Finite Group Databasebeta

← #10855259

Subgroups — #10855259

Order 768.

944 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4844C211yesmaximal, characteristic
2C2×C4×D24C211yesmaximal, characteristic
3#73781C211yesmaximal, characteristic, complemented
4C4⋊C4×D12C211yesmaximal, characteristic, complemented
5C3×#1916C211yesmaximal, characteristic, complemented
6#70463C211yesmaximal, characteristic, complemented
7#69767C211yesmaximal, characteristic, complemented
8#68691C211yesmaximal, characteristic, complemented
9#68673C211yesmaximal, characteristic
10#68672C211yesmaximal, characteristic, complemented
11#68672C211yesmaximal, characteristic, complemented
12#68670C211yesmaximal, characteristic, complemented
13#68670C211yesmaximal, characteristic, complemented
14#68635C211yesmaximal, characteristic, complemented
15#68630C211yesmaximal, characteristic, complemented
16#26164—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C22×#204C2211yescharacteristic
19C2×D3×C4⋊C4—21no—
20C2×#875C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×C4×C8C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×#818C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38#4992—21no—
39#4989—21no—
40#4987—21no—
41#4939C2211yescharacteristic
42#4935C2211yescharacteristic
43#4929C2211yescharacteristic
44#4929C2211yescharacteristic
45#4928C2211yescharacteristic
46#4928C2211yescharacteristic
47#4928C2211yescharacteristic
48#4844—24no—
49C4×D24—24no—
50#4675—21no—
51#4668—21no—
52#4668—21no—
53#4664—21no—
54C3×#336C2211yescharacteristic
55C3×#336C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#463—31no—
63C2×C4×D8—31no—
64#2447—31no—
65D4×C4⋊C4—31no—
66#2095—31no—
67#2087—31no—
68#2071—31no—
69#2042—31no—
70#1933—31no—
71#1916D311yescharacteristic, complemented
72#1879—31no—
73#1877—31no—
74#1877—31no—
75#1862—31no—
76#1862—31no—
77C22×D12C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×C4×D3—21no—
83C22×Dic6C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
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)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×D24—22no—
112C2×D24—22no—
113C2×D24D414yes—
114C2×#204—22no—
115C2×#204—22no—
116C2×#204—22no—
117D3×C4⋊C4—44no—
118#875—22no—
119#875—41no—
120C4×D12—22no—
121C4×D12—22no—
122C4×D12—22no—
123C4×D12—22no—
124C4×D12—22no—
125C4×D12—41no—
126C4×D12—41no—
127C4×D12—41no—
128C3×C4⋊C8—22no—
129C3×C4×C8—22no—
130C3×#95D411yescharacteristic
131C3×#95D411yescharacteristic
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139#821—22no—
140#821—22no—
141#821—22no—
142#821—22no—
143#821—22no—
144#818—22no—
145#818—41no—
146#816—22no—
147#816—22no—
148#816—41no—
149C22×SD16—31no—
150C22×D8—31no—
151C2×#124—31no—
152C2×C4×D4—31no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155C2×C4×D4—31no—
156C22×C4⋊C4—61no—
157#463—64no—
158C4×D8—64no—
159C2×#107—31no—
160C2×#107—31no—
161C2×C4⋊C8C2×D311yescharacteristic
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×C4×C8C2×D311yescharacteristic
171#397—61no—
172#397—61no—
173#391—31no—
174#391—31no—
175#390—31no—
176#389—31no—
177#389—31no—
178#389—31no—
179#385—61no—
180#385—61no—
181#385—61no—
182#384—31no—
183#381—61no—
184#380—61no—
185C4×C4⋊C4C2×D311yescharacteristic
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#336C2×D311yescharacteristic
191#336C2×D311yescharacteristic
192#336C2×D311yescharacteristic
193#336C2×D311yescharacteristic
194C23×D3—21no—
195C23×D3—21no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2411yescharacteristic
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×D12—21no—
216C2×D12—21no—
217C2×D12C2×D412yes—
218C2×D12C2×D412yes—
219C2×D12#4112yes—
