Finite Group Databasebeta

← #10798404

Subgroups — #10798404

Order 768.

800 conjugacy classes of subgroups.

The subgroup lattice has 682 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#78818C211yesmaximal, characteristic
4C3×#1866C211yesmaximal, characteristic, complemented
5#72010C211yesmaximal, characteristic
6#70699C211yesmaximal, characteristic, complemented
7Dic3×#108C211yesmaximal, characteristic, complemented
8#70534C211yesmaximal, characteristic, complemented
9#70377C211yesmaximal, characteristic
10Dic3×#104C211yesmaximal, characteristic, complemented
11#70292C211yesmaximal, characteristic, complemented
12#68738C211yesmaximal, characteristic, complemented
13#68728C211yesmaximal, characteristic, complemented
14#68646C211yesmaximal, characteristic, complemented
15#68596C211yesmaximal, characteristic, complemented
16#21137—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×C8×Dic3—21no—
40C2×C8×Dic3C2211yescharacteristic
41C2×C8×Dic3C2211yescharacteristic
42#4999C2211yescharacteristic
43#4998C2211yescharacteristic
44C42×Dic3C2211yescharacteristic
45C2×#805C2211yescharacteristic
46#4840—21no—
47#4840—21no—
48#4833—21no—
49#4833—21no—
50#4790C411yescharacteristic, complemented
51#4790C411yescharacteristic, complemented
52#4788C411yescharacteristic, complemented
53#4788C411yescharacteristic, complemented
54#4758C411yescharacteristic, complemented
55#4758C411yescharacteristic, complemented
56#4756C411yescharacteristic, complemented
57#4756C411yescharacteristic, complemented
58C3×#339C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#4558C2211yescharacteristic
61#4558C2211yescharacteristic
62#4555C2211yescharacteristic
63#4550C2211yescharacteristic
64#4550C2211yescharacteristic
65#4532C2211yescharacteristic
66#4524C2211yescharacteristic
67#4523C2211yescharacteristic
68C4×#135—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
79C4×#108—31no—
80C4×#104—31no—
81C4×#104—31no—
82#1744—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×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3C2311yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C2×#208—21no—
117C2×#208C2311yescharacteristic
118C2×#208C2311yescharacteristic
119C22×C3⋊C8—21no—
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122#893C2×C411yescharacteristic
123#893C2×C411yescharacteristic
124#887C2×C411yescharacteristic
125#887C2×C411yescharacteristic
126#67⋊C4—21no—
127#67⋊C4—21no—
128#872—21no—
129C42×D3—21no—
130C3×#108—22no—
131C3×C4⋊C8—21no—
132C3×C4⋊C8—21no—
133C3×#104—22no—
134C3×#103C2×C411yescharacteristic
135C3×#103C2×C411yescharacteristic
136C3×#102C2×C411yescharacteristic
137C3×#102C2×C411yescharacteristic
138#837—22no—
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—
153C8×Dic3—21no—
154C8×Dic3—21no—
155C8×Dic3—21no—
156C8×Dic3—21no—
157C8×Dic3C2×C411yescharacteristic
158C8×Dic3C2×C411yescharacteristic
159C8×Dic3C2×C411yescharacteristic
160C8×Dic3C2×C411yescharacteristic
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—
177C4×Q16—31no—
178C4×Q16—31no—
179C4×SD16—31no—
180C4×SD16—31no—
181C4×SD16—31no—
182C4×SD16—31no—
183C4×D8—31no—
184C4×D8—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×C4×C8—31no—
210C2×C4×C8—31no—
211C2×C4×C8—61no—
212C43—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—24no—
286C4×Dic3—21no—
287C4×Dic3—21no—
288C4×Dic3—21no—
289C4×Dic3—21no—
290C4×Dic3—21no—
291C4×Dic3—21no—
292C4×Dic3—21no—
293C4×Dic3—42no—
294C4×Dic3C22×C411yescharacteristic
295C4×Dic3C22×C411yescharacteristic
296C4×Dic3C2×D411yescharacteristic
297C4×Dic3C2×D411yescharacteristic
298#208—24no—
299#208—21no—
300#208—21no—
