Finite Group Databasebeta

← #10845409

Subgroups — #10845409

Order 768.

946 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#474C211yesmaximal, characteristic, complemented
2#83240C211yesmaximal, characteristic, complemented
3#82417C211yesmaximal, characteristic
4C2×#4838C211yesmaximal, characteristic, complemented
5#71787C211yesmaximal, characteristic, complemented
6C4×#832C211yesmaximal, characteristic, complemented
7#70637C212yesmaximal, complemented
8#70631C212yesmaximal, complemented
9#70454C211yesmaximal, characteristic, complemented
10#70203C212yesmaximal, complemented
11#70176C212yesmaximal, complemented
12#25347—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×#947—21no—
16C2×#938C2212yes—
17C2×#930—21no—
18C22×ΓL(1,𝔽25)C2212yes—
19C2×C4×D12C2211yescharacteristic
20C3×#474—24no—
21C2×C3×#107C2212yes—
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#102C2212yes—
24#5229—21no—
25C2×#832C2211yescharacteristic
26C2×#832C2211yescharacteristic
27#5212—21no—
28C2×(C2×D3)⋊C8C2212yes—
29C2×#817C2212yes—
30#4992—22no—
31#4987C2212yes—
32#4987C2212yes—
33#4987C2212yes—
34#4961—22no—
35#4935C2211yescharacteristic
36C4×C4⋊Dic3C2211yescharacteristic
37C2×C4×C3⋊C8C2211yescharacteristic
38#4838—24no—
39#4551C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4529C2211yescharacteristic
43#4529C2211yescharacteristic
44C2×#474D311yescharacteristic, complemented
45C2×#436—31no—
46#2547—31no—
47#2537—31no—
48#2076—32no—
49#2034—32no—
50#1990—31no—
51#1948—31no—
52#1895—32no—
53#1809—32no—
54C4×#102—31no—
55C22×C3×Q8—21no—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4—21no—
58C22×#67—22no—
59C22×D12D411yescharacteristic
60C22×C4×D3C2312yes—
61C22×C3×C8C2312yes—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8—22no—
64C3×C4×D4—22no—
65C3×C4×D4—22no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—21no—
70C2×C3×C42C2311yescharacteristic
71C2×#217—22no—
72#947—44no—
73#938—24no—
74#938—42no—
75#930—44no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4D412yes—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3D411yescharacteristic
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2312yes—
86C2×C4×Dic3—21no—
87C22×C3⋊C8C2311yescharacteristic
88C22×C3⋊C8C2311yescharacteristic
89C2×ΓL(1,𝔽25)—24no—
90C2×ΓL(1,𝔽25)—24no—
91C2×ΓL(1,𝔽25)—42no—
92C4×D12—22no—
93C4×D12—22no—
94C3×#107—24no—
95C3×C4⋊C8D414yescomplemented
96C3×#102—24no—
97#832—22no—
98#832—22no—
99C3⋊#95—21no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102C3⋊#95—22no—
103(C2×D3)⋊C8—24no—
104#817—42no—
105C4×C3⋊C8—22no—
106C22×M16—32no—
107C2×C4⋊Q8C2×D311yescharacteristic
108C2×#124—61no—
109C2×#121—32no—
110C2×#121—61no—
111C2×C4×D4—31no—
112C2×C4×D4C2×D311yescharacteristic
113#474—24no—
114#436—64no—
115C2×#107C2×D312yes—
116C2×#106—32no—
117C2×C4⋊C8C2×D311yescharacteristic
118C2×#102—31no—
119C2×#102—31no—
120C2×#102C2×D312yes—
121C2×C22⋊C8—32no—
122C2×C4×C8—31no—
123#397—62no—
124#394—62no—
125#391—61no—
126#384—31no—
127#380—32no—
128#380—32no—
129#380—32no—
130#380—61no—
131C4×C4⋊C4—31no—
132#340—32no—
133#340—31no—
134#340—32no—
135#340—31no—
136#340—32no—
137C24×C3—21no—
138C23×D3C2×D412yes—
139C2×C3×Q8—42no—
140C2×C3×Q8—42no—
141C2×C3×Q8—42no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4#4112yes—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4—42no—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2411yescharacteristic
156C2×#67—44no—
157C2×#67—44no—
158C2×#67—82no—
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—22no—
