Finite Group Databasebeta

← #10844917

Subgroups — #10844917

Order 768.

856 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2592C211yesmaximal, characteristic, complemented
2C2×#5057C211yesmaximal, characteristic
3#82500C211yesmaximal, characteristic, complemented
4C2×#5024C211yesmaximal, characteristic, complemented
5#71726C211yesmaximal, characteristic, complemented
6#69682C211yesmaximal, characteristic, complemented
7#69680C211yesmaximal, characteristic, complemented
8#69676C211yesmaximal, characteristic, complemented
9#69675C211yesmaximal, characteristic, complemented
10#69652C211yesmaximal, characteristic, complemented
11#69650C211yesmaximal, characteristic
12#69648C211yesmaximal, characteristic
13#69644C211yesmaximal, characteristic, complemented
14#69474C211yesmaximal, characteristic, complemented
15#69467C211yesmaximal, characteristic, complemented
16#25309—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×#946—21no—
20C22×#215C2211yescharacteristic
21C22×#213C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C3×#397—21no—
24C3×#393—21no—
25C3×#391C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#390C2211yescharacteristic
29C3×#388—21no—
30C3×#388—21no—
31C3×C4×C4⋊C4C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#832C2211yescharacteristic
38#5057—24no—
39#5024—24no—
40#4994—21no—
41#4994—21no—
42#4987—21no—
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#808C2211yescharacteristic
47#4939C2211yescharacteristic
48#4939C2211yescharacteristic
49#4935C2211yescharacteristic
50#4928C2211yescharacteristic
51#4928C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×#803C2211yescharacteristic
56C2×#803C2211yescharacteristic
57C2×C4×C3⋊C8C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#463—31no—
63C2×#446—31no—
64#2592D311yescharacteristic, complemented
65#2538—31no—
66#2066—31no—
67#2065—31no—
68#2055—31no—
69#2054—31no—
70#1972—31no—
71#1968—31no—
72#1959—31no—
73#1948—31no—
74#1935—31no—
75#1934—31no—
76#1762—31no—
77C22×C3×Q8D411yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×C3×D4C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C3×#124—22no—
83C3×#124—22no—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101#946—44no—
102C2×#215—22no—
103C2×#215—22no—
104C2×#215—41no—
105C2×#213—22no—
106C2×#213—22no—
107C2×#213—41no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124C4×D12—22no—
125C4×D12—22no—
126C3×#95—21no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95—21no—
130C3×#95—21no—
131#835—22no—
132#835—22no—
133#835—41no—
134#832—22no—
135#832—22no—
136#832—41no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139#809—22no—
140#809—22no—
141#808—41no—
142#808—41no—
143C4⋊(C3⋊C8)—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)D414yes—
146#803—22no—
147#803—22no—
148C4×C3⋊C8—22no—
149C22×SD16—31no—
150C22×D8—31no—
151C2×#124C2×D311yescharacteristic
152C2×C22⋊Q8—61no—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×D4—31no—
155#463—64no—
156#446—64no—
157C2×#106—31no—
158C2×#106—31no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×#97—31no—
171C2×#97—31no—
172C2×C4×C8—31no—
173#397—21no—
174#396—61no—
175#396—61no—
176#393—21no—
177#391C2×D311yescharacteristic
178#391C2×D311yescharacteristic
179#390—31no—
180#390—31no—
181#390C2×D311yescharacteristic
182#390C2×D311yescharacteristic
183#389—31no—
184#389—31no—
185#388—21no—
186#388—21no—
187#384—31no—
188#380—61no—
189C4×C4⋊C4C2×D311yescharacteristic
190#340—31no—
