Finite Group Databasebeta

← #10855239

Subgroups — #10855239

Order 768.

918 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#5027C211yesmaximal, characteristic
3C2×C4×#215C211yesmaximal, characteristic, complemented
4#73780C211yesmaximal, characteristic
5#71633C211yesmaximal, characteristic, complemented
6#71633C211yesmaximal, characteristic, complemented
7#71633C211yesmaximal, characteristic, complemented
8#71629C211yesmaximal, characteristic, complemented
9#69627C211yesmaximal, characteristic
10#69626C211yesmaximal, characteristic, complemented
11#69615C211yesmaximal, characteristic, complemented
12#69610C211yesmaximal, characteristic, complemented
13#69587C211yesmaximal, characteristic, complemented
14#69585C211yesmaximal, characteristic, complemented
15#69482C211yesmaximal, characteristic, complemented
16#26163—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C22×#215C2211yescharacteristic
22C2×#894—21no—
23C2×#874C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C3×#389—21no—
26C3×#389—21no—
27C3×#389C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C3×#384—21no—
30C3×#384C2211yescharacteristic
31C3×C4×C4⋊C4—21no—
32C3×C4×C4⋊C4C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35#5027—24no—
36C4×#215—24no—
37#4991—21no—
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#808C2211yescharacteristic
46#4939C2211yescharacteristic
47#4939C2211yescharacteristic
48#4939C2211yescharacteristic
49#4928C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C2×C4×C3⋊C8C2211yescharacteristic
52#4672—21no—
53#4665—21no—
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#460—31no—
63C2×C4×SD16—31no—
64#2445—31no—
65Q8×C4⋊C4D311yescharacteristic, complemented
66#2094—31no—
67#2088—31no—
68#2071—31no—
69#2041—31no—
70#1934—31no—
71#1916—31no—
72#1880—31no—
73#1880—31no—
74#1880—31no—
75#1865—31no—
76#1864—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12—21no—
79C22×D12—21no—
80C22×D12C2311yescharacteristic
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C3×C4×Q8—41no—
88C3×C4×Q8—41no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C42—21no—
102C2×C3×C42—21no—
103C2×C3×C42—21no—
104C2×C3×C42C2311yescharacteristic
105C2×C3×C42C2311yescharacteristic
106C2×C3×C42C2311yescharacteristic
107C2×#215—22no—
108C2×#215—22no—
109C2×#215—22no—
110C2×#215—22no—
111C2×#215—22no—
112C2×#215D414yes—
113C2×(C2×D3)⋊C4—21no—
114C2×(C2×D3)⋊C4—21no—
115C2×(C2×D3)⋊C4—21no—
116C2×(C2×D3)⋊C4—21no—
117C2×(C2×D3)⋊C4—21no—
118C2×(C2×D3)⋊C4—21no—
119C2×(C2×D3)⋊C4—21no—
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122C2×Dic3⋊C4—21no—
123C22×C3⋊C8C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127#894—44no—
128#874—22no—
129#874—41no—
130C4×D12—22no—
131C4×D12—41no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95D411yescharacteristic
135C3×#95D411yescharacteristic
136#835—22no—
137#835—22no—
138C3⋊#95—21no—
139#809—22no—
140#809—22no—
141#809—22no—
142#809—22no—
143#809—22no—
144#809—41no—
145#808—22no—
146#808—41no—
147C4⋊(C3⋊C8)—22no—
148C4×C3⋊C8—22no—
149C22×SD16—31no—
150C22×SD16—31no—
151C2×#127—31no—
152C2×#123—61no—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×Q8C2×D311yescharacteristic
155C2×C4×Q8C2×D311yescharacteristic
156C2×C4×D4—31no—
157#460—64no—
158C4×SD16—64no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102—31no—
170C2×C4×C8—31no—
171#394—61no—
172#394—61no—
173#390—31no—
174#390—31no—
175#390—31no—
176#389—21no—
177#389—21no—
178#389—31no—
179#389C2×D311yescharacteristic
180#389C2×D311yescharacteristic
181#386—61no—
182#384—21no—
183#384C2×D311yescharacteristic
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2×D311yescharacteristic
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#336—31no—
