Finite Group Databasebeta

← #10763257

Subgroups — #10763257

Order 768.

962 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#125C211yesmaximal, characteristic
2#79331C211yesmaximal, characteristic
3#78517C211yesmaximal, characteristic
4C3×#1528C211yesmaximal, characteristic, complemented
5#71735C211yesmaximal, characteristic, complemented
6#70812C211yesmaximal, characteristic, complemented
7#69863C211yesmaximal, characteristic, complemented
8#68367C214yesmaximal, complemented
9#68363C214yesmaximal, complemented
10#17667—31nomaximal, Sylow
11C2×C3×#125C2211yescharacteristic
12D3×#125—21no—
13D3×#125—21no—
14D3×#125—21no—
15D3×#125—21no—
16D3×#125C412yes—
17D3×#125C412yes—
18D3×#125C2212yes—
19D3×#125C2212yes—
20C2×D3×C4⋊C4—21no—
21C2×D3×C4⋊C4—21no—
22C2×D3×C4⋊C4C411yescharacteristic
23C2×D3×C4⋊C4C411yescharacteristic
24C2×#890C411yescharacteristic
25C2×#890C411yescharacteristic
26C2×#889—21no—
27C2×#889—21no—
28C2×C42×D3C2211yescharacteristic
29C2×#870C2211yescharacteristic
30C3×#432C2211yescharacteristic
31C3×C4×M16C2211yescharacteristic
32#5126—21no—
33#5118—21no—
34#4926C2211yescharacteristic
35C4×#208C2211yescharacteristic
36#4837—21no—
37#4833—21no—
38#4726—21no—
39#4718—21no—
40#4707C2212yes—
41#4705C2212yes—
42#4705C2212yes—
43#4705C2212yes—
44#4685—21no—
45#4685—21no—
46C3×#330C2214yescomplemented
47#4543C2214yescomplemented
48#4465C2214yescomplemented
49#4465C2214yescomplemented
50C22×#125—31no—
51#2920—31no—
52#2920—31no—
53#1534—34no—
54#1528—31no—
55#1528—31no—
56#1528—34no—
57#1528—31no—
58#1528D311yescharacteristic, complemented
59C22×C4×D3—21no—
60C22×C4×D3—21no—
61C22×C4×D3C2×C411yescharacteristic
62C22×C4×D3D411yescharacteristic
63C22×C4×D3D411yescharacteristic
64C2×C3×M16—21no—
65C3×#125—21no—
66C3×#125—21no—
67C3×#125C2×C412yes—
68C3×#125C2312yes—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2×C411yescharacteristic
72C2×C3×C4⋊C4C2×C411yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3C2×C411yescharacteristic
77C2×C4⋊Dic3C2×C411yescharacteristic
78C2×Dic3⋊C4—21no—
79C2×Dic3⋊C4—21no—
80C2×Dic3⋊C4—21no—
81C2×Dic3⋊C4—21no—
82C2×Dic3⋊C4—21no—
83C2×Dic3⋊C4—21no—
84C2×C4×Dic3C2×C411yescharacteristic
85C2×C4×Dic3D411yescharacteristic
86C2×C4×Dic3D411yescharacteristic
87C2×#208—21no—
88C2×ΓL(1,𝔽25)—22no—
89D3×C4⋊C4—21no—
90D3×C4⋊C4—21no—
91D3×C4⋊C4—21no—
92D3×C4⋊C4—21no—
93D3×C4⋊C4—22no—
94D3×C4⋊C4—22no—
95D3×C4⋊C4—22no—
96D3×C4⋊C4—22no—
97D3×C4⋊C4—22no—
98D3×C4⋊C4—22no—
99D3×C4⋊C4—22no—
100D3×C4⋊C4—22no—
101D3×C4⋊C4C2×C412yes—
102D3×C4⋊C4C2×C412yes—
103D3×C4⋊C4C2×C412yes—
104D3×C4⋊C4C2×C412yes—
105#890—21no—
106#890—21no—
107#890—21no—
108#890—21no—
109#890C2×C412yes—
110#890C2×C412yes—
111#890C2×C412yes—
112#890C2×C412yes—
113#889—22no—
114#889—22no—
115#889—22no—
116#889—22no—
117#889—22no—
118#889—22no—
119#889—22no—
120#889—22no—
121C42×D3C2×C411yescharacteristic
122C42×D3C2×C411yescharacteristic
123C42×D3D411yescharacteristic
124C42×D3D411yescharacteristic
125C42×D3D411yescharacteristic
126C42×D3D411yescharacteristic
127C42×D3C2311yescharacteristic
128C42×D3C2311yescharacteristic
129#870—21no—
