Finite Group Databasebeta

← #10844756

Subgroups — #10844756

Order 768.

828 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#446C211yesmaximal, characteristic, complemented
2#82501C211yesmaximal, characteristic, complemented
3#82484C211yesmaximal, characteristic, complemented
4C2×#4844C211yesmaximal, characteristic
5#70488C211yesmaximal, characteristic, complemented
6#70484C211yesmaximal, characteristic, complemented
7#70482C211yesmaximal, characteristic, complemented
8#70481C211yesmaximal, characteristic
9#70455C211yesmaximal, characteristic, complemented
10#69826C211yesmaximal, characteristic, complemented
11#69825C211yesmaximal, characteristic
12#69818C211yesmaximal, characteristic, complemented
13#69817C211yesmaximal, characteristic, complemented
14#69773C211yesmaximal, characteristic, complemented
15C4×#821C211yesmaximal, characteristic, complemented
16#25298—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C22×D24C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C2×#875C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C3×#446—24no—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#97C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29#5231—21no—
30#5231—21no—
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#818C2211yescharacteristic
37C2×#817C2211yescharacteristic
38C2×#816C2211yescharacteristic
39C2×#816C2211yescharacteristic
40C2×#816C2211yescharacteristic
41#4993—21no—
42#4992—21no—
43#4990—21no—
44#4990—21no—
45#4989—21no—
46#4989—21no—
47#4939C2211yescharacteristic
48#4939C2211yescharacteristic
49#4939C2211yescharacteristic
50#4939C2211yescharacteristic
51#4936C2211yescharacteristic
52C4×C4⋊Dic3C2211yescharacteristic
53#4929C2211yescharacteristic
54#4929C2211yescharacteristic
55#4928C2211yescharacteristic
56#4928C2211yescharacteristic
57#4844—24no—
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4551C2211yescharacteristic
62C2×#463—31no—
63C2×#446D311yescharacteristic, complemented
64#2592—31no—
65#2580—31no—
66#2067—31no—
67#2066—31no—
68#2055—31no—
69#2054—31no—
70#1973—31no—
71#1972—31no—
72#1958—31no—
73#1949—31no—
74#1936—31no—
75#1935—31no—
76C4×#102—31no—
77C22×C3×Q8D411yescharacteristic
78C22×D12C2311yescharacteristic
79C22×D12C2311yescharacteristic
80C22×C4×D3—21no—
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
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×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—21no—
113C2×D24—22no—
114C2×D24—22no—
115C2×D24—41no—
116C2×#204—22no—
117C2×#204—22no—
118C2×#204—41no—
119#875—22no—
120#875—22no—
121C4×D12—22no—
122C4×D12—22no—
123C3×C4⋊C8—22no—
124C3×C4⋊C8—22no—
125C3×C4⋊C8D414yes—
126C3×#97—22no—
127C3×#97—22no—
128C3×C4×C8—22no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138#821—22no—
139#821—22no—
140#821—22no—
141#821—22no—
142#821—41no—
143#818—41no—
144#817—41no—
145#816—22no—
146#816—22no—
147#816—41no—
148C22×SD16—31no—
149C22×D8—31no—
150C2×#124—31no—
151C2×C4×Q8C2×D311yescharacteristic
152C2×C4×D4—31no—
153#463—64no—
154#446—24no—
155C2×#107—31no—
156C2×#106—31no—
157C2×C4⋊C8C2×D311yescharacteristic
158C2×C4⋊C8C2×D311yescharacteristic
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#102—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97C2×D311yescharacteristic
169C2×#97C2×D311yescharacteristic
170C2×C4×C8C2×D311yescharacteristic
171#399—61no—
172#397—61no—
173#393—61no—
174#393—61no—
175#391—31no—
176#391—31no—
177#390—31no—
178#390—31no—
179#390—31no—
180#390—31no—
181#389—31no—
182#389—31no—
183#388—61no—
184#388—61no—
185#385—61no—
186#385—61no—
187#382—31no—
