Finite Group Databasebeta

← #10883581

Subgroups — #10883581

Order 768.

812 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83799C211yesmaximal, characteristic, complemented
2#83792C211yesmaximal, characteristic, complemented
3C2×#4897C211yesmaximal, characteristic, complemented
4#79167C211yesmaximal, characteristic, complemented
5C3×#1814C211yesmaximal, characteristic, complemented
6#70698C211yesmaximal, characteristic, complemented
7#70664C211yesmaximal, characteristic, complemented
8#70409C212yesmaximal, complemented
9#70408C212yesmaximal, complemented
10#69027C212yesmaximal, complemented
11#69021C212yesmaximal, complemented
12#28183—31nomaximal, Sylow
13C2×C3×#131C2211yescharacteristic
14#5830C2212yes—
15#5822C2211yescharacteristic
16C2×#908C2212yes—
17C22×D24C2211yescharacteristic
18C2×#901C2211yescharacteristic
19#5521C2212yes—
20C2×C3×#108C2211yescharacteristic
21C3×#427C2211yescharacteristic
22C3×#421C2212yes—
23#5236C2211yescharacteristic
24#5145—22no—
25#5144—22no—
26#5139C2211yescharacteristic
27#5135—22no—
28#5131C2212yes—
29C2×#823C2212yes—
30#5124C2211yescharacteristic
31#5115—22no—
32#5106C2211yescharacteristic
33#5101C2211yescharacteristic
34#5006C2212yes—
35#5001C2212yes—
36#4897—22no—
37#4897—22no—
38#4872—22no—
39#4869—22no—
40C3×#344C2212yes—
41#4559C2212yes—
42#4557C2212yes—
43#4531C2212yes—
44C2×#472—31no—
45#3058—31no—
46#2955—31no—
47#2948—31no—
48#2081—32no—
49#2008—32no—
50#1911—32no—
51#1904—32no—
52#1833—31no—
53#1814D311yescharacteristic, complemented
54#1812—31no—
55C2×C3×#41C2311yescharacteristic
56C2×#235C2311yescharacteristic
57C22×D12C2312yes—
58C3×#131—21no—
59C3×#131—21no—
60C3×#131—21no—
61C3×#131—21no—
62C2×C3×M16—21no—
63C2×C3×M16C2311yescharacteristic
64C2×C3×M16C2312yes—
65C22×C3×C8—21no—
66C22×C3×C8C2311yescharacteristic
67C3×#117C2312yes—
68#951—21no—
69#950—22no—
70#949—21no—
71#947—22no—
72#942—21no—
73#940—22no—
74#937—42no—
75#932—21no—
76#930—24no—
77#926C2312yes—
78#926C2311yescharacteristic
79C2×#208C2311yescharacteristic
80C2×#208C2311yescharacteristic
81#908—24no—
82#908—42no—
83C2×D24—21no—
84C2×D24—21no—
85C2×D24—21no—
86C2×D24—21no—
87C2×D24—22no—
88C2×D24D412yes—
89C2×D24D412yes—
90C2×#204—22no—
91#901—21no—
92#901—21no—
93#901—21no—
94#901—21no—
95C2×ΓL(1,𝔽25)—21no—
96C2×D3×C8—21no—
97#893—24no—
98D3×C22⋊C4—42no—
99C4×D12—42no—
100C3×#108—21no—
101C3×#108—21no—
102C3×#107—21no—
103C3×#106—21no—
104C3×#104—42no—
105#837—22no—
106#825—22no—
107#823—24no—
108#822—22no—
109#821—24no—
110#818—21no—
111#817—21no—
112#815D412yescomplemented
113C8×Dic3D412yescomplemented
114#809—24no—
115#806—42no—
116#575—31no—
117C2×AGL(1,ℤ/8)—32no—
118C22×D8—31no—
119C2×#131—31no—
120C2×#131C2×D311yescharacteristic
121#535—32no—
122#518—32no—
123#472—62no—
124#472—62no—
125#469—62no—
126#466—62no—
127#444—62no—
128#443—62no—
129#442—62no—
130C4×D8—62no—
131#430—31no—
132C2×#108C2×D311yescharacteristic
133#427—31no—
134#427C2×D311yescharacteristic
135#421—31no—
136#421—31no—
137#421C2×D312yes—
138#418—32no—
139#418—32no—
140C2×#100—32no—
141#410—32no—
142#405—31no—
143#344—32no—
144#344C2×D312yes—
145H(3,ℤ/4)—32no—
146H(3,ℤ/4)—32no—
147C23×D3—22no—
148C3×#41—21no—
149C3×#41—21no—
150C3×#41—21no—
151C3×#41—21no—
152C2×C3×Q8#4111yescharacteristic
153C2×C3×D4—21no—
154C2×C3×D4#4111yescharacteristic
155C22×C3×C4—21no—
156C22×C3×C4C2411yescharacteristic
157C2×#67—21no—
158C2×#67—44no—
159#235—21no—
160#235—21no—
161#235—21no—
162#235—21no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—22no—