130#870—21no—
131#870C2×C412yes—
132#870C2312yes—
133C3×C4⋊C8—21no—
134C3×C22⋊C8—21no—
135C3×#97C2312yes—
136C3×#97C2312yes—
137C3×C4×C8—21no—
138#830—21no—
139(C2×D3)⋊C8—22no—
140(C2×D3)⋊C8—22no—
141(C2×D3)⋊C8—22no—
142Dic3⋊C8—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145C8×Dic3—22no—
146C4⋊(C3⋊C8)—21no—
147#803C2312yes—
148#803C2312yes—
149C4×C3⋊C8—21no—
150C2×#125—32no—
151C2×#125—32no—
152C2×#125—32no—
153C2×#125—32no—
154C2×#125—31no—
155C2×#125—31no—
156C2×#125—31no—
157C2×#125—61no—
158C2×#125—61no—
159C2×#125—61no—
160C2×#125—61no—
161C2×#125—61no—
162C2×#125—61no—
163C2×#125C2×D311yescharacteristic
164C22×C4⋊C4—31no—
165C22×C4⋊C4—31no—
166C22×C4⋊C4—61no—
167C22×C4⋊C4—61no—
168C22×C42—31no—
169#432—31no—
170#432—32no—
171#432—32no—
172#432—32no—
173#432C2×D311yescharacteristic
174#431—61no—
175#431—61no—
176#431—61no—
177#431—61no—
178#423—61no—
179#423—61no—
180#407—61no—
181#407—61no—
182C4×M16—31no—
183C4×M16—32no—
184C4×M16C2×D311yescharacteristic
185#330—34no—
186#330—34no—
187#330—34no—
188#330C2×D314yescomplemented
189C23×D3C22⋊C411yescharacteristic
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4C22×C411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3C22⋊C411yescharacteristic
198C22×Dic3C22⋊C411yescharacteristic
199C22×Dic3C22⋊C411yescharacteristic
200C2×C4×D3—24no—
201C2×C4×D3—22no—
202C2×C4×D3—22no—
203C2×C4×D3—22no—
204C2×C4×D3—22no—
205C2×C4×D3—42no—
206C2×C4×D3—42no—
207C2×C4×D3—44no—
208C2×C4×D3—44no—
209C2×C4×D3C22⋊C411yescharacteristic
210C2×C4×D3C22⋊C411yescharacteristic
211C2×C4×D3C22⋊C411yescharacteristic
212C2×C4×D3C22⋊C411yescharacteristic
213C2×C4×D3C22⋊C411yescharacteristic
214C2×C4×D3C22⋊C411yescharacteristic
215C2×C4×D3C22×C411yescharacteristic
216C2×C4×D3C22×C411yescharacteristic
217C2×C4×D3C2×D411yescharacteristic
218C2×C4×D3C2×D411yescharacteristic
219C2×C4×D3C2×D411yescharacteristic
220C2×C4×D3C2×D411yescharacteristic
221C3×M16—42no—
222C2×C3×C8—22no—
223C2×C3×C8—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4C22×C412yes—
231C3×C4⋊C4C22×C412yes—
232C3×C42C22×C411yescharacteristic
233C3×C42C2×D411yescharacteristic
234C3×C42C2×D411yescharacteristic
235C3×C42C2411yescharacteristic
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—22no—
239C4⋊Dic3—22no—
240C4⋊Dic3—22no—
241C4⋊Dic3—22no—
242C4⋊Dic3C22×C412yes—
243C4⋊Dic3C22×C412yes—
244Dic3⋊C4—22no—
245Dic3⋊C4—22no—
246Dic3⋊C4—22no—
247Dic3⋊C4—22no—
248Dic3⋊C4—22no—
249Dic3⋊C4—22no—
250Dic3⋊C4—22no—
251Dic3⋊C4—22no—
252Dic3⋊C4—22no—
253Dic3⋊C4—22no—
254Dic3⋊C4—22no—
255Dic3⋊C4—22no—
256C4×Dic3C22⋊C411yescharacteristic
257C4×Dic3C22⋊C411yescharacteristic
258C4×Dic3C22⋊C411yescharacteristic
259C4×Dic3C22⋊C411yescharacteristic
260C4×Dic3C22⋊C411yescharacteristic
261C4×Dic3C22⋊C411yescharacteristic
262C4×Dic3C22×C411yescharacteristic
263C4×Dic3C22×C411yescharacteristic
264C4×Dic3C2×D411yescharacteristic
265C4×Dic3C2×D411yescharacteristic
266C4×Dic3C2×D411yescharacteristic
267C4×Dic3C2×D411yescharacteristic
268#208—42no—
269C2×C3⋊C8—22no—
270C2×C3⋊C8—22no—
271ΓL(1,𝔽25)—44no—
272C23×C4—31no—
273C23×C4—31no—
274C23×C4—31no—
