Finite Group Databasebeta

← #10858308

Subgroups — #10858308

Order 768.

828 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4829C211yesmaximal, characteristic, complemented
2C2×C4×#204C211yesmaximal, characteristic
3#73677C211yesmaximal, characteristic
4C4⋊C4×Dic6C211yesmaximal, characteristic, complemented
5C3×#1916C211yesmaximal, characteristic, complemented
6#70484C211yesmaximal, characteristic
7#69770C211yesmaximal, characteristic, complemented
8#68686C211yesmaximal, characteristic
9#68686C211yesmaximal, characteristic
10#68683C211yesmaximal, characteristic, complemented
11#68683C211yesmaximal, characteristic, complemented
12#68681C211yesmaximal, characteristic
13#68671C211yesmaximal, characteristic
14#68640C211yesmaximal, characteristic, complemented
15#68634C211yesmaximal, characteristic, complemented
16#26379—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×C4×D12C2211yescharacteristic
19C2×#870C2211yescharacteristic
20C2×C4×Dic6C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×#818C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37#4993—21no—
38#4989—21no—
39#4975—21no—
40Dic3×C4⋊C4—21no—
41#4939C2211yescharacteristic
42#4939C2211yescharacteristic
43#4935C2211yescharacteristic
44#4928C2211yescharacteristic
45#4928C2211yescharacteristic
46#4928C2211yescharacteristic
47#4928C2211yescharacteristic
48#4829—24no—
49C4×#204—24no—
50#4676—21no—
51#4667—21no—
52#4647—21no—
53#4647—21no—
54C3×#336C2211yescharacteristic
55C3×#336C2211yescharacteristic
56C3×#336C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#479—31no—
63C2×C4×SD16—31no—
64#2454—31no—
65Q8×C4⋊C4—31no—
66#2093—31no—
67#2084—31no—
68#2067—31no—
69#2059—31no—
70#1942—31no—
71#1916D311yescharacteristic, complemented
72#1880—31no—
73#1880—31no—
74#1864—31no—
75#1863—31no—
76#1863—31no—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×#204—22no—
115C2×#204—22no—
116C2×#204Q814yes—
117C4×D12—22no—
118C4×D12—41no—
119#870—22no—
120#870—41no—
121C4×Dic6—22no—
122C4×Dic6—22no—
123C4×Dic6—22no—
124C4×Dic6—22no—
125C4×Dic6—41no—
126C4×Dic6—41no—
127C3×C4⋊C8—22no—
128C3×C4×C8—22no—
129C3×#95D411yescharacteristic
130C3×#95D411yescharacteristic
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135#821—22no—
136#821—22no—
137#821—41no—
138#818—22no—
139#818—22no—
140#818—41no—
141#817—22no—
142#817—22no—
143#816—22no—
144#816—22no—
145#816D414yes—
146C22×SD16—31no—
147C2×#125—31no—
148C2×C4×Q8—31no—
149C2×C4×Q8—31no—
150C2×C4×Q8—31no—
151C2×C4×D4—31no—
152#479—64no—
153C4×SD16—64no—
154C2×#107—31no—
155C2×#107—31no—
156C2×#107—31no—
157C2×#106—31no—
158C2×#106—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×C4×C8C2×D311yescharacteristic
167#399—61no—
168#396—61no—
169#390—31no—
170#390—31no—
171#389—31no—
172#389—31no—
173#389—31no—
174#389—31no—
175#389—61no—
176#389—61no—
177#385—61no—
178#384—31no—
179#384—61no—
180#380—61no—
181C4×C4⋊C4—61no—
182C4×C4⋊C4C2×D311yescharacteristic
183#340—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#336C2×D311yescharacteristic
188#336C2×D311yescharacteristic
189#336C2×D311yescharacteristic
190#336C2×D311yescharacteristic
191C23×D3—21no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12C2×Q812yes—
212C2×C4×D3—42no—
213C2×C4×D3—42no—