168C2×D12C2×D414yes—
169C2×D12#4111yescharacteristic
170C2×C4×D3—21no—
171C2×C4×D3—44no—
172C2×Dic6#4111yescharacteristic
173C3×M16—21no—
174C3×M16—22no—
175C3×M16—21no—
176C3×M16—22no—
177C3×M16—41no—
178C3×M16—41no—
179C2×C3×C8—21no—
180C2×C3×C8—22no—
181C2×C3×C8—41no—
182C2×C3×C8—41no—
183C2×C3×C8C2×D411yescharacteristic
184C2×C3×C8C2×D411yescharacteristic
185C2×C3×C8C2×D411yescharacteristic
186C2×C3×C8C2×D411yescharacteristic
187C2×C3×C8#4111yescharacteristic
188C2×C3×C8#4111yescharacteristic
189C3×C4⋊C4—22no—
190C3×C22⋊C4—22no—
191C3×C42—22no—
192#217—21no—
193#217—22no—
194#216—22no—
195#215—22no—
196#214—22no—
197#213—22no—
198(C2×D3)⋊C4—44no—
199(C2×D3)⋊C4—44no—
200C4⋊Dic3—21no—
201C4⋊Dic3—22no—
202C4×Dic3—22no—
203C4×Dic3C2×D412yes—
204#208—21no—
205#208—21no—
206#208—21no—
207#208—21no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8C2×D412yes—
210D24—22no—
211D24—22no—
212D24—22no—
213D24—22no—
214D24—44no—
215#204—44no—
216ΓL(1,𝔽25)—41no—
217ΓL(1,𝔽25)—41no—
218D3×C8—41no—
219D3×C8—41no—
220C2×#41—31no—
221C2×#41C22×D311yescharacteristic
222C22×D4—32no—
223#137—61no—
224#137—61no—
225AGL(1,ℤ/8)—64no—
226AGL(1,ℤ/8)—61no—
227AGL(1,ℤ/8)—61no—
228AGL(1,ℤ/8)—122no—
229#135—62no—
230C2×SD16—62no—
231C2×D8—32no—
232C2×D8—32no—
233C2×D8—62no—
234C2×D8—61no—
235C2×D8—61no—
236C2×D8—61no—
237C2×D8—61no—
238#131—21no—
239#131—21no—
240#131—21no—
241#131—21no—
242#131—61no—
243#131—61no—
244#131—61no—
245#131—61no—
246C2×M16—21no—
247C2×M16—31no—
248C2×M16—31no—
249C2×M16—61no—
250C2×M16C22×D311yescharacteristic
251C2×M16C22×D312yes—
252C22×C8—21no—
253C22×C8—61no—
254C22×C8C22×D311yescharacteristic
255#127—62no—
256#124—62no—
257#121—64no—
258#120—122no—
259C4×D4—64no—
260C4×D4—122no—
261#117—31no—
262#117—32no—
263#117C22×D312yes—
264C2×C22⋊C4—122no—
265#108—21no—
266#108—21no—
267#108—62no—
268#107—21no—
269#107—61no—
270#106—21no—
271#106—61no—
272#104—42no—
273#104—62no—
274#104—62no—
275#102—64no—
276#102—64no—
277#100—64no—
278#99—122no—
279#97—32no—
280C4×C8—32no—
281C23×C3—21no—
282C23×C3H(5,𝔽2)11yescharacteristic
283C22×D3—24no—
284C22×D3—21no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3—162no—
288C3×Q8—21no—
289C3×Q8—21no—
290C3×D4—21no—
291C3×D4—21no—
292C3×D4—41no—
293C3×D4—41no—
294C2×C3×C4—21no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C2×#4111yescharacteristic
301C2×C3×C4C2×#4111yescharacteristic
302C2×C3×C4C2×#4111yescharacteristic
303C2×C3×C4H(5,𝔽2)11yescharacteristic
304#67—41no—
305#67—41no—
306#67—164no—
307C2×Dic3—22no—
308C2×Dic3—21no—
309C2×Dic3—42no—
310D12—24no—
311D12—21no—
312D12—24no—
313D12—21no—
314D12—42no—
315D12—42no—
316C4×D3—41no—
317C4×D3—41no—
318C4×D3—164no—
319Dic6—21no—
320Dic6—21no—
321C3×C8—41no—
322C3×C8—42no—
323C3×C8#12111yescharacteristic
324C3×C8#12111yescharacteristic
325C3×C8#12111yescharacteristic
326C3×C8#12111yescharacteristic
327C3⋊C8—22no—
328C3⋊C8—41no—
329C24—62no—
330#41—21no—
331#41—21no—
332#41—21no—
333#41—21no—
334#41—61no—
335#41—61no—
336#41—61no—
337#41—61no—
338C2×Q8—31no—
339C2×Q8#23911yescharacteristic
340C2×D4—21no—
341C2×D4—34no—
342C2×D4—31no—
343C2×D4—61no—
344C2×D4—62no—
345C2×D4—62no—
346C2×D4—62no—
347C2×D4—62no—
348C2×D4—62no—
349C2×D4—124no—
350C2×D4#23911yescharacteristic
351C22×C4—21no—
352C22×C4—61no—
353C22×C4—124no—
354C22×C4C23×D311yescharacteristic
355Q16—62no—
356SD16—62no—
357SD16—62no—
358SD16—124no—
359D8—62no—
360D8—62no—
361D8—62no—
362D8—62no—
363D8—62no—
