Finite Group Databasebeta

← #10897652

Subgroups — #10897652

Order 768.

662 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83811C211yesmaximal, characteristic, complemented
2#79039C211yesmaximal, characteristic, complemented
3C3×#1861C211yesmaximal, characteristic, complemented
4#72029C211yesmaximal, characteristic, complemented
5#70729C211yesmaximal, characteristic, complemented
6#70727C211yesmaximal, characteristic, complemented
7#70530C211yesmaximal, characteristic, complemented
8#70406C211yesmaximal, characteristic
9#70401C211yesmaximal, characteristic, complemented
10#70383C211yesmaximal, characteristic, complemented
11#70379C211yesmaximal, characteristic, complemented
12#69036C211yesmaximal, characteristic, complemented
13#69017C211yesmaximal, characteristic, complemented
14#68757C211yesmaximal, characteristic, complemented
15#68751C211yesmaximal, characteristic, complemented
16#29385—31nomaximal, Sylow
17C3×#568C2211yescharacteristic
18#5828C2211yescharacteristic
19#5822C2211yescharacteristic
20#5745C2211yescharacteristic
21#5524C2211yescharacteristic
22C2×#872C2211yescharacteristic
23C3×#430C2211yescharacteristic
24C3×#423C2211yescharacteristic
25C3×#421C2211yescharacteristic
26C2×C3×#104C2211yescharacteristic
27#5237C2211yescharacteristic
28C2×#837C2211yescharacteristic
29#5145—22no—
30#5144—22no—
31#5139C2211yescharacteristic
32C2×#825C2211yescharacteristic
33#5132C2211yescharacteristic
34#5126C2211yescharacteristic
35#5126C2211yescharacteristic
36#5126C2211yescharacteristic
37#5126C2211yescharacteristic
38#5124C2211yescharacteristic
39#5124C2211yescharacteristic
40Dic3×M16C2211yescharacteristic
41Dic3×M16C2211yescharacteristic
42C2×#805C2211yescharacteristic
43#4924C2211yescharacteristic
44#4840—22no—
45#4840—22no—
46C3×#344C2211yescharacteristic
47C3×#339C2211yescharacteristic
48C3⋊#338C2211yescharacteristic
49#4556C2211yescharacteristic
50#4556C2211yescharacteristic
51#4555C2211yescharacteristic
52#4533C2211yescharacteristic
53#4532C2211yescharacteristic
54#4531C2211yescharacteristic
55#4523C2211yescharacteristic
56#2956—31no—
57#2921—31no—
58#2102—31no—
59#2091—31no—
60#2050—31no—
61#2010—31no—
62#1932—31no—
63#1932—31no—
64#1897—31no—
65#1897—31no—
66#1891—31no—
67#1891—31no—
68#1861—31no—
69#1861D311yescharacteristic, complemented
70#1841—31no—
71C2×C3×#41C2311yescharacteristic
72C2×#235C2311yescharacteristic
73C22×C4×D3C2311yescharacteristic
74C3×#131—22no—
75C3×#131—41no—
76C2×C3×M16—41no—
77C2×C3×M16C2311yescharacteristic
78C2×C3×M16C2311yescharacteristic
79C2×C3×M16C2311yescharacteristic
80C2×C3×M16C2311yescharacteristic
81C3×#117C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83#951—21no—
84#950—22no—
85#949—21no—
86#946—21no—
87#942—21no—
88#938—21no—
89#937—21no—
90#935—21no—
91#932—21no—
92C4×#67—41no—
93C2×(C2×D3)⋊C4—21no—
94#926C2311yescharacteristic
95C2×Dic3⋊C4—21no—
96C2×C4×Dic3C2311yescharacteristic
97C2×#208C2311yescharacteristic
98C2×#208C2311yescharacteristic
