Finite Group Databasebeta

← #10818144

Subgroups — #10818144

Order 768.

824 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4830C211yesmaximal, characteristic, complemented
2C2×#4705C211yesmaximal, characteristic
3C4×#896C211yesmaximal, characteristic
4C3×#1917C211yesmaximal, characteristic, complemented
5#70450C211yesmaximal, characteristic
6#69811C211yesmaximal, characteristic, complemented
7#69478C211yesmaximal, characteristic, complemented
8#68658C211yesmaximal, characteristic
9#68657C211yesmaximal, characteristic
10#68656C211yesmaximal, characteristic, complemented
11#68652C211yesmaximal, characteristic, complemented
12#68619C211yesmaximal, characteristic, complemented
13#68617C211yesmaximal, characteristic, complemented
14#68615C211yesmaximal, characteristic
15#68599C211yesmaximal, characteristic
16#23051—31nomaximal, Sylow
17C2×#896C411yescharacteristic
18C2×C42×D3C2211yescharacteristic
19C2×C4×Dic6C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×#97C2211yescharacteristic
22C3×C4×C4⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C8C2211yescharacteristic
24C2×(C2×D3)⋊C8C2211yescharacteristic
25C2×(C2×D3)⋊C8C2211yescharacteristic
26C2×(C2×D3)⋊C8C2211yescharacteristic
27C2×#815C2211yescharacteristic
28C2×Dic3⋊C8C2211yescharacteristic
29C2×Dic3⋊C8C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32C2×C8×Dic3C2211yescharacteristic
33C2×C8×Dic3C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35#4989C411yescharacteristic
36#4980C411yescharacteristic
37#4976C411yescharacteristic
38C4×(C2×D3)⋊C4C2211yescharacteristic
39C4×(C2×D3)⋊C4C2211yescharacteristic
40C4×Dic3⋊C4C2211yescharacteristic
41C4×Dic3⋊C4C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43C2×#803C2211yescharacteristic
44C2×#803C2211yescharacteristic
45C2×C4×C3⋊C8C2211yescharacteristic
46#4830—24no—
47#4705—24no—
48#4668C411yescharacteristic
49#4667C411yescharacteristic
50#4652C411yescharacteristic
51#4646C411yescharacteristic
52C3×#336C2211yescharacteristic
53C3×#336C2211yescharacteristic
54C3×#336C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×C8×Q8—31no—
63C2×#432—31no—
64C4×C22⋊Q8—31no—
65#1984—31no—
66#1939—31no—
67#1924—31no—
68#1924—31no—
69#1923—31no—
70#1923—31no—
71#1917—31no—
72#1917D311yescharacteristic, complemented
73#1771—31no—
74#1771—31no—
75#1770—31no—
76C4×C22⋊C8—31no—
77C22×C4×D3C2×C411yescharacteristic
78C22×C4×D3C2×C411yescharacteristic
79C22×C4×D3C2311yescharacteristic
80C22×Dic6C2×C411yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C2×C3×C4⋊C4C2×C411yescharacteristic
86C2×C3×C4⋊C4C2×C411yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×(C2×D3)⋊C4C2×C411yescharacteristic
91C2×(C2×D3)⋊C4C2×C411yescharacteristic
92C2×(C2×D3)⋊C4C2×C411yescharacteristic
93C2×(C2×D3)⋊C4C2×C411yescharacteristic
94C2×C4⋊Dic3C2×C411yescharacteristic
95C2×Dic3⋊C4C2×C411yescharacteristic
96C2×Dic3⋊C4C2×C411yescharacteristic
97C2×Dic3⋊C4C2×C411yescharacteristic
98C2×Dic3⋊C4C2×C411yescharacteristic
99C2×Dic3⋊C4C2×C411yescharacteristic
100C2×Dic3⋊C4C2×C411yescharacteristic
101C2×C4×Dic3C2×C411yescharacteristic
