Finite Group Databasebeta

← #10840293

Subgroups — #10840293

Order 768.

638 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#471C211yesmaximal, characteristic
2#83843C211yesmaximal, characteristic, complemented
3#72104C211yesmaximal, characteristic, complemented
4#72101C211yesmaximal, characteristic, complemented
5#72092C211yesmaximal, characteristic
6#72089C211yesmaximal, characteristic
7#72052C211yesmaximal, characteristic, complemented
8#72013C211yesmaximal, characteristic, complemented
9#72011C211yesmaximal, characteristic, complemented
10#70706C211yesmaximal, characteristic, complemented
11#70698C211yesmaximal, characteristic, complemented
12#70112C211yesmaximal, characteristic, complemented
13#70101C211yesmaximal, characteristic, complemented
14#70078C211yesmaximal, characteristic, complemented
15#70067C211yesmaximal, characteristic, complemented
16#24987—31nomaximal, Sylow
17C2×C3×#137C2211yescharacteristic
18C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
19C22×C3×SD16C2211yescharacteristic
20C2×C3×#131C2211yescharacteristic
21#5812C2211yescharacteristic
22#5798C2211yescharacteristic
23C3×#471—22no—
24C3×#471—22no—
25C2×C3×#108C2211yescharacteristic
26C2×C3×#101C2211yescharacteristic
27C2×C3×#100C2211yescharacteristic
28#5240C2211yescharacteristic
29#5237C2211yescharacteristic
30#5236C2211yescharacteristic
31#5236C2211yescharacteristic
32C2×#836C2211yescharacteristic
33#5225C2211yescharacteristic
34#5225C2211yescharacteristic
35C2×#833C2211yescharacteristic
36#5215C2211yescharacteristic
37#5215C2211yescharacteristic
38#5164—22no—
39Dic3×SD16—22no—
40#5124C2211yescharacteristic
41#5124C2211yescharacteristic
42C2×#819C2211yescharacteristic
43#5106C2211yescharacteristic
44#5101C2211yescharacteristic
45#5101C2211yescharacteristic
46#4559C2211yescharacteristic
47#4559C2211yescharacteristic
48#4559C2211yescharacteristic
49#4559C2211yescharacteristic
50#4557C2211yescharacteristic
51#4557C2211yescharacteristic
52#4556C2211yescharacteristic
53#4556C2211yescharacteristic
54#4556C2211yescharacteristic
55#4556C2211yescharacteristic
56C2×#471D311yescharacteristic
57#2949—31no—
58#2082—31no—
59#2081—31no—
60#2049—31no—
61#2048—31no—
62#1977—31no—
63#1953—31no—
64#1905—31no—
65#1904—31no—
66#1833—31no—
67#1830—31no—
68#1829—31no—
69#1814—31no—
70#1757—31no—
71C2×C3×#41C2311yescharacteristic
72C22×C3×Q8C2311yescharacteristic
73C22×C3×D4C2311yescharacteristic
74C3×#137—22no—
75C3×#137—41no—
76C3×AGL(1,ℤ/8)—22no—
77C3×AGL(1,ℤ/8)—41no—
78C2×C3×Q16—21no—
79C2×C3×SD16—21no—
80C2×C3×SD16—21no—
81C2×C3×SD16—21no—
82C2×C3×SD16—21no—
83C2×C3×SD16—21no—
84C2×C3×SD16—21no—
85C2×C3×SD16D412yes—
86C2×C3×SD16D412yes—
87C2×C3×D8—21no—
88C3×#131—21no—
89C3×#131—21no—
90C3×#131—21no—
91C3×#131—21no—
92C2×C3×M16—21no—
93C2×C3×M16C2311yescharacteristic
94C2×C3×M16C2311yescharacteristic
95C2×C3×M16C2311yescharacteristic
96C22×C3×C8—21no—
97C22×C3×C8C2311yescharacteristic
98C2×#217—22no—
99Q8×Dic3—22no—
100Q8×Dic3—41no—
101D4×Dic3—22no—
102D4×Dic3—41no—
103#926—22no—
104#926C2311yescharacteristic
105#926C2311yescharacteristic
106#926C2311yescharacteristic
107#926C2311yescharacteristic
108C2×#208C2311yescharacteristic
109C2×#208C2311yescharacteristic
110C2×#208C2311yescharacteristic
111C2×#208C2311yescharacteristic
112C3×#108—21no—
113C3×#108—21no—
114C3×#101—22no—
115C3×#100—22no—
116#837—41no—
117#837—41no—
118#836—21no—
119#836—41no—
120#835—22no—
121#835—41no—
122#834—22no—
123#833—21no—
124#833—41no—
125#832—22no—
126#832—41no—
127#822—41no—
128#822—41no—
129#819—21no—
130#819—21no—
131#818—21no—
132#817—21no—
133#815—21no—
134#815—21no—
135C8×Dic3—21no—
136C8×Dic3—21no—
137C2×#137C2×D311yescharacteristic
138C2×AGL(1,ℤ/8)C2×D311yescharacteristic
139C22×SD16C2×D311yescharacteristic
140C2×#131C2×D311yescharacteristic