99C2×#208C2311yescharacteristic
100C2×#208C2311yescharacteristic
101#907—41no—
102D3×M16—41no—
103#901—22no—
104#897—21no—
105#895—21no—
106#886—21no—
107#882—21no—
108C4×D12—41no—
109#872—22no—
110#872—21no—
111#872—21no—
112C3×#108—41no—
113C3×C4⋊C8—22no—
114C3×#104—22no—
115C3×#104—41no—
116#837—21no—
117#837—21no—
118#836—21no—
119#833—21no—
120#825—21no—
121#825—21no—
122#825—41no—
123#824—21no—
124#823—21no—
125#822—22no—
126#822—41no—
127(C2×D3)⋊C8—22no—
128(C2×D3)⋊C8—22no—
129(C2×D3)⋊C8—41no—
130(C2×D3)⋊C8—41no—
131#815—21no—
132#815D412yescomplemented
133C8×Dic3—21no—
134C8×Dic3D412yescomplemented
135#805—22no—
136C4⋊(C3⋊C8)—22no—
137#575—31no—
138#568—31no—
139#568C2×D311yescharacteristic
140#525—31no—
141#525—31no—
142C2×#117—31no—
143#444—62no—
144#443—62no—
145#433—62no—
146#433—62no—
147#430—31no—
148#430—31no—
149#430C2×D311yescharacteristic
150#423—31no—
151#423C2×D311yescharacteristic
152#421—31no—
153#421C2×D311yescharacteristic
154C2×#104—31no—
155C2×#104—31no—
156C2×#104—31no—
157C2×#104C2×D311yescharacteristic
158#413—31no—
159#413—31no—
160#407—31no—
161#407—31no—
162#407—31no—
163#407—31no—
164C4×M16—31no—
165C4×M16—31no—
166#344—31no—
167#344—31no—
168#344—31no—
169#344C2×D311yescharacteristic
170#339—31no—
171#339—31no—
172#339—31no—
173#339C2×D311yescharacteristic
174#338—31no—
175H(3,ℤ/4)—31no—
176C23×D3#4111yescharacteristic
177C3×#41—21no—
178C3×#41—21no—
179C3×#41—21no—
180C3×#41—21no—
181C2×C3×Q8#4111yescharacteristic
182C2×C3×D4—21no—
183C2×C3×D4#4111yescharacteristic
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4C2411yescharacteristic
187C2×#67—21no—
188C2×#67—41no—
189C22×Dic3—21no—
190C22×Dic3#4111yescharacteristic
191#235—21no—
192#235—21no—
193#235—21no—
194#235—21no—
195C2×D12—41no—
196C2×D12#4111yescharacteristic
197C2×C4×D3—21no—
198C2×C4×D3—21no—
199C2×C4×D3—21no—
200C2×C4×D3—21no—
201C2×C4×D3—21no—
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×Dic6#4111yescharacteristic
205C3×M16—21no—
206C3×M16—21no—
207C3×M16—21no—
208C3×M16—21no—
209C3×M16—21no—
210C3×M16—21no—
211C3×M16—21no—
212C3×M16—21no—
213C3×M16—42no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—42no—
218C2×C3×C8C2×D412yes—
219C3×C4⋊C4—21no—
220C3×C22⋊C4—21no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42#4112yes—
224#217—21no—
225#217—41no—
226#216—22no—
227#215—22no—
228#214—22no—
229#213—22no—
230(C2×D3)⋊C4—22no—
231(C2×D3)⋊C4—22no—
232(C2×D3)⋊C4—41no—
233(C2×D3)⋊C4—41no—
234(C2×D3)⋊C4—41no—
235C4⋊Dic3—21no—
236C4⋊Dic3—41no—
237Dic3⋊C4—22no—
238Dic3⋊C4—22no—
239Dic3⋊C4—41no—
240C4×Dic3—21no—
241C4×Dic3—21no—
242C4×Dic3C2×D411yescharacteristic
243C4×Dic3C2×D411yescharacteristic
244#208—21no—
245#208—21no—
246#208—21no—
247#208—21no—
248#208—21no—
249#208—21no—
250#208—21no—
251#208—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8C2×D412yes—
256ΓL(1,𝔽25)—42no—
257D3×C8—42no—
258C2×#41—31no—
259C2×#41C22×D311yescharacteristic
260C23×C4—31no—
261#137—61no—
262#137—61no—
263AGL(1,ℤ/8)—61no—
