Finite Group Databasebeta

← #10855109

Subgroups — #10855109

Order 768.

846 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4861C211yesmaximal, characteristic, complemented
2C2×#4868C211yesmaximal, characteristic, complemented
3#76210C211yesmaximal, characteristic
4#73771C211yesmaximal, characteristic
5C3×#1920C211yesmaximal, characteristic, complemented
6#69609C211yesmaximal, characteristic, complemented
7#69465C211yesmaximal, characteristic, complemented
8#68884C211yesmaximal, characteristic, complemented
9#68878C211yesmaximal, characteristic, complemented
10#68876C211yesmaximal, characteristic, complemented
11#68876C211yesmaximal, characteristic, complemented
12#68831C211yesmaximal, characteristic, complemented
13#68824C211yesmaximal, characteristic, complemented
14#68693C211yesmaximal, characteristic, complemented
15#68669C211yesmaximal, characteristic
16#26154—31nomaximal, Sylow
17C22×ΓL(1,𝔽25)C2211yescharacteristic
18C2×#897C2211yescharacteristic
19C2×#897C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×#892C2211yescharacteristic
22C2×D3×C4⋊C4C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×#817C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34#4991—21no—
35#4987C2211yescharacteristic
36#4987C2211yescharacteristic
37#4987C2211yescharacteristic
38#4984—21no—
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#807C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4940—21no—
45C4×(C2×D3)⋊C4—21no—
46#4936C2211yescharacteristic
47#4929C2211yescharacteristic
48#4868—24no—
49#4861—24no—
50#4674—21no—
51#4670—21no—
52#4665—21no—
53#4653—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#485—31no—
63C2×#428—31no—
64#2415—31no—
65#2352—31no—
66#2097—31no—
67#2085—31no—
68#2076—31no—
69#2035—31no—
70#1928—31no—
71#1920D311yescharacteristic, complemented
72#1894—31no—
73#1872—31no—
74#1870—31no—
75#1854—31no—
76#1854—31no—
77C22×D12D411yescharacteristic
78C22×C4×D3—21no—
79C22×C4×D3C2311yescharacteristic
80C22×C4×D3C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
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×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4D411yescharacteristic
99C2×(C2×D3)⋊C4D411yescharacteristic
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3D411yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C2×ΓL(1,𝔽25)—22no—
112C2×ΓL(1,𝔽25)—22no—
113C2×ΓL(1,𝔽25)—22no—
114#897—22no—
115#897—22no—
116#897—22no—
117#897—41no—
118#892—22no—
119#892—22no—
120#892—41no—
121#892—41no—
122D3×C4⋊C4—22no—
123D3×C4⋊C4—41no—
124C4×D12—22no—
125C4×D12—22no—
126C3×#106D414yescomplemented
127C3×#95—21no—
128C3×#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136(C2×D3)⋊C8—22no—
137(C2×D3)⋊C8—22no—
138(C2×D3)⋊C8—22no—
139#818—22no—
140#817—41no—
141#816—22no—
142#816—41no—
143#810—22no—
144#810—22no—
145#808—22no—
146#807—22no—
147#807—22no—
148C22×M16—31no—
149C2×C22⋊Q8—31no—
150C2×C22⋊Q8—31no—
151C2×C4×D4—31no—
152C2×#117—31no—
153C2×#117—31no—
154C22×C4⋊C4—31no—
155#485—64no—
156#428—64no—
157C2×#107—31no—
158C2×#107—31no—
159C2×#107—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×#106C2×D311yescharacteristic
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×C22⋊C8—31no—
170#397—61no—
171#394—61no—
172#391—31no—
173#389—61no—
174#389C2×D311yescharacteristic
175#388—61no—
176#388—61no—
177#387—61no—
178#386—61no—
179#382—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183C4×C4⋊C4C2×D311yescharacteristic
184C4×C22⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340C2×D311yescharacteristic
189#340C2×D311yescharacteristic
190#340C2×D311yescharacteristic
191#336—31no—
192#336C2×D311yescharacteristic