220C2×D12#4112yes—
221C2×C4×D3—24no—
222C2×C4×D3—42no—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×C4×D3—42no—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×C4×D3—81no—
231C2×C4×D3—81no—
232C2×C4×D3—81no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8C2×D412yes—
251C3×C4⋊C4—22no—
252C3×C4⋊C4—22no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—21no—
259(C2×D3)⋊C4—24no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—42no—
266(C2×D3)⋊C4—81no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—42no—
277C4⋊Dic3—81no—
278C4⋊Dic3#4112yes—
279Dic3⋊C4—42no—
280Dic3⋊C4—42no—
281Dic3⋊C4—81no—
282D24—28no—
283#204—44no—
284C22×Q8—31no—
285C22×D4—31no—
286C22×D4—31no—
287C23×C4—61no—
288C23×C4—61no—
289C23×C4—61no—
290C23×C4—61no—
291C2×SD16—62no—
292C2×SD16—62no—
293C2×SD16—62no—
294C2×D8—34no—
295C2×D8—62no—
296C2×D8—62no—
297C22×C8C22×D311yescharacteristic
298C22×C8C22×D311yescharacteristic
299C22×C8C22×D311yescharacteristic
300C22×C8C22×D311yescharacteristic
301#124—62no—
302#124—121no—
303C4×D4—62no—
304C4×D4—62no—
305C4×D4—62no—
306C4×D4—62no—
307C4×D4—62no—
308C4×D4—121no—
309C4×D4—121no—
310C4×D4—121no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—124no—
322C2×C4⋊C4D1211yescharacteristic
323C2×C4⋊C4D1211yescharacteristic
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C22⋊C4—61no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C42C22×D311yescharacteristic
333C2×C42C22×D311yescharacteristic
334C2×C42C22×D311yescharacteristic
335#107—62no—
336#107—121no—
337C4⋊C8—22no—
338#103—62no—
339#103—62no—
340#103—121no—
341#102—62no—
342#102—62no—
343#102—62no—
344#102—62no—
345#102—62no—
346C4×C8—22no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95D1211yescharacteristic
355#95D1211yescharacteristic
356C23×C3C22×D411yescharacteristic
357C22×D3—22no—
358C22×D3—22no—
359C22×D3—22no—
360C22×D3—22no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
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#13512yes—
388C2×C3×C4AGL(1,ℤ/8)12yes—
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—22no—
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—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413D12—22no—
414D12—22no—
415D12—22no—
416D12—22no—
417D12—22no—
418D12—41no—
419D12—41no—
420D12#12314yes—
421C4×D3—48no—
422C4×D3—84no—
423C4×D3—84no—
424C4×D3—84no—
425Dic6—41no—
426Dic6—41no—
427Dic6—41no—
428Dic6—41no—
429C3×C8—22no—
430C3×C8—41no—
431C3×C8—41no—
432C3×C8—41no—
433C24—61no—
434C24—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×D4—32no—
442C2×D4—32no—
443C2×D4—32no—
444C2×D4—32no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C22×C4—64no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—122no—
461C22×C4—122no—
462C22×C4—122no—
463C22×C4—122no—
464C22×C4—122no—
465C22×C4—122no—
466C22×C4—122no—
467C22×C4—122no—
468C22×C4—241no—
469C22×C4—241no—
470C22×C4—241no—
471C22×C4C2×D1211yescharacteristic
472C22×C4C2×D1211yescharacteristic
473C22×C4C2×D1211yescharacteristic
474C22×C4C2×D1211yescharacteristic
475C22×C4C2×D1211yescharacteristic
476C22×C4C2×D1211yescharacteristic
477C22×C4C23×D311yescharacteristic
478SD16—124no—
479D8—68no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8D3×D412yes—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—32no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—122no—
505C4⋊C4—122no—
506C4⋊C4—122no—
507C4⋊C4—241no—
508C4⋊C4—241no—
509C22⋊C4—64no—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C22⋊C4—122no—
515C22⋊C4—122no—
516C22⋊C4—241no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C22×C3#52511yescharacteristic
524C22×C3D4211yescharacteristic
525C22×C3#54811yescharacteristic
