Finite Group Databasebeta

← #10701546

Subgroups — #10701546

Order 768.

1054 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5131C211yesmaximal, characteristic
2C22×#823C211yesmaximal, characteristic
3C2×#5122C211yesmaximal, characteristic, complemented
4C22×#809C211yesmaximal, characteristic
5C2×C3×#343C211yesmaximal, characteristic, complemented
6C2×#4558C211yesmaximal, characteristic, complemented
7#71558C211yesmaximal, characteristic, complemented
8#10943—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C23×C3⋊C8C2211yescharacteristic
16#5131—22no—
17#5131C414yescomplemented
18C2×#823—22no—
19C2×#823—22no—
20C2×#823C414yescomplemented
21#5122—22no—
22#5122—22no—
23C2×#821—22no—
24C2×#815—21no—
25C2×C8×Dic3—21no—
26C2×#809—22no—
27C2×#809—21no—
28C2×#809—21no—
29C2×#809C414yescomplemented
30C3×#343—22no—
31#4558—22no—
32#4529—22no—
33C2×#418—31no—
34C22×#102—31no—
35C22×#100—31no—
36C2×#405—31no—
37C2×#343D311yescharacteristic, complemented
38C2×#341—31no—
39#1736—31no—
40C24×D3—21no—
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C22×D12—21no—
44C22×D12—21no—
45C22×D12—21no—
46C22×D12—21no—
47C22×D12—21no—
48C22×D12—21no—
49C22×D12—21no—
50C22×D12—21no—
51C22×D12—21no—
52C22×D12—21no—
53C22×D12—21no—
54C22×D12C2×C412yes—
55C22×D12C2×C412yes—
56C22×D12C2×C412yes—
57C2×C3×M16—21no—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
63C2×C3×M16—21no—
64C2×C3×M16—21no—
65C2×C3×M16C2×C412yes—
66C2×C3×M16C2×C412yes—
67C2×C3×M16D412yes—
68C2×C3×M16D412yes—
69C22×C3×C8—21no—
70C22×C3×C8—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4C2×C412yes—
77C2×#217—21no—
78#926—21no—
79#926—21no—
80#926C2×C412yes—
81#926D412yes—
82C2×C4⋊Dic3D412yes—
83C2×C4×Dic3—21no—
84C2×#208—21no—
85C2×#208—21no—
86C2×#208—21no—
87C2×#208—21no—
88C2×#208—21no—
89C2×#208C2×C412yes—
90C22×C3⋊C8—21no—
91C22×C3⋊C8—21no—
92C22×C3⋊C8—21no—
93C22×C3⋊C8—21no—
94C22×C3⋊C8—21no—
95C22×C3⋊C8C2×C412yes—
96C22×C3⋊C8D412yes—
97C22×C3⋊C8D411yescharacteristic
98C22×C3⋊C8D411yescharacteristic
99#823—28no—
100#821—28no—
101#815—24no—
102C8×Dic3—24no—
103#809—24no—
104#809—24no—
105C23×D4—31no—
106C22×M16—31no—
107C22×M16C2×D311yescharacteristic
108C22×M16C2×D311yescharacteristic
109C23×C8—31no—
110C2×#117—31no—
111C22×C4⋊C4C2×D311yescharacteristic
112#418—34no—
113#418—62no—
114C2×#102—34no—
115C2×#102—62no—
116C2×#102—61no—
117C2×#102—61no—
118C2×#102—62no—
119C2×#100—34no—
120C2×#100—62no—
121C2×#100—62no—
122#405—62no—
123#405—62no—
124C2×#97—61no—
125C2×C4×C8—61no—
126#343—22no—
127#341—62no—
128#340—62no—
129C24×C3#4111yescharacteristic
130C23×D3—22no—
131C23×D3—22no—
132C23×D3—22no—
133C23×D3—41no—
134C23×D3—41no—
135C23×D3—41no—
136C23×D3—41no—
137C23×D3—82no—
138C23×D3—82no—
139C22×C3×C4—22no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4—82no—
143C22×C3×C4C22×C411yescharacteristic
144C22×C3×C4C22×C411yescharacteristic
145C22×C3×C4C22×C411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4#4111yescharacteristic
153C22×C3×C4#4111yescharacteristic
154C22×C3×C4#4111yescharacteristic
155C22×C3×C4#4111yescharacteristic
156C22×C3×C4#4111yescharacteristic
157C22×Dic3—22no—
158C2×D12—22no—
159C2×D12—22no—
160C2×D12—22no—
161C2×D12—22no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—22no—
