Finite Group Databasebeta

← #10710939

Subgroups — #10710939

Order 768.

888 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3584C211yesmaximal, characteristic
2#72081C211yesmaximal, characteristic, complemented
3#72080C211yesmaximal, characteristic, complemented
4C2×#4566C211yesmaximal, characteristic, complemented
5C2×#4566C211yesmaximal, characteristic, complemented
6#72015C211yesmaximal, characteristic, complemented
7#72006C211yesmaximal, characteristic, complemented
8#12752—31nomaximal, Sylow
9C3×#585—21no—
10C3×#578—21no—
11C3×#578C412yescomplemented
12C3×#577—22no—
13C3×#577—22no—
14C3×#576—21no—
15C3×#576C412yescomplemented
16C3×#575—22no—
17C3×#575—21no—
18C2×C3×#135—22no—
19C2×C3×#135C2211yescharacteristic
20C2×C3×#135C2211yescharacteristic
21C2×C3×#131C2211yescharacteristic
22#5232—21no—
23C2×#819C2211yescharacteristic, complemented
24C2×#817C2211yescharacteristic, complemented
25C2×#812C2211yescharacteristic, complemented
26C2×#812C2211yescharacteristic, complemented
27#4567—22no—
28#4566—22no—
29#4566—22no—
30#4565—21no—
31#4563—21no—
32#4561—22no—
33#4554—21no—
34#3584D311yescharacteristic
35#2148—31no—
36C2×#360—31no—
37C2×#360—31no—
38#2143—31no—
39#2117—31no—
40#1815—31no—
41C3×2−1+4—22no—
42C3×2−1+4—22no—
43C3×2−1+4—41no—
44C3×H(5,𝔽2)—22no—
45C3×H(5,𝔽2)—22no—
46C3×H(5,𝔽2)—42no—
47C3×H(5,𝔽2)—41no—
48C2×C3×#41—24no—
49C2×C3×#41—21no—
50C2×C3×#41—21no—
51C2×C3×#41—42no—
52C2×C3×#41—42no—
53C2×C3×#41D411yescharacteristic
54C3×#137—22no—
55C3×#137—24no—
56C3×#137—24no—
57C3×#137—22no—
58C3×AGL(1,ℤ/8)—22no—
59C3×AGL(1,ℤ/8)—24no—
60C3×AGL(1,ℤ/8)—24no—
61C3×AGL(1,ℤ/8)—22no—
62C3×#135—24no—
63C3×#135—24no—
64C3×#135—22no—
65C3×#135—22no—
66C3×#135—21no—
67C3×#135—21no—
68C3×#135—21no—
69C3×#135—21no—
70C3×#135C2×C412yes—
71C3×#135D412yes—
72C3×#135D412yes—
73C3×#135D412yes—
74C2×C3×Q16—22no—
75C2×C3×Q16C2×C411yescharacteristic
76C2×C3×Q16D411yescharacteristic
77C2×C3×SD16—24no—
78C2×C3×SD16—21no—
79C2×C3×SD16—21no—
80C2×C3×D8—22no—
81C2×C3×D8C2×C411yescharacteristic
82C2×C3×D8D411yescharacteristic
83C3×#131—21no—
84C3×#131—21no—
85C3×#131C2×C412yes—
86C3×#131D412yes—
87C2×C3×M16—21no—
88C2×C3×M16D411yescharacteristic
89C22×C3×C8—21no—
90C22×C3×C8C2311yescharacteristic
91C2×C4⋊Dic3—21no—
92C2×#208—21no—
93#835—41no—
94#832—41no—
95#819—21no—
96#819—21no—
97#817—21no—
98#817—21no—
99#812—21no—
100#812—21no—
101#812—21no—
102#812—21no—
103C2×C3⋊C16—21no—
104C2×C3⋊C16—21no—
105#585—21no—
106#578—21no—
107#578Dic312yescomplemented
108#577—22no—
109#577—22no—
110#576—21no—
111#576Dic312yescomplemented
112#575—22no—
113#575—21no—
114C2×#135—22no—
115C2×#135C2×D311yescharacteristic
116C2×#135C2×D311yescharacteristic