214C2×C4×D3—81no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6C2×D412yes—
219C2×Dic6C2×Q812yes—
220C2×Dic6#4112yes—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8C2×D412yes—
232C2×C3×C8C2×Q812yes—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—21no—
240C3×C42—21no—
241(C2×D3)⋊C4—42no—
242(C2×D3)⋊C4—42no—
243(C2×D3)⋊C4—81no—
244(C2×D3)⋊C4—81no—
245(C2×D3)⋊C4—81no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—42no—
257C4⋊Dic3—81no—
258C4⋊Dic3C2×D412yes—
259C4⋊Dic3#4112yes—
260Dic3⋊C4—24no—
261Dic3⋊C4—42no—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264Dic3⋊C4—42no—
265Dic3⋊C4—81no—
266Dic3⋊C4—81no—
267Dic3⋊C4—81no—
268Dic3⋊C4—81no—
269C4×Dic3—24no—
270C4×Dic3—42no—
271C4×Dic3—42no—
272#204—28no—
273C22×Q8—31no—
274C22×D4—31no—
275C23×C4—61no—
276C2×SD16—34no—
277C2×SD16—62no—
278C2×SD16—62no—
279C22×C8C22×D311yescharacteristic
280C22×C8C22×D311yescharacteristic
281C22×C8C22×D311yescharacteristic
282C22×C8C22×D311yescharacteristic
283#125—62no—
284#125—121no—
285C4×Q8—62no—
286C4×Q8—62no—
287C4×Q8—62no—
288C4×Q8—62no—
289C4×Q8—121no—
290C4×Q8—121no—
291C4×D4—62no—
292C4×D4—121no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—61no—
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⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4D1211yescharacteristic
311C2×C4⋊C4D1211yescharacteristic
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C42—61no—
318C2×C42—61no—
319C2×C42—61no—
320C2×C42C22×D311yescharacteristic
321C2×C42C22×D311yescharacteristic
322C2×C42C22×D311yescharacteristic
323#107—62no—
324#107—62no—
325#107—121no—
326#106—62no—
327#106—62no—
328C4⋊C8—22no—
329#103—34no—
330#103—62no—
331#103—62no—
332#102—62no—
333#102—62no—
334#102—121no—
335C4×C8—22no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95D1211yescharacteristic
341#95D1211yescharacteristic
342C23×C3C22×D411yescharacteristic
343C22×D3—22no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
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#13512yes—
367C2×C3×C4#13712yes—
368C2×C3×C4C22×D411yescharacteristic
369C2×C3×C4C22×Q811yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4H(5,𝔽2)11yescharacteristic
373C2×C3×C42−1+411yescharacteristic
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
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—
395D12—22no—
396D12—22no—
397D12—41no—
398D12—41no—
399C4×D3—84no—
400Dic6—22no—
401Dic6—22no—
402Dic6—22no—
403Dic6C22⋊Q814yes—
404C3×C8—22no—
405C3×C8—22no—
406C3×C8—41no—
407C3×C8—41no—
408C24—61no—
409C2×Q8—32no—
410C2×Q8—32no—
411C2×Q8—32no—
412C2×Q8—61no—
413C2×Q8—61no—
414C2×Q8—61no—
415C2×D4—32no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—122no—
429C22×C4—122no—
430C22×C4—241no—
431C22×C4C2×D1211yescharacteristic
432C22×C4C2×D1211yescharacteristic
433C22×C4C2×D1211yescharacteristic
434C22×C4C2×D1211yescharacteristic
435C22×C4C2×D1211yescharacteristic
436C22×C4C2×D1211yescharacteristic
437C22×C4C23×D311yescharacteristic
438SD16—68no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8D3×D412yes—
450C2×C8D3×Q812yes—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—32no—
454C4⋊C4—32no—
455C4⋊C4—64no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—61no—
466C4⋊C4—122no—
467C4⋊C4—122no—
468C4⋊C4—122no—
469C4⋊C4—122no—
470C4⋊C4—122no—