141#519—31no—
142#518—31no—
143#471—22no—
144#471—22no—
145#440—62no—
146C4×SD16—62no—
147#430—31no—
148#430—31no—
149C2×#108—31no—
150C2×#108C2×D311yescharacteristic
151#427—31no—
152#421—31no—
153#421—31no—
154#419—31no—
155#419—31no—
156#418—31no—
157#418—31no—
158#413—31no—
159C2×#101—31no—
160C2×#101C2×D311yescharacteristic
161C2×#100—31no—
162C2×#100C2×D311yescharacteristic
163#410—31no—
164#405—31no—
165#405—31no—
166#344—31no—
167#344—31no—
168#344—31no—
169#344—31no—
170#344—31no—
171#344—31no—
172#344—31no—
173#344—31no—
174H(3,ℤ/4)—31no—
175H(3,ℤ/4)—31no—
176C24×C3—21no—
177C3×#41—21no—
178C3×#41—21no—
179C3×#41—21no—
180C3×#41—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8C2×D412yes—
187C2×C3×Q8#4111yescharacteristic
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4C2×D412yes—
195C2×C3×D4#4111yescharacteristic
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—42no—
200C3×Q16—42no—
201C3×SD16—22no—
202C3×SD16—22no—
203C3×SD16—22no—
204C3×SD16—22no—
205C3×SD16—42no—
206C3×SD16—42no—
207C3×D8—42no—
208C3×M16—21no—
209C3×M16—21no—
210C3×M16—21no—
211C3×M16—21no—
212C3×M16—21no—
213C3×M16—21no—
214C3×M16—41no—
215C3×M16—41no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—41no—
220C2×C3×C8—41no—
221C2×C3×C8C2×D411yescharacteristic
222C2×C3×C8C2×D411yescharacteristic
223C2×C3×C8#4111yescharacteristic
224C2×C3×C8#4111yescharacteristic
225C2×C3×C8#4111yescharacteristic
226C2×C3×C8#4111yescharacteristic
227#217—21no—
228#217—21no—
229#217—21no—
230#217—42no—
231#217#4112yes—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—42no—
237C4×Dic3—21no—
238C4×Dic3—21no—
239C4×Dic3—21no—
240C4×Dic3—21no—
241C4×Dic3—42no—
242#208—21no—
243#208—21no—
244#208—21no—
245#208—21no—
246#208—21no—
247#208—21no—
248#208—21no—
249#208—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×#41C22×D311yescharacteristic
255C22×Q8C22×D311yescharacteristic
256C22×D4C22×D311yescharacteristic
257#137—22no—
258#137—41no—
259AGL(1,ℤ/8)—22no—
260AGL(1,ℤ/8)—41no—
261C2×Q16—21no—
262C2×SD16—21no—
263C2×SD16—21no—
264C2×SD16—21no—
265C2×SD16—21no—
266C2×SD16—21no—
267C2×SD16—21no—
268C2×SD16#6712yes—
269C2×SD16#6712yes—
270C2×D8—21no—
271#131—21no—
272#131—21no—
273#131—21no—
274#131—21no—
275C2×M16—21no—
276C2×M16—31no—
277C2×M16—31no—
278C2×M16—31no—
279C2×M16—31no—
280C2×M16C22×D311yescharacteristic
281C2×M16C22×D311yescharacteristic
282C2×M16C22×D311yescharacteristic
283C22×C8—21no—
284C22×C8C22×D311yescharacteristic
285C4×Q8—62no—
286C4×Q8—121no—
287C4×D4—62no—
288C4×D4—121no—
289#117—31no—
290#117—31no—
291#117—31no—
292#117—31no—
293#117—62no—
294C2×C22⋊C4—62no—
295#108—21no—
296#108—21no—
297#108—61no—
298#108—61no—
299#108—121no—
300#108—121no—
301#107—61no—
302#106—61no—
303#104—121no—
304#104—121no—
305#103—62no—
306#103—121no—
307#102—62no—
308#102—121no—
309#101—22no—
310#101—61no—
311#101—121no—
312#100—22no—
313#100—61no—
314#100—121no—
315#99—62no—
316#97—61no—
317#97—61no—
318C4×C8—61no—
319C4×C8—61no—
320C23×C3—21no—
321C23×C3—22no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—161no—
325C23×C3C2×#4111yescharacteristic
326C3×Q8—22no—
327C3×Q8—21no—
328C3×Q8—22no—
329C3×Q8—21no—
330C3×Q8—41no—
331C3×Q8—41no—
332C3×D4—22no—
333C3×D4—21no—
334C3×D4—22no—
335C3×D4—21no—
336C3×D4—41no—
337C3×D4—41no—
338C3×D4—41no—
339C3×D4—41no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4H(5,𝔽2)11yescharacteristic
351C2×C3×C4H(5,𝔽2)11yescharacteristic
352C2×Dic3—22no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—162no—
357C3×C8—41no—
358C3×C8—41no—
359C3×C8—41no—
360C3×C8#12311yescharacteristic
361C3×C8#12311yescharacteristic
362C3×C8#12311yescharacteristic
363C3×C8#12311yescharacteristic
364C3⋊C8—41no—
365C3⋊C8—41no—
366C3⋊C8—41no—
367C3⋊C8—41no—
368C24—21no—
