Finite Group Databasebeta

← #10723608

Subgroups — #10723608

Order 768.

1390 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4976C212yesmaximal
2C2×#4928C211yesmaximal, characteristic
3C2×#4653C212yesmaximal
4C3×C4⋊#95C211yesmaximal, characteristic
5#71566C212yesmaximal
6#71535C211yesmaximal, characteristic
7#69420C212yesmaximal
8#69411C212yesmaximal
9#69407C212yesmaximal
10#14441—31nomaximal, Sylow
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C42C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×Dic3⋊C4C2212yes—
19C22×Dic3⋊C4C2212yes—
20C22×Dic3⋊C4C2212yes—
21C22×C4×Dic3C2212yes—
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32#4976—28no—
33#4928C418yescomplemented
34#4653—28no—
35C2×#389—31no—
36C2×#389—32no—
37C2×#387—32no—
38#1444—32no—
39#1441—32no—
40#1441—32no—
41#1436—32no—
42C4⋊#95—31no—
43C4⋊#95D311yescharacteristic
44C23×C3×C4C2312yes—
45C23×C3×C4C2312yes—
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C23×Dic3C2312yes—
52C23×Dic3C2312yes—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4—41no—
55C2×C3×C4⋊C4—41no—
56C2×C3×C4⋊C4—41no—
57C2×C3×C4⋊C4—41no—
58C2×C3×C4⋊C4D414yes—
59C2×C3×C42—41no—
60C2×C3×C42—41no—
61C2×C3×C42C2×C414yes—
62C2×C4⋊Dic3—22no—
63C2×C4⋊Dic3—41no—
64C2×C4⋊Dic3—41no—
65C2×C4⋊Dic3—41no—
66C2×C4⋊Dic3—41no—
67C2×C4⋊Dic3—41no—
68C2×C4⋊Dic3—41no—
69C2×C4⋊Dic3C2×C414yes—
70C2×C4⋊Dic3C2×C414yes—
71C2×C4⋊Dic3D414yes—
72C2×C4⋊Dic3Q814yes—
73C2×C4⋊Dic3Q814yes—
74C2×Dic3⋊C4—24no—
75C2×Dic3⋊C4—24no—
76C2×Dic3⋊C4—24no—
77C2×Dic3⋊C4—24no—
78C2×Dic3⋊C4—24no—
79C2×Dic3⋊C4—42no—
80C2×Dic3⋊C4—42no—
81C2×Dic3⋊C4—42no—
82C2×Dic3⋊C4C2×C418yes—
83C2×C4×Dic3—24no—
84C2×C4×Dic3—24no—
85C2×C4×Dic3—42no—
86C3×#95—24no—
87C3×#95—24no—
88C3⋊#95—24no—
89C3⋊#95—24no—
90C3⋊#95—24no—
91C3⋊#95—24no—
92C3⋊#95—24no—
93C3⋊#95—42no—
94C3⋊#95—42no—
95C3⋊#95C2×C418yes—
96C22×C4⋊C4—31no—
97C22×C4⋊C4—32no—
98C22×C4⋊C4—31no—
99C22×C4⋊C4—31no—
100C22×C4⋊C4—32no—
101C22×C4⋊C4—32no—
102C22×C4⋊C4—31no—
103C22×C4⋊C4C2×D311yescharacteristic
104C22×C4⋊C4C2×D311yescharacteristic
105C22×C42—32no—
106C22×C42C2×D311yescharacteristic
107#389—38no—
108#389—68no—
109#387—68no—
110C2×#95—32no—
111C2×#95—32no—
112C2×#95—32no—
113C2×#95—32no—
114C2×#95—32no—
115C2×#95—32no—
116C2×#95—32no—
117C2×#95—32no—
118C2×#95C2×D312yes—
119C2×#95C2×D312yes—
120C24×C3C2411yescharacteristic
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4C22×C412yes—
147C22×C3×C4C22×C412yes—
148C22×C3×C4C22×C412yes—
149C22×C3×C4C2×D414yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×Q812yes—
153C22×C3×C4C2×Q812yes—
154C22×C3×C4#4112yes—
155C22×C3×C4#4112yes—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3C22×C414yes—
180C22×Dic3C22×C414yes—
181C22×Dic3C2×D414yes—
182C22×Dic3C2×Q814yes—
183C22×Dic3C2×Q814yes—
184C3×C4⋊C4—44no—
185C3×C4⋊C4—44no—
186C3×C42—44no—
187C4⋊Dic3—44no—
188C4⋊Dic3—44no—
189C4⋊Dic3—44no—
190C4⋊Dic3C4⋊C4116yescomplemented
191Dic3⋊C4—216no—
192Dic3⋊C4—48no—
193Dic3⋊C4—48no—
194C4×Dic3—48no—
195C23×C4—32no—
196C23×C4—32no—
197C23×C4—32no—
198C23×C4—32no—
199C23×C4C22×D312yes—
200C23×C4C22×D312yes—
201C23×C4C22×D311yescharacteristic
202C23×C4C22×D311yescharacteristic
203C23×C4C22×D311yescharacteristic
204C2×C4⋊C4—22no—
205C2×C4⋊C4—34no—
206C2×C4⋊C4—34no—
207C2×C4⋊C4—34no—
208C2×C4⋊C4—38no—
209C2×C4⋊C4—34no—
210C2×C4⋊C4—34no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—64no—
216C2×C4⋊C4—64no—
217C2×C4⋊C4—64no—