162C22×Dic3C2×D412yes—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—22no—
166C2×C4×D3—22no—
167C2×C4×D3—22no—
168C2×C4×D3—22no—
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3—22no—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8C2×D414yes—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—42no—
180C3×C22⋊C4—42no—
181C3×C42C2×D412yes—
182C3×C42#4112yes—
183#217—44no—
184(C2×D3)⋊C4—24no—
185(C2×D3)⋊C4—44no—
186(C2×D3)⋊C4—44no—
187(C2×D3)⋊C4—44no—
188C4⋊Dic3—22no—
189C4⋊Dic3—21no—
190C4⋊Dic3—21no—
191C4⋊Dic3—22no—
192C4⋊Dic3—21no—
193C4⋊Dic3—22no—
194C4⋊Dic3—21no—
195C4⋊Dic3—42no—
196C4⋊Dic3—81no—
197C4×Dic3—42no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8#4112yes—
204ΓL(1,𝔽25)—48no—
205C22×Q8—21no—
206C22×D4—31no—
207C22×D4—62no—
208C22×D4C22×D311yescharacteristic
209C23×C4—21no—
210C23×C4—32no—
211C2×M16—64no—
212C2×M16—64no—
213C2×M16—122no—
214C22×C8—31no—
215C22×C8—31no—
216C22×C8C22×D312yes—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219#124—124no—
220#121—64no—
221#121—122no—
222#121—124no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—62no—
226C4×D4—62no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—31no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—31no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—61no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4C22×D311yescharacteristic
235C2×C4⋊C4C22×D312yes—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—32no—
238C2×C22⋊C4—62no—
239C2×C22⋊C4—62no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4—62no—
242C2×C42—61no—
243C2×C42C22×D311yescharacteristic
244#107—24no—
245#106—122no—
246C4⋊C8#6714yescomplemented
247#102—24no—
248#102—62no—
249#102—62no—
250C22⋊C8—64no—
251C4×C8—62no—
252#95—62no—
253#95—62no—
254#95—62no—
255#95—61no—
256C23×C3—22no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3C22×D411yescharacteristic
264C22×D3—22no—
265C22×D3—22no—
266C22×D3—22no—
267C22×D3—22no—
268C22×D3—22no—
269C22×D3—22no—
270C22×D3—22no—
271C3×Q8—84no—
272C3×D4—22no—
273C3×D4—22no—
274C3×D4—41no—
275C3×D4—41no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—41no—
279C2×C3×C4—42no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—84no—
285C2×C3×C4#12112yes—
286C2×C3×C4#12112yes—
287C2×C3×C4AGL(1,ℤ/8)12yes—
288C2×C3×C4AGL(1,ℤ/8)12yes—
289C2×C3×C4C22×D411yescharacteristic
290C2×C3×C4C2×#4111yescharacteristic
291C2×C3×C4C2×#4111yescharacteristic
292C2×C3×C4C2×#4111yescharacteristic
293C2×C3×C4H(5,𝔽2)11yescharacteristic
294C2×C3×C42−1+411yescharacteristic
295#67—88no—
296C2×Dic3—22no—
297C2×Dic3—22no—
298C2×Dic3—22no—
299C2×Dic3—42no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308D12—44no—
309C4×D3—42no—
310C4×D3—42no—
311C4×D3—42no—
312C4×D3—42no—
313C3×C8—24no—
314C3⋊C8—22no—
315C3⋊C8—41no—
316C24—21no—
317C24—32no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—62no—
327C2×D4—62no—
328C2×D4—62no—
329C2×D4—124no—
330C2×D4—124no—
331C2×D4—242no—
332C2×D4#23912yes—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—32no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—62no—
340C22×C4—62no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4—62no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4—62no—
348C22×C4D3×D411yescharacteristic
349C22×C4D3×D411yescharacteristic
350C22×C4C23×D311yescharacteristic
351M16—128no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—32no—
355C2×C8—61no—
356C2×C8—61no—
357C2×C8—61no—
358C2×C8—61no—
359C2×C8—61no—
360C2×C8C2×#6714yes—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—42no—
366C4⋊C4—62no—