369#41—21no—
370#41—21no—
371#41—21no—
372#41—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8#23711yescharacteristic
379C2×Q8C2×#6712yes—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4#23711yescharacteristic
387C2×D4C2×#6712yes—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—122no—
391C22×C4C23×D311yescharacteristic
392Q16—42no—
393SD16—22no—
394SD16—22no—
395SD16—22no—
396SD16—22no—
397SD16—42no—
398SD16—42no—
399D8—42no—
400M16—21no—
401M16—21no—
402M16—21no—
403M16—21no—
404M16—21no—
405M16—21no—
406M16—41no—
407M16—41no—
408M16—61no—
409M16—61no—
410M16—61no—
411M16—61no—
412M16—61no—
413M16—61no—
414M16—61no—
415M16—61no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8#23711yescharacteristic
426C2×C8#23711yescharacteristic
427C2×C8#23711yescharacteristic
428C2×C8#23711yescharacteristic
429C2×C8C2×#6711yescharacteristic
430C2×C8C2×#6711yescharacteristic
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—122no—
436C22⋊C4—32no—
437C22⋊C4—61no—
438C22⋊C4—61no—
439C22⋊C4—61no—
440C22⋊C4—122no—
441C42—61no—
442C42—61no—
443C42—61no—
444C42—61no—
445C42—122no—
446C22×C3—41no—
447C22×C3—41no—
448C22×C3—42no—
449C22×C3—82no—
450C22×C3—81no—
451C22×C3—161no—
452C22×C3—161no—
453C22×C3#52911yescharacteristic
454C22×C3#54611yescharacteristic
455C22×C3#55511yescharacteristic
456C3×C4—41no—
457C3×C4—42no—
458C3×C4C2×#12311yescharacteristic
459C3×C4#54011yescharacteristic
460C3×C4#54611yescharacteristic
461C3×C4#55311yescharacteristic
462Dic3—42no—
463Dic3—81no—
464Dic3—81no—
465Dic3—81no—
466C23—22no—
467C23—21no—
468C23—41no—
469C23—41no—
470C23—161no—
471C23D3×#4111yescharacteristic
472Q8—22no—
473Q8—21no—
474Q8—22no—
475Q8—21no—
476Q8—41no—
477Q8—41no—
478D4—22no—
479D4—21no—
480D4—22no—
481D4—21no—
482D4—41no—
483D4—41no—
484D4—41no—
485D4—41no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—62no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4—121no—
496C2×C4—482no—
497C2×C4C2×#23711yescharacteristic
498C2×C4C2×#23711yescharacteristic
499C2×C4C2×#23711yescharacteristic
500C2×C4#100411yescharacteristic
501C2×C4#100911yescharacteristic
502C2×C4C22×#6711yescharacteristic
503C8—41no—
504C8—41no—
505C8—41no—
506C8—121no—
507C8—121no—
508C8—121no—
509C8—121no—
510C8#93511yescharacteristic
511C8#93511yescharacteristic
512C8#93511yescharacteristic
513C8#93511yescharacteristic
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—81no—
518C2×C3—82no—
519C2×C3#242011yescharacteristic
520C22—41no—
521C22—41no—
522C22—42no—
523C22—82no—
524C22—81no—
525C22—161no—
526C22—161no—
527C22#560011yescharacteristic
528C22#561611yescharacteristic
529C22#582911yescharacteristic
530C4—41no—
531C4—42no—
532C4—122no—
533C4—241no—
534C4—241no—
535C4—241no—
536C4#561811yescharacteristic
537C4#561911yescharacteristic
538C4C2×#93511yescharacteristic
539C4#580611yescharacteristic
540C3#2498711yescharacteristic, complemented, Sylow
541C2—21no—
542C2—21no—
543C2—21no—
544C2—82no—
545C2—81no—
546C2#8264311yescharacteristic
subgroupG/H
centerC2#82643
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#471C2
socleC2×C3#2420
prime radicalC2×C3×AGL(1,ℤ/8)C22
hypercenterC2×#471D3
hypocenterC3#24987
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#471

Complements

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

NH ≅ G/Ntype
2. #83843C2semidirect
3. #72104C2semidirect
4. #72101C2semidirect
7. #72052C2semidirect
8. #72013C2semidirect
9. #72011C2semidirect
10. #70706C2semidirect
11. #70698C2semidirect
12. #70112C2semidirect
13. #70101C2semidirect
14. #70078C2semidirect
15. #70067C2semidirect
540. C3#24987semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4712
#838432
#721042
#721012
#720922
#720892
#720522
#720132
#720112
#707062
#706982
#701122
#701012
#700782
#700672
#249873