Finite Group Databasebeta

← #10831796

Subgroups — #10831796

Order 768.

1064 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×#5569C211yesmaximal, characteristic, complemented
2C2×#5026C211yesmaximal, characteristic
3C2×#4847C211yesmaximal, characteristic
4#74369C211yesmaximal, characteristic
5C3×#1803C211yesmaximal, characteristic, complemented
6#70473C211yesmaximal, characteristic, complemented
7#70240C211yesmaximal, characteristic
8#70240C211yesmaximal, characteristic
9#70238C211yesmaximal, characteristic, complemented
10#70236C211yesmaximal, characteristic, complemented
11#69583C211yesmaximal, characteristic, complemented
12#68899C211yesmaximal, characteristic
13#68898C211yesmaximal, characteristic
14#68828C211yesmaximal, characteristic, complemented
15#68826C211yesmaximal, characteristic, complemented
16#24346—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C22×#215C2211yescharacteristic
21#5569—24no—
22#5569—24no—
23C2×#897—21no—
24C2×#890—21no—
25C2×#890C2211yescharacteristic
26C2×#888—21no—
27C2×#888C2211yescharacteristic
28C2×C4×D12C2211yescharacteristic
29C2×#870—21no—
30C2×#870C2211yescharacteristic
31C2×C4×Dic6—21no—
32C2×C4×Dic6C2211yescharacteristic
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×C3×#103C2211yescharacteristic
35C2×C3×#103C2211yescharacteristic
36C3×#384C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#821C2211yescharacteristic
40C2×#818C2211yescharacteristic
41C2×#817C2211yescharacteristic
42#5026—24no—
43#4994—21no—
44#4994—21no—
45#4988C2211yescharacteristic
46#4988C2211yescharacteristic
47#4987—21no—
48#4984C2211yescharacteristic
49#4984C2211yescharacteristic
50C2×#809C2211yescharacteristic
51C2×#807C2211yescharacteristic
52C2×#803C2211yescharacteristic
53#4847—24no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4529C2211yescharacteristic
64C2×#557—31no—
65C2×#478—31no—
66C2×#440—31no—
67#2538—31no—
68#2059—31no—
69#2057—31no—
70#2025—31no—
71#2023—31no—
72#1992—31no—
73#1985—31no—
74#1880—31no—
75#1880—31no—
76#1865—31no—
77#1863—31no—
78#1803D311yescharacteristic, complemented
79C22×C3×Q8C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×Dic6—21no—
83C22×Dic6D411yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4×Q8—22no—
87C3×C4×Q8D414yes—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C42—21no—
95C2×C3×C42C2311yescharacteristic
96Q8×Dic3—22no—
97Q8×Dic3—22no—
98Q8×Dic3—22no—
99Q8×Dic3—22no—
100C2×#215—22no—
101C2×#215—22no—
102C2×#215Q814yescomplemented
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3D411yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4⋊Dic3C2311yescharacteristic
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4D411yescharacteristic
122C2×Dic3⋊C4D411yescharacteristic
123C2×C4×Dic3—21no—
124C2×C4×Dic3—21no—
125C2×C4×Dic3C2311yescharacteristic
126C2×C4×Dic3C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128C22×C3⋊C8C2311yescharacteristic
129#897—44no—
130#890—28no—
131#890—22no—
132#890—22no—
133#888—28no—
134#888—22no—
135#888—22no—
136C4×D12—22no—
137C4×D12—22no—
138#870—28no—
139#870—22no—
140#870D414yes—
141C4×Dic6—28no—
142C4×Dic6—22no—
143C4×Dic6—22no—
144C3×C4⋊C8—22no—
145C3×#103—22no—
146C3×#103—22no—
147#835—22no—
148C3⋊#95—21no—
149C3⋊#95—21no—
150#821—22no—
151#821—22no—
152#818—22no—
153#817—22no—
154#809—22no—
155#807—22no—
156#803—22no—
157C22×SD16—31no—
158#557—64no—
159#557—64no—
160C2×C4⋊Q8—31no—
161C2×C4⋊Q8—61no—
162C2×#125—31no—
163C2×#125—31no—
164C2×#125—61no—
165C2×#125—61no—
166C2×C22⋊Q8—61no—
167C2×C4×Q8—31no—
168C2×C4×Q8—31no—
169C2×C4×Q8—31no—
170C2×C4×Q8—61no—
171C2×C4×Q8C2×D311yescharacteristic
172C2×C4×D4—31no—
173#478—64no—
174#440—64no—
175C2×#107—31no—
176C2×#107—31no—
177C2×#106—31no—
178C2×C4⋊C8C2×D311yescharacteristic
179C2×#103—31no—
180C2×#103C2×D311yescharacteristic
181C2×#103C2×D311yescharacteristic
182C2×#102—31no—
183C2×#102—31no—
184C2×#102—31no—
185C2×#97—31no—
186#396—61no—
187#396—61no—
188#390—31no—
189#390—31no—
190#389—31no—