193C23×D3C2×D411yescharacteristic
194C23×D3C2×D411yescharacteristic
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2411yescharacteristic
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C2×D12—22no—
209C2×D12—41no—
210C2×D12—41no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—21no—
214C2×C4×D3—21no—
215C2×C4×D3—21no—
216C2×C4×D3—21no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—42no—
222C2×C4×D3—42no—
223C2×C4×D3—81no—
224C2×C4×D3#4112yes—
225C2×C4×D3#4112yes—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8C2×D412yes—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—42no—
245C3×C4⋊C4C2×D412yes—
246C3×C4⋊C4C2×D412yes—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—81no—
250(C2×D3)⋊C4—22no—
251(C2×D3)⋊C4—22no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4—42no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—81no—
259C4⋊Dic3—22no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—42no—
267C4⋊Dic3—42no—
268Dic3⋊C4—42no—
269C4×Dic3—42no—
270C4×Dic3—42no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8#4112yes—
277ΓL(1,𝔽25)—28no—
278C22×Q8—31no—
279C22×D4—31no—
280C23×C4—31no—
281C23×C4—31no—
282C23×C4—61no—
283C2×M16—62no—
284C2×M16—62no—
285C2×M16—62no—
286C22×C8—31no—
287C22×C8—31no—
288C22×C8C22×D311yescharacteristic
289C22×C8C22×D311yescharacteristic
290C22⋊Q8—62no—
291C22⋊Q8—62no—
292C22⋊Q8—62no—
293C22⋊Q8—121no—
294C4×D4—62no—
295C4×D4—62no—
296#117—62no—
297#117—62no—
298#117—121no—
299#117—121no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—62no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—121no—
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C22⋊C4—31no—
315C2×C22⋊C4—31no—
316C2×C22⋊C4—61no—
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×C42—21no—
324C2×C42—61no—
325C2×C42—61no—
326C2×C42C22×D311yescharacteristic
327#107—62no—
328#107—62no—
329#107—62no—
330#106—62no—
331#106—121no—
332#106D1214yescomplemented
333#103—62no—
334#103—62no—
335#103—62no—
336#103—121no—
337C22⋊C8—62no—
338C22⋊C8—62no—
339C22⋊C8—62no—
340#95—21no—
341#95—21no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349C23×C3C22×D411yescharacteristic
350C22×D3—21no—
351C22×D3—21no—
352C22×D3—21no—
353C22×D3—21no—
354C22×D3—21no—
355C22×D3—21no—
356C22×D3—21no—
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—21no—
363C22×D3#12312yes—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4H(5,𝔽2)11yescharacteristic
387C2×C3×C42−1+411yescharacteristic
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3#12312yes—
406D12—44no—
407C4×D3—22no—
408C4×D3—22no—
409C4×D3—22no—
410C4×D3—22no—
411C4×D3—41no—
412C4×D3—41no—
413C4×D3—41no—
414C4×D3—41no—
415C4×D3—84no—
416Dic6—41no—
417Dic6—41no—
418Dic6—41no—
419Dic6—41no—
420C3×C8—22no—
421C3×C8—41no—
422C3⋊C8—22no—
423C3⋊C8—41no—
424C24—31no—
425C24—31no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×D4—62no—
433C2×D4—121no—
434C2×D4—121no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—31no—
440C22×C4—32no—
441C22×C4—32no—
442C22×C4—31no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—122no—
458C22×C4—122no—
459C22×C4—241no—
460C22×C4D3×D411yescharacteristic
461C22×C4D3×D411yescharacteristic
462C22×C4C23×D311yescharacteristic
463M16—68no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—32no—
470C2×C8—61no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8C2×D1212yes—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—42no—
483C4⋊C4—62no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—122no—
491C4⋊C4—122no—
492C4⋊C4—122no—
493C4⋊C4C2×D1212yes—
494C4⋊C4C2×D1212yes—
495C22⋊C4—62no—
496C22⋊C4—62no—
497C22⋊C4—122no—