471C4⋊C4—241no—
472C4⋊C4—241no—
473C4⋊C4—241no—
474C4⋊C4—241no—
475C4⋊C4—241no—
476C22⋊C4—122no—
477C22⋊C4—122no—
478C22⋊C4—241no—
479C22⋊C4—241no—
480C22⋊C4—241no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—64no—
488C42—122no—
489C42—122no—
490C22×C3#52511yescharacteristic
491C22×C3D4211yescharacteristic
492C22×C3D4×Q811yescharacteristic
493C22×C3C2×#13511yescharacteristic
494C22×C3C2×#13711yescharacteristic
495C22×C3#57611yescharacteristic
496C22×C3#57711yescharacteristic
497C2×D3—42no—
498C2×D3—42no—
499C2×D3—42no—
500C2×D3—81no—
501C2×D3—81no—
502C2×D3—81no—
503C2×D3—81no—
504C3×C4—22no—
505C3×C4—22no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—41no—
509C3×C4—41no—
510C3×C4C2×C22⋊Q811yescharacteristic
511C3×C4#53911yescharacteristic
512C3×C4#54711yescharacteristic
513C3×C4#55411yescharacteristic
514Dic3—24no—
515Dic3—42no—
516Dic3—42no—
517Dic3—81no—
518Dic3—81no—
519Dic3—81no—
520Dic3—81no—
521C23—62no—
522C23—121no—
523C23—121no—
524C23—121no—
525C23—121no—
526C23—121no—
527C23—121no—
528C23C22×D1211yescharacteristic
529Q8—34no—
530Q8—62no—
531Q8—62no—
532Q8—62no—
533D4—62no—
534D4—62no—
535D4—121no—
536D4—121no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—244no—
575C2×C4#90412yes—
576C2×C4#90912yes—
577C2×C4C2×D3×D411yescharacteristic
578C2×C4C2×D3×Q811yescharacteristic
579C2×C4D3×#4111yescharacteristic
580C2×C4D3×#4111yescharacteristic
581C2×C4#100911yescharacteristic
582C2×C4#101011yescharacteristic
583C8—22no—
584C8—22no—
585C8—41no—
586C8—41no—
587C2×C3#244911yescharacteristic
588C2×C3#310811yescharacteristic
589C2×C3#311911yescharacteristic
590C2×C3#326611yescharacteristic
591C2×C3#328711yescharacteristic
592C2×C3#338111yescharacteristic
593C2×C3#338511yescharacteristic
594D3—82no—
595C22—122no—
596C22—122no—
597C22—122no—
598C22—241no—
599C22—241no—
600C22—241no—
601C22—241no—
602C22#552411yescharacteristic
603C22D4×D1211yescharacteristic
604C22Q8×D1211yescharacteristic
605C22C2×#90411yescharacteristic
606C22C2×#90911yescharacteristic
607C22#575111yescharacteristic
608C22#575211yescharacteristic
609C4—22no—
610C4—22no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—64no—
616C4—122no—
617C4—122no—
618C4—241no—
619C4—241no—
620C4—241no—
621C4—241no—
622C4D3×C22⋊Q811yescharacteristic
623C4#565511yescharacteristic
624C4#569011yescharacteristic
625C4#570711yescharacteristic
626C3#2637911yescharacteristic, complemented, Sylow
627C2—242no—
628C2#7532111yescharacteristic
629C2#7931111yescharacteristic
630C2#7938411yescharacteristic
631C2#7944611yescharacteristic
632C2#7949511yescharacteristic
633C2#7993111yescharacteristic
634C2#7997611yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1916C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1916D3
hypocenterC3#26379
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
1. C2×#4829C2semidirect
4. C4⋊C4×Dic6C2semidirect
5. C3×#1916C2semidirect
7. #69770C2semidirect
10. #68683C2semidirect
11. #68683C2semidirect
14. #68640C2semidirect
15. #68634C2semidirect
71. #1916D3semidirect
626. C3#26379semidirect

Maximal subgroups

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

subgroupindex
C2×#48292
C2×C4×#2042
#736772
C4⋊C4×Dic62
C3×#19162
#704842
#697702
#686862
#686832
#686812
#686712
#686402
#686342
#263793