Finite Group Databasebeta

← #10764393

Subgroups — #10764393

Order 768.

572 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2989C211yesmaximal, characteristic
2#82521C211yesmaximal, characteristic
3#82513C211yesmaximal, characteristic
4#70797C211yesmaximal, characteristic, complemented
5#70796C211yesmaximal, characteristic, complemented
6#70795C211yesmaximal, characteristic, complemented
7#70794C211yesmaximal, characteristic, complemented
8#70748C211yesmaximal, characteristic, complemented
9#70747C211yesmaximal, characteristic, complemented
10#70495C211yesmaximal, characteristic, complemented
11#70493C211yesmaximal, characteristic, complemented
12#70492C211yesmaximal, characteristic, complemented
13#70490C211yesmaximal, characteristic, complemented
14#70423C211yesmaximal, characteristic, complemented
15#70421C211yesmaximal, characteristic, complemented
16#17824—31nomaximal, Sylow
17C3×C4×#41C2211yescharacteristic
18C3×#446—21no—
19C3×#436—21no—
20C3×#435—21no—
21C3×#435—21no—
22C3×#432C2211yescharacteristic
23C3×#432C2211yescharacteristic
24C3×#431C2211yescharacteristic
25C3×#423C2211yescharacteristic
26C3×#423C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28#5234C412yes—
29#5024—21no—
30#5012—21no—
31#5011—21no—
32#5011—21no—
33#5000C412yes—
34#5000C412yes—
35#4999C412yes—
36#4926C2211yescharacteristic
37#4926C2211yescharacteristic
38#4925C2211yescharacteristic
39#4925C2211yescharacteristic
40#4925C2211yescharacteristic
41#4925C2211yescharacteristic
42#4925C2211yescharacteristic
43#4924C2211yescharacteristic
44#4924C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46C4×#208C2211yescharacteristic
47C2×C4×C3⋊C8C2211yescharacteristic
48#4480C2211yescharacteristic
49#4480C2211yescharacteristic
50#4480C2211yescharacteristic
51#4480C2211yescharacteristic
52#4480C2211yescharacteristic
53#4480C2211yescharacteristic
54#4480C2211yescharacteristic
55#4480C2211yescharacteristic
56#4480C2211yescharacteristic
57#4480C2211yescharacteristic
58#4479C2211yescharacteristic
59#4479C2211yescharacteristic
60#4479C2211yescharacteristic
61#4479C2211yescharacteristic
62#4479C2211yescharacteristic
63#4479C2211yescharacteristic
64#2989—31no—
65#2989D311yescharacteristic
66#2922—31no—
67#1574—31no—
68#1574—31no—
69#1573—31no—
70#1573—31no—
71#1572—31no—
72#1572—31no—
73#1570—31no—
74#1570—31no—
75#1569—31no—
76#1569—31no—
77#1568—31no—
78#1566—31no—
79C2×C3×#41C2×C411yescharacteristic
80C2×C3×M16—21no—
81C2×C3×M16—21no—
82C22×C3×C8—21no—
83C3×C4×Q8D411yescharacteristic
84C3×C4×Q8Q811yescharacteristic
85C3×C4×D4D411yescharacteristic
86C3×C4×D4D411yescharacteristic
87C3×C4×D4D411yescharacteristic
88C3×C4×D4Q811yescharacteristic
89C3×C4×D4Q811yescharacteristic
90C3×C4×D4Q811yescharacteristic
91C3×#117C2×C411yescharacteristic
92C3×#117C2×C411yescharacteristic
93C3×#117C2×C411yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×#208—21no—
98C2×#208—21no—
99C2×#208C2×C412yes—
100C22×C3⋊C8—21no—
101C22×C3⋊C8C2×C412yes—
102C3×C4⋊C8—21no—
103C3×C4⋊C8—21no—
104C3×C4⋊C8—21no—
105C3×C4⋊C8C2311yescharacteristic
106C3×C4⋊C8C2311yescharacteristic