191#340—31no—
192#340—31no—
193#340—31no—
194C24×C3—21no—
195C23×D3—21no—
196C2×C3×Q8—22no—
197C2×C3×Q8—22no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—41no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12#4112yes—
230C2×C4×D3—42no—
231C2×C4×D3—42no—
232C2×C4×D3—42no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C22⋊C4—42no—
241C3×C22⋊C4—42no—
242C3×C22⋊C4—42no—
243C3×C22⋊C4—42no—
244C3×C22⋊C4—81no—
245C3×C22⋊C4—81no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—81no—
251C3×C42C2×D412yes—
252C3×C42#4112yes—
253#215—44no—
254#213—44no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—81no—
259(C2×D3)⋊C4—81no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—42no—
267C4⋊Dic3—81no—
268Dic3⋊C4—42no—
269Dic3⋊C4—42no—
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—
279C2×C3⋊C8C2×D412yes—
280C2×C3⋊C8C2×D412yes—
281C2×C3⋊C8#4112yes—
282C22×Q8#6711yescharacteristic
283C22×Q8C22×D311yescharacteristic
284C22×D4—31no—
285C22×D4C22×D311yescharacteristic
286C23×C4—61no—
287C2×SD16—62no—
288C2×SD16—62no—
289C2×SD16—121no—
290C2×D8—62no—
291C2×D8—62no—
292C2×D8—121no—
293C22×C8—31no—
294C22×C8—31no—
295C22×C8—31no—
296C22×C8—31no—
297#124—22no—
298#124—22no—
299C22⋊Q8—124no—
300C4×Q8—22no—
301C4×Q8—22no—
302C4×D4—62no—
303C4×D4—62no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4#6711yescharacteristic
312C2×C4⋊C4#6711yescharacteristic
313C2×C4⋊C4#6711yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C42—21no—
328C2×C42C22×D311yescharacteristic
329C2×C42C22×D311yescharacteristic
330C2×C42C22×D311yescharacteristic
331#106—121no—
332#106—121no—
333C4⋊C8—34no—
334C4⋊C8—62no—
335C4⋊C8—62no—
336#103—62no—
337#103—62no—
338#103—121no—
339#102—62no—
340#102—62no—
341#102—62no—
342#102—62no—
343#102—121no—
344#97—62no—
345#97—62no—
346C4×C8—62no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—61no—
353#95—61no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3C22×D411yescharacteristic
362C22×D3—22no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C3×Q8—44no—
370C3×Q8—41no—
371C3×Q8—41no—
372C3×Q8—41no—
373C3×Q8—41no—
374C3×D4—41no—
375C3×D4—41no—
376C3×D4—41no—
377C3×D4—41no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4#12112yes—
403C2×C3×C4#12112yes—
404C2×C3×C4AGL(1,ℤ/8)12yes—
405C2×C3×C4C22×D411yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4H(5,𝔽2)11yescharacteristic
410C2×C3×C42−1+411yescharacteristic
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420D12—22no—
421D12—22no—
422D12—41no—
423D12—41no—
424C4×D3—84no—
425C3⋊C8—22no—
426C3⋊C8—22no—
427C3⋊C8—22no—
428C3⋊C8—41no—
429C24—21no—
430C24—61no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—41no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—32no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—122no—
459C22×C4—122no—
460C22×C4—122no—
461C22×C4C2×#6711yescharacteristic
462C22×C4C2×#6711yescharacteristic
463C22×C4C2×#6711yescharacteristic
464C22×C4C2×#6711yescharacteristic
465C22×C4C2×#6711yescharacteristic
466C22×C4C2×#6711yescharacteristic
467C22×C4C23×D311yescharacteristic
468SD16—124no—
469D8—124no—
470C2×C8—32no—
471C2×C8—32no—
472C2×C8—32no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—122no—
496C4⋊C4—122no—
497C4⋊C4—122no—
498C4⋊C4—241no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—81no—
504C22⋊C4—81no—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C22⋊C4—122no—
508C22⋊C4—241no—
509C22⋊C4—241no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—81no—
