C3×C4—41no—
562C3×C4—41no—
563C3×C4—81no—
564C3×C4C2×#11711yescharacteristic
565C3×C4C4×#4111yescharacteristic
566C3×C4#52511yescharacteristic
567C3×C4#52511yescharacteristic
568Dic3—21no—
569Dic3—21no—
570Dic3—21no—
571Dic3—21no—
572Dic3—41no—
573Dic3—41no—
574Dic3—41no—
575Dic3—81no—
576C23—21no—
577C23—61no—
578C23D3×#4111yescharacteristic
579Q8—21no—
580Q8—61no—
581Q8—61no—
582Q8#89212yes—
583D4—22no—
584D4—22no—
585D4—21no—
586D4—61no—
587D4—61no—
588D4—121no—
589D4—121no—
590D4#89212yes—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—31no—
597C2×C4—31no—
598C2×C4—31no—
599C2×C4—31no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—122no—
607C2×C4—122no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—122no—
611C2×C4—122no—
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×#23711yescharacteristic
620C2×C4C2×#23711yescharacteristic
621C2×C4C2×#23911yescharacteristic
622C2×C4D3×#4111yescharacteristic
623C8—21no—
624C8—21no—
625C8—21no—
626C8—22no—
627C8—21no—
628C8—41no—
629C8—61no—
630C8—61no—
631C8—61no—
632C8—61no—
633C8—62no—
634C8—121no—
635C2×C3—21no—
636C2×C3—21no—
637C2×C3—42no—
638C2×C3#229011yescharacteristic
639C2×C3#295511yescharacteristic
640C2×C3#295511yescharacteristic
641D3—81no—
642C22—21no—
643C22—21no—
644C22—22no—
645C22—41no—
646C22—41no—
647C22—121no—
648C22—121no—
649C22—121no—
650C22C4×D3×D411yescharacteristic
651C22#562411yescharacteristic
652C22#564911yescharacteristic
653C4—22no—
654C4—41no—
655C4—41no—
656C4—61no—
657C4—61no—
658C4—61no—
659C4—61no—
660C4—81no—
661C4—121no—
662C4—121no—
663C4—121no—
664C4—241no—
665C4C2×#89211yescharacteristic
666C4C4×#23711yescharacteristic
667C4#558611yescharacteristic
668C4#562711yescharacteristic
669C3#2113811yescharacteristic, complemented, Sylow
670C2—21no—
671C2—21no—
672C2—42no—
673C2—241no—
674C2#7437711yescharacteristic
675C2#7988811yescharacteristic
676C2#7989711yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1866C2
socleC22×C3C2×C4×D4
prime radicalC2×#950C22
cosocleC22×C3×C4C24
hypercenter#1866D3
hypocenterC3#21138
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1866

Complements

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

NH ≅ G/Ntype
1. Dic3×#131C2semidirect
2. #79070C2semidirect
4. C3×#1866C2semidirect
6. #70699C2semidirect
7. #70692C2semidirect
8. #70534C2semidirect
10. #70354C2semidirect
11. #70292C2semidirect
12. #68738C2semidirect
13. #68729C2semidirect
14. #68646C2semidirect
15. #68598C2semidirect
50. #4817C4semidirect
51. #4817C4semidirect
52. #4789C4semidirect
53. #4789C4semidirect
54. #4785C4semidirect
55. #4785C4semidirect
56. #4757C4semidirect
57. #4757C4semidirect
78. #1866D3semidirect
669. C3#21138semidirect

Maximal subgroups

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

subgroupindex
Dic3×#1312
#790702
#788192
C3×#18662
#720102
#706992
#706922
#705342
#703772
#703542
#702922
#687382
#687292
#686462
#685982
#211383