364D8—124no—
365M16—21no—
366M16—22no—
367M16—21no—
368M16—22no—
369M16—41no—
370M16—41no—
371M16—61no—
372M16—61no—
373M16—61no—
374M16—61no—
375M16—121no—
376M16—121no—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—32no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—61no—
383C2×C8—121no—
384C2×C8—121no—
385C2×C8#23511yescharacteristic
386C2×C8#23511yescharacteristic
387C2×C8D3×D411yescharacteristic
388C2×C8D3×D411yescharacteristic
389C2×C8D3×D411yescharacteristic
390C2×C8D3×D411yescharacteristic
391C4⋊C4—22no—
392C4⋊C4—61no—
393C4⋊C4—62no—
394C22⋊C4—22no—
395C22⋊C4—61no—
396C22⋊C4—62no—
397C22⋊C4—124no—
398C22⋊C4—124no—
399C42—22no—
400C42—32no—
401C42—62no—
402C22×C3—41no—
403C22×C3—42no—
404C22×C3—41no—
405C22×C3#53511yescharacteristic
406C22×C3#54611yescharacteristic
407C22×C3#55311yescharacteristic
408C2×D3—41no—
409C2×D3—44no—
410C2×D3—84no—
411C2×D3—84no—
412C2×D3—81no—
413C2×D3—162no—
414C3×C4—41no—
415C3×C4—82no—
416C3×C4C2×#12111yescharacteristic
417C3×C4#52511yescharacteristic
418C3×C4#54611yescharacteristic
419C3×C4#55311yescharacteristic
420Dic3—42no—
421Dic3—41no—
422Dic3—82no—
423C23—21no—
424C23—61no—
425C23—64no—
426C23—122no—
427C23—122no—
428C23—482no—
429C23#100911yescharacteristic
430Q8—21no—
431Q8—21no—
432Q8—61no—
433Q8—61no—
434D4—21no—
435D4—21no—
436D4—41no—
437D4—41no—
438D4—64no—
439D4—61no—
440D4—64no—
441D4—61no—
442D4—121no—
443D4—121no—
444D4—122no—
445D4—122no—
446D4—484no—
447C2×C4—21no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—61no—
452C2×C4—62no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—122no—
456C2×C4—484no—
457C2×C4C2×#23511yescharacteristic
458C2×C4C2×D3×D411yescharacteristic
459C2×C4C2×D3×D411yescharacteristic
460C2×C4#100411yescharacteristic
461C2×C4C2×#23911yescharacteristic
462C2×C4D3×#4111yescharacteristic
463C8—41no—
464C8—42no—
465C8—62no—
466C8—121no—
467C8#88411yescharacteristic
468C8#88411yescharacteristic
469C8#88411yescharacteristic
470C8#88411yescharacteristic
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—81no—
475C2×C3#253511yescharacteristic
476D3—81no—
477D3—84no—
478C22—41no—
479C22—42no—
480C22—41no—
481C22—121no—
482C22—124no—
483C22—244no—
484C22—241no—
485C22—244no—
486C22—482no—
487C22#553011yescharacteristic
488C22#560711yescharacteristic
489C22#563011yescharacteristic
490C4—41no—
491C4—82no—
492C4—122no—
493C4—121no—
494C4—242no—
495C4C2×#88411yescharacteristic
496C4#551411yescharacteristic
497C4#561211yescharacteristic
498C4#562711yescharacteristic
499C3#2818311yescharacteristic, complemented, Sylow
500C2—21no—
501C2—21no—
502C2—21no—
503C2—81no—
504C2—244no—
505C2—241no—
506C2#7393311yescharacteristic
subgroupG/H
centerC2#73933
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1814C2
socleC2×C3#2535
hypercenter#1814D3
hypocenterC3#28183
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1814

Complements

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

NH ≅ G/Ntype
1. #83799C2semidirect
2. #83792C2semidirect
3. C2×#4897C2semidirect
4. #79167C2semidirect
5. C3×#1814C2semidirect
6. #70698C2semidirect
7. #70664C2semidirect
8. #70409C2semidirect
9. #70408C2semidirect
10. #69027C2semidirect
11. #69021C2semidirect
53. #1814D3semidirect
112. #815D4semidirect
113. C8×Dic3D4semidirect
499. C3#28183semidirect

Maximal subgroups

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

subgroupindex
#837992
#837922
C2×#48972
#791672
C3×#18142
#706982
#706642
#704092
#704082
#690272
#690212
#281833