102C2×C4×Dic3C2×C411yescharacteristic
103C2×C4×Dic3C2311yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C2×C4×Dic3C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110#896—24no—
111C42×D3—22no—
112C42×D3—22no—
113C4×Dic6—22no—
114C4×Dic6—22no—
115C3×C4⋊C8—22no—
116C3×#97—22no—
117C3×#95C2×C411yescharacteristic
118C3×#95C2×C411yescharacteristic
119C3⋊#95C2×C411yescharacteristic
120C3⋊#95C2×C411yescharacteristic
121C3⋊#95C2×C411yescharacteristic
122C3⋊#95C2×C411yescharacteristic
123C3⋊#95C2×C411yescharacteristic
124C3⋊#95C2×C411yescharacteristic
125C3⋊#95C2×C411yescharacteristic
126C3⋊#95C2×C411yescharacteristic
127(C2×D3)⋊C8—22no—
128(C2×D3)⋊C8—22no—
129(C2×D3)⋊C8—22no—
130(C2×D3)⋊C8—22no—
131#815—22no—
132Dic3⋊C8—22no—
133Dic3⋊C8—22no—
134Dic3⋊C8—22no—
135Dic3⋊C8D414yes—
136C8×Dic3—22no—
137C8×Dic3—22no—
138C8×Dic3—22no—
139C4⋊(C3⋊C8)—22no—
140#803—22no—
141#803—22no—
142C4×C3⋊C8—22no—
143C2×C22⋊Q8—31no—
144C2×C4×Q8—31no—
145C22×C42—31no—
146C8×Q8—64no—
147#432—64no—
148C2×C4⋊C8—31no—
149C2×C4⋊C8—31no—
150C2×C4⋊C8—31no—
151C2×C4⋊C8—31no—
152C2×C4⋊C8—31no—
153C2×C4⋊C8C2×D311yescharacteristic
154C2×C22⋊C8—31no—
155C2×C22⋊C8—31no—
156C2×C22⋊C8—31no—
157C2×C22⋊C8—31no—
158C2×#97—31no—
159C2×#97—31no—
160C2×#97—31no—
161C2×#97C2×D311yescharacteristic
162C2×C4×C8—31no—
163C2×C4×C8—31no—
164C2×C4×C8—31no—
165C2×C4×C8—31no—
166#391—31no—
167#389—31no—
168#387—31no—
169#387—31no—
170#385—31no—
171#385—31no—
172#380—31no—
173C4×C4⋊C4—31no—
174C4×C4⋊C4—31no—
175C4×C4⋊C4C2×D311yescharacteristic
176C4×C22⋊C4—31no—
177C4×C22⋊C4—31no—
178#336—31no—
179#336—31no—
180#336—31no—
181#336—31no—
182#336—31no—
183#336—31no—
184#336C2×D311yescharacteristic
185#336C2×D311yescharacteristic
186#336C2×D311yescharacteristic
187#336C2×D311yescharacteristic
188C23×D3C22×C411yescharacteristic
189C22×C3×C4C22×C411yescharacteristic
190C22×C3×C4C22×C411yescharacteristic
191C22×C3×C4C22×C411yescharacteristic
192C22×C3×C4C22×C411yescharacteristic
193C22×C3×C4C22×C411yescharacteristic
194C22×C3×C4C22×C411yescharacteristic
195C22×C3×C4C2411yescharacteristic
196C22×Dic3C22×C411yescharacteristic
197C22×Dic3C22×C411yescharacteristic
198C22×Dic3C22×C411yescharacteristic
199C22×Dic3C22×C411yescharacteristic
200C22×Dic3C22×C411yescharacteristic
201C22×Dic3C22×C411yescharacteristic
202C22×Dic3C22×C411yescharacteristic
203C2×C4×D3—22no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—22no—
211C2×C4×D3—22no—
212C2×C4×D3—22no—
213C2×C4×D3—22no—
214C2×C4×D3—41no—
215C2×Dic6—22no—
216C2×Dic6—22no—
217C2×Dic6M1614yes—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
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×C8C2×D412yes—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—41no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—21no—
238(C2×D3)⋊C4—22no—
239(C2×D3)⋊C4—22no—
240(C2×D3)⋊C4—22no—
241(C2×D3)⋊C4—22no—
242C4⋊Dic3—22no—
243Dic3⋊C4—22no—
244Dic3⋊C4—22no—
245Dic3⋊C4—22no—
246Dic3⋊C4—22no—
247Dic3⋊C4—22no—
248Dic3⋊C4—41no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—22no—
254C4×Dic3—22no—
255C4×Dic3C2×D412yes—