172C2×D12—22no—
173C2×D12—41no—
174C2×D12—41no—
175C2×D12—41no—
176C2×D12—41no—
177C2×D12—41no—
178C2×D12—41no—
179C2×D12—41no—
180C2×D12—41no—
181C2×D12—41no—
182C2×D12—41no—
183C2×D12—41no—
184C2×D12—41no—
185C2×D12C4214yes—
186C3×M16—22no—
187C3×M16—22no—
188C3×M16—22no—
189C3×M16—22no—
190C3×M16—22no—
191C3×M16—41no—
192C3×M16—41no—
193C3×M16C22⋊C414yes—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—41no—
199C2×C3×C8—41no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—41no—
203C3×C4⋊C4—41no—
204#217—22no—
205#217—22no—
206C4⋊Dic3—22no—
207C4⋊Dic3C22⋊C414yes—
208C4×Dic3—22no—
209C4×Dic3—22no—
210#208—22no—
211#208—22no—
212#208—41no—
213#208—41no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—41no—
224C2×C3⋊C8—41no—
225C2×C3⋊C8C22⋊C412yes—
226C2×C3⋊C8C22⋊C412yes—
227C25—61no—
228C22×D4—32no—
229C22×D4—32no—
230C22×D4—32no—
231C22×D4—61no—
232C22×D4—61no—
233C22×D4—61no—
234C22×D4—61no—
235C22×D4—61no—
236C22×D4—61no—
237C22×D4—61no—
238C22×D4—61no—
239C22×D4—61no—
240C22×D4—61no—
241C22×D4—61no—
242C23×C4—21no—
243C23×C4C22×D311yescharacteristic
244C2×M16—21no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—32no—
253C2×M16—61no—
254C2×M16—61no—
255C2×M16—61no—
256C2×M16—61no—
257C2×M16—61no—
258C2×M16C4×D312yes—
259C2×M16C4×D312yes—
260C2×M16D1212yes—
261C2×M16#6712yes—
262C22×C8—21no—
263C22×C8—21no—
264C22×C8—32no—
265C22×C8—32no—
266C22×C8—31no—
267C22×C8—31no—
268C22×C8—61no—
269C22×C8—61no—
270C22×C8—61no—
271C22×C8—61no—
272C22×C8—61no—
273#117—32no—
274#117—32no—
275#117—61no—
276#117—61no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—32no—
283C2×C4⋊C4C4×D312yes—
284C2×C22⋊C4—61no—
285C2×C42—61no—
286#102—64no—
287#102—64no—
288#102—68no—
289#100—68no—
290#97—64no—
291C4×C8—64no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3#11711yescharacteristic
295C23×C3C4×D411yescharacteristic
296C23×C3C4×D411yescharacteristic
297C23×C3#12111yescharacteristic
298C23×C3#12311yescharacteristic
299C23×C3#12411yescharacteristic
300C23×C3#12611yescharacteristic
301C22×D3—24no—
302C22×D3—42no—
303C22×D3—42no—
304C22×D3—42no—
305C22×D3—42no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—84no—
309C22×D3—84no—
310C22×D3—82no—
311C22×D3—82no—
312C22×D3—82no—
313C22×D3—82no—
314C22×D3—84no—
315C22×D3—84no—
316C22×D3—82no—
317C22×D3—82no—
318C2×C3×C4—42no—
319C2×C3×C4—82no—
320C2×C3×C4—82no—
321C2×C3×C4—82no—
322C2×C3×C4C2×C4211yescharacteristic
323C2×C3×C4C2×C22⋊C411yescharacteristic
324C2×C3×C4C2×C22⋊C411yescharacteristic
325C2×C3×C4C2×C22⋊C411yescharacteristic
326C2×C3×C4#11711yescharacteristic
327C2×C3×C4#11711yescharacteristic
328C2×C3×C4C4×D411yescharacteristic
329C2×C3×C4C4×D411yescharacteristic
330C2×C3×C4C4×D411yescharacteristic
331C2×C3×C4C4×D411yescharacteristic
332C2×C3×C4C4×D411yescharacteristic
333C2×C3×C4C4×D411yescharacteristic
334C2×C3×C4C4×D411yescharacteristic
335C2×C3×C4C4×D411yescharacteristic
336C2×C3×C4C4×D411yescharacteristic
337C2×C3×C4C4×D411yescharacteristic
338C2×C3×C4#12011yescharacteristic
339C2×C3×C4#12111yescharacteristic
340C2×C3×C4#12111yescharacteristic
341C2×C3×C4#12111yescharacteristic
342C2×C3×C4#12111yescharacteristic
343C2×C3×C4#12111yescharacteristic
344C2×C3×C4#12311yescharacteristic