218C2×C4⋊C4—64no—
219C2×C4⋊C4—64no—
220C2×C4⋊C4—62no—
221C2×C4⋊C4—122no—
222C2×C4⋊C4—122no—
223C2×C4⋊C4—122no—
224C2×C4⋊C4—121no—
225C2×C4⋊C4—121no—
226C2×C4⋊C4—121no—
227C2×C4⋊C4—121no—
228C2×C4⋊C4—121no—
229C2×C4⋊C4—121no—
230C2×C4⋊C4D1214yes—
231C2×C42—41no—
232C2×C42—41no—
233C2×C42—64no—
234C2×C42—64no—
235C2×C42—122no—
236C2×C42C4×D314yes—
237#95—24no—
238#95—24no—
239#95—38no—
240#95—64no—
241#95—64no—
242#95—64no—
243#95—64no—
244#95—64no—
245#95—122no—
246#95—122no—
247C23×C3C23×C411yescharacteristic
248C23×C3C22×D411yescharacteristic
249C23×C3C22×D411yescharacteristic
250C23×C3C22×Q811yescharacteristic
251C23×C3C22×Q811yescharacteristic
252C23×C3C2×#4111yescharacteristic
253C23×C3C2×#4111yescharacteristic
254C23×C3H(5,𝔽2)11yescharacteristic
255C23×C3H(5,𝔽2)11yescharacteristic
256C23×C3H(5,𝔽2)11yescharacteristic
257C23×C3H(5,𝔽2)11yescharacteristic
258C23×C32−1+411yescharacteristic
259C23×C32−1+411yescharacteristic
260C23×C32−1+411yescharacteristic
261C23×C32−1+411yescharacteristic
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4C2×C4⋊C414yes—
295C2×C3×C4C4×D414yes—
296C2×C3×C4C4×Q814yes—
297C2×Dic3—24no—
298C2×Dic3—24no—
299C2×Dic3—24no—
300C2×Dic3—24no—
301C2×Dic3—24no—
302C2×Dic3—24no—
303C2×Dic3—24no—
304C2×Dic3—24no—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3—24no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3C2×C4⋊C418yes—
325C24C23×D311yescharacteristic
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—34no—
352C22×C4—34no—
353C22×C4—34no—
354C22×C4—34no—
355C22×C4—34no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4C2×C4×D312yes—
380C22×C4C2×C4×D312yes—
381C22×C4C2×C4×D312yes—
382C22×C4C2×D1214yes—
383C22×C4C2×D1212yes—
384C22×C4#23512yes—
385C22×C4D3×D412yes—
386C22×C4#23712yes—
387C22×C4D3×Q812yes—
388C22×C4D3×Q812yes—
389C4⋊C4—316no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—616no—
393C4⋊C4—124no—
394C4⋊C4—124no—
395C4⋊C4—128no—
396C4⋊C4—128no—
397C4⋊C4—124no—
398C42—44no—
399C42—128no—
400C22×C3C22×C4⋊C411yescharacteristic
401C22×C3C2×C4×D411yescharacteristic
402C22×C3C2×C4×Q811yescharacteristic
403C22×C3#51811yescharacteristic
404C22×C3#51911yescharacteristic
405C22×C3#52011yescharacteristic
406C22×C3#52011yescharacteristic
407C22×C3#53411yescharacteristic
408C22×C3#53711yescharacteristic
409C22×C3#53711yescharacteristic
410C22×C3#53711yescharacteristic
411C22×C3#53811yescharacteristic
412C22×C3#53811yescharacteristic
413C22×C3#53911yescharacteristic
414C22×C3#53911yescharacteristic
415C22×C3#54311yescharacteristic
416C22×C3#54411yescharacteristic
417C22×C3#54411yescharacteristic
418C22×C3#54411yescharacteristic
419C22×C3D4211yescharacteristic
420C22×C3#54711yescharacteristic
421C22×C3#54711yescharacteristic
422C22×C3D4×Q811yescharacteristic
423C22×C3D4×Q811yescharacteristic
424C22×C3D4×Q811yescharacteristic
425C22×C3#55011yescharacteristic
426C22×C3#55411yescharacteristic
427C22×C3#55611yescharacteristic
428C22×C3#55711yescharacteristic
429C22×C3#55711yescharacteristic
430C22×C3Q8211yescharacteristic
431C22×C3#56211yescharacteristic
432C22×C3#56311yescharacteristic
433C22×C3#56311yescharacteristic
434C22×C3#56411yescharacteristic
435C3×C4—24no—
436C3×C4—24no—
437C3×C4—24no—
438C3×C4—44no—
439C3×C4—82no—
440Dic3—28no—
441Dic3—44no—
442Dic3—44no—
443Dic3—82no—
444C23C22×C4×D311yescharacteristic
445C23C22×D1211yescharacteristic
446C23C2×#23511yescharacteristic
447C23C2×D3×D411yescharacteristic