256C4×Dic3#4112yes—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8C2×D412yes—
267C2×C3⋊C8#4112yes—
268C2×C3⋊C8#4112yes—
269C22×Q8—31no—
270C23×C4—31no—
271C23×C4—31no—
272C23×C4—31no—
273C22×C8—31no—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8—31no—
277C22×C8C22×D311yescharacteristic
278C22×C8C22×D311yescharacteristic
279C22×C8C22×D311yescharacteristic
280C22×C8C22×D311yescharacteristic
281C22⋊Q8—64no—
282C4×Q8—62no—
283C4×Q8—62no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4C4×D311yescharacteristic
292C2×C4⋊C4C4×D311yescharacteristic
293C2×C22⋊C4—31no—
294C2×C22⋊C4—31no—
295C2×C22⋊C4—31no—
296C2×C22⋊C4—31no—
297C2×C42—31no—
298C2×C42—31no—
299C2×C42—31no—
300C2×C42—31no—
301C2×C42—31no—
302C2×C42—62no—
303C2×C42—62no—
304C2×C42C22×D311yescharacteristic
305C2×C42C22×D311yescharacteristic
306C2×C42C22×D311yescharacteristic
307C4⋊C8—22no—
308C4⋊C8—34no—
309C4⋊C8—62no—
310C4⋊C8—62no—
311C4⋊C8—62no—
312C4⋊C8—62no—
313C22⋊C8—62no—
314C22⋊C8—62no—
315C22⋊C8—62no—
316C22⋊C8—62no—
317#97—22no—
318#97—62no—
319#97—62no—
320#97—62no—
321C4×C8—62no—
322C4×C8—62no—
323C4×C8—62no—
324C4×C8—62no—
325#95—31no—
326#95—31no—
327#95—31no—
328#95—31no—
329#95—31no—
330#95—31no—
331#95—31no—
332#95—31no—
333#95C4×D311yescharacteristic
334#95C4×D311yescharacteristic
335C23×C3C23×C411yescharacteristic
336C22×D3—21no—
337C22×D3—21no—
338C22×D3—21no—
339C22×D3—21no—
340C22×D3—21no—
341C22×D3—21no—
342C22×D3#13112yes—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
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×C4C2×M1612yes—
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C42−1+411yescharacteristic
366C2×C3×C42−1+411yescharacteristic
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3C4×D412yes—
384C2×Dic3C4×D412yes—
385C2×Dic3C2×M1612yes—
386C2×Dic3C2×M1612yes—
387C2×Dic3#13112yes—
388C4×D3—44no—
389C4×D3—41no—
390C4×D3—41no—
391C4×D3—41no—
392C4×D3—41no—
393C4×D3—44no—
394Dic6—28no—
395C3×C8—22no—
396C3×C8—41no—
397C3×C8—41no—
398C3×C8—41no—
399C3⋊C8—22no—
400C3⋊C8—22no—
401C3⋊C8—22no—
402C3⋊C8—41no—
403C24—31no—
404C2×Q8—34no—
405C2×Q8—62no—
406C2×Q8—62no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—121no—
426C22×C4C2×C4×D311yescharacteristic
427C22×C4C2×C4×D311yescharacteristic
428C22×C4C2×C4×D311yescharacteristic
429C22×C4C2×C4×D311yescharacteristic
430C22×C4C2×C4×D311yescharacteristic
431C22×C4C2×C4×D311yescharacteristic
432C22×C4C23×D311yescharacteristic
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—32no—
445C2×C8—32no—
446C2×C8—32no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—61no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8D3×D412yes—
457C4⋊C4—22no—
458C4⋊C4—41no—
459C4⋊C4—62no—
460C4⋊C4—62no—
461C4⋊C4—62no—
462C4⋊C4—62no—
463C4⋊C4—62no—
464C4⋊C4—62no—
465C4⋊C4—121no—
466C22⋊C4—62no—
467C22⋊C4—62no—
468C22⋊C4—62no—
469C22⋊C4—62no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—32no—
477C42—32no—
478C42—62no—
479C42—62no—
480C42—61no—
481C42—61no—
482C42—61no—
483C42—61no—
484C22×C3C2×C4×D411yescharacteristic
485C22×C3C4×#4111yescharacteristic
486C22×C3#51911yescharacteristic
487C22×C3C22×M1611yescharacteristic