191#389—31no—
192#384C2×D311yescharacteristic
193#380—61no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340C2×D311yescharacteristic
199#340C2×D311yescharacteristic
200#336—31no—
201#336—31no—
202#336—31no—
203#336—31no—
204C23×D3—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8C2×D412yes—
209C2×C3×Q8C2×Q812yes—
210C2×C3×Q8#4112yes—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2411yescharacteristic
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C22×Dic3C2×D411yescharacteristic
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12C2×Q812yes—
231C2×C4×D3—42no—
232C2×C4×D3—42no—
233C2×C4×D3—42no—
234C2×Dic6—24no—
235C2×Dic6—22no—
236C2×Dic6—22no—
237C2×Dic6—41no—
238C2×Dic6—42no—
239C2×Dic6—42no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—42no—
253C3×C4⋊C4—81no—
254C3×C4⋊C4C2×D412yes—
255C3×C42—24no—
256C3×C42C2×D412yes—
257C3×C42C2×D412yes—
258#215—28no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—81no—
263(C2×D3)⋊C4—81no—
264C4⋊Dic3—24no—
265C4⋊Dic3—24no—
266C4⋊Dic3—24no—
267C4⋊Dic3—22no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—42no—
273C4⋊Dic3—42no—
274C4⋊Dic3C2×D412yes—
275C4⋊Dic3C2×D412yes—
276Dic3⋊C4—24no—
277Dic3⋊C4—24no—
278Dic3⋊C4—22no—
279Dic3⋊C4—22no—
280Dic3⋊C4—24no—
281Dic3⋊C4—24no—
282C4×Dic3—24no—
283C4×Dic3—24no—
284C4×Dic3—21no—
285C4×Dic3—21no—
286C4×Dic3—21no—
287C4×Dic3—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8C2×Q812yes—
294C22×Q8—31no—
295C22×Q8—61no—
296C22×Q8C22×D311yescharacteristic
297C22×D4—31no—
298C23×C4—61no—
299C2×SD16—34no—
300C2×SD16—62no—
301C2×SD16—62no—
302C22×C8—31no—
303C22×C8—31no—
304C22×C8C22×D311yescharacteristic
305C22×C8C22×D311yescharacteristic
306C4⋊Q8—62no—
307C4⋊Q8—68no—
308C4⋊Q8—62no—
309#125—34no—
310#125—68no—
311#125—62no—
312#125—68no—
313#125—62no—
314#125—62no—
315C22⋊Q8—124no—
316C4×Q8—22no—
317C4×Q8—68no—
318C4×Q8—62no—
319C4×Q8—62no—
320C4×Q8—62no—
321C4×Q8—62no—
322C4×Q8—62no—
323C4×Q8—62no—
324C4×Q8D1214yes—
325C4×D4—62no—
326C4×D4—62no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—31no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4—61no—
344C2×C4⋊C4—61no—
345C2×C4⋊C4C22×D311yescharacteristic
346C2×C4⋊C4C22×D311yescharacteristic
347C2×C4⋊C4C22×D311yescharacteristic
348C2×C22⋊C4—61no—
349C2×C22⋊C4—61no—
350C2×C22⋊C4—61no—
351C2×C22⋊C4—61no—
352C2×C22⋊C4—61no—
353C2×C42—21no—
354C2×C42—31no—
355C2×C42—31no—
356C2×C42—61no—
357C2×C42—61no—
358C2×C42C22×D311yescharacteristic
359#107—62no—
360#107—62no—
361#106—62no—
362C4⋊C8—22no—
363#103—22no—
364#103—22no—
365#103—62no—
366#102—62no—
367#102—62no—
368#102—62no—
369#97—62no—
370#95—61no—
371#95—61no—
372C23×C3C22×D411yescharacteristic
373C22×D3—22no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C3×Q8—22no—
381C3×Q8—22no—
382C3×Q8—22no—
383C3×Q8C22⋊Q814yescomplemented
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—22no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4#12012yes—
401C2×C3×C4#12012yes—
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C22×D411yescharacteristic
404C2×C3×C4C22×Q811yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C42−1+411yescharacteristic
407C2×C3×C42−1+411yescharacteristic
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—22no—
413C2×Dic3—21no—
414C2×Dic3—22no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3AGL(1,ℤ/8)12yes—
429D12—22no—
430D12—22no—
431D12—41no—
432D12—41no—
433C4×D3—84no—
434Dic6—44no—
435Dic6—48no—
436C3×C8—41no—
437C3×C8—41no—
438C3⋊C8—22no—
439C3⋊C8—41no—
440C24—61no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—62no—
445C2×Q8—64no—
446C2×Q8—62no—
447C2×Q8—121no—
448C2×Q8—122no—
449C2×Q8—122no—
450C2×Q8C2×D1212yes—
451C2×Q8#23712yes—
452C2×Q8D3×Q812yes—
453C2×D4—32no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—61no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—31no—
466C22×C4—31no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—122no—
471C22×C4—122no—
472C22×C4—122no—
