Finite Group Databasebeta

← #10844596

Subgroups — #10844596

Order 768.

822 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4850C211yesmaximal, characteristic, complemented
2C2×#4714C211yesmaximal, characteristic
3#73710C211yesmaximal, characteristic
4#73650C211yesmaximal, characteristic, complemented
5C3×#1940C211yesmaximal, characteristic, complemented
6#70489C211yesmaximal, characteristic, complemented
7#70484C211yesmaximal, characteristic
8#70458C211yesmaximal, characteristic
9#70458C211yesmaximal, characteristic
10#70456C211yesmaximal, characteristic, complemented
11#69813C211yesmaximal, characteristic, complemented
12#68631C211yesmaximal, characteristic, complemented
13#68630C211yesmaximal, characteristic
14#68628C211yesmaximal, characteristic, complemented
15#68625C211yesmaximal, characteristic
16#25286—31nomaximal, Sylow
17C2×#887—21no—
18C2×#875C2211yescharacteristic
19C2×#875C2211yescharacteristic
20C2×#869C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#97C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#818C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×#816C2211yescharacteristic
42#4994—21no—
43#4994—21no—
44#4973—21no—
45#4939C2211yescharacteristic
46C4×C4⋊Dic3C2211yescharacteristic
47C4×C4⋊Dic3C2211yescharacteristic
48C4×C4⋊Dic3C2211yescharacteristic
49#4929C2211yescharacteristic
50#4928C2211yescharacteristic
51#4928C2211yescharacteristic
52#4850—24no—
53#4714—24no—
54#4670—21no—
55#4670—21no—
56#4653—21no—
57#4653—21no—
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#489—31no—
63C2×#477—31no—
64#2545—31no—
65#2365—31no—
66#2068—31no—
67#2067—31no—
68#2043—31no—
69#2042—31no—
70#1983—31no—
71#1983—31no—
72#1981—31no—
73#1940D311yescharacteristic, complemented
74#1928—31no—
75#1925—31no—
76#1763—31no—
77C22×D12C2311yescharacteristic
78C22×Dic6—21no—
79C22×Dic6C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C4⋊C4D411yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111#887—44no—
112#875—22no—
113#875—22no—
114#875—22no—
115#875—41no—
116#869—22no—
117#869—22no—
118#869—41no—
119#869D414yes—
120C4×Dic6—22no—
121C4×Dic6—22no—
122C3×C4⋊C8—22no—
123C3×C4⋊C8—41no—
124C3×C4⋊C8—41no—
125C3×#97—22no—
126C3×#95D411yescharacteristic
127C3×#95D411yescharacteristic
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135#821—22no—
136#821—22no—
137#821—22no—
138#821—41no—
139#818—22no—
140#818—22no—
141#816—22no—
142#816—22no—
143#816—22no—
144#816—22no—
145#816—22no—
146#816—22no—
147#816—22no—
148#816—41no—
149C2×C4⋊Q8—31no—
150C2×C4⋊Q8—31no—
151C2×#124—31no—
152C2×#124—31no—
153C2×C4×Q8—31no—
154C2×C4×Q8—61no—
155#489—64no—
156#477—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×C4⋊C8C2×D311yescharacteristic
161C2×C4⋊C8C2×D311yescharacteristic
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#102—31no—
171C2×#102—31no—
172C2×#102—31no—
173C2×#102—31no—
174C2×#97C2×D311yescharacteristic
175#396—61no—
176#396—61no—
177#391—31no—
178#391—61no—
179#390—31no—
180#389—31no—
181#389—31no—
182#388—61no—
183#388—61no—
184#387—61no—
185#387—61no—
186C4×C4⋊C4—31no—
187C4×C4⋊C4—31no—
188C4×C4⋊C4—31no—
189C4×C4⋊C4C2×D311yescharacteristic
190#340—31no—
191#340—31no—
192#336C2×D311yescharacteristic
193#336C2×D311yescharacteristic
194C23×D3—21no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2411yescharacteristic
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—24no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—42no—
228C2×Dic6—42no—
229C2×Dic6#4112yes—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—41no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42C2×D412yes—
249C3×C42#4112yes—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256C4⋊Dic3—24no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—42no—
265C4⋊Dic3—42no—
266C4⋊Dic3C2×D412yes—
267C4⋊Dic3C2×D412yes—
268C4⋊Dic3#4112yes—
269Dic3⋊C4—42no—
270Dic3⋊C4—42no—
271Dic3⋊C4—42no—
272C4×Dic3—42no—
273C4×Dic3—42no—
274C4×Dic3—42no—
275C4×Dic3—42no—
276C22×Q8—31no—
277C22×Q8—31no—
278C22×Q8—61no—
279C22×D4—31no—
280C22×C8C22×D311yescharacteristic
281C22×C8C22×D311yescharacteristic
282C22×C8C22×D311yescharacteristic
283C22×C8C22×D311yescharacteristic
284C4⋊Q8—34no—
285C4⋊Q8—62no—
286C4⋊Q8—62no—
287C4⋊Q8—121no—
288#124—62no—
289#124—62no—
290#124—62no—
291#124—121no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×Q8—124no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4D1211yescharacteristic
307C2×C4⋊C4D1211yescharacteristic
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42—61no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319C2×C42C22×D311yescharacteristic
320C2×C42C22×D311yescharacteristic
321#107—62no—
322#107—62no—
323C4⋊C8—22no—
324C4⋊C8—41no—
325C4⋊C8—41no—
326#103—62no—
327#103—62no—
328#103—62no—
329#103—62no—
330#103—62no—
331#103—62no—
332#103—62no—
333#103—121no—
334#102—62no—
335#102—62no—
336#102—62no—
