Finite Group Databasebeta

← #10823424

Subgroups — #10823424

Order 768.

1228 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5470C211yesmaximal, characteristic, complemented
2C4×#930C211yesmaximal, characteristic
3C2×#4710C212yesmaximal, complemented
4C3×#1749C211yesmaximal, characteristic, complemented
5#69962C212yesmaximal
6#69937C212yesmaximal, complemented
7#69813C212yesmaximal, complemented
8#68687C211yesmaximal, characteristic, complemented
9#68678C211yesmaximal, characteristic, complemented
10#68632C211yesmaximal, characteristic, complemented
11#68631C211yesmaximal, characteristic, complemented
12#23574—31nomaximal, Sylow
13C22×C3×C42C2211yescharacteristic
14C2×#930C2211yescharacteristic
15C2×C4×#67—21no—
16C2×#926—21no—
17C2×#924—21no—
18C2×#924C2212yes—
19C22×#204C2212yes—
20C2×C4×D12C2211yescharacteristic
21#5470—28no—
22C2×#870—21no—
23C2×#869C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C4×Dic6—21no—
26C2×C4×Dic6C2212yes—
27C2×C3×C22⋊C8C2212yes—
28C2×C3×#97C2212yes—
29#5213—21no—
30#5212C2211yescharacteristic
31C2×#821C2212yes—
32C2×#818C2212yes—
33C2×#816C2212yes—
34C2×#816C2212yes—
35C2×#816C2212yes—
36#4940—21no—
37#4939C2211yescharacteristic
38C4×(C2×D3)⋊C4—21no—
39C4×C4⋊Dic3C2211yescharacteristic
40#4928C2211yescharacteristic
41#4710—28no—
42C3×#336C2211yescharacteristic
43C3×#336C2211yescharacteristic
44#4551C2212yes—
45#4551C2212yes—
46C2×#533—31no—
47C2×#440—32no—
48C4×#121—31no—
49#2057—31no—
50#2043—31no—
51#2036—32no—
52#2023—31no—
53#2013—31no—
54#2000—32no—
55#1763—32no—
56#1749D311yescharacteristic, complemented
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67—21no—
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C22×Dic6—21no—
64C22×Dic6C2312yes—
65C22×C3×C8C2312yes—
66C22×C3×C8C2312yes—
67C2×C3×C42—24no—
68C2×C3×C42D411yescharacteristic
69C2×C3×C42D411yescharacteristic
70C2×C3×C42C2311yescharacteristic
71C2×C3×C42C2311yescharacteristic
72C2×#217—22no—
73C2×#217—21no—
74#930—22no—
75#930—22no—
76C4×#67—44no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—21no—
81#926—28no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2312yes—
88C2×C4⋊Dic3C2311yescharacteristic
89#924—28no—
90#924—24no—
91#924—24no—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—21no—
95C2×C4×Dic3—22no—
96C2×C4×Dic3—21no—
97C2×#204—24no—
98C2×#204—24no—
99C2×#204—24no—
100C4×D12—24no—
101#870—28no—
102#869—24no—
103#869—24no—
104C4×Dic6—28no—
105C4×Dic6—24no—
106C4×Dic6D418yescomplemented
107C3×C22⋊C8—24no—
108C3×#97—24no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112#821—24no—
113#818—24no—
114#816—24no—
115#816—24no—
116#816—24no—
117C22×SD16—32no—
118#533—68no—
119C2×C4⋊Q8—31no—
120C2×C4⋊Q8—31no—
121C2×#125—61no—
122C2×C22⋊Q8—32no—
123C2×C22⋊Q8—61no—
124C2×#121—31no—
125C2×C4×Q8—32no—
126C2×C4×Q8—61no—
127C2×C4×D4—31no—
128C2×C4×D4—61no—
129C2×#117—61no—
130C22×C42C2×D311yescharacteristic
131#440—68no—
132C2×#107—32no—
133C2×#103—32no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#102—32no—
137C2×C22⋊C8C2×D312yes—
138C2×#97C2×D312yes—
139#390—31no—
140#389—31no—
141#388—61no—
142#386—61no—
143#380—31no—
144C4×C4⋊C4—31no—
145C4×C22⋊C4—61no—
146#340—32no—
147#340—32no—
148#336C2×D311yescharacteristic
149#336C2×D311yescharacteristic
150C24×C3C2×D411yescharacteristic
151C23×D3—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—41no—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C2×#67—24no—
169C2×#67—42no—
170C2×#67—42no—