301#208—21no—
302#208#4112yes—
303C2×C3⋊C8—24no—
304C2×C3⋊C8—21no—
305C2×C3⋊C8—21no—
306C2×C3⋊C8—21no—
307C2×C3⋊C8—21no—
308C2×C3⋊C8—21no—
309C2×C3⋊C8—21no—
310C2×C3⋊C8C22×C411yescharacteristic
311C2×C3⋊C8C22×C411yescharacteristic
312C2×C3⋊C8C2×D411yescharacteristic
313C2×C3⋊C8C2×D411yescharacteristic
314C2×#41—31no—
315C2×#41C22×D311yescharacteristic
316#135—62no—
317#135—62no—
318C2×Q16—31no—
319C2×SD16—31no—
320C2×SD16—31no—
321C2×D8—31no—
322#131—22no—
323#131—22no—
324#131—62no—
325#131—62no—
326C2×M16—21no—
327C2×M16—31no—
328C2×M16—31no—
329C2×M16—61no—
330C2×M16C22×D311yescharacteristic
331C2×M16C22×D311yescharacteristic
332C22×C8—21no—
333C22×C8—31no—
334C22×C8—31no—
335C22×C8—61no—
336C22×C8C22×D311yescharacteristic
337C22×C8C22×D311yescharacteristic
338C4×Q8—31no—
339C4×Q8—31no—
340C4×Q8—31no—
341C4×Q8—31no—
342C4×D4—31no—
343C4×D4—31no—
344C4×D4—31no—
345C4×D4—31no—
346C4×D4—61no—
347C4×D4—61no—
348C4×D4—61no—
349C4×D4—61no—
350#117—31no—
351#117—61no—
352#117—61no—
353#117C22×D311yescharacteristic
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—61no—
357C2×C42—61no—
358C2×C42—61no—
359C2×C42—61no—
360C2×C42C22×D311yescharacteristic
361#108—22no—
362#108—62no—
363#107—31no—
364#107—31no—
365#106—31no—
366#106—31no—
367C4⋊C8—21no—
368C4⋊C8—21no—
369C4⋊C8—61no—
370C4⋊C8—61no—
371#104—22no—
372#104—62no—
373#104—62no—
374#104—62no—
375#103—31no—
376#103—31no—
377#103C4×D311yescharacteristic
378#103C4×D311yescharacteristic
379#102—31no—
380#102—31no—
381#102C4×D311yescharacteristic
382#102C4×D311yescharacteristic
383C22⋊C8—61no—
384C22⋊C8—61no—
385C22⋊C8—61no—
386C22⋊C8—61no—
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—31no—
396C4×C8—31no—
397C4×C8—31no—
398C4×C8—31no—
399C4×C8—61no—
400C4×C8—61no—
401C4×C8—61no—
402C4×C8—61no—
403C23×C3—21no—
404C23×C3C2×#4111yescharacteristic
405C22×D3—21no—
406C3×Q8—21no—
407C3×Q8#11712yes—
408C3×D4—22no—
409C3×D4—22no—
410C3×D4—21no—
411C3×D4#11712yes—
412C2×C3×C4—21no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—21no—
417C2×C3×C4—21no—
418C2×C3×C4—41no—
419C2×C3×C4—41no—
420C2×C3×C4C23×C411yescharacteristic
421C2×C3×C4C22×D411yescharacteristic
422C2×C3×C4C2×#4111yescharacteristic
423C2×C3×C4C2×#4111yescharacteristic
424C2×C3×C4C2×#4111yescharacteristic
425C2×C3×C4C2×#4111yescharacteristic
426#67—41no—
427#67—41no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—21no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—42no—
437C2×Dic3—42no—
438C2×Dic3C4×D411yescharacteristic
439C2×Dic3C4×D411yescharacteristic
440C2×Dic3C4×D411yescharacteristic
441C2×Dic3C4×D411yescharacteristic
442D12—21no—
443D12—21no—
444C4×D3—41no—
445C4×D3—41no—
446C4×D3—42no—
447C4×D3—42no—
448Dic6—21no—
449Dic6—21no—
450C3×C8—21no—
451C3×C8—21no—
452C3×C8—21no—
453C3×C8—22no—
454C3×C8—21no—
455C3×C8—41no—
456C3⋊C8—21no—
457C3⋊C8—21no—
458C3⋊C8—21no—
459C3⋊C8—22no—
460C3⋊C8—21no—
461C3⋊C8—41no—
462#41—21no—
463#41—21no—
464#41—61no—
465#41—61no—
466#41—61no—
467#41—61no—
468#41#23712yes—
469#41#23912yes—
470C2×Q8—31no—
471C2×Q8C2×C4×D311yescharacteristic
472C2×D4—21no—
473C2×D4—31no—
474C2×D4—61no—
475C2×D4C2×C4×D311yescharacteristic
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—61no—
486C22×C4C23×D311yescharacteristic
487Q16—62no—
488SD16—62no—
489SD16—62no—
490D8—62no—
491M16—24no—
492M16—21no—
493M16—21no—
494M16—21no—
495M16—32no—
496M16—64no—
497M16—61no—
498M16—61no—
499M16—61no—
500M16#23712yes—
501C2×C8—24no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—31no—
509C2×C8—31no—
510C2×C8—31no—
511C2×C8—31no—
512C2×C8—64no—
513C2×C8—61no—
514C2×C8—61no—
515C2×C8—61no—