264AGL(1,ℤ/8)—61no—
265#135—62no—
266#131—22no—
267#131—41no—
268#131—62no—
269#131—121no—
270C2×M16—31no—
271C2×M16—31no—
272C2×M16—31no—
273C2×M16—31no—
274C2×M16—41no—
275C2×M16—121no—
276C2×M16C22×D311yescharacteristic
277C2×M16C22×D311yescharacteristic
278C2×M16C22×D311yescharacteristic
279C2×M16C22×D311yescharacteristic
280#123—61no—
281#123—61no—
282C22⋊Q8—61no—
283C22⋊Q8—61no—
284#121—61no—
285#121—61no—
286#120—61no—
287#120—61no—
288C4×D4—121no—
289C4×D4—121no—
290#117—31no—
291#117—61no—
292#117—61no—
293#117—62no—
294#117C22×D311yescharacteristic
295C2×C4⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—31no—
298C2×C42C22×D311yescharacteristic
299#108—41no—
300#108—62no—
301#108—121no—
302C4⋊C8—22no—
303C4⋊C8—62no—
304#104—22no—
305#104—41no—
306#104—61no—
307#104—61no—
308#104—61no—
309#104—61no—
310#104—62no—
311#104—121no—
312#101—61no—
313#101—61no—
314#100—61no—
315#100—61no—
316C22⋊C8—62no—
317C22⋊C8—62no—
318C22⋊C8—121no—
319C22⋊C8—121no—
320#97—32no—
321#97—61no—
322C4×C8—32no—
323C4×C8—61no—
324C23×C3—21no—
325C23×C3C2×#4111yescharacteristic
326C22×D3—21no—
327C22×D3—21no—
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—81no—
331C3×Q8—21no—
332C3×Q8—21no—
333C3×D4—21no—
334C3×D4—21no—
335C3×D4—41no—
336C3×D4—41no—
337C2×C3×C4—22no—
338C2×C3×C4—21no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—81no—
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4H(5,𝔽2)11yescharacteristic
348C2×C3×C4H(5,𝔽2)11yescharacteristic
349#67—41no—
350#67—41no—
351#67—82no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—41no—
359C2×Dic3—81no—
360D12—21no—
361D12—21no—
362D12—82no—
363C4×D3—41no—
364C4×D3—41no—
365C4×D3—41no—
366C4×D3—41no—
367C4×D3—41no—
368C4×D3—41no—
369Dic6—21no—
370Dic6—21no—
371C3×C8—22no—
372C3×C8—41no—
373C3×C8—41no—
374C3×C8—41no—
375C3⋊C8—22no—
376C3⋊C8—41no—
377C3⋊C8—41no—
378C3⋊C8—41no—
379C24—31no—
380#41—21no—
381#41—21no—
382#41—21no—
383#41—21no—
384#41—61no—
385#41—61no—
386#41—61no—
387#41—61no—
388C2×Q8—31no—
389C2×Q8#23911yescharacteristic
390C2×D4—21no—
391C2×D4—31no—
392C2×D4—61no—
393C2×D4—121no—
394C2×D4—121no—
395C2×D4#23911yescharacteristic
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—31no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4C23×D311yescharacteristic
408Q16—62no—
409SD16—62no—
410SD16—62no—
411D8—62no—
412M16—21no—
413M16—21no—
414M16—21no—
415M16—21no—
416M16—21no—
417M16—21no—
418M16—21no—
419M16—21no—
420M16—42no—
421M16—61no—
422M16—61no—
423M16—61no—
424M16—61no—
425M16—61no—
426M16—61no—
427M16—61no—
428M16—61no—
429M16—122no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—32no—
434C2×C8—42no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—122no—
439C2×C8D3×D412yes—
440C4⋊C4—21no—
441C4⋊C4—62no—
442C4⋊C4—62no—
443C4⋊C4—61no—
444C4⋊C4—121no—
445C4⋊C4—121no—
446C22⋊C4—21no—
447C22⋊C4—62no—
448C22⋊C4—62no—
449C22⋊C4—61no—
450C22⋊C4—121no—
451C22⋊C4—121no—
452C22⋊C4—121no—
453C22⋊C4—121no—
454C42—21no—
455C42—21no—