171C22×Dic3—21no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12—21no—
179C2×D12#4112yes—
180C2×C4×D3—42no—
181C2×C4×D3—42no—
182C2×C4×D3—42no—
183C2×Dic6—22no—
184C2×Dic6—22no—
185C2×Dic6—22no—
186C2×Dic6—22no—
187C2×Dic6—24no—
188C2×Dic6—22no—
189C2×Dic6—42no—
190C2×Dic6—42no—
191C2×Dic6C2×D414yes—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8#4114yes—
198C3×C42—22no—
199C3×C42—22no—
200C3×C42—22no—
201C3×C42C2×D414yes—
202#217—28no—
203#217—42no—
204(C2×D3)⋊C4—24no—
205(C2×D3)⋊C4—42no—
206(C2×D3)⋊C4—42no—
207(C2×D3)⋊C4—42no—
208C4⋊Dic3—24no—
209C4⋊Dic3—24no—
210C4⋊Dic3—24no—
211C4⋊Dic3—22no—
212C4⋊Dic3—21no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3C2×D414yes—
216C4⋊Dic3#4112yes—
217Dic3⋊C4—28no—
218Dic3⋊C4—28no—
219Dic3⋊C4—42no—
220C4×Dic3—28no—
221C4×Dic3—42no—
222#204—216no—
223C22×Q8—32no—
224C22×Q8—61no—
225C22×D4—31no—
226C22×D4—61no—
227C23×C4—61no—
228C23×C4D1211yescharacteristic
229C23×C4D1211yescharacteristic
230C23×C4C22×D311yescharacteristic
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—64no—
234C22×C8C22×D312yes—
235C22×C8C22×D312yes—
236C4⋊Q8—64no—
237C4⋊Q8—64no—
238#125—68no—
239C22⋊Q8—64no—
240C22⋊Q8—64no—
241C22⋊Q8—68no—
242#121—62no—
243#121—62no—
244C4×Q8—38no—
245C4×Q8—68no—
246C4×Q8—64no—
247C4×D4—64no—
248C4×D4—124no—
249#117—68no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4—61no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4—61no—
263C2×C22⋊C4—61no—
264C2×C22⋊C4—61no—
265C2×C22⋊C4—61no—
266C2×C42—24no—
267C2×C42—62no—
268C2×C42—61no—
269C2×C42D1211yescharacteristic
270C2×C42D1211yescharacteristic
271C2×C42C22×D311yescharacteristic
272C2×C42C22×D311yescharacteristic
273#107—64no—
274#103—64no—
275#103—64no—
276#103—64no—
277#102—64no—
278C22⋊C8—24no—
279#97—24no—
280#95—61no—
281#95—61no—
282#95—61no—
283C23×C3—21no—
284C23×C3—21no—
285C23×C3—21no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3#13712yes—
290C23×C3C22×D411yescharacteristic
291C22×D3—22no—
292C22×D3—41no—
293C22×D3—41no—
294C22×D3—41no—
295C22×D3—41no—
296C22×D3—41no—
297C22×D3—41no—
298C2×C3×C4—28no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—44no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4#12012yes—
314C2×C3×C4#12012yes—
315C2×C3×C4AGL(1,ℤ/8)12yes—
316C2×C3×C4#13712yes—
317C2×C3×C4C22×D412yes—
318C2×C3×C4C2×#4112yes—
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321#67—48no—
322C2×Dic3—24no—
323C2×Dic3—24no—
324C2×Dic3—22no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—41no—
328C2×Dic3—42no—
329C2×Dic3—41no—
330C2×Dic3—42no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337D12—24no—
338D12—42no—
339C4×D3—84no—
340Dic6—24no—
341Dic6—24no—
342Dic6—48no—
343Dic6—42no—
344Dic6—42no—
345C3×C8—24no—
346C3×C8—42no—
347C24—61no—
348C24C2×D1211yescharacteristic
349C2×Q8—34no—
350C2×Q8—64no—
351C2×Q8—62no—
352C2×Q8—62no—
353C2×Q8—62no—
354C2×Q8—62no—
355C2×Q8—62no—
356C2×Q8—122no—
357C2×Q8—122no—
358C2×D4—32no—
359C2×D4—64no—
360C2×D4—62no—
361C2×D4—62no—
362C2×D4—61no—
363C2×D4—122no—
364C2×D4—122no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—41no—
375C22×C4—61no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—122no—
381C22×C4—122no—
382C22×C4—122no—
383C22×C4C2×D1211yescharacteristic
384C22×C4C2×D1211yescharacteristic
385C22×C4C2×D1211yescharacteristic
386C22×C4C2×D1211yescharacteristic
387C22×C4C2×D1211yescharacteristic
388C22×C4C23×D311yescharacteristic
389SD16—616no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8#23514yes—
396C4⋊C4—34no—
397C4⋊C4—32no—