367C4⋊C4—61no—
368C4⋊C4—62no—
369C4⋊C4—61no—
370C4⋊C4—61no—
371C4⋊C4—61no—
372C4⋊C4—62no—
373C4⋊C4—122no—
374C4⋊C4—241no—
375C22⋊C4—42no—
376C22⋊C4—64no—
377C22⋊C4—124no—
378C22⋊C4—124no—
379C22⋊C4—124no—
380C22⋊C4—124no—
381C42—122no—
382C42#23712yes—
383C42C2×#6712yes—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3—81no—
390C22×C3—81no—
391C22×C3C2×#12111yescharacteristic
392C22×C3#54611yescharacteristic
393C22×C3#54711yescharacteristic
394C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
395C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
396C22×C3#57711yescharacteristic
397C22×C3#57811yescharacteristic
398C2×D3—42no—
399C2×D3—42no—
400C2×D3—42no—
401C2×D3—42no—
402C2×D3—42no—
403C2×D3—42no—
404C2×D3—42no—
405C3×C4—22no—
406C3×C4—81no—
407C3×C4—82no—
408C3×C4#46914yes—
409C3×C4#52911yescharacteristic
410C3×C4#54211yescharacteristic
411C3×C4#54611yescharacteristic
412C3×C4#54811yescharacteristic
413Dic3—42no—
414Dic3—81no—
415Dic3—81no—
416Dic3—82no—
417C23—22no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—62no—
425C23—62no—
426C23—62no—
427C23—62no—
428C23—62no—
429C23—62no—
430C23—62no—
431C23C2×D3×D411yescharacteristic
432Q8—84no—
433D4—22no—
434D4—22no—
435D4—41no—
436D4—41no—
437D4—124no—
438D4—248no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—84no—
451C2×C4—122no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—122no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4#91112yes—
465C2×C4#91112yes—
466C2×C4#93812yes—
467C2×C4#93812yes—
468C2×C4C2×#23711yescharacteristic
469C2×C4C2×#23911yescharacteristic
470C2×C4#100711yescharacteristic
471C2×C4D3×#4111yescharacteristic
472C2×C4#100911yescharacteristic
473C2×C4C22×#6711yescharacteristic
474C8—24no—
475C8—62no—
476C8—121no—
477C2×C3—82no—
478C2×C3#242711yescharacteristic
479C2×C3C2×#46911yescharacteristic
480C2×C3#306311yescharacteristic
481C2×C3#332411yescharacteristic
482C2×C3#333011yescharacteristic
483C2×C3#335311yescharacteristic
484C2×C3#335411yescharacteristic
485D3—82no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—122no—
494C22—122no—
495C22—122no—
496C22—122no—
497C22—122no—
498C22—122no—
499C22—122no—
500C22#563011yescharacteristic
501C22#563111yescharacteristic
502C22C2×#91111yescharacteristic
503C22C2×#91111yescharacteristic
504C22#577811yescharacteristic
505C22#577911yescharacteristic
506C22C2×#93811yescharacteristic
507C4—22no—
508C4—81no—
509C4—82no—
510C4—122no—
511C4—242no—
512C4—241no—
513C4—241no—
514C4#515914yes—
515C4#562411yescharacteristic
516C4#564311yescharacteristic
517C4#581711yescharacteristic
518C4#582911yescharacteristic
519C3#2534711yescharacteristic, complemented, Sylow
520C2—82no—
521C2—242no—
522C2#8098411yescharacteristic
523C2#8098511yescharacteristic
524C2#8099011yescharacteristic
525C2#8099111yescharacteristic
526C2#8275611yescharacteristic
527C2C2×#515911yescharacteristic
528C2#8398611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#474C2
socleC23×C3C22×D4
prime radical#82417C2
hypercenterC2×#474D3
hypocenterC3#25347
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×C3×#474C2semidirect
2. #83240C2semidirect
4. C2×#4838C2semidirect
5. #71787C2semidirect
6. C4×#832C2semidirect
7. #70637C2semidirect
8. #70631C2semidirect
9. #70454C2semidirect
10. #70203C2semidirect
11. #70176C2semidirect
44. C2×#474D3semidirect
95. C3×C4⋊C8D4semidirect
246. C4⋊C8#67semidirect
519. C3#25347semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4742
#832402
#824172
C2×#48382
#717872
C4×#8322
#706372
#706312
#704542
#702032
#701762
#253473