526C22×C3C2×#13511yescharacteristic
527C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
528C22×C3#57611yescharacteristic
529C22×C3#57811yescharacteristic
530C2×D3—24no—
531C2×D3—42no—
532C2×D3—42no—
533C2×D3—42no—
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—42no—
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—42no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C3×C4—22no—
545C3×C4—22no—
546C3×C4—41no—
547C3×C4—41no—
548C3×C4—41no—
549C3×C4—41no—
550C3×C4C2×#12311yescharacteristic
551C3×C4#53911yescharacteristic
552C3×C4#54711yescharacteristic
553C3×C4#55311yescharacteristic
554Dic3—42no—
555Dic3—81no—
556Dic3—81no—
557Dic3—81no—
558Dic3—81no—
559Dic3—81no—
560C23—62no—
561C23—62no—
562C23—62no—
563C23—62no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23—121no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23C22×D1211yescharacteristic
575Q8—121no—
576Q8—121no—
577Q8—121no—
578Q8—121no—
579D4—34no—
580D4—62no—
581D4—62no—
582D4—62no—
583D4—62no—
584D4—62no—
585D4—121no—
586D4—121no—
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—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—62no—
604C2×C4—128no—
605C2×C4—121no—
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—121no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—244no—
623C2×C4—244no—
624C2×C4—244no—
625C2×C4#90412yes—
626C2×C4#90812yes—
627C2×C4C2×D3×D411yescharacteristic
628C2×C4#100711yescharacteristic
629C2×C4D3×#4111yescharacteristic
630C2×C4D3×#4111yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4#100911yescharacteristic
633C8—22no—
634C8—41no—
635C8—41no—
636C8—41no—
637C2×C3#244311yescharacteristic
638C2×C3#310511yescharacteristic
639C2×C3#312411yescharacteristic
640C2×C3#326411yescharacteristic
641C2×C3#328311yescharacteristic
642C2×C3#336111yescharacteristic
643C2×C3#336711yescharacteristic
644D3—44no—
645D3—82no—
646C22—64no—
647C22—122no—
648C22—122no—
649C22—122no—
650C22—122no—
651C22—122no—
652C22—122no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—241no—
657C22—241no—
658C22—241no—
659C22—241no—
660C22#552411yescharacteristic
661C22D4×D1211yescharacteristic
662C22#557611yescharacteristic
663C22C2×#90411yescharacteristic
664C22C2×#90811yescharacteristic
665C22#575211yescharacteristic
666C22#575311yescharacteristic
667C4—22no—
668C4—22no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—122no—
674C4—241no—
675C4—241no—
676C4—241no—
677C4—241no—
678C4—241no—
679C4#563111yescharacteristic
680C4D3×#12311yescharacteristic
681C4#569211yescharacteristic
682C4#570911yescharacteristic
683C3#2616411yescharacteristic, complemented, Sylow
684C2—124no—
685C2—242no—
686C2#7531911yescharacteristic
687C2#7930011yescharacteristic
688C2#7938811yescharacteristic
689C2#7943611yescharacteristic
690C2#7949311yescharacteristic
691C2#7992711yescharacteristic
692C2#7997011yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1916C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1916D3
hypocenterC3#26164
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1916

Complements

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

NH ≅ G/Ntype
3. #73781C2semidirect
4. C4⋊C4×D12C2semidirect
5. C3×#1916C2semidirect
6. #70463C2semidirect
7. #69767C2semidirect
8. #68691C2semidirect
10. #68672C2semidirect
11. #68672C2semidirect
12. #68670C2semidirect
13. #68670C2semidirect
14. #68635C2semidirect
15. #68630C2semidirect
71. #1916D3semidirect
683. C3#26164semidirect

Maximal subgroups

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

subgroupindex
C2×#48442
C2×C4×D242
#737812
C4⋊C4×D122
C3×#19162
#704632
#697672
#686912
#686732
#686722
#686702
#686352
#686302
#261643