398C4⋊C4—64no—
399C4⋊C4—64no—
400C4⋊C4—68no—
401C4⋊C4—64no—
402C4⋊C4—68no—
403C4⋊C4—62no—
404C4⋊C4—61no—
405C4⋊C4—62no—
406C4⋊C4—62no—
407C4⋊C4—122no—
408C22⋊C4—64no—
409C22⋊C4—68no—
410C22⋊C4—122no—
411C22⋊C4—122no—
412C22⋊C4—122no—
413C22⋊C4—122no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—68no—
418C42—122no—
419C42D3×D414yes—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—41no—
427C22×C3C2×#12011yescharacteristic
428C22×C3#52511yescharacteristic
429C22×C3#53211yescharacteristic
430C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
431C22×C3C2×#13711yescharacteristic
432C22×C3C2×#13711yescharacteristic
433C22×C3#57511yescharacteristic
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—81no—
438C2×D3—81no—
439C2×D3—81no—
440C2×D3—81no—
441C3×C4—22no—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—22no—
445C3×C4—41no—
446C3×C4#52512yes—
447C3×C4#53212yes—
448Dic3—42no—
449Dic3—44no—
450Dic3—44no—
451Dic3—81no—
452Dic3—81no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—62no—
460C23—121no—
461C23—121no—
462C23—121no—
463C23—121no—
464C23—121no—
465C23—121no—
466C23#90912yes—
467C23C22×D1211yescharacteristic
468Q8—64no—
469Q8—64no—
470Q8—128no—
471Q8—122no—
472Q8—122no—
473D4—64no—
474D4—128no—
475D4—122no—
476C2×C4—28no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—64no—
492C2×C4—64no—
493C2×C4—62no—
494C2×C4—122no—
495C2×C4—121no—
496C2×C4—122no—
497C2×C4—121no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4—121no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—244no—
507C2×C4#88212yes—
508C2×C4#88212yes—
509C2×C4#90812yes—
510C2×C4#90912yes—
511C2×C4C2×#23512yes—
512C2×C4C2×D3×D412yes—
513C2×C4D3×#4111yescharacteristic
514C2×C4D3×#4111yescharacteristic
515C8—24no—
516C8—42no—
517C2×C3—42no—
518C2×C3#238811yescharacteristic
519C2×C3#300411yescharacteristic
520C2×C3#300611yescharacteristic
521C2×C3#311011yescharacteristic
522C2×C3#311611yescharacteristic
523C2×C3#318111yescharacteristic
524C2×C3#318211yescharacteristic
525D3—82no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—122no—
534C22—122no—
535C22—122no—
536C22—241no—
537C22—241no—
538C22—241no—
539C22—241no—
540C22#547711yescharacteristic
541C22C2×#88211yescharacteristic
542C22#552411yescharacteristic
543C22C2×#90811yescharacteristic
544C22C2×#90911yescharacteristic
545C22C2×#90911yescharacteristic
546C22#574811yescharacteristic
547C4—22no—
548C4—22no—
549C4—22no—
550C4—22no—
551C4—41no—
552C4—124no—
553C4—124no—
554C4—122no—
555C4—241no—
556C4—241no—
557C4#554712yes—
558C4#554812yes—
559C3#2357411yescharacteristic, complemented, Sylow
560C2—42no—
561C2—242no—
562C2#7358011yescharacteristic
563C2#7850411yescharacteristic
564C2#7853311yescharacteristic
565C2#7863911yescharacteristic
566C2#7864511yescharacteristic
567C2#7934411yescharacteristic
568C2#7934811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1749C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1749D3
hypocenterC3#23574
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
1. C2×#5470C2semidirect
3. C2×#4710C2semidirect
4. C3×#1749C2semidirect
6. #69937C2semidirect
7. #69813C2semidirect
8. #68687C2semidirect
9. #68678C2semidirect
10. #68632C2semidirect
11. #68631C2semidirect
56. #1749D3semidirect
106. C4×Dic6D4semidirect
559. C3#23574semidirect

Maximal subgroups

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

subgroupindex
C2×#54702
C4×#9302
C2×#47102
C3×#17492
#699622
#699372
#698132
#686872
#686782
#686322
#686312
#235743