345C2×C3×C4#12311yescharacteristic
346C2×C3×C4#12411yescharacteristic
347C2×C3×C4#12411yescharacteristic
348C2×C3×C4#12611yescharacteristic
349C2×C3×C4#12711yescharacteristic
350C2×Dic3—24no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353D12—24no—
354D12—24no—
355D12—42no—
356D12—42no—
357D12—42no—
358D12—42no—
359D12—42no—
360D12—42no—
361C3×C8—24no—
362C3×C8—42no—
363C3⋊C8—22no—
364C3⋊C8—22no—
365C3⋊C8—42no—
366C24—62no—
367C24—62no—
368C24—62no—
369C24—121no—
370C24—121no—
371C24—121no—
372C24—121no—
373C24—242no—
374C24—242no—
375C24#23511yescharacteristic
376C2×D4—34no—
377C2×D4—62no—
378C2×D4—62no—
379C2×D4—62no—
380C2×D4—62no—
381C2×D4—62no—
382C2×D4—62no—
383C2×D4—62no—
384C2×D4—62no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—62no—
389C2×D4—62no—
390C2×D4—62no—
391C2×D4—62no—
392C2×D4—121no—
393C2×D4—121no—
394C2×D4—121no—
395C2×D4—121no—
396C2×D4—121no—
397C2×D4—121no—
398C2×D4—121no—
399C2×D4—121no—
400C2×D4—121no—
401C2×D4—121no—
402C2×D4—121no—
403C2×D4—121no—
404C22×C4—22no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—62no—
408C22×C4—82no—
409C22×C4C2×C4×D311yescharacteristic
410C22×C4C2×C4×D311yescharacteristic
411C22×C4C2×C4×D311yescharacteristic
412C22×C4C2×D1211yescharacteristic
413C22×C4#23511yescharacteristic
414C22×C4D3×D411yescharacteristic
415C22×C4D3×D411yescharacteristic
416C22×C4D3×D411yescharacteristic
417C22×C4D3×D411yescharacteristic
418C22×C4#23711yescharacteristic
419C22×C4#23711yescharacteristic
420C22×C4#23911yescharacteristic
421C22×C4#23911yescharacteristic
422C22×C4C2×#6711yescharacteristic
423M16—22no—
424M16—22no—
425M16—22no—
426M16—22no—
427M16—22no—
428M16—41no—
429M16—41no—
430M16—62no—
431M16—62no—
432M16—121no—
433M16—121no—
434M16(C2×D3)⋊C414yes—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—32no—
440C2×C8—32no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—62no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—62no—
447C2×C8—62no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—62no—
451C2×C8—62no—
452C2×C8—121no—
453C2×C8—121no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—34no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—62no—
460C22⋊C4—62no—
461C22⋊C4—62no—
462C42—62no—
463C42—62no—
464C22×C3—22no—
465C22×C3—22no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3C4×C22⋊C411yescharacteristic
473C22×C3#38611yescharacteristic
474C22×C3#38811yescharacteristic
475C22×C3#38811yescharacteristic
476C22×C3#38811yescharacteristic
477C22×C3#38811yescharacteristic
478C22×C3#39011yescharacteristic
479C22×C3#47112yes—
480C22×C3#47212yes—
481C2×D3—44no—
482C2×D3—84no—
483C2×D3—84no—
484C2×D3—84no—
485C2×D3—84no—
486C2×D3—84no—
487C2×D3—82no—
488C2×D3—82no—
489C2×D3—84no—
490C3×C4—82no—
491C3×C4C4×C22⋊C411yescharacteristic
492C3×C4C4×C22⋊C411yescharacteristic
493C3×C4#38611yescharacteristic
494C3×C4#38611yescharacteristic
495C3×C4#38811yescharacteristic
496C3×C4#38811yescharacteristic
497C3×C4#39011yescharacteristic
498C3×C4#39011yescharacteristic
499Dic3—44no—
500C23—22no—
501C23—22no—
502C23—64no—
503C23—122no—
504C23—122no—
505C23—122no—
506C23—122no—
507C23—122no—
508C23—122no—
509C23—244no—
510C23—244no—
511C23—242no—
512C23—242no—
513C23—244no—
514C23—244no—
