Finite Group Databasebeta

← #10845264

Subgroups — #10845264

Order 768.

840 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2567C211yesmaximal, characteristic, complemented
2C2×#5089C211yesmaximal, characteristic
3C2×#5028C211yesmaximal, characteristic
4#73655C211yesmaximal, characteristic
5#71780C211yesmaximal, characteristic, complemented
6#71631C211yesmaximal, characteristic, complemented
7#69690C211yesmaximal, characteristic, complemented
8#69687C211yesmaximal, characteristic
9#69681C211yesmaximal, characteristic, complemented
10#69680C211yesmaximal, characteristic
11#69589C211yesmaximal, characteristic, complemented
12#69588C211yesmaximal, characteristic, complemented
13#69585C211yesmaximal, characteristic
14#69584C211yesmaximal, characteristic
15#69485C211yesmaximal, characteristic, complemented
16#25336—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×#875C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C3×#395—21no—
25C3×#395—21no—
26C3×#391C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#389—21no—
29C3×#389—21no—
30C3×#384C2211yescharacteristic
31C3×#384C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35#5089—24no—
36#5028—24no—
37#4990—21no—
38#4989—21no—
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45#4939C2211yescharacteristic
46C4×C4⋊Dic3C2211yescharacteristic
47#4929C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C2×#803C2211yescharacteristic
52#4669—21no—
53#4666—21no—
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#489—31no—
63C2×#464—31no—
64#2567D311yescharacteristic, complemented
65#2367—31no—
66#2072—31no—
67#2071—31no—
68#2056—31no—
69#2055—31no—
70#1988—31no—
71#1940—31no—
72#1878—31no—
73#1877—31no—
74#1803—31no—
75#1802—31no—
76#1763—31no—
77C22×C3×Q8—21no—
78C22×C3×Q8C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×Dic6C2311yescharacteristic
82C3×C4⋊Q8—22no—
83C3×C4⋊Q8—22no—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#216—22no—
102C2×#216—22no—
103C2×#216—41no—
104C2×#215—22no—
105C2×#215—22no—
106C2×#215—41no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#875—22no—
122#875D414yes—
123C4×D12—22no—
124C4×D12—41no—
125C4×Dic6—22no—
126C4×Dic6—41no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95D411yescharacteristic
132C3×#95D411yescharacteristic
133#835—22no—
134#835—22no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138#810—22no—
139#810—22no—
140#810—22no—
141#809—22no—
142#809—22no—
143#809—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—41no—
146C4⋊(C3⋊C8)—41no—
147#803—22no—
148C22×Q16—31no—
149C22×SD16—31no—
150C2×C4⋊Q8C2×D311yescharacteristic
151C2×#124—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×D4—31no—
155#489—64no—
156#464—64no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169#395—21no—
170#395—21no—
171#391—31no—
172#391C2×D311yescharacteristic
173#391C2×D311yescharacteristic
174#390—31no—
175#389—21no—
176#389—21no—
177#388—61no—
178#388—61no—
179#385—61no—
180#384C2×D311yescharacteristic
181#384C2×D311yescharacteristic
182C4×C4⋊C4—31no—
183C4×C4⋊C4C2×D311yescharacteristic
184C4×C22⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C23×D3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—42no—
200C2×C3×Q8—81no—
201C2×C3×Q8—81no—
202C2×C3×Q8#4112yes—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12C2×D412yes—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×C4×D3—81no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6C2×D412yes—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4—81no—
243C3×C4⋊C4—81no—
244C3×C4⋊C4—81no—
245C3×C4⋊C4—81no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—42no—
251C3×C42C2×D412yes—
252C3×C42#4112yes—
253#216—44no—
254#215—44no—
255(C2×D3)⋊C4—24no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—81no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3#4112yes—
264Dic3⋊C4—42no—
265C4×Dic3—42no—
266C4×Dic3—81no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C22×Q8—21no—
280C22×Q8—31no—
281C22×Q8C22×D311yescharacteristic
282C22×D4—31no—
283C23×C4—61no—
284C2×Q16—62no—
285C2×Q16—62no—
286C2×Q16—121no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—121no—
290C22×C8—31no—
291C22×C8—31no—
292C22×C8—31no—
293C22×C8—31no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296#124—34no—
297#124—62no—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8—62no—
301C4×Q8—121no—
302C4×D4—62no—
303C4×D4—121no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4#6711yescharacteristic
314C2×C4⋊C4#6711yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C42—21no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327C2×C42C22×D311yescharacteristic
328C2×C42C22×D311yescharacteristic
329C4⋊C8—62no—
330C4⋊C8—121no—
331C4⋊C8—121no—
332#103—62no—
333#103—62no—
334#103—62no—
335#103—62no—
336#103—62no—
337#102—62no—
338#102—62no—
339#102—62no—
340#97—62no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95#6711yescharacteristic