107C3×C4⋊C8C2311yescharacteristic
108C3×C4⋊C8C2311yescharacteristic
109C3×C22⋊C8—21no—
110C3×C22⋊C8—21no—
111C3×C22⋊C8—21no—
112C3×#97—21no—
113C3×#97—21no—
114C3×C4×C8—21no—
115#830—21no—
116#830—21no—
117#830—21no—
118#830C2×C412yes—
119#830C2×C412yes—
120#830C2×C412yes—
121#830C2×C412yes—
122C4⋊(C3⋊C8)—21no—
123C4⋊(C3⋊C8)—21no—
124C4⋊(C3⋊C8)—21no—
125C4⋊(C3⋊C8)C2×C412yes—
126C4⋊(C3⋊C8)C2×C412yes—
127C4⋊(C3⋊C8)C2×C412yes—
128C4⋊(C3⋊C8)C2×C412yes—
129C4⋊(C3⋊C8)C2311yescharacteristic
130C4⋊(C3⋊C8)C2311yescharacteristic
131C4⋊(C3⋊C8)C2311yescharacteristic
132C4⋊(C3⋊C8)C2311yescharacteristic
133#803—21no—
134#803—21no—
135#803C2×C412yes—
136C4×C3⋊C8—21no—
137C4×C3⋊C8C2×C412yes—
138C4×C3⋊C8C2311yescharacteristic
139C4×C3⋊C8C2311yescharacteristic
140C4×C3⋊C8C2311yescharacteristic
141C4×C3⋊C8C2311yescharacteristic
142C4×#41C2×D311yescharacteristic
143#446—21no—
144#446—61no—
145#436—21no—
146#436—61no—
147#435—21no—
148#435—21no—
149#435—61no—
150#435—61no—
151#433—32no—
152#433—32no—
153#432—31no—
154#432—31no—
155#432C2×D311yescharacteristic
156#432C2×D311yescharacteristic
157#431—31no—
158#431—31no—
159#431—31no—
160#431—31no—
161#431—31no—
162#431C2×D311yescharacteristic
163#424—32no—
164#423—31no—
165#423—31no—
166#423C2×D311yescharacteristic
167#423C2×D311yescharacteristic
168C2×C4⋊C8—31no—
169C2×C4⋊C8C2×D311yescharacteristic
170#408—32no—
171C4×M16—31no—
172C2×C4×C8—31no—
173#335—31no—
174#335—31no—
175#335—31no—
176#335—31no—
177#335—31no—
178#335—31no—
179#334—31no—
180#334—31no—
181#334—31no—
182#334—31no—
183#334—31no—
184#334—31no—
185#334—31no—
186#334—31no—
187#334—31no—
188#334—31no—
189C3×#41—42no—
190C2×C3×Q8C4⋊C411yescharacteristic
191C2×C3×D4C4⋊C411yescharacteristic
192C2×C3×D4C4⋊C411yescharacteristic
193C2×C3×D4C4⋊C411yescharacteristic
194C22×C3×C4C22×C411yescharacteristic
195C22×C3×C4C22×C411yescharacteristic
196C22×C3×C4C22×C411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×Q811yescharacteristic
201C22×C3×C4C2×Q811yescharacteristic
202C22×C3×C4C2×Q811yescharacteristic
203C3×M16—81no—
204C3×M16—81no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—81no—
210C3×C4⋊C4C4⋊C411yescharacteristic
211C3×C4⋊C4C4⋊C411yescharacteristic
212C3×C4⋊C4C4⋊C411yescharacteristic
213C3×C4⋊C4C4⋊C411yescharacteristic
214C3×C4⋊C4C4⋊C411yescharacteristic
215C3×C4⋊C4C4⋊C411yescharacteristic
216C3×C22⋊C4C4⋊C411yescharacteristic
217C3×C22⋊C4C4⋊C411yescharacteristic
218C3×C22⋊C4C4⋊C411yescharacteristic
219C3×C22⋊C4C4⋊C411yescharacteristic
220C3×C22⋊C4C4⋊C411yescharacteristic
221C3×C22⋊C4C4⋊C411yescharacteristic
222C3×C42C22×C411yescharacteristic
223C3×C42C22×C411yescharacteristic
224C3×C42C22×C411yescharacteristic
225C3×C42C2×D411yescharacteristic
226C3×C42C2×D411yescharacteristic
227C3×C42C2×D411yescharacteristic
228C3×C42C2×Q811yescharacteristic
229C3×C42C2×Q811yescharacteristic
230C3×C42C2×Q811yescharacteristic
231C3×C42C2411yescharacteristic
232#208—42no—
233#208—81no—
234#208—81no—
235C2×C3⋊C8—21no—
236C2×C3⋊C8—21no—
237C2×C3⋊C8—21no—
238C2×C3⋊C8—21no—
239C2×C3⋊C8—42no—
240C2×C3⋊C8—81no—
