Finite Group Databasebeta

← #10690461

Subgroups — #10690461

Order 768.

822 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5073C211yesmaximal, characteristic
2#73653C211yesmaximal, characteristic, complemented
3C3×#1982C211yesmaximal, characteristic, complemented
4#71761C211yesmaximal, characteristic, complemented
5C2×#4485C211yesmaximal, characteristic, complemented
6#67704C211yesmaximal, characteristic, complemented
7#67619C211yesmaximal, characteristic, complemented
8#9061—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C22×#213—22no—
11C2×#893—21no—
12C2×#874C2211yescharacteristic
13C2×C4×D12—21no—
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×#102—21no—
16C2×C3×#102—21no—
17C3×C4×C4⋊C4C2211yescharacteristic
18C2×#832—21no—
19C2×#832—21no—
20#5073—22no—
21#5073C414yescomplemented
22#4988—21no—
23#4939—21no—
24C4×C4⋊Dic3C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27#4665—22no—
28#4485—22no—
29C2×#493—31no—
30#2366—31no—
31#1982—31no—
32#1982D311yescharacteristic, complemented
33C2×#331—31no—
34#1279—31no—
35#1279—31no—
36C22×C3×D4—21no—
37C22×C3×D4—21no—
38C22×C3×D4—21no—
39C22×D12—21no—
40C22×D12—21no—
41C22×D12D412yes—
42C22×C4×D3—22no—
43C22×C3×C8—21no—
44C3×#127—21no—
45C3×#127—21no—
46C3×#127—21no—
47C3×#127C2×C412yescomplemented
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#213—28no—
53C2×#213—44no—
54C2×#213—44no—
55C2×(C2×D3)⋊C4—22no—
56C2×(C2×D3)⋊C4—22no—
57C2×(C2×D3)⋊C4—22no—
58C2×C4⋊Dic3D411yescharacteristic
59C2×C4⋊Dic3D411yescharacteristic
60C2×C4×Dic3—21no—
61C22×C3⋊C8—21no—
62C22×C3⋊C8—21no—
63#893—42no—
64#893—42no—
65#874—21no—
66#874—21no—
67#874C2×C412yes—
68#874D412yes—
69C4×D12—42no—
70C4×D12—42no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×#102—42no—
74C3×#102—42no—
75C3×#95—21no—
76#832—42no—
77#832—42no—
78C3⋊#95—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4×C3⋊C8—21no—
82C4×C3⋊C8C2×C412yescomplemented
83C22×D8—62no—
84C2×#127—31no—
85C2×#127C2×D311yescharacteristic
86C2×C4×D4—61no—
87C2×C4×D4—61no—
88#493—34no—
89#493—62no—
90C2×C4⋊C8—31no—
91C2×C4⋊C8C2×D311yescharacteristic
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—61no—
95C2×#102—61no—
96C2×C4×C8—31no—
97#390—61no—
98#390—61no—
99#386—62no—
100C4×C4⋊C4—31no—
101C4×C4⋊C4C2×D311yescharacteristic
102#331—62no—
103C24×C3—41no—
104C23×D3—22no—
105C2×C3×D4—22no—
106C2×C3×D4—22no—
107C2×C3×D4—22no—
108C2×C3×D4—42no—
109C2×C3×D4—42no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×Dic3—21no—
127C22×Dic3—21no—
128C2×D12—22no—
129C2×D12—22no—
130C2×D12—22no—
131C2×D12—22no—
132C2×D12—22no—
133C2×D12—41no—
134C2×D12—41no—
135C2×D12—41no—
136C2×D12—41no—
137C2×D12—41no—
138C2×D12—41no—
139C2×D12—41no—
140C2×D12—41no—
141C2×D12—42no—
142C2×D12—42no—
143C2×D12C22⋊C414yes—
144C2×C4×D3—44no—
145C2×C4×D3—44no—
146C2×C4×D3—82no—
147C2×C3×C8—41no—
148C2×C3×C8—41no—
149C2×C3×C8—41no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4D812yescomplemented
154C3×C42—41no—
155C3×C42—41no—
156C3×C42C22×C411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42#4111yescharacteristic
159C3×C42#4111yescharacteristic
160#213—416no—
161(C2×D3)⋊C4—44no—
162(C2×D3)⋊C4—44no—
163(C2×D3)⋊C4—82no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3D812yescomplemented
168C4×Dic3—41no—
169C4×Dic3—41no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—41no—
174C2×C3⋊C8—41no—
175C2×C3⋊C8—41no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—32no—
180C22×D4—61no—
181C22×D4—61no—
182C23×C4—62no—
183C2×D8—68no—
184C2×D8—124no—
185C2×D8—124no—
186C22×C8—21no—
187C22×C8—61no—