473C22×C4D3×D411yescharacteristic
474C22×C4D3×D411yescharacteristic
475C22×C4C23×D311yescharacteristic
476SD16—68no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—32no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—32no—
496C4⋊C4—32no—
497C4⋊C4—42no—
498C4⋊C4—64no—
499C4⋊C4—64no—
500C4⋊C4—64no—
501C4⋊C4—64no—
502C4⋊C4—64no—
503C4⋊C4—62no—
504C4⋊C4—62no—
505C4⋊C4—62no—
506C4⋊C4—64no—
507C4⋊C4—64no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—81no—
513C4⋊C4—122no—
514C4⋊C4—122no—
515C4⋊C4C2×D1212yes—
516C22⋊C4—122no—
517C22⋊C4—122no—
518C22⋊C4—122no—
519C22⋊C4—241no—
520C22⋊C4—241no—
521C42—24no—
522C42—64no—
523C42—64no—
524C42—61no—
525C42—61no—
526C42—61no—
527C42—61no—
528C42C2×D1212yes—
529C42D3×D412yes—
530C22×C3C2×#12011yescharacteristic
531C22×C3#54811yescharacteristic
532C22×C3D4×Q811yescharacteristic
533C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
534C22×C3#57511yescharacteristic
535C22×C3#57711yescharacteristic
536C22×C3#57811yescharacteristic
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—42no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C3×C4—24no—
545C3×C4—22no—
546C3×C4—22no—
547C3×C4—42no—
548C3×C4—81no—
549C3×C4—81no—
550C3×C4C2×C22⋊Q811yescharacteristic
551C3×C4#53611yescharacteristic
552C3×C4#54811yescharacteristic
553C3×C4D4×Q811yescharacteristic
554Dic3—22no—
555Dic3—41no—
556Dic3—42no—
557Dic3—41no—
558Dic3—41no—
559Dic3—42no—
560Dic3—81no—
561C23—62no—
562C23—121no—
563C23—121no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23C2×D3×D411yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—22no—
572Q8—124no—
573Q8—128no—
574Q8#89714yescomplemented
575D4—62no—
576D4—62no—
577D4—121no—
578D4—121no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—32no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4—121no—
615C2×C4—121no—
616C2×C4—244no—
617C2×C4#88212yes—
618C2×C4#88212yes—
619C2×C4C22×D1211yescharacteristic
620C2×C4C2×D3×D411yescharacteristic
621C2×C4C2×#23711yescharacteristic
622C2×C4C2×D3×Q811yescharacteristic
623C2×C4#101011yescharacteristic
624C2×C4#101011yescharacteristic
625C8—41no—
626C8—41no—
627C8—62no—
628C8—121no—
629C2×C3#240911yescharacteristic
630C2×C3#300611yescharacteristic
631C2×C3#301311yescharacteristic
632C2×C3#336811yescharacteristic
633C2×C3#337711yescharacteristic
634C2×C3#338611yescharacteristic
635C2×C3#338811yescharacteristic
636D3—82no—
637C22—122no—
638C22—122no—
639C22—122no—
640C22—241no—
641C22—241no—
642C22—241no—
643C22—241no—
644C22C2×#88211yescharacteristic
645C22#559811yescharacteristic
646C22#563211yescharacteristic
647C22#576211yescharacteristic
648C22#576611yescharacteristic
649C22D3×AGL(1,ℤ/8)11yescharacteristic
650C22#577711yescharacteristic
651C4—24no—
652C4—22no—
653C4—22no—
654C4—42no—
655C4—62no—
656C4—81no—
657C4—81no—
658C4—121no—
659C4—122no—
660C4—121no—
661C4—121no—
662C4—122no—
663C4—241no—
664C4C2×#89711yescharacteristic
665C4#552611yescharacteristic
666C4#559811yescharacteristic
667C4#563211yescharacteristic
668C3#2434611yescharacteristic, complemented, Sylow
669C2—242no—
670C2#7401711yescharacteristic
671C2#7879511yescharacteristic
672C2#7888411yescharacteristic
673C2#8037211yescharacteristic
674C2#8046711yescharacteristic
675C2#8100811yescharacteristic
676C2#8110311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1803C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1803D3
hypocenterC3#24346
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
1. C2×#5569C2semidirect
5. C3×#1803C2semidirect
6. #70473C2semidirect
9. #70238C2semidirect
10. #70236C2semidirect
11. #69583C2semidirect
14. #68828C2semidirect
15. #68826C2semidirect
78. #1803D3semidirect
102. C2×#215Q8semidirect
383. C3×Q8C22⋊Q8semidirect
574. Q8#897semidirect
668. C3#24346semidirect

Maximal subgroups

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

subgroupindex
C2×#55692
C2×#50262
C2×#48472
#743692
C3×#18032
#704732
#702402
#702382
#702362
#695832
#688992
#688982
#688282
#688262
#243463