448C23C2×#23711yescharacteristic
449C23#100411yescharacteristic
450C23#100411yescharacteristic
451C23#100411yescharacteristic
452C23C2×D3×Q811yescharacteristic
453C23C2×D3×Q811yescharacteristic
454C23#100711yescharacteristic
455C23#100911yescharacteristic
456C23#101011yescharacteristic
457C23#101011yescharacteristic
458C23#101011yescharacteristic
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—38no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—64no—
493C2×C4—64no—
494C2×C4—64no—
495C2×C4—64no—
496C2×C4—64no—
497C2×C4—64no—
498C2×C4—64no—
499C2×C4—64no—
500C2×C4—64no—
501C2×C4—64no—
502C2×C4—64no—
503C2×C4—64no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4C4×D1214yes—
520C2×C4#88714yes—
521C2×C4D3×C4⋊C414yes—
522C2×C3#231611yescharacteristic
523C2×C3#232511yescharacteristic
524C2×C3#232911yescharacteristic
525C2×C3D4×C4⋊C411yescharacteristic
526C2×C3Q8×C4⋊C411yescharacteristic
527C2×C3#236511yescharacteristic
528C2×C3#238211yescharacteristic
529C2×C3#268211yescharacteristic
530C2×C3#268311yescharacteristic
531C2×C3#272611yescharacteristic
532C2×C3#272611yescharacteristic
533C2×C3#272711yescharacteristic
534C2×C3#273411yescharacteristic
535C2×C3#280511yescharacteristic
536C2×C3#280611yescharacteristic
537C22C2×C4×D1211yescharacteristic
538C22#548611yescharacteristic
539C22#549311yescharacteristic
540C22#549411yescharacteristic
541C22C2×#88711yescharacteristic
542C22C2×D3×C4⋊C411yescharacteristic
543C22#550411yescharacteristic
544C22#551011yescharacteristic
545C22#551511yescharacteristic
546C22#551611yescharacteristic
547C22#552311yescharacteristic
548C22#554211yescharacteristic
549C22D4×D1211yescharacteristic
550C22#555111yescharacteristic
551C22#555511yescharacteristic
552C22#556711yescharacteristic
553C22Q8×D1211yescharacteristic
554C22Q8×D1211yescharacteristic
555C22#557911yescharacteristic
556C22#559711yescharacteristic
557C22#559711yescharacteristic
558C22#560211yescharacteristic
559C22#561711yescharacteristic
560C22#562311yescharacteristic
561C22#562311yescharacteristic
562C22#562311yescharacteristic
563C22#563411yescharacteristic
564C22#567211yescharacteristic
565C22#568411yescharacteristic
566C22C3⋊#56411yescharacteristic
567C22#569911yescharacteristic
568C22#571811yescharacteristic
569C22#572111yescharacteristic
570C22#572211yescharacteristic
571C22#572711yescharacteristic
572C4—24no—
573C4—24no—
574C4—24no—
575C4—44no—
576C4—68no—
577C4—82no—
578C4—124no—
579C4—124no—
580C4—242no—
581C3#1444111yescharacteristic, complemented, Sylow
582C2#7333311yescharacteristic
583C2#7365011yescharacteristic
584C2C4⋊C4×D1211yescharacteristic
585C2#7386911yescharacteristic
586C2#7387411yescharacteristic
587C2#7416111yescharacteristic
588C2#7432311yescharacteristic
589C2#7612711yescharacteristic
590C2#7612911yescharacteristic
591C2#7618111yescharacteristic
592C2#7626111yescharacteristic
593C2#7637911yescharacteristic
594C2#7639911yescharacteristic
595C2#7657411yescharacteristic
596C2#7659811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×C4⋊#95C2
socle, prime radical, cosocleC24×C3C24
hypercenterC4⋊#95D3
hypocenterC3#14441
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
33. #4928C4semidirect
190. C4⋊Dic3C4⋊C4semidirect
581. C3#14441semidirect

Maximal subgroups

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

subgroupindex
C2×#49762
C2×#49282
C2×#46532
C3×C4⋊#952
#715662
#715352
#694202
#694112
#694072
#144413