191#336—31no—
192#336—31no—
193#336—31no—
194C23×D3—21no—
195C23×D3—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8C2×D412yes—
200C2×C3×Q8#4112yes—
201C2×C3×Q8#4112yes—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—42no—
227C2×D12—42no—
228C2×D12—42no—
229C2×D12—42no—
230C2×D12—81no—
231C2×D12—81no—
232C2×D12C2×D412yes—
233C2×C4×D3—42no—
234C2×C4×D3—42no—
235C2×C4×D3—81no—
236C3×C4⋊C4—24no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—41no—
243C3×C4⋊C4—42no—
244C3×C4⋊C4—42no—
245C3×C4⋊C4—42no—
246C3×C4⋊C4—81no—
247C3×C4⋊C4—81no—
248C3×C4⋊C4—81no—
249C3×C4⋊C4SD1614yescomplemented
250C3×C4⋊C4#4112yes—
251C3×C42—24no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—42no—
259C3×C42—42no—
260#215—28no—
261#215—44no—
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—
267(C2×D3)⋊C4—81no—
268(C2×D3)⋊C4—81no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273Dic3⋊C4—42no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8C2×D412yes—
286C22×Q8C22×D311yescharacteristic
287C22×D4—31no—
288C22×D4—31no—
289C22×D4—61no—
290C22×D4—61no—
291C23×C4—61no—
292C2×SD16—34no—
293C2×SD16—62no—
294C2×SD16—62no—
295C2×SD16—62no—
296C2×SD16—62no—
297C2×SD16—62no—
298C22×C8—31no—
299C22×C8—31no—
300C22×C8—31no—
301C22×C8—31no—
302#127—62no—
303#127—121no—
304#123—124no—
305C4×Q8—22no—
306C4×Q8—22no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—41no—
310C4×Q8—41no—
311C4×D4—62no—
312C4×D4—121no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4#6711yescharacteristic
324C2×C4⋊C4#6711yescharacteristic
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42C22×D311yescharacteristic
339C2×C42C22×D311yescharacteristic
340C2×C42C22×D311yescharacteristic
341#106—62no—
342#106—121no—
343C4⋊C8—62no—
344#103—62no—
345#103—62no—
346#102—62no—
347#102—62no—
348#102—62no—
349#102—62no—
350#102—62no—
351#102—121no—
352C4×C8—62no—
353#95—21no—
354#95—21no—
355#95—61no—
356#95#6711yescharacteristic
357#95#6711yescharacteristic
358C23×C3C22×D411yescharacteristic
359C22×D3—22no—
360C22×D3—41no—
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—
371C22×D3—41no—
372C22×D3—41no—
373C3×Q8—22no—
374C3×Q8—22no—
375C3×Q8—22no—
376C3×Q8#12314yescomplemented
377C2×C3×C4—21no—
378C2×C3×C4—22no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4C2×SD1612yes—
405C2×C3×C4C2×SD1612yes—
406C2×C3×C4C2×SD1612yes—
407C2×C3×C4C22×D411yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×C3×C4H(5,𝔽2)11yescharacteristic
412C2×C3×C42−1+411yescharacteristic
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419D12—22no—
420D12—22no—
421D12—41no—
422D12—41no—
423D12—41no—
424D12—41no—
425D12—41no—
426D12—41no—
427D12—84no—
428D12—84no—
429C4×D3—84no—
430C3⋊C8—22no—
431C3⋊C8—41no—
432C3⋊C8—41no—
433C3⋊C8—41no—
434C24—61no—
435C24—61no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8#23512yes—
440C2×Q8D3×D412yes—
441C2×Q8#23912yes—
442C2×D4—32no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—122no—
455C2×D4—122no—
456C2×D4—122no—
457C2×D4—122no—
458C2×D4—241no—
459C2×D4—241no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—122no—
467C22×C4—122no—
468C22×C4—241no—
469C22×C4C2×#6711yescharacteristic
470C22×C4C2×#6711yescharacteristic
471C22×C4C2×#6711yescharacteristic
472C22×C4C2×#6711yescharacteristic
473C22×C4C2×#6711yescharacteristic
474C22×C4C2×#6711yescharacteristic
475C22×C4C23×D311yescharacteristic
476SD16—68no—
477SD16—124no—
478C2×C8—32no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C4⋊C4—24no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—41no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4—81no—
507C4⋊C4—122no—
508C4⋊C4#21514yescomplemented
509C4⋊C4#23512yes—
510C22⋊C4—122no—
511C22⋊C4—122no—
512C22⋊C4—122no—
513C22⋊C4—122no—
