Finite Group Databasebeta

← #10844911

Subgroups — #10844911

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
1#82376C211yesmaximal, characteristic, complemented
2C2×#5016C211yesmaximal, characteristic
3C2×#4701C211yesmaximal, characteristic, complemented
4#73735C211yesmaximal, characteristic, complemented
5C3×#1762C211yesmaximal, characteristic, complemented
6#70217C211yesmaximal, characteristic, complemented
7#70212C211yesmaximal, characteristic
8#70191C211yesmaximal, characteristic
9#70189C211yesmaximal, characteristic, complemented
10#69821C211yesmaximal, characteristic, complemented
11#69467C211yesmaximal, characteristic, complemented
12#68912C211yesmaximal, characteristic, complemented
13#68900C211yesmaximal, characteristic, complemented
14#68854C211yesmaximal, characteristic, complemented
15#68852C211yesmaximal, characteristic, complemented
16#25309—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C22×#214C2211yescharacteristic
19C22×#213C2211yescharacteristic
20C2×#924—21no—
21C2×#875C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27#5221C2211yescharacteristic
28#5221C2211yescharacteristic
29#5212—21no—
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#815C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38#5016—24no—
39#4988C2211yescharacteristic
40#4988C2211yescharacteristic
41#4973C2211yescharacteristic
42#4973C2211yescharacteristic
43#4972C2211yescharacteristic
44#4972C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#808C2211yescharacteristic
49#4948—21no—
50#4948—21no—
51#4940—21no—
52#4940—21no—
53#4935C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55#4701—24no—
56#4674—21no—
57#4673—21no—
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#463—31no—
63C2×#446—31no—
64#2592—31no—
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#1762D311yescharacteristic, complemented
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C22×D12C2311yescharacteristic
80C22×Dic6D411yescharacteristic
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#214—22no—
97C2×#214—22no—
98C2×#214—41no—
99C2×#213—22no—
100C2×#213—22no—
101C2×#213—41no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3D411yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113#924—44no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4D411yescharacteristic
117C2×Dic3⋊C4D411yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122#875—22no—
123#875—22no—
124C4×Dic6—22no—
125C4×Dic6—22no—
126C3×#102—22no—
127C3×#102—22no—
128C3×#97—22no—
129C3×#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136#821—22no—
137#821—41no—
138#816—22no—
139#816—41no—
140#815—22no—
141Dic3⋊C8—22no—
142Dic3⋊C8—22no—
143C8×Dic3—22no—
144#810—22no—
145#809—22no—
146#808—41no—
147#808—41no—
148C4⋊(C3⋊C8)D414yes—
149C22×SD16—31no—
150C22×D8—31no—
151C2×#124—31no—
152C2×C22⋊Q8—61no—
153C2×C4×Q8—31no—
154C2×C4×D4C2×D311yescharacteristic
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×#102C2×D311yescharacteristic
169C2×#102C2×D311yescharacteristic
170C2×#97—31no—
171C2×#97C2×D311yescharacteristic
172C2×C4×C8—31no—
173#397—61no—
174#396—61no—
175#396—61no—
176#393—61no—
177#391—31no—
178#391—31no—
179#390—31no—
180#390—31no—
181#390—31no—
182#390—31no—
183#389—31no—
184#389—31no—
185#388—61no—
186#388—61no—
187#384—31no—
188#380—61no—
189C4×C4⋊C4C2×D311yescharacteristic
190#340—31no—
191#340—31no—
192#340C2×D311yescharacteristic
193#340C2×D311yescharacteristic
194C24×C3—21no—
195C23×D3—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4#4112yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—42no—
206C22×C3×C4—42no—
207C22×C3×C4—42no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×Dic6—22no—
225C2×Dic6—22no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—41no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8#4112yes—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C22⋊C4—42no—
246C3×C42—81no—
247C3×C42C2×D412yes—
248C3×C42#4112yes—
249#217—42no—
250#217—42no—
251#217—81no—
252#217—81no—
253#214—44no—
254#213—44no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—81no—
260(C2×D3)⋊C4—81no—
261C4⋊Dic3—22no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—42no—
267C4⋊Dic3—81no—
268Dic3⋊C4—22no—
269Dic3⋊C4—22no—
270Dic3⋊C4—42no—
271Dic3⋊C4—42no—
272C4×Dic3—21no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8C2×D412yes—
281C2×C3⋊C8C2×D412yes—
282C22×Q8—31no—
283C22×Q8—31no—
284C22×D4—31no—
285C22×D4C22×D311yescharacteristic
286C23×C4—21no—
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×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297#124—62no—
298#124—62no—
299C22⋊Q8—124no—
300C4×Q8—62no—
301C4×Q8—62no—
302C4×D4—22no—
303C4×D4—22no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C42—21no—
328C2×C42—31no—
329C2×C42—31no—
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—22no—
340#102—22no—
341#102—62no—
342#102—62no—
343#102—121no—
344#97—22no—
345#97—62no—
346C4×C8—62no—