241C2×C3⋊C8C22×C412yes—
242C2×C3⋊C8C22×C412yes—
243C2×C3⋊C8C22×C412yes—
244C2×C3⋊C8C22×C412yes—
245C2×#41C4×D311yescharacteristic
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—32no—
249C2×M16—61no—
250C2×M16—61no—
251C22×C8—21no—
252C22×C8—32no—
253C22×C8—61no—
254C4×Q8Dic611yescharacteristic
255C4×Q8#6711yescharacteristic
256C4×D4Dic611yescharacteristic
257C4×D4Dic611yescharacteristic
258C4×D4Dic611yescharacteristic
259C4×D4#6711yescharacteristic
260C4×D4#6711yescharacteristic
261C4×D4#6711yescharacteristic
262#117C4×D311yescharacteristic
263#117C4×D311yescharacteristic
264#117C4×D311yescharacteristic
265C2×C42C22×D311yescharacteristic
266C2×C42C22×D311yescharacteristic
267C2×C42C22×D311yescharacteristic
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—32no—
272C4⋊C8—32no—
273C4⋊C8—32no—
274C4⋊C8—32no—
275C4⋊C8—31no—
276C4⋊C8—31no—
277C4⋊C8—31no—
278C4⋊C8—31no—
279C4⋊C8—61no—
280C4⋊C8—61no—
281C4⋊C8—61no—
282C4⋊C8C22×D311yescharacteristic
283C4⋊C8C22×D311yescharacteristic
284C4⋊C8C22×D311yescharacteristic
285C4⋊C8C22×D311yescharacteristic
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—32no—
290C22⋊C8—32no—
291C22⋊C8—32no—
292C22⋊C8—32no—
293C22⋊C8—61no—
294C22⋊C8—61no—
295C22⋊C8—61no—
296#97—21no—
297#97—21no—
298#97—32no—
299#97—61no—
300#97—61no—
301C4×C8—21no—
302C4×C8—32no—
303C4×C8—31no—
304C4×C8—31no—
305C4×C8—31no—
306C4×C8—31no—
307C4×C8—61no—
308C23×C3C2×C4⋊C411yescharacteristic
309C23×C3C2×C4⋊C411yescharacteristic
310C23×C3C2×C4⋊C411yescharacteristic
311C3×Q8—41no—
312C3×D4—41no—
313C3×D4—41no—
314C3×D4—41no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4C2×C4⋊C411yescharacteristic
328C2×C3×C4C2×C4⋊C411yescharacteristic
329C2×C3×C4C2×C4⋊C411yescharacteristic
330C2×C3×C4C2×C4⋊C411yescharacteristic
331C2×C3×C4C2×C4⋊C411yescharacteristic
332C2×C3×C4C2×C4⋊C411yescharacteristic
333C2×C3×C4C2×C4⋊C411yescharacteristic
334C2×C3×C4C2×C4⋊C411yescharacteristic
335C2×C3×C4C2×C4⋊C411yescharacteristic
336C2×C3×C4C23×C411yescharacteristic
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C22×Q811yescharacteristic
339C3×C8—41no—
340C3×C8—41no—
341C3×C8—41no—
342C3×C8—41no—
343C3⋊C8—22no—
344C3⋊C8—22no—
345C3⋊C8—22no—
346C3⋊C8—22no—
347C3⋊C8—41no—
348C3⋊C8—41no—
349C3⋊C8—41no—
350C3⋊C8—41no—
351#41—42no—
352C2×Q8Dic3⋊C411yescharacteristic
353C2×D4Dic3⋊C411yescharacteristic
354C2×D4Dic3⋊C411yescharacteristic
355C2×D4Dic3⋊C411yescharacteristic
356C22×C4C2×Dic611yescharacteristic
357C22×C4C2×Dic611yescharacteristic
358C22×C4C2×Dic611yescharacteristic
359C22×C4C2×C4×D311yescharacteristic
360C22×C4C2×C4×D311yescharacteristic
361C22×C4C2×C4×D311yescharacteristic
362C22×C4C2×#6711yescharacteristic
363C22×C4C2×#6711yescharacteristic
364C22×C4C2×#6711yescharacteristic
365M16—81no—
366M16—81no—
367M16—122no—
368M16—241no—
369M16—241no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—32no—
375C2×C8—32no—
376C2×C8—32no—
377C2×C8—32no—
378C2×C8—61no—
379C2×C8—61no—
380C2×C8—61no—
381C2×C8—61no—
382C2×C8—81no—
383C2×C8—122no—
384C2×C8—241no—
385C4⋊C4Dic3⋊C411yescharacteristic
386C4⋊C4Dic3⋊C411yescharacteristic