337#102—121no—
338#97—22no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95D1211yescharacteristic
347#95D1211yescharacteristic
348C23×C3C22×D411yescharacteristic
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
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#12112yes—
372C2×C3×C4#12112yes—
373C2×C3×C4AGL(1,ℤ/8)12yes—
374C2×C3×C4C22×D411yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4H(5,𝔽2)11yescharacteristic
379C2×C3×C42−1+411yescharacteristic
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401D12—41no—
402D12—41no—
403D12—41no—
404D12—41no—
405Dic6—22no—
406Dic6—22no—
407Dic6—48no—
408Dic6—41no—
409Dic6—41no—
410Dic6—41no—
411Dic6—41no—
412Dic6—41no—
413Dic6—41no—
414C3×C8—41no—
415C3×C8—41no—
416C3×C8—41no—
417C3×C8—41no—
418C24—61no—
419C2×Q8—32no—
420C2×Q8—64no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—122no—
433C2×Q8—122no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4C2×D1211yescharacteristic
448C22×C4C2×D1211yescharacteristic
449C22×C4C2×D1211yescharacteristic
450C22×C4C2×D1211yescharacteristic
451C22×C4C2×D1211yescharacteristic
452C22×C4C2×D1211yescharacteristic
453C22×C4C23×D311yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C4⋊C4—22no—
467C4⋊C4—32no—
468C4⋊C4—32no—
469C4⋊C4—32no—
470C4⋊C4—41no—
471C4⋊C4—64no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—122no—
480C4⋊C4—122no—
481C4⋊C4—122no—
482C4⋊C4—122no—
483C4⋊C4—122no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C22⋊C4—122no—
489C22⋊C4—122no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—122no—
495C42—122no—
496C42—122no—
497C42—122no—
498C42D3×D412yes—
499C42#23912yes—
500C22×C3C2×#12111yescharacteristic
501C22×C3#54611yescharacteristic
502C22×C3#54811yescharacteristic
503C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
504C22×C3#57511yescharacteristic
505C22×C3#57811yescharacteristic
506C22×C3#57811yescharacteristic
507C2×D3—81no—
508C2×D3—81no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C2×D3—81no—
513C2×D3—81no—
514C3×C4—22no—
515C3×C4—22no—
516C3×C4—22no—
517C3×C4—41no—
518C3×C4—41no—
519C3×C4—41no—
520C3×C4#52911yescharacteristic
521C3×C4#54211yescharacteristic
522C3×C4#54811yescharacteristic
523C3×C4#55011yescharacteristic
524Dic3—42no—
525Dic3—42no—
526Dic3—42no—
527Dic3—42no—
528Dic3—81no—
529Dic3—81no—
530Dic3—81no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23C22×D1211yescharacteristic
539Q8—62no—
540Q8—62no—
541Q8—128no—
542Q8—121no—
543Q8—121no—
544Q8—121no—
545Q8—121no—
546Q8—121no—
547Q8—121no—
548D4—121no—
549D4—121no—
550D4—121no—
551D4—121no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
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—62no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4#89512yes—
589C2×C4#89512yes—
590C2×C4#90812yes—
591C2×C4C2×D3×D411yescharacteristic
592C2×C4#100411yescharacteristic
593C2×C4C2×#23911yescharacteristic
594C2×C4#100711yescharacteristic
595C2×C4D3×#4111yescharacteristic
596C2×C4D3×#4111yescharacteristic
597C8—41no—
598C8—41no—
599C8—41no—
600C8—41no—
601C2×C3#242611yescharacteristic
602C2×C3#303611yescharacteristic
603C2×C3#304411yescharacteristic
604C2×C3#330811yescharacteristic
605C2×C3#331311yescharacteristic
606C2×C3#336711yescharacteristic
607C2×C3#337711yescharacteristic
608D3—161no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22—241no—
615C22—241no—
616C22C2×#89511yescharacteristic
617C22#555411yescharacteristic
618C22#557611yescharacteristic
619C22C2×#90811yescharacteristic
620C22#574811yescharacteristic
621C22#575311yescharacteristic
622C22#575311yescharacteristic
623C4—22no—
624C4—22no—
625C4—22no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—122no—
630C4—122no—
631C4—122no—
632C4—122no—
633C4—241no—
634C4—241no—
635C4—241no—
636C4#559911yescharacteristic
637C4#563311yescharacteristic
638C4#569111yescharacteristic
639C4#573311yescharacteristic
640C3#2528611yescharacteristic, complemented, Sylow
641C2—481no—
642C2#7621811yescharacteristic
643C2#7905511yescharacteristic
644C2#7910011yescharacteristic
645C2#7950911yescharacteristic
646C2#7951211yescharacteristic
647C2#7997011yescharacteristic
648C2#7998511yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1940C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1940D3
hypocenterC3#25286
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1940

Complements

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

NH ≅ G/Ntype
1. C2×#4850C2semidirect
4. #73650C2semidirect
5. C3×#1940C2semidirect
6. #70489C2semidirect
10. #70456C2semidirect
11. #69813C2semidirect
12. #68631C2semidirect
14. #68628C2semidirect
73. #1940D3semidirect
640. C3#25286semidirect

Maximal subgroups

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

subgroupindex
C2×#48502
C2×#47142
#737102
#736502
C3×#19402
#704892
#704842
#704582
#704562
#698132
#686312
#686302
#686282
#686252
#252863