488C22×C3C2×#13111yescharacteristic
489C22×C3C2×#13111yescharacteristic
490C22×C3#56811yescharacteristic
491C2×D3—22no—
492C2×D3—22no—
493C2×D3—22no—
494C2×D3—41no—
495C2×D3—41no—
496C2×D3—41no—
497C2×D3—41no—
498C3×C4—22no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4—41no—
502C3×C4—41no—
503C3×C4—41no—
504C3×C4#54711yescharacteristic
505C3×C4#54811yescharacteristic
506C3×C4#55211yescharacteristic
507C3×C4#55611yescharacteristic
508Dic3—22no—
509Dic3—22no—
510Dic3—41no—
511Dic3—41no—
512Dic3—41no—
513Dic3—41no—
514Dic3#43514yes—
515C23—32no—
516C23—61no—
517C23—61no—
518C23—61no—
519C23—61no—
520C23—61no—
521C23—61no—
522C23C22×C4×D311yescharacteristic
523Q8—68no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—32no—
542C2×C4—32no—
543C2×C4—32no—
544C2×C4—32no—
545C2×C4—32no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—124no—
563C2×C4—124no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4D3×M1612yes—
569C2×C4C2×D3×D411yescharacteristic
570C2×C4#100711yescharacteristic
571C2×C4D3×#4111yescharacteristic
572C2×C4D3×#4111yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4#101011yescharacteristic
575C8—22no—
576C8—41no—
577C8—41no—
578C8—41no—
579C8—62no—
580C8—62no—
581C8—62no—
582C8—121no—
583C2×C3#236211yescharacteristic
584C2×C3C2×#43511yescharacteristic
585C2×C3#293311yescharacteristic
586C2×C3#296611yescharacteristic
587C2×C3#297111yescharacteristic
588C2×C3#298411yescharacteristic
589C2×C3#298511yescharacteristic
590D3—42no—
591C22—62no—
592C22—62no—
593C22—62no—
594C22—121no—
595C22—121no—
596C22—121no—
597C22—121no—
598C22#552211yescharacteristic
599C22C4×D3×D411yescharacteristic
600C22#557211yescharacteristic
601C22C2×D3×M1611yescharacteristic
602C22#574511yescharacteristic
603C22D3×#13111yescharacteristic
604C22D3×#13111yescharacteristic
605C4—22no—
606C4—34no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—62no—
613C4—62no—
614C4—121no—
615C4—121no—
616C4—121no—
617C4—121no—
618C4#563311yescharacteristic
619C4#565511yescharacteristic
620C4#569311yescharacteristic
621C4#570211yescharacteristic
622C3#2305111yescharacteristic, complemented, Sylow
623C2—122no—
624C2#7496611yescharacteristic
625C2#7932911yescharacteristic
626C2#7936611yescharacteristic
627C2D3×#43511yescharacteristic
628C2#7953911yescharacteristic
629C2#7996411yescharacteristic
630C2#7996611yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×#1917C2
prime radicalC23×D3C22×C4
cosocleC22×C3×C4C24
hypercenter#1917D3
hypocenterC3#23051
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #1917

Complements

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

NH ≅ G/Ntype
1. C2×#4830C2semidirect
4. C3×#1917C2semidirect
6. #69811C2semidirect
7. #69478C2semidirect
10. #68656C2semidirect
11. #68652C2semidirect
12. #68619C2semidirect
13. #68617C2semidirect
72. #1917D3semidirect
622. C3#23051semidirect

Maximal subgroups

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

subgroupindex
C2×#48302
C2×#47052
C4×#8962
C3×#19172
#704502
#698112
#694782
#686582
#686572
#686562
#686522
#686192
#686172
#686152
#685992
#230513