188C4×C4⋊C4—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193C23×D3—21no—
194C23×D3—21no—
195C2×C3×Q8—22no—
196C2×C3×Q8—41no—
197C2×C3×Q8—41no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12#4112yes—
223C2×C4×D3—42no—
224C2×C4×D3—81no—
225C2×C4×D3—81no—
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×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8C2×D412yes—
242C2×C3×C8C2×D412yes—
243C2×C3×C8#4112yes—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—22no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42C2×D412yes—
252C3×C42#4112yes—
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—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—81no—
261(C2×D3)⋊C4—81no—
262(C2×D3)⋊C4—81no—
263(C2×D3)⋊C4—81no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274Dic3⋊C4—81no—
275Dic3⋊C4—81no—
276C4×Dic3—81no—
277D24—44no—
278#204—44no—
279C22×Q8—31no—
280C22×Q8D1211yescharacteristic
281C22×D4—31no—
282C22×D4—31no—
283C23×C4—61no—
284C2×SD16—62no—
285C2×SD16—62no—
286C2×SD16—121no—
287C2×D8—62no—
288C2×D8—62no—
289C2×D8—121no—
290C22×C8C22×D311yescharacteristic
291C22×C8C22×D311yescharacteristic
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294#124—62no—
295#124—62no—
296C4×Q8—22no—
297C4×Q8—22no—
298C4×D4—62no—
299C4×D4—62no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4D1211yescharacteristic
308C2×C4⋊C4D1211yescharacteristic
309C2×C4⋊C4D1211yescharacteristic
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C22⋊C4—61no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323C2×C42C22×D311yescharacteristic
324C2×C42C22×D311yescharacteristic
325#107—121no—
326#106—121no—
327C4⋊C8—22no—
328C4⋊C8—22no—
329C4⋊C8#6714yes—
330#103—62no—
331#103—62no—
332#103—121no—
333#102—62no—
334#102—62no—
335#102—62no—
336#102—62no—
337#102—121no—
338#97—22no—
339#97—22no—
340C4×C8—22no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3C22×D411yescharacteristic
351C22×D3—22no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—41no—
359C22×D3—41no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—41no—
363C22×D3—41no—
364C22×D3—41no—
365C3×Q8—44no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
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#12112yes—
383C2×C3×C4#12112yes—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4H(5,𝔽2)11yescharacteristic
389C2×C3×C42−1+411yescharacteristic
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—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408D12—22no—
409D12—22no—
410D12—41no—
411D12—41no—
412D12—41no—
413D12—41no—
414D12—41no—
415D12—41no—
416C4×D3—84no—
417Dic6—41no—
418Dic6—41no—
419Dic6—41no—
420Dic6—41no—
421C3×C8—22no—
422C3×C8—22no—
423C3×C8—22no—
424C3×C8—41no—
425C24—61no—
426C24—61no—
427C2×Q8—22no—
428C2×Q8—41no—
429C2×Q8—41no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×D4—32no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—61no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—122no—
455C22×C4—241no—
456C22×C4—241no—
457C22×C4C2×D1211yescharacteristic
458C22×C4C2×D1211yescharacteristic
459C22×C4C2×D1211yescharacteristic
460C22×C4C2×D1211yescharacteristic
461C22×C4C2×D1211yescharacteristic
462C22×C4C2×D1211yescharacteristic
463C22×C4C23×D311yescharacteristic
464SD16—124no—
465D8—124no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8#23912yes—
476C2×C8C2×#6712yes—
477C2×C8C2×#6712yes—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—241no—
492C4⋊C4—241no—
493C22⋊C4—122no—
494C22⋊C4—122no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—122no—
498C22⋊C4—122no—
499C22⋊C4—122no—
500C22⋊C4—241no—