275C23×C4—61no—
276C23×C4—61no—
277C2×M16—21no—
278C2×M16—62no—
279C2×M16—61no—
280#125—21no—
281#125—21no—
282#125—32no—
283#125—32no—
284#125—32no—
285#125—32no—
286#125—32no—
287#125—32no—
288#125—62no—
289#125—62no—
290#125—62no—
291#125—62no—
292#125—62no—
293#125—62no—
294#125—62no—
295#125—62no—
296#125—61no—
297#125—61no—
298#125—61no—
299#125—61no—
300#125—61no—
301#125—61no—
302#125C4×D312yes—
303#125C22×D312yes—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—32no—
309C2×C4⋊C4—32no—
310C2×C4⋊C4—32no—
311C2×C4⋊C4—32no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—62no—
317C2×C4⋊C4—62no—
318C2×C4⋊C4—62no—
319C2×C4⋊C4—62no—
320C2×C4⋊C4—62no—
321C2×C4⋊C4—62no—
322C2×C4⋊C4—62no—
323C2×C4⋊C4—62no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4—61no—
328C2×C4⋊C4—61no—
329C2×C4⋊C4—61no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4C4×D311yescharacteristic
333C2×C4⋊C4C4×D311yescharacteristic
334C2×C42—31no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—31no—
339C2×C42—31no—
340C2×C42—31no—
341C2×C42—31no—
342C2×C42—31no—
343C2×C42—31no—
344C2×C42—31no—
345C2×C42C22×D311yescharacteristic
346C4⋊C8—21no—
347C4⋊C8—62no—
348C4⋊C8—62no—
349C4⋊C8—62no—
350C4⋊C8—61no—
351C22⋊C8—21no—
352C22⋊C8—62no—
353C22⋊C8—62no—
354C22⋊C8—62no—
355C22⋊C8—61no—
356#97—32no—
357#97—32no—
358#97C22×D312yes—
359#97C22×D312yes—
360C4×C8—21no—
361C4×C8—62no—
362C4×C8—61no—
363C23×C3C2×C22⋊C411yescharacteristic
364C22×D3—22no—
365C22×D3—22no—
366C22×D3—22no—
367C22×D3C2×C22⋊C411yescharacteristic
368C22×D3C2×C22⋊C411yescharacteristic
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—41no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4C2×C22⋊C411yescharacteristic
377C2×C3×C4C2×C22⋊C411yescharacteristic
378C2×C3×C4C2×C22⋊C411yescharacteristic
379C2×C3×C4C23×C411yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C22×D411yescharacteristic
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—41no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3C2×C22⋊C411yescharacteristic
390C2×Dic3C2×C22⋊C411yescharacteristic
391C2×Dic3C2×C22⋊C411yescharacteristic
392C2×Dic3C2×C22⋊C411yescharacteristic
393C2×Dic3C2×C22⋊C411yescharacteristic
394C2×Dic3C2×C22⋊C411yescharacteristic
395C4×D3—22no—
396C4×D3—22no—
397C4×D3—22no—
398C4×D3—22no—
399C4×D3—22no—
400C4×D3—41no—
401C4×D3—41no—
402C4×D3—41no—
403C4×D3—41no—
404C4×D3—41no—
405C4×D3—41no—
406C4×D3—44no—
407C4×D3—44no—
408C4×D3—44no—
409C4×D3—44no—
410C4×D3#10114yes—
411C3×C8—42no—
412C3×C8—42no—
413C3⋊C8—42no—
414C3⋊C8—42no—
415C24—31no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—64no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—122no—
441C22×C4—122no—
442C22×C4—124no—
443C22×C4—124no—
444C22×C4C2×C4×D311yescharacteristic
445C22×C4D3×D411yescharacteristic
446C22×C4D3×D411yescharacteristic
447M16—42no—
448M16—124no—
449M16—122no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—62no—
453C2×C8—62no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—32no—
461C4⋊C4—32no—
462C4⋊C4—62no—
463C4⋊C4—62no—
464C4⋊C4—62no—
465C4⋊C4—62no—
466C4⋊C4—62no—
467C4⋊C4—62no—
468C4⋊C4—62no—