188C22×C8—61no—
189#127—21no—
190#127—21no—
191#127—21no—
192#127—32no—
193#127—32no—
194#127—61no—
195#127—61no—
196#127C4×D312yescomplemented
197C4×D4—122no—
198C4×D4—122no—
199C4×D4—122no—
200C4×D4—122no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4#6711yescharacteristic
204C2×C4⋊C4#6711yescharacteristic
205C2×C22⋊C4—62no—
206C2×C22⋊C4—62no—
207C2×C22⋊C4—62no—
208C2×C42—21no—
209C2×C42—61no—
210C2×C42C22×D311yescharacteristic
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—61no—
214C4⋊C8—61no—
215#102—42no—
216#102—42no—
217#102—122no—
218#102—122no—
219C4×C8—32no—
220C4×C8—61no—
221#95—21no—
222#95—61no—
223C23×C3—42no—
224C23×C3—81no—
225C23×C3—81no—
226C23×C3—81no—
227C23×C3—81no—
228C23×C3—81no—
229C23×C3—81no—
230C23×C3#12011yescharacteristic
231C22×D3—24no—
232C22×D3—42no—
233C22×D3—42no—
234C22×D3—42no—
235C22×D3—42no—
236C22×D3—42no—
237C22×D3—42no—
238C3×D4—44no—
239C3×D4—42no—
240C3×D4—42no—
241C3×D4—42no—
242C3×D4—42no—
243C3×D4—82no—
244C3×D4—81no—
245C3×D4—81no—
246C3×D4—81no—
247C3×D4—81no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—22no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4C2×C22⋊C411yescharacteristic
257C2×C3×C4C4×D411yescharacteristic
258C2×C3×C4C4×D411yescharacteristic
259C2×C3×C4#12111yescharacteristic
260C2×C3×C4#12111yescharacteristic
261C2×C3×C4#12311yescharacteristic
262C2×C3×C4C2×D811yescharacteristic
263C2×C3×C4C2×D811yescharacteristic
264C2×C3×C4#13511yescharacteristic
265C2×C3×C4#13511yescharacteristic
266C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
267C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
268C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
269C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
270C2×Dic3—22no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C2×Dic3—41no—
276D12—44no—
277D12—42no—
278D12—42no—
279D12—42no—
280D12—42no—
281D12—42no—
282D12—42no—
283D12—42no—
284D12—42no—
285D12—82no—
286C4×D3—88no—
287C3×C8—81no—
288C3⋊C8—24no—
289C3⋊C8—81no—
290C24—41no—
291C24—62no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—34no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—62no—
310C2×D4—62no—
311C2×D4—62no—
312C2×D4—62no—
313C2×D4—62no—
314C2×D4—122no—
315C2×D4—122no—
316C2×D4—121no—
317C2×D4—121no—
318C2×D4—121no—
319C2×D4—121no—
320C2×D4—121no—
321C2×D4—121no—
322C2×D4—121no—
323C2×D4—121no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—61no—
327C22×C4—61no—
328C22×C4—124no—
329C22×C4—124no—
330C22×C4—242no—
331C22×C4D3×D411yescharacteristic
332C22×C4D3×D411yescharacteristic
333C22×C4C2×#6711yescharacteristic
334D8—1216no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—62no—
339C2×C8—62no—
340C2×C8—62no—
341C2×C8—121no—
342C2×C8—121no—
343C2×C8—121no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—32no—
348C4⋊C4—61no—
349C4⋊C4—61no—
350C4⋊C4—61no—
351C4⋊C4#21312yescomplemented
352C22⋊C4—124no—
353C22⋊C4—124no—
354C22⋊C4—242no—
355C42—41no—
356C42—41no—
357C42—121no—
358C42—121no—
359C42C2×C4×D311yescharacteristic
360C42#23511yescharacteristic
361C42D3×D411yescharacteristic
362C42#23711yescharacteristic
363C22×C3—82no—
364C22×C3—82no—
365C22×C3—82no—
366C22×C3—161no—
367C22×C3—161no—
368C22×C3—161no—
369C22×C3—161no—
370C22×C3#38611yescharacteristic
371C22×C3#44711yescharacteristic
372C22×C3#44711yescharacteristic
373C22×C3#44911yescharacteristic
374C22×C3#44911yescharacteristic
375C22×C3#45311yescharacteristic
376C22×C3#45311yescharacteristic
377C2×D3—44no—
378C2×D3—44no—
379C2×D3—44no—
380C2×D3—82no—
381C2×D3—82no—
382C2×D3—82no—