501C22⋊C4—241no—
502C22⋊C4—241no—
503C22⋊C4—241no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—241no—
509C42#23512yes—
510C42C2×#6712yes—
511C22×C3C2×#12111yescharacteristic
512C22×C3#54811yescharacteristic
513C22×C3#55011yescharacteristic
514C22×C3#57511yescharacteristic
515C22×C3#57511yescharacteristic
516C22×C3#57611yescharacteristic
517C22×C3#57711yescharacteristic
518C2×D3—42no—
519C2×D3—42no—
520C2×D3—42no—
521C2×D3—81no—
522C2×D3—81no—
523C2×D3—81no—
524C2×D3—81no—
525C2×D3—81no—
526C2×D3—81no—
527C2×D3—81no—
528C2×D3—81no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—81no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—41no—
535C3×C4—41no—
536C3×C4—41no—
537C3×C4—41no—
538C3×C4#52911yescharacteristic
539C3×C4#54211yescharacteristic
540C3×C4#54611yescharacteristic
541C3×C4#54811yescharacteristic
542Dic3—81no—
543Dic3—81no—
544Dic3—81no—
545Dic3—81no—
546Dic3—81no—
547Dic3—81no—
548C23—62no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23—121no—
557C23—121no—
558C23—121no—
559C23—121no—
560C23—121no—
561C23—121no—
562C23C22×D1211yescharacteristic
563Q8—44no—
564Q8—121no—
565Q8—121no—
566Q8—121no—
567Q8—121no—
568D4—62no—
569D4—62no—
570D4—121no—
571D4—121no—
572D4—121no—
573D4—121no—
574D4—121no—
575D4—121no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—121no—
593C2×C4—121no—
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×C4#93012yes—
612C2×C4#93012yes—
613C2×C4C2×#23511yescharacteristic
614C2×C4C2×#23911yescharacteristic
615C2×C4#100711yescharacteristic
616C2×C4D3×#4111yescharacteristic
617C2×C4#100911yescharacteristic
618C2×C4C22×#6711yescharacteristic
619C8—22no—
620C8—22no—
621C8—22no—
622C8—41no—
623C2×C3#242611yescharacteristic
624C2×C3#305411yescharacteristic
625C2×C3#306211yescharacteristic
626C2×C3#337011yescharacteristic
627C2×C3#337611yescharacteristic
628C2×C3#340111yescharacteristic
629C2×C3#340311yescharacteristic
630D3—82no—
631D3—161no—
632C22—122no—
633C22—122no—
634C22—122no—
635C22—241no—
636C22—241no—
637C22—241no—
638C22—241no—
639C22—241no—
640C22—241no—
641C22—241no—
642C22—241no—
643C22—241no—
644C22—241no—
645C22—241no—
646C22#557611yescharacteristic
647C22#557711yescharacteristic
648C22#574811yescharacteristic
649C22#574811yescharacteristic
650C22#575111yescharacteristic
651C22#575211yescharacteristic
652C22C2×#93011yescharacteristic
653C4—22no—
654C4—22no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—241no—
660C4—241no—
661C4—241no—
662C4—241no—
663C4—241no—
664C4—241no—
665C4#558011yescharacteristic
666C4#558211yescharacteristic
667C4#581711yescharacteristic
668C4#582911yescharacteristic
669C3#2529811yescharacteristic, complemented, Sylow
670C2—242no—
671C2—481no—
672C2#7995411yescharacteristic
673C2#7995611yescharacteristic
674C2#7997111yescharacteristic
675C2#7997411yescharacteristic
676C2#8249511yescharacteristic
677C2#8377111yescharacteristic
678C2#8380211yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#446C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenterC2×#446D3
hypocenterC3#25298
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
1. C2×C3×#446C2semidirect
2. #82501C2semidirect
3. #82484C2semidirect
5. #70488C2semidirect
6. #70484C2semidirect
7. #70482C2semidirect
9. #70455C2semidirect
10. #69826C2semidirect
12. #69818C2semidirect
13. #69817C2semidirect
14. #69773C2semidirect
15. C4×#821C2semidirect
63. C2×#446D3semidirect
669. C3#25298semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4462
#825012
#824842
C2×#48442
#704882
#704842
#704822
#704812
#704552
#698262
#698252
#698182
#698172
#697732
C4×#8212
#252983