117C2×#131C2×D311yescharacteristic
118C2×M32—31no—
119C2×M32—31no—
120C2×#108—31no—
121C2×#106—31no—
122#417—61no—
123#362—61no—
124#361—61no—
125#360—62no—
126#360—62no—
127#359—62no—
128#350—62no—
129#343—61no—
130C3×#41—21no—
131C3×#41—21no—
132C3×#41—24no—
133C3×#41—24no—
134C3×#41—22no—
135C3×#41—22no—
136C3×#41—22no—
137C3×#41—22no—
138C3×#41—44no—
139C3×#41—44no—
140C3×#41—44no—
141C3×#41—44no—
142C3×#41—42no—
143C3×#41—42no—
144C3×#41—42no—
145C3×#41—42no—
146C3×#41—41no—
147C3×#41—41no—
148C3×#41—41no—
149C3×#41—41no—
150C3×#41C22⋊C412yes—
151C3×#41SD1612yes—
152C2×C3×Q8—24no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—42no—
156C2×C3×Q8—42no—
157C2×C3×Q8D811yescharacteristic
158C2×C3×D4—21no—
159C2×C3×D4—24no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—44no—
163C2×C3×D4—44no—
164C2×C3×D4—42no—
165C2×C3×D4—44no—
166C2×C3×D4—41no—
167C2×C3×D4—41no—
168C2×C3×D4—42no—
169C2×C3×D4D811yescharacteristic
170C22×C3×C4—21no—
171C22×C3×C4—44no—
172C22×C3×C4—41no—
173C22×C3×C4—41no—
174C22×C3×C4C2×D411yescharacteristic
175C22×Dic3—41no—
176C3×Q16—24no—
177C3×Q16—21no—
178C3×Q16—21no—
179C3×Q16C22⋊C412yes—
180C3×Q16C22⋊C412yes—
181C3×SD16—24no—
182C3×SD16—24no—
183C3×SD16—22no—
184C3×SD16—22no—
185C3×SD16—22no—
186C3×SD16—22no—
187C3×D8—24no—
188C3×D8—21no—
189C3×D8—21no—
190C3×D8C22⋊C412yes—
191C3×D8C22⋊C412yes—
192C3×M16—21no—
193C3×M16—22no—
194C3×M16—22no—
195C3×M16C22⋊C412yes—
196C2×C3×C8—21no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8C22×C411yescharacteristic
200C2×C3×C8C2×D411yescharacteristic
201C2×C3×C8C2×D411yescharacteristic
202C2×C3×C8C2×D411yescharacteristic
203C2×C3×C8C2×D411yescharacteristic
204C2×C3×C8C2×D411yescharacteristic
205C4⋊Dic3—41no—
206C4⋊Dic3—41no—
207#208—41no—
208#208—41no—
209C2×C3⋊C8—41no—
210C3⋊C16—41no—
211C3⋊C16—41no—
2122−1+4—22no—
2132−1+4—22no—
2142−1+4—41no—
215H(5,𝔽2)—22no—
216H(5,𝔽2)—22no—
217H(5,𝔽2)—42no—
218H(5,𝔽2)—41no—
219C2×#41—24no—
220C2×#41—21no—
221C2×#41—21no—
222C2×#41—42no—
223C2×#41—42no—
224C2×#41#6711yescharacteristic
225#137—22no—
226#137—24no—
227#137—24no—
228#137—22no—
229AGL(1,ℤ/8)—22no—
230AGL(1,ℤ/8)—24no—
231AGL(1,ℤ/8)—24no—
232AGL(1,ℤ/8)—22no—
233#135—24no—
234#135—24no—
235#135—22no—
236#135—22no—
237#135—21no—
238#135—21no—
239#135—21no—
240#135—21no—
241#135C2×Dic312yes—
242#135#6712yes—
243#135#6712yes—
244#135#6712yes—
245C2×Q16—22no—
246C2×Q16C2×Dic311yescharacteristic
247C2×Q16#6711yescharacteristic
248C2×SD16—24no—
249C2×SD16—21no—
250C2×SD16—21no—
251C2×D8—22no—
252C2×D8C2×Dic311yescharacteristic
253C2×D8#6711yescharacteristic
254#131—21no—
255#131—21no—
256#131C2×Dic312yes—
257#131#6712yes—
258C2×M16—21no—
259C2×M16—61no—