456C42—31no—
457C42—31no—
458C42—61no—
459C42—61no—
460C42#23512yes—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—81no—
464C22×C3#52511yescharacteristic
465C22×C3#55111yescharacteristic
466C22×C3#55111yescharacteristic
467C2×D3—41no—
468C2×D3—41no—
469C2×D3—42no—
470C2×D3—41no—
471C2×D3—41no—
472C2×D3—41no—
473C2×D3—41no—
474C2×D3—81no—
475C3×C4—42no—
476C3×C4—41no—
477C3×C4—81no—
478C3×C4#52511yescharacteristic
479C3×C4#53511yescharacteristic
480C3×C4#55311yescharacteristic
481C3×C4#55311yescharacteristic
482Dic3—41no—
483Dic3—41no—
484Dic3—41no—
485Dic3—41no—
486Dic3—81no—
487C23—21no—
488C23—61no—
489C23—61no—
490C23—61no—
491C23—61no—
492C23—241no—
493C23D3×#4111yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—61no—
497Q8—61no—
498D4—21no—
499D4—21no—
500D4—41no—
501D4—41no—
502D4—61no—
503D4—61no—
504D4—121no—
505D4—121no—
506D4—242no—
507D4—242no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—81no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—241no—
528C2×C4C2×#23511yescharacteristic
529C2×C4C2×D3×D411yescharacteristic
530C2×C4#100411yescharacteristic
531C2×C4#100411yescharacteristic
532C2×C4C2×#23911yescharacteristic
533C2×C4D3×#4111yescharacteristic
534C8—22no—
535C8—41no—
536C8—41no—
537C8—41no—
538C8—62no—
539C8—121no—
540C8—121no—
541C8—121no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—81no—
546C2×C3#259811yescharacteristic
547D3—81no—
548D3—81no—
549C22—41no—
550C22—41no—
551C22—81no—
552C22—121no—
553C22—121no—
554C22—122no—
555C22—121no—
556C22—121no—
557C22—121no—
558C22—121no—
559C22—241no—
560C22#554711yescharacteristic
561C22#563711yescharacteristic
562C22#565911yescharacteristic
563C4—42no—
564C4—41no—
565C4—81no—
566C4—121no—
567C4—121no—
568C4—121no—
569C4—121no—
570C4—241no—
571C4#551411yescharacteristic
572C4#555811yescharacteristic
573C4#559211yescharacteristic
574C4#562711yescharacteristic
575C3#2938511yescharacteristic, complemented, Sylow
576C2—21no—
577C2—21no—
578C2—21no—
579C2—81no—
580C2—241no—
581C2—241no—
582C2#7449611yescharacteristic
subgroupG/H
centerC4#5592
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1861C2
socleC2×C3#2598
prime radical#70406C2
hypercenter#1861D3
hypocenterC3#29385
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1861

Complements

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

NH ≅ G/Ntype
1. #83811C2semidirect
2. #79039C2semidirect
3. C3×#1861C2semidirect
4. #72029C2semidirect
5. #70729C2semidirect
6. #70727C2semidirect
7. #70530C2semidirect
9. #70401C2semidirect
10. #70383C2semidirect
11. #70379C2semidirect
12. #69036C2semidirect
13. #69017C2semidirect
14. #68757C2semidirect
15. #68751C2semidirect
69. #1861D3semidirect
132. #815D4semidirect
134. C8×Dic3D4semidirect
575. C3#29385semidirect

Maximal subgroups

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

subgroupindex
#838112
#790392
C3×#18612
#720292
#707292
#707272
#705302
#704062
#704012
#703832
#703792
#690362
#690172
#687572
#687512
#293853