515C42D3×D412yes—
516C42#23712yes—
517C22×C3—81no—
518C22×C3—81no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3C2×#12111yescharacteristic
525C22×C3#54811yescharacteristic
526C22×C3#55011yescharacteristic
527C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
528C22×C3#57511yescharacteristic
529C22×C3#57711yescharacteristic
530C22×C3#57711yescharacteristic
531C2×D3—42no—
532C2×D3—42no—
533C2×D3—42no—
534C2×D3—81no—
535C2×D3—81no—
536C2×D3—81no—
537C2×D3—81no—
538C3×C4—22no—
539C3×C4—22no—
540C3×C4—22no—
541C3×C4—41no—
542C3×C4—41no—
543C3×C4—41no—
544C3×C4—81no—
545C3×C4—81no—
546C3×C4—81no—
547C3×C4#52911yescharacteristic
548C3×C4#54211yescharacteristic
549C3×C4#54611yescharacteristic
550C3×C4#54811yescharacteristic
551Dic3—81no—
552Dic3—81no—
553Dic3—81no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—62no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23C22×#6711yescharacteristic
569Q8—44no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573Q8—41no—
574D4—41no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—62no—
579D4—62no—
580D4—121no—
581D4—121no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—244no—
616C2×C4#93912yes—
617C2×C4#93912yes—
618C2×C4#94912yes—
619C2×C4C2×D3×D411yescharacteristic
620C2×C4C2×#23711yescharacteristic
621C2×C4D3×#4111yescharacteristic
622C2×C4D3×#4111yescharacteristic
623C2×C4#100911yescharacteristic
624C2×C4#101011yescharacteristic
625C8—62no—
626C8—62no—
627C8—62no—
628C8—121no—
629C2×C3—161no—
630C2×C3#242611yescharacteristic
631C2×C3#305511yescharacteristic
632C2×C3#306011yescharacteristic
633C2×C3#337311yescharacteristic
634C2×C3#337611yescharacteristic
635C2×C3#339711yescharacteristic
636C2×C3#340111yescharacteristic
637D3—82no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—122no—
646C22—122no—
647C22—122no—
648C22—241no—
649C22—241no—
650C22—241no—
651C22—241no—
652C22C2×#93911yescharacteristic
653C22C2×#94911yescharacteristic
654C22#582211yescharacteristic
655C22#583111yescharacteristic
656C22#583411yescharacteristic
657C22#583611yescharacteristic
658C22#583711yescharacteristic
659C4—22no—
660C4—22no—
661C4—22no—
662C4—41no—
663C4—41no—
664C4—41no—
665C4—81no—
666C4—81no—
667C4—81no—
668C4—241no—
669C4—241no—
670C4—241no—
671C4#567611yescharacteristic
672C4#567711yescharacteristic
673C4#568011yescharacteristic
674C4#568211yescharacteristic
675C3#2530911yescharacteristic, complemented, Sylow
676C2—161no—
677C2—242no—
678C2#8292411yescharacteristic
679C2#8299511yescharacteristic
680C2#8342011yescharacteristic
681C2#8366911yescharacteristic
682C2#8367111yescharacteristic
683C2#8371211yescharacteristic
684C2#8371511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2592C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#2592D3
hypocenterC3#25309
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2592

Complements

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

NH ≅ G/Ntype
1. C3×#2592C2semidirect
3. #82500C2semidirect
4. C2×#5024C2semidirect
5. #71726C2semidirect
6. #69682C2semidirect
7. #69680C2semidirect
8. #69676C2semidirect
9. #69675C2semidirect
10. #69652C2semidirect
13. #69644C2semidirect
14. #69474C2semidirect
15. #69467C2semidirect
64. #2592D3semidirect
675. C3#25309semidirect

Maximal subgroups

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

subgroupindex
C3×#25922
C2×#50572
#825002
C2×#50242
#717262
#696822
#696802
#696762
#696752
#696522
#696502
#696482
#696442
#694742
#694672
#253093