347#95—21no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354C23×C3—22no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3C22×D411yescharacteristic
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C3×D4—22no—
370C3×D4—22no—
371C3×D4—41no—
372C3×D4—41no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—41no—
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—84no—
387C2×C3×C4#12112yes—
388C2×C3×C4#12112yes—
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4H(5,𝔽2)11yescharacteristic
394C2×C3×C42−1+411yescharacteristic
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3AGL(1,ℤ/8)12yes—
416D12—41no—
417D12—41no—
418D12—41no—
419D12—41no—
420Dic6—44no—
421Dic6—41no—
422Dic6—41no—
423Dic6—41no—
424Dic6—41no—
425C3×C8—22no—
426C3×C8—41no—
427C3⋊C8—22no—
428C3⋊C8—22no—
429C24—21no—
430C24—61no—
431C2×Q8—62no—
432C2×Q8—62no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8—121no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4#23512yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4D3×D411yescharacteristic
466C22×C4D3×D411yescharacteristic
467C22×C4C23×D311yescharacteristic
468SD16—124no—
469D8—124no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—32no—
476C2×C8—32no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8#23512yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—62no—
489C4⋊C4—62no—
490C4⋊C4—62no—
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—122no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—122no—
506C22⋊C4—241no—
507C22⋊C4—241no—
508C22⋊C4—241no—
509C22⋊C4—241no—
510C42—61no—
511C42—61no—
512C42—61no—
513C42—61no—
514C42—81no—
515C42#23512yes—
516C42D3×D412yes—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—42no—
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—81no—
532C2×D3—81no—
533C2×D3—81no—
534C2×D3—81no—
535C2×D3—81no—
536C2×D3—81no—
537C2×D3—81no—
538C3×C4—22no—
539C3×C4—22no—
540C3×C4—81no—
541C3×C4—81no—
542C3×C4—81no—
543C3×C4#52911yescharacteristic
544C3×C4#54211yescharacteristic
545C3×C4#54611yescharacteristic
546C3×C4#54811yescharacteristic
547Dic3—22no—
548Dic3—41no—
549Dic3—41no—
550Dic3—41no—
551Dic3—81no—
552Dic3—81no—
553Dic3—81no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—121no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23C2×D3×D411yescharacteristic
569Q8—124no—
570Q8—121no—
571Q8—121no—
572Q8—121no—
573Q8—121no—
574D4—22no—
575D4—22no—
576D4—41no—
577D4—41no—
578D4—121no—
579D4—121no—
580D4—121no—
581D4—121no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—32no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—61no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—84no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4#88412yes—
618C2×C4#88412yes—
619C2×C4C2×#23511yescharacteristic
620C2×C4C2×#23511yescharacteristic
621C2×C4C2×#23511yescharacteristic
622C2×C4C2×D3×D411yescharacteristic
623C2×C4#100911yescharacteristic
624C2×C4#101011yescharacteristic
625C8—22no—
626C8—41no—
627C8—62no—
628C8—62no—
629C2×C3—82no—
630C2×C3#242611yescharacteristic
631C2×C3#305511yescharacteristic
632C2×C3#306011yescharacteristic
633C2×C3#337311yescharacteristic
634C2×C3#337611yescharacteristic
635C2×C3#339711yescharacteristic
636C2×C3#340111yescharacteristic
637D3—161no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—241no—
646C22—241no—
647C22—241no—
648C22—241no—
649C22—241no—
650C22—241no—
651C22—241no—
652C22C2×#88411yescharacteristic
653C22#555211yescharacteristic
654C22#555311yescharacteristic
655C22D3×AGL(1,ℤ/8)11yescharacteristic
656C22#577311yescharacteristic
657C22#577511yescharacteristic
658C22#577811yescharacteristic
659C4—22no—
660C4—22no—
661C4—62no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—121no—
666C4—121no—
667C4—121no—
668C4—241no—
669C4—241no—
670C4—241no—
671C4#547911yescharacteristic
672C4#553411yescharacteristic
673C4#555011yescharacteristic
674C4#555211yescharacteristic
675C3#2530911yescharacteristic, complemented, Sylow
676C2—82no—
677C2—481no—
678C2#7348511yescharacteristic
679C2#7889011yescharacteristic
680C2#7893011yescharacteristic
681C2#7975711yescharacteristic
682C2#7976311yescharacteristic
683C2#7985311yescharacteristic
684C2#7985811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1762C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#1762D3
hypocenterC3#25309
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
1. #82376C2semidirect
3. C2×#4701C2semidirect
4. #73735C2semidirect
5. C3×#1762C2semidirect
6. #70217C2semidirect
9. #70189C2semidirect
10. #69821C2semidirect
11. #69467C2semidirect
12. #68912C2semidirect
13. #68900C2semidirect
14. #68854C2semidirect
15. #68852C2semidirect
76. #1762D3semidirect
675. C3#25309semidirect

Maximal subgroups

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

subgroupindex
#823762
C2×#50162
C2×#47012
#737352
C3×#17622
#702172
#702122
#701912
#701892
#698212
#694672
#689122
#689002
#688542
#688522
#253093