349#95#6711yescharacteristic
350C23×C3C22×D411yescharacteristic
351C22×D3—22no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—41no—
358C3×Q8—22no—
359C3×Q8—22no—
360C3×Q8—41no—
361C3×Q8—41no—
362C3×Q8—84no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—22no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4#12112yes—
390C2×C3×C4#12112yes—
391C2×C3×C4#13712yes—
392C2×C3×C4#13712yes—
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4H(5,𝔽2)11yescharacteristic
398C2×C3×C42−1+411yescharacteristic
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408D12—22no—
409D12—22no—
410D12—41no—
411D12—41no—
412C4×D3—84no—
413Dic6—22no—
414Dic6—22no—
415Dic6—41no—
416Dic6—41no—
417C3⋊C8—41no—
418C3⋊C8—41no—
419C3⋊C8—41no—
420C3⋊C8—41no—
421C24—61no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—32no—
428C2×Q8—42no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—81no—
435C2×Q8—81no—
436C2×Q8#23912yes—
437C2×D4—32no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—122no—
451C22×C4—122no—
452C22×C4—241no—
453C22×C4C2×#6711yescharacteristic
454C22×C4C2×#6711yescharacteristic
455C22×C4C2×#6711yescharacteristic
456C22×C4C2×#6711yescharacteristic
457C22×C4C2×#6711yescharacteristic
458C22×C4C2×#6711yescharacteristic
459C22×C4C23×D311yescharacteristic
460Q16—124no—
461SD16—124no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—32no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—122no—
493C22⋊C4—64no—
494C22⋊C4—64no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—241no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—42no—
503C42—122no—
504C42—241no—
505C42D3×D412yes—
506C42#23712yes—
507C22×C3C2×#12111yescharacteristic
508C22×C3#54611yescharacteristic
509C22×C3#54811yescharacteristic
510C22×C3C2×#13711yescharacteristic
511C22×C3C2×#13711yescharacteristic
512C22×C3#57611yescharacteristic
513C22×C3#57711yescharacteristic
514C2×D3—42no—
515C2×D3—42no—
516C2×D3—42no—
517C2×D3—81no—
518C2×D3—81no—
519C2×D3—81no—
520C2×D3—81no—
521C3×C4—22no—
522C3×C4—22no—
523C3×C4—22no—
524C3×C4—22no—
525C3×C4—42no—
526C3×C4—41no—
527C3×C4—41no—
528C3×C4—81no—
529C3×C4—81no—
530C3×C4—81no—
531C3×C4#52911yescharacteristic
532C3×C4#54211yescharacteristic
533C3×C4#54611yescharacteristic
534C3×C4#54711yescharacteristic
535Dic3—42no—
536Dic3—42no—
537Dic3—81no—
538C23—62no—
539C23—121no—
540C23—121no—
541C23—121no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23C22×#6711yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—62no—
551Q8—62no—
552Q8—84no—
553Q8—121no—
554Q8—121no—
555D4—62no—
556D4—62no—
557D4—121no—
558D4—121no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—244no—
595C2×C4#93812yes—
596C2×C4#93812yes—
597C2×C4#94212yes—
598C2×C4#94212yes—
599C2×C4C2×D3×D411yescharacteristic
600C2×C4C2×#23711yescharacteristic
601C2×C4C2×#23911yescharacteristic
602C2×C4#100711yescharacteristic
603C2×C4D3×#4111yescharacteristic
604C2×C4#100911yescharacteristic
605C8—121no—
606C8—121no—
607C8—121no—
608C8—121no—
609C2×C3#242711yescharacteristic
610C2×C3#303911yescharacteristic
611C2×C3#304511yescharacteristic
612C2×C3#331411yescharacteristic
613C2×C3#331611yescharacteristic
614C2×C3#336911yescharacteristic
615C2×C3#337511yescharacteristic
616D3—82no—
617C22—122no—
618C22—122no—
619C22—122no—
620C22—241no—
621C22—241no—
622C22—241no—
623C22—241no—
624C22C2×#93811yescharacteristic
625C22C2×#94211yescharacteristic
626C22C2×#94211yescharacteristic
627C22#581711yescharacteristic
628C22#582911yescharacteristic
629C22#583511yescharacteristic
630C22#583611yescharacteristic
631C4—22no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—42no—
636C4—41no—
637C4—41no—
638C4—81no—
639C4—81no—
640C4—81no—
641C4—122no—
642C4—122no—
643C4—241no—
644C4#562411yescharacteristic
645C4#563011yescharacteristic
646C4#563111yescharacteristic
647C4#564311yescharacteristic
648C3#2533611yescharacteristic, complemented, Sylow
649C2—242no—
650C2#8275611yescharacteristic
651C2#8320911yescharacteristic
652C2#8338511yescharacteristic
653C2#8363511yescharacteristic
654C2#8364111yescharacteristic
655C2#8370411yescharacteristic
656C2#8370811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2567C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2567D3
hypocenterC3#25336
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2567

Complements

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

NH ≅ G/Ntype
1. C3×#2567C2semidirect
5. #71780C2semidirect
6. #71631C2semidirect
7. #69690C2semidirect
9. #69681C2semidirect
11. #69589C2semidirect
12. #69588C2semidirect
15. #69485C2semidirect
64. #2567D3semidirect
648. C3#25336semidirect

Maximal subgroups

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

subgroupindex
C3×#25672
C2×#50892
C2×#50282
#736552
#717802
#716312
#696902
#696872
#696812
#696802
#695892
#695882
#695852
#695842
#694852
#253363