387C4⋊C4Dic3⋊C411yescharacteristic
388C4⋊C4Dic3⋊C411yescharacteristic
389C4⋊C4Dic3⋊C411yescharacteristic
390C4⋊C4Dic3⋊C411yescharacteristic
391C22⋊C4Dic3⋊C411yescharacteristic
392C22⋊C4Dic3⋊C411yescharacteristic
393C22⋊C4Dic3⋊C411yescharacteristic
394C22⋊C4Dic3⋊C411yescharacteristic
395C22⋊C4Dic3⋊C411yescharacteristic
396C22⋊C4Dic3⋊C411yescharacteristic
397C42C2×Dic611yescharacteristic
398C42C2×Dic611yescharacteristic
399C42C2×Dic611yescharacteristic
400C42C2×C4×D311yescharacteristic
401C42C2×C4×D311yescharacteristic
402C42C2×C4×D311yescharacteristic
403C42C2×#6711yescharacteristic
404C42C2×#6711yescharacteristic
405C42C2×#6711yescharacteristic
406C42C23×D311yescharacteristic
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3C22×C4⋊C411yescharacteristic
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4#56812yes—
428C3×C4#57711yescharacteristic
429C3×C4#57711yescharacteristic
430C23C2×Dic3⋊C411yescharacteristic
431C23C2×Dic3⋊C411yescharacteristic
432C23C2×Dic3⋊C411yescharacteristic
433Q8—41no—
434D4—41no—
435D4—41no—
436D4—41no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4C2×Dic3⋊C411yescharacteristic
450C2×C4C2×Dic3⋊C411yescharacteristic
451C2×C4C2×Dic3⋊C411yescharacteristic
452C2×C4C2×Dic3⋊C411yescharacteristic
453C2×C4C2×Dic3⋊C411yescharacteristic
454C2×C4C2×Dic3⋊C411yescharacteristic
455C2×C4C2×Dic3⋊C411yescharacteristic
456C2×C4C2×Dic3⋊C411yescharacteristic
457C2×C4C2×Dic3⋊C411yescharacteristic
458C2×C4C22×Dic611yescharacteristic
459C2×C4C22×C4×D311yescharacteristic
460C2×C4C22×#6711yescharacteristic
461C8—41no—
462C8—41no—
463C8—41no—
464C8—41no—
465C8—62no—
466C8—62no—
467C8—62no—
468C8—62no—
469C8—121no—
470C8—121no—
471C8—121no—
472C8—121no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3—41no—
476C2×C3#290511yescharacteristic
477C2×C3#291211yescharacteristic
478C2×C3#291611yescharacteristic
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22C22×Dic3⋊C411yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4#574512yes—
500C4#583611yescharacteristic
501C4#583711yescharacteristic
502C3#1782411yescharacteristic, complemented, Sylow
503C2—41no—
504C2—41no—
505C2—41no—
506C2#8256111yescharacteristic
507C2#8375011yescharacteristic
508C2#8381011yescharacteristic
subgroupG/H
centerC2×C4C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#2989C2
socleC22×C3C22×C4⋊C4
prime radicalC2×C3×#41C2×C4
cosocleC3×C42C24
hypercenter#2989D3
hypocenterC3#17824
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2989

Complements

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

NH ≅ G/Ntype
4. #70797C2semidirect
5. #70796C2semidirect
6. #70795C2semidirect
7. #70794C2semidirect
8. #70748C2semidirect
9. #70747C2semidirect
10. #70495C2semidirect
11. #70493C2semidirect
12. #70492C2semidirect
13. #70490C2semidirect
14. #70423C2semidirect
15. #70421C2semidirect
502. C3#17824semidirect

Maximal subgroups

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

subgroupindex
C3×#29892
#825212
#825132
#707972
#707962
#707952
#707942
#707482
#707472
#704952
#704932
#704922
#704902
#704232
#704212
#178243