260C2×M16#6711yescharacteristic
261C22×C8—21no—
262C22×C8C22×D311yescharacteristic
263C2×C4⋊C4—61no—
264M32—61no—
265M32—61no—
266M32—61no—
267M32—61no—
268C2×C16—61no—
269C2×C16—61no—
270#108—61no—
271#108—61no—
272#106—61no—
273#106—61no—
274#103—121no—
275#102—121no—
276C23×C3—21no—
277C23×C3—44no—
278C23×C3—41no—
279C23×C3—41no—
280C23×C3AGL(1,ℤ/8)11yescharacteristic
281C3×Q8—21no—
282C3×Q8—22no—
283C3×Q8—22no—
284C3×Q8—44no—
285C3×Q8—41no—
286C3×Q8—41no—
287C3×Q8#10212yes—
288C3×D4—22no—
289C3×D4—22no—
290C3×D4—22no—
291C3×D4—22no—
292C3×D4—21no—
293C3×D4—44no—
294C3×D4—44no—
295C3×D4—44no—
296C3×D4—42no—
297C3×D4—42no—
298C3×D4—42no—
299C3×D4—42no—
300C3×D4—41no—
301C3×D4—41no—
302C3×D4#10212yes—
303C2×C3×C4—21no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—44no—
307C2×C3×C4—44no—
308C2×C3×C4—44no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4C2×C22⋊C411yescharacteristic
316C2×C3×C4#12011yescharacteristic
317C2×C3×C4#12011yescharacteristic
318C2×C3×C4C2×D811yescharacteristic
319C2×C3×C4C2×SD1611yescharacteristic
320C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
321C2×Dic3—81no—
322C2×Dic3—161no—
323C3×C8—22no—
324C3×C8C2×C22⋊C411yescharacteristic
325C3×C8C2×C22⋊C411yescharacteristic
326C3×C8#12011yescharacteristic
327C3×C8#12011yescharacteristic
328C3⋊C8—81no—
329#41—21no—
330#41—21no—
331#41—24no—
332#41—24no—
333#41—22no—
334#41—22no—
335#41—22no—
336#41—22no—
337#41—44no—
338#41—44no—
339#41—44no—
340#41—44no—
341#41—42no—
342#41—42no—
343#41—42no—
344#41—42no—
345#41—41no—
346#41—41no—
347#41—41no—
348#41—41no—
349#41#21412yes—
350#41#21712yes—
351C2×Q8—24no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8#21311yescharacteristic
357C2×D4—21no—
358C2×D4—24no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—42no—
364C2×D4—44no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—42no—
368C2×D4#21311yescharacteristic
369C22×C4—21no—
370C22×C4—44no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—121no—
374C22×C4C2×#6711yescharacteristic
375Q16—24no—
376Q16—21no—
377Q16—21no—
378Q16#21712yes—
379Q16#21712yes—
380SD16—24no—
381SD16—24no—
382SD16—22no—
383SD16—22no—
384SD16—22no—
385SD16—22no—
386D8—24no—
387D8—21no—
388D8—21no—
389D8#21712yes—
390D8#21712yes—
391M16—21no—
392M16—22no—
393M16—22no—
394M16—121no—
395M16—121no—
396M16#21712yes—
397C2×C8—21no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—121no—
401C2×C8C22×Dic311yescharacteristic
402C2×C8C2×#6711yescharacteristic
403C2×C8C2×#6711yescharacteristic
404C2×C8C2×#6711yescharacteristic
405C2×C8C2×#6711yescharacteristic
406C2×C8C2×#6711yescharacteristic
407C4⋊C4—121no—
408C4⋊C4—121no—
409C16—121no—
410C16—121no—
411C22×C3—22no—