469C4⋊C4—62no—
470C4⋊C4—62no—
471C4⋊C4—62no—
472C4⋊C4—62no—
473C4⋊C4—62no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—62no—
477C4⋊C4—62no—
478C4⋊C4—62no—
479C4⋊C4—62no—
480C4⋊C4C2×C4×D312yes—
481C4⋊C4C2×C4×D312yes—
482C42—31no—
483C42—31no—
484C42—31no—
485C42—31no—
486C42—31no—
487C42—31no—
488C42—31no—
489C42—31no—
490C42—31no—
491C42—31no—
492C42—31no—
493C42—31no—
494C42C2×C4×D311yescharacteristic
495C42D3×D411yescharacteristic
496C42D3×D411yescharacteristic
497C42C23×D311yescharacteristic
498C22×C3—21no—
499C22×C3—21no—
500C22×C3#42112yes—
501C22×C3C22×C22⋊C411yescharacteristic
502C2×D3—28no—
503C2×D3—21no—
504C2×D3—21no—
505C2×D3—21no—
506C2×D3—21no—
507C2×D3C2×#10112yes—
508C2×D3#42112yes—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—42no—
515C3×C4—42no—
516C3×C4C2×#10112yes—
517Dic3—21no—
518Dic3—21no—
519Dic3—21no—
520Dic3—21no—
521Dic3—21no—
522Dic3—42no—
523Dic3—42no—
524Dic3C2×#10112yes—
525C23—31no—
526C23—31no—
527C23—62no—
528C23—62no—
529C23—62no—
530C23D3×C22⋊C411yescharacteristic
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—31no—
536C2×C4—34no—
537C2×C4—31no—
538C2×C4—31no—
539C2×C4—31no—
540C2×C4—31no—
541C2×C4—31no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—62no—
546C2×C4—62no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—124no—
561C2×C4—124no—
562C2×C4—124no—
563C2×C4—124no—
564C2×C4—121no—
565C2×C4—122no—
566C2×C4—122no—
567C2×C4D3×C22⋊C411yescharacteristic
568C2×C4D3×C22⋊C411yescharacteristic
569C2×C4D3×C22⋊C411yescharacteristic
570C2×C4C22×C4×D311yescharacteristic
571C2×C4C2×D3×D411yescharacteristic
572C2×C4C2×D3×D411yescharacteristic
573C8—42no—
574C8—42no—
575C8—122no—
576C8—122no—
577C2×C3—22no—
578C2×C3C22×#10111yescharacteristic
579C2×C3C2×#42111yescharacteristic
580C2×C3C2×#42111yescharacteristic
581D3—22no—
582D3—22no—
583C22—21no—
584C22—21no—
585C22—32no—
586C22—32no—
587C22—68no—
588C22—61no—
589C22—61no—
590C22—61no—
591C22—61no—
592C22#482112yes—
593C22C2×D3×C22⋊C411yescharacteristic
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—32no—
600C4—42no—
601C4—42no—
602C4—61no—
603C4—61no—
604C4—61no—
605C4—61no—
606C4—61no—
607C4—122no—
608C4—122no—
609C4D3×#10112yes—
610C3#1766711yescharacteristic, complemented, Sylow
611C2—22no—
612C2—62no—
613C2—62no—
614C2C2×D3×#10111yescharacteristic
615C2C2×#482111yescharacteristic
616C2D3×#42111yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1528C2
socleC22×C3C22×C22⋊C4
prime radicalC23×D3C22⋊C4
cosocleC3×C42C24
hypercenter#1528D3
hypocenterC3#17667
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1528

Complements

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

NH ≅ G/Ntype
4. C3×#1528C2semidirect
5. #71735C2semidirect
6. #70812C2semidirect
7. #69863C2semidirect
8. #68367C2semidirect
9. #68363C2semidirect
46. C3×#330C22semidirect
47. #4543C22semidirect
48. #4465C22semidirect
49. #4465C22semidirect
58. #1528D3semidirect
188. #330C2×D3semidirect
610. C3#17667semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#1252
#793312
#785172
C3×#15282
#717352
#708122
#698632
#683672
#683632
#176673