516C2×C8—61no—
517C2×C8—61no—
518C2×C8—61no—
519C2×C8C2×C4×D311yescharacteristic
520C2×C8C2×C4×D311yescharacteristic
521C2×C8D3×D411yescharacteristic
522C2×C8D3×D411yescharacteristic
523C4⋊C4—31no—
524C4⋊C4—31no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C4⋊C4C2×C4×D311yescharacteristic
530C4⋊C4C2×C4×D311yescharacteristic
531C22⋊C4—21no—
532C22⋊C4—61no—
533C22⋊C4—61no—
534C22⋊C4—61no—
535C22⋊C4—61no—
536C22⋊C4—61no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—31no—
541C42—31no—
542C42—31no—
543C42—31no—
544C42—61no—
545C42—61no—
546C42—61no—
547C42—61no—
548C42—64no—
549C42—61no—
550C42—61no—
551C42—61no—
552C42—122no—
553C22×C3—21no—
554C22×C3—21no—
555C22×C3—22no—
556C22×C3—41no—
557C22×C3—41no—
558C22×C3C2×C4×D411yescharacteristic
559C22×C3#52911yescharacteristic
560C22×C3#52911yescharacteristic
561C2×D3—41no—
562C2×D3—41no—
563C2×D3—41no—
564C3×C4—22no—
565C3×C4—42no—
566C3×C4—41no—
567C3×C4—41no—
568C3×C4C2×#11711yescharacteristic
569C3×C4C4×#4111yescharacteristic
570C3×C4#52511yescharacteristic
571C3×C4#52511yescharacteristic
572Dic3—21no—
573Dic3—21no—
574Dic3—42no—
575Dic3—41no—
576Dic3—41no—
577Dic3—41no—
578Dic3#44312yes—
579Dic3#44412yes—
580C23—21no—
581C23—61no—
582C23D3×#4111yescharacteristic
583Q8—21no—
584Q8—61no—
585Q8—61no—
586Q8#89212yes—
587D4—22no—
588D4—22no—
589D4—21no—
590D4—61no—
591D4—61no—
592D4—121no—
593D4—121no—
594D4#89212yes—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—31no—
602C2×C4—31no—
603C2×C4—31no—
604C2×C4—31no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—122no—
614C2×C4—122no—
615C2×C4—122no—
616C2×C4—122no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4C22×C4×D311yescharacteristic
622C2×C4C2×D3×D411yescharacteristic
623C2×C4C2×#23711yescharacteristic
624C2×C4C2×#23711yescharacteristic
625C2×C4C2×#23911yescharacteristic
626C2×C4D3×#4111yescharacteristic
627C8—21no—
628C8—21no—
629C8—21no—
630C8—22no—
631C8—21no—
632C8—41no—
633C8—61no—
634C8—61no—
635C8—61no—
636C8—61no—
637C8—62no—
638C8—121no—
639C2×C3—21no—
640C2×C3—21no—
641C2×C3—42no—
642C2×C3#229011yescharacteristic
643C2×C3C2×#44311yescharacteristic
644C2×C3C2×#44411yescharacteristic
645D3—81no—
646C22—21no—
647C22—21no—
648C22—22no—
649C22—41no—
650C22—41no—
651C22—121no—
652C22—121no—
653C22—121no—
654C22C4×D3×D411yescharacteristic
655C22#562411yescharacteristic
656C22#564911yescharacteristic
657C4—22no—
658C4—32no—
659C4—32no—
660C4—42no—
661C4—41no—
662C4—41no—
663C4—61no—
664C4—61no—
665C4—122no—
666C4—121no—
667C4—121no—
668C4—121no—
669C4C2×#89211yescharacteristic
670C4C4×#23711yescharacteristic
671C4#558611yescharacteristic
672C4#562711yescharacteristic
673C3#2113711yescharacteristic, complemented, Sylow
674C2—21no—
675C2—21no—
676C2—42no—
677C2—241no—
678C2#7437711yescharacteristic
679C2D3×#44311yescharacteristic
680C2D3×#44411yescharacteristic
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#21137
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. Dic3×#108C2semidirect
8. #70534C2semidirect
10. Dic3×#104C2semidirect
11. #70292C2semidirect
12. #68738C2semidirect
13. #68728C2semidirect
14. #68646C2semidirect
15. #68596C2semidirect
50. #4790C4semidirect
51. #4790C4semidirect
52. #4788C4semidirect
53. #4788C4semidirect
54. #4758C4semidirect
55. #4758C4semidirect
56. #4756C4semidirect
57. #4756C4semidirect
78. #1866D3semidirect
673. C3#21137semidirect

Maximal subgroups

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

subgroupindex
Dic3×#1312
#790702
#788182
C3×#18662
#720102
#706992
Dic3×#1082
#705342
#703772
Dic3×#1042
#702922
#687382
#687282
#686462
#685962
#211373