383C2×D3—82no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—42no—
387C3×C4—81no—
388C3×C4C4×D811yescharacteristic
389C3×C4C4×D811yescharacteristic
390C3×C4#44211yescharacteristic
391C3×C4#44211yescharacteristic
392C3×C4#45911yescharacteristic
393C3×C4#45911yescharacteristic
394C3×C4#46311yescharacteristic
395C3×C4#46311yescharacteristic
396Dic3—42no—
397Dic3—81no—
398C23—42no—
399C23—64no—
400C23—81no—
401C23—81no—
402C23—81no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—122no—
407C23—122no—
408C23—122no—
409C23—122no—
410C23—122no—
411C23—122no—
412C23#93711yescharacteristic
413D4—44no—
414D4—42no—
415D4—42no—
416D4—42no—
417D4—42no—
418D4—82no—
419D4—81no—
420D4—81no—
421D4—81no—
422D4—81no—
423D4—124no—
424D4—122no—
425D4—122no—
426D4—122no—
427D4—122no—
428D4—122no—
429D4—122no—
430D4—122no—
431D4—122no—
432D4—242no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—62no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4—248no—
448C2×C4D3×C22⋊C411yescharacteristic
449C2×C4#67⋊C411yescharacteristic
450C2×C4#88311yescharacteristic
451C2×C4#88411yescharacteristic
452C2×C4D3×D811yescharacteristic
453C2×C4#91111yescharacteristic
454C2×C4#91411yescharacteristic
455C2×C4#91611yescharacteristic
456C2×C4C4×#6711yescharacteristic
457C2×C4C2×#21311yescharacteristic
458C2×C4#93211yescharacteristic
459C2×C4#93811yescharacteristic
460C2×C4#94911yescharacteristic
461C2×C4#95011yescharacteristic
462C8—64no—
463C8—81no—
464C8—241no—
465C2×C3—162no—
466C2×C3#161911yescharacteristic
467C2×C3#162011yescharacteristic
468C2×C3#162111yescharacteristic
469C2×C3#162111yescharacteristic
470C2×C3#187911yescharacteristic
471C2×C3#187911yescharacteristic
472C2×C3#188711yescharacteristic
473D3—84no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—124no—
478C22—124no—
479C22—124no—
480C22—161no—
481C22—161no—
482C22—161no—
483C22—161no—
484C22—242no—
485C22—242no—
486C22—242no—
487C22—242no—
488C22#495611yescharacteristic
489C22#503611yescharacteristic
490C22#503711yescharacteristic
491C22#515611yescharacteristic
492C22#517211yescharacteristic
493C22#519811yescharacteristic
494C22#519811yescharacteristic
495C4—21no—
496C4—21no—
497C4—42no—
498C4—81no—
499C4—122no—
500C4—241no—
501C4#475611yescharacteristic
502C4#478511yescharacteristic
503C4#478611yescharacteristic
504C4#478711yescharacteristic
505C4C4×#21311yescharacteristic
506C4#501411yescharacteristic
507C4#505711yescharacteristic
508C4#507211yescharacteristic
509C3#906111yescharacteristic, complemented, Sylow
510C2—162no—
511C2—244no—
512C2#6965711yescharacteristic
513C2#7020811yescharacteristic
514C2#7032311yescharacteristic
515C2#7101311yescharacteristic
516C2#7101511yescharacteristic
517C2#7103111yescharacteristic
518C2#7103211yescharacteristic
subgroupG/H
centerC23#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1982C2
socleC23×C3#120
prime radicalC2×#5073C2
cosocleC2×C3×C42C23
hypercenter#1982D3
hypocenterC3#9061
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1982

Complements

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

NH ≅ G/Ntype
2. #73653C2semidirect
3. C3×#1982C2semidirect
4. #71761C2semidirect
5. C2×#4485C2semidirect
6. #67704C2semidirect
7. #67619C2semidirect
21. #5073C4semidirect
32. #1982D3semidirect
47. C3×#127C2×C4semidirect
82. C4×C3⋊C8C2×C4semidirect
153. C3×C4⋊C4D8semidirect
167. C4⋊Dic3D8semidirect
196. #127C4×D3semidirect
351. C4⋊C4#213semidirect
509. C3#9061semidirect

Maximal subgroups

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

subgroupindex
C2×#50732
#736532
C3×#19822
#717612
C2×#44852
#677042
#676192
#90613