498C22⋊C4—122no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—241no—
504C42—21no—
505C42—21no—
506C42—81no—
507C42—122no—
508C42—122no—
509C22×C3C2×#12311yescharacteristic
510C22×C3D4211yescharacteristic
511C22×C3#54711yescharacteristic
512C22×C3#57511yescharacteristic
513C22×C3#57511yescharacteristic
514C22×C3#57711yescharacteristic
515C22×C3#57811yescharacteristic
516C2×D3—22no—
517C2×D3—22no—
518C2×D3—22no—
519C2×D3—41no—
520C2×D3—41no—
521C2×D3—41no—
522C2×D3—41no—
523C2×D3—41no—
524C2×D3—41no—
525C2×D3—41no—
526C2×D3—41no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—41no—
530C3×C4—41no—
531C3×C4—41no—
532C3×C4—42no—
533C3×C4—42no—
534C3×C4—81no—
535C3×C4—81no—
536C3×C4#52511yescharacteristic
537C3×C4#53911yescharacteristic
538C3×C4#54811yescharacteristic
539C3×C4#55311yescharacteristic
540Dic3—22no—
541Dic3—41no—
542Dic3—81no—
543Dic3—81no—
544Dic3—81no—
545Dic3—81no—
546C23—32no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23—61no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23—61no—
556C23—61no—
557C23—61no—
558C23—61no—
559C23—61no—
560C23C2×D3×D411yescharacteristic
561Q8—121no—
562Q8—121no—
563Q8—121no—
564Q8—121no—
565D4—124no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—32no—
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—61no—
586C2×C4—61no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—244no—
611C2×C4C22×D1211yescharacteristic
612C2×C4#100711yescharacteristic
613C2×C4D3×#4111yescharacteristic
614C2×C4D3×#4111yescharacteristic
615C2×C4D3×#4111yescharacteristic
616C2×C4#100911yescharacteristic
617C8—22no—
618C8—41no—
619C8—62no—
620C8—121no—
621C2×C3#244311yescharacteristic
622C2×C3#309311yescharacteristic
623C2×C3#309711yescharacteristic
624C2×C3#326911yescharacteristic
625C2×C3#327511yescharacteristic
626C2×C3#335011yescharacteristic
627C2×C3#335211yescharacteristic
628D3—42no—
629D3—81no—
630C22—62no—
631C22—62no—
632C22—62no—
633C22—121no—
634C22—121no—
635C22—121no—
636C22—121no—
637C22—121no—
638C22—121no—
639C22—121no—
640C22—121no—
641C22—121no—
642C22—121no—
643C22—121no—
644C22D4×D1211yescharacteristic
645C22#563111yescharacteristic
646C22D3×#12311yescharacteristic
647C22#576211yescharacteristic
648C22#576811yescharacteristic
649C22#577311yescharacteristic
650C22#577911yescharacteristic
651C4—41no—
652C4—42no—
653C4—42no—
654C4—41no—
655C4—62no—
656C4—81no—
657C4—81no—
658C4—121no—
659C4—241no—
660C4—241no—
661C4—241no—
662C4—241no—
663C4#552411yescharacteristic
664C4#557611yescharacteristic
665C4#569211yescharacteristic
666C4#570911yescharacteristic
667C3#2615411yescharacteristic, complemented, Sylow
668C2—122no—
669C2—241no—
670C2#7531911yescharacteristic
671C2#7967111yescharacteristic
672C2#7979711yescharacteristic
673C2#8108411yescharacteristic
674C2#8126011yescharacteristic
675C2#8137211yescharacteristic
676C2#8142911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1920C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenter#1920D3
hypocenterC3#26154
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
1. C2×#4861C2semidirect
2. C2×#4868C2semidirect
5. C3×#1920C2semidirect
6. #69609C2semidirect
7. #69465C2semidirect
8. #68884C2semidirect
9. #68878C2semidirect
10. #68876C2semidirect
11. #68876C2semidirect
12. #68831C2semidirect
13. #68824C2semidirect
14. #68693C2semidirect
71. #1920D3semidirect
126. C3×#106D4semidirect
332. #106D12semidirect
667. C3#26154semidirect

Maximal subgroups

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

subgroupindex
C2×#48612
C2×#48682
#762102
#737712
C3×#19202
#696092
#694652
#688842
#688782
#688762
#688312
#688242
#686932
#686692
#261543