514C22⋊C4—241no—
515C22⋊C4—241no—
516C22⋊C4—241no—
517C42—24no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—42no—
525C42—42no—
526C22×C3#52511yescharacteristic
527C22×C3D4211yescharacteristic
528C22×C3#54811yescharacteristic
529C22×C3C22×SD1611yescharacteristic
530C22×C3#57511yescharacteristic
531C22×C3#57611yescharacteristic
532C22×C3#57711yescharacteristic
533C2×D3—42no—
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—81no—
537C2×D3—81no—
538C2×D3—81no—
539C2×D3—81no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C2×D3—81no—
544C2×D3—81no—
545C2×D3—81no—
546C2×D3—81no—
547C3×C4—22no—
548C3×C4—24no—
549C3×C4—22no—
550C3×C4—22no—
551C3×C4—41no—
552C3×C4—42no—
553C3×C4—41no—
554C3×C4—41no—
555C3×C4—81no—
556C3×C4—81no—
557C3×C4C2×#12311yescharacteristic
558C3×C4#53911yescharacteristic
559C3×C4#54711yescharacteristic
560C3×C4#55311yescharacteristic
561Dic3—81no—
562Dic3—81no—
563C23—62no—
564C23—121no—
565C23—121no—
566C23—121no—
567C23—121no—
568C23—121no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23—121no—
575C23—121no—
576C23—121no—
577C23C22×#6711yescharacteristic
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8#89414yescomplemented
582D4—62no—
583D4—62no—
584D4—121no—
585D4—121no—
586D4—121no—
587D4—121no—
588D4—121no—
589D4—121no—
590D4—244no—
591D4—244no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—244no—
626C2×C4C2×#21512yes—
627C2×C4C2×#21512yes—
628C2×C4C2×#21512yes—
629C2×C4C2×#23511yescharacteristic
630C2×C4C2×#23511yescharacteristic
631C2×C4C2×D3×D411yescharacteristic
632C2×C4#100411yescharacteristic
633C2×C4C2×#23911yescharacteristic
634C2×C4#100711yescharacteristic
635C8—62no—
636C8—121no—
637C8—121no—
638C8—121no—
639C2×C3#244311yescharacteristic
640C2×C3#310311yescharacteristic
641C2×C3#311911yescharacteristic
642C2×C3#326811yescharacteristic
643C2×C3D4×SD1611yescharacteristic
644C2×C3#336211yescharacteristic
645C2×C3#337111yescharacteristic
646D3—82no—
647D3—161no—
648C22—122no—
649C22—122no—
650C22—122no—
651C22—241no—
652C22—241no—
653C22—241no—
654C22—241no—
655C22—241no—
656C22—241no—
657C22—241no—
658C22—241no—
659C22—241no—
660C22—241no—
661C22—241no—
662C22#579111yescharacteristic
663C22D4×#6711yescharacteristic
664C22C22×#21511yescharacteristic
665C22#581711yescharacteristic
666C22#582211yescharacteristic
667C22#583511yescharacteristic
668C22#583711yescharacteristic
669C4—22no—
670C4—24no—
671C4—22no—
672C4—22no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—42no—
677C4—81no—
678C4—81no—
679C4—241no—
680C4—241no—
681C4C2×#89411yescharacteristic
682C4#550711yescharacteristic
683C4#551411yescharacteristic
684C4#551511yescharacteristic
685C3#2616311yescharacteristic, complemented, Sylow
686C2—242no—
687C2—481no—
688C2#8219511yescharacteristic
689C2#8244211yescharacteristic
690C2#8256811yescharacteristic
691C2D4×#21511yescharacteristic
692C2#8357711yescharacteristic
693C2#8369111yescharacteristic
694C2#8370311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×Q8×C4⋊C4C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenterQ8×C4⋊C4D3
hypocenterC3#26163
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×Q8×C4⋊C4C2semidirect
3. C2×C4×#215C2semidirect
5. #71633C2semidirect
6. #71633C2semidirect
7. #71633C2semidirect
8. #71629C2semidirect
10. #69626C2semidirect
11. #69615C2semidirect
12. #69610C2semidirect
13. #69587C2semidirect
14. #69585C2semidirect
15. #69482C2semidirect
65. Q8×C4⋊C4D3semidirect
249. C3×C4⋊C4SD16semidirect
376. C3×Q8#123semidirect
508. C4⋊C4#215semidirect
581. Q8#894semidirect
685. C3#26163semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C4⋊C42
C2×#50272
C2×C4×#2152
#737802
#716332
#716292
#696272
#696262
#696152
#696102
#695872
#695852
#694822
#261633