515C23—242no—
516C23—242no—
517C23—242no—
518C23—242no—
519C23#87211yescharacteristic
520C23C4×D1211yescharacteristic
521C23#88411yescharacteristic
522C23#88511yescharacteristic
523C23#89411yescharacteristic
524C23#89811yescharacteristic
525C23C4×#6711yescharacteristic
526D4—64no—
527D4—64no—
528D4—122no—
529D4—122no—
530D4—122no—
531D4—122no—
532D4—122no—
533D4—122no—
534C2×C4—42no—
535C2×C4—64no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—122no—
540C2×C4—122no—
541C2×C4C42×D311yescharacteristic
542C2×C4C4×D1211yescharacteristic
543C2×C4D3×C22⋊C411yescharacteristic
544C2×C4D3×C22⋊C411yescharacteristic
545C2×C4#67⋊C411yescharacteristic
546C2×C4#67⋊C411yescharacteristic
547C2×C4#67⋊C411yescharacteristic
548C2×C4#67⋊C411yescharacteristic
549C2×C4#88211yescharacteristic
550C2×C4#88411yescharacteristic
551C2×C4#88511yescharacteristic
552C2×C4#88611yescharacteristic
553C2×C4#89211yescharacteristic
554C2×C4#89211yescharacteristic
555C2×C4#89311yescharacteristic
556C2×C4#89311yescharacteristic
557C2×C4#89311yescharacteristic
558C2×C4#89311yescharacteristic
559C2×C4#89411yescharacteristic
560C2×C4#89511yescharacteristic
561C2×C4#89511yescharacteristic
562C2×C4#89811yescharacteristic
563C2×C4C2×(C2×D3)⋊C411yescharacteristic
564C2×C4C4×#6711yescharacteristic
565C2×C4#93911yescharacteristic
566C2×C4#93911yescharacteristic
567C2×C4#94011yescharacteristic
568C2×C4#94711yescharacteristic
569C8—24no—
570C8—42no—
571C8—62no—
572C8—62no—
573C8—122no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3#143711yescharacteristic
579C2×C3#190712yes—
580C2×C3#193112yes—
581C2×C3#197711yescharacteristic
582C2×C3#197811yescharacteristic
583D3—84no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—124no—
593C22—244no—
594C22—244no—
595C22—244no—
596C22—244no—
597C22—244no—
598C22—244no—
599C22—242no—
600C22—242no—
601C22#466511yescharacteristic
602C22#466911yescharacteristic
603C22#466911yescharacteristic
604C22#467011yescharacteristic
605C22#474912yes—
606C22#475412yes—
607C22C4×(C2×D3)⋊C411yescharacteristic
608C22#495711yescharacteristic
609C22#499011yescharacteristic
610C4—82no—
611C4—124no—
612C4#466511yescharacteristic
613C4#466611yescharacteristic
614C4#466611yescharacteristic
615C4#467011yescharacteristic
616C4#495711yescharacteristic
617C4#495811yescharacteristic
618C4#498811yescharacteristic
619C4#499011yescharacteristic
620C3#1094311yescharacteristic, complemented, Sylow
621C2—21no—
622C2—21no—
623C2—21no—
624C2—21no—
625C2—244no—
626C2#6898312yes—
627C2#6900612yes—
628C2#6940811yescharacteristic
629C2#7007411yescharacteristic
630C2#7010711yescharacteristic
subgroupG/H
centerC23#884
commutatorC2×C3×C4C2×C42
FrattiniC23×C4C22×D3
FittingC2×C3×#343C2
socleC23×C3#121
prime radicalC23×D12C22
cosocleC23×C3×C4C23
hypercenterC2×#343D3
hypocenterC3#10943
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
3. C2×#5122C2semidirect
5. C2×C3×#343C2semidirect
6. C2×#4558C2semidirect
7. #71558C2semidirect
17. #5131C4semidirect
20. C2×#823C4semidirect
29. C2×#809C4semidirect
37. C2×#343D3semidirect
620. C3#10943semidirect

Maximal subgroups

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

subgroupindex
C2×#51312
C22×#8232
C2×#51222
C22×#8092
C2×C3×#3432
C2×#45582
#715582
#109433