412C22×C3—41no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—81no—
416C22×C3—161no—
417C22×C3—161no—
418C22×C3—164no—
419C22×C3#41811yescharacteristic
420C22×C3#44711yescharacteristic
421C22×C3#45011yescharacteristic
422C3×C4—22no—
423C3×C4—42no—
424C3×C4—42no—
425C3×C4#37911yescharacteristic
426C3×C4C2×#10211yescharacteristic
427C3×C4#44711yescharacteristic
428C3×C4#45011yescharacteristic
429Dic3—161no—
430C23—21no—
431C23—44no—
432C23—41no—
433C23—41no—
434C23#93211yescharacteristic
435Q8—21no—
436Q8—22no—
437Q8—22no—
438Q8—44no—
439Q8—41no—
440Q8—41no—
441Q8#83212yes—
442D4—22no—
443D4—22no—
444D4—21no—
445D4—22no—
446D4—22no—
447D4—44no—
448D4—44no—
449D4—44no—
450D4—42no—
451D4—42no—
452D4—42no—
453D4—42no—
454D4—41no—
455D4—41no—
456D4#83212yes—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—241no—
470C2×C4—481no—
471C2×C4C2×#21311yescharacteristic
472C2×C4#93211yescharacteristic
473C2×C4C2×#21411yescharacteristic
474C2×C4C2×#21711yescharacteristic
475C2×C4#95311yescharacteristic
476C2×C4#95311yescharacteristic
477C8—22no—
478C8—241no—
479C8C2×#21711yescharacteristic
480C8C2×#21711yescharacteristic
481C8#95311yescharacteristic
482C8#95311yescharacteristic
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—42no—
487C2×C3—82no—
488C2×C3—82no—
489C2×C3#178611yescharacteristic
490C22—22no—
491C22—41no—
492C22—42no—
493C22—42no—
494C22—81no—
495C22—161no—
496C22—161no—
497C22—164no—
498C22#521511yescharacteristic
499C22#521711yescharacteristic
500C22#521811yescharacteristic
501C4—22no—
502C4—42no—
503C4—42no—
504C4—481no—
505C4C2×#83211yescharacteristic
506C4#521711yescharacteristic
507C4#521811yescharacteristic
508C4#524711yescharacteristic
509C3#1275211yescharacteristic, complemented, Sylow
510C2—21no—
511C2—21no—
512C2—21no—
513C2—42no—
514C2—82no—
515C2—82no—
516C2#7212811yescharacteristic
subgroupG/H
centerC2#72128
commutatorC2×C3×C8C22×C4
FrattiniC22×C8C22×D3
Fitting, prime radicalC3×#3584C2
socleC2×C3#1786
cosocleC22×C3×C8C23
hypercenter#3584D3
hypocenterC3#12752
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3584

Complements

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

NH ≅ G/Ntype
2. #72081C2semidirect
3. #72080C2semidirect
4. C2×#4566C2semidirect
5. C2×#4566C2semidirect
6. #72015C2semidirect
7. #72006C2semidirect
11. C3×#578C4semidirect
15. C3×#576C4semidirect
23. C2×#819C22semidirect
24. C2×#817C22semidirect
25. C2×#812C22semidirect
26. C2×#812C22semidirect
107. #578Dic3semidirect
111. #576Dic3semidirect
509. C3#12752semidirect

Maximal subgroups

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

subgroupindex
C3×#35842
#720812
#720802
C2×#45662
#720152
#720062
#127523