Finite Group Databasebeta

← #10845402

Subgroups — #10845402

Order 768.

926 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82417C211yesmaximal, characteristic
2C2×#5008C211yesmaximal, characteristic, complemented
3C2×#4688C211yesmaximal, characteristic, complemented
4#73807C211yesmaximal, characteristic, complemented
5C3×C4×#102C211yesmaximal, characteristic, complemented
6#70206C212yesmaximal, complemented
7#70203C212yesmaximal, complemented
8#69777C211yesmaximal, characteristic, complemented
9#69466C211yesmaximal, characteristic, complemented
10#68870C212yesmaximal, complemented
11#68833C212yesmaximal, complemented
12#25346—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#938C2212yes—
15C2×#930—21no—
16C22×ΓL(1,𝔽25)C2212yes—
17C2×C4×D12C2211yescharacteristic
18C2×#869C2211yescharacteristic
19C2×C3×#102C2211yescharacteristic
20C2×C3×#102C2211yescharacteristic
21C2×C3×C4×C8C2211yescharacteristic
22C3×C4×C4⋊C4C2211yescharacteristic
23C2×#832C2212yes—
24#5212—21no—
25C2×(C2×D3)⋊C8C2212yes—
26#5008—24no—
27#4987C2212yes—
28#4987C2212yes—
29#4987C2212yes—
30C2×#808C2212yes—
31C2×#807C2212yes—
32#4961—22no—
33#4939—21no—
34#4936—21no—
35#4935C2211yescharacteristic
36C2×C4⋊(C3⋊C8)C2211yescharacteristic
37#4688—24no—
38#4673—22no—
39C3×#340C2211yescharacteristic
40C3×#340C2211yescharacteristic
41#4551C2212yes—
42#4551C2212yes—
43#4529C2212yes—
44C2×#476—31no—
45C2×#436—31no—
46#2563—31no—
47#2537—31no—
48#2077—32no—
49#2033—32no—
50#1990—31no—
51#1947—31no—
52#1895—32no—
53#1809—32no—
54C4×#102D311yescharacteristic, complemented
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4—21no—
57C22×#67—22no—
58C22×D12D411yescharacteristic
59C22×C4×D3C2312yes—
60C22×Dic6—21no—
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×C4×D4—22no—
64C3×C4×D4—22no—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C22⋊C4—21no—
69C2×C3×C42—21no—
70C2×C3×C42C2311yescharacteristic
71C2×#217—22no—
72#938—24no—
73#938—42no—
74#930—44no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4D412yes—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3D411yescharacteristic
83C2×C4⋊Dic3C2312yes—
84C2×C4⋊Dic3C2312yes—
85C22×C3⋊C8C2312yes—
86C2×ΓL(1,𝔽25)—24no—
87C2×ΓL(1,𝔽25)—24no—
88C2×ΓL(1,𝔽25)—42no—
89C4×D12—22no—
90C4×D12—22no—
91#869—22no—
92#869—22no—
93C3×#102—22no—
94C3×#102—22no—
95C3×C4×C8—22no—
96C3×#95—21no—
97#832—24no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101(C2×D3)⋊C8—24no—
102#808—24no—
103#807—42no—
104C4⋊(C3⋊C8)D414yescomplemented
105C22×M16—32no—
106C2×C4⋊Q8—31no—
107C2×#121—32no—
108C2×#121—61no—
109C2×C4×D4—31no—
110C2×C4×D4C2×D311yescharacteristic
111#476—64no—
112#436—64no—
113C2×#107—32no—
114C2×#106—32no—
115C2×C4⋊C8—31no—
116C2×#102—32no—
117C2×#102C2×D311yescharacteristic
118C2×#102C2×D311yescharacteristic
119C2×C22⋊C8—32no—
120C2×C4×C8C2×D311yescharacteristic
121#394—62no—
122#393—62no—
123#390—61no—
124#384—31no—
125#382—61no—
126#380—32no—
127#380—32no—
128#380—32no—
129#380—61no—
130C4×C4⋊C4C2×D311yescharacteristic
131#340—32no—
132#340—32no—
133#340—32no—
134#340C2×D311yescharacteristic
135#340C2×D311yescharacteristic
136C24×C3—21no—
137C23×D3C2×D412yes—
138C2×C3×D4—21no—
139C2×C3×D4—21no—
140C2×C3×D4—21no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4#4112yes—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—42no—
148C22×C3×C4—42no—
149C22×C3×C4—42no—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2411yescharacteristic
153C2×#67—44no—
154C2×#67—44no—
155C2×#67—82no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3C2×D412yes—
159C2×D12—22no—
160C2×D12—22no—
161C2×D12—22no—
162C2×C4×D3—22no—
163C2×C4×D3—22no—
164C2×C4×D3—22no—
165C2×C4×D3—22no—
166C2×C4×D3—22no—
167C2×C4×D3—22no—
168C2×Dic6—42no—
169C2×Dic6—81no—
170C2×Dic6—81no—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8#4112yes—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C22⋊C4—42no—
184C3×C42—42no—
185C3×C42C2×D412yes—
186C3×C42#4112yes—
187#217—44no—
188(C2×D3)⋊C4—24no—
189(C2×D3)⋊C4—44no—
190(C2×D3)⋊C4—82no—
191(C2×D3)⋊C4—82no—
192C4⋊Dic3—22no—
193C4⋊Dic3—22no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3—22no—
197C4⋊Dic3—42no—
198C4⋊Dic3—42no—
199C4⋊Dic3—81no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8C2×D414yes—
203ΓL(1,𝔽25)—48no—
204C22×Q8—61no—
205C22×D4—31no—
206C22×D4—62no—
207C22×D4C22×D311yescharacteristic
208C23×C4—21no—
209C23×C4—32no—
210C2×M16—64no—
211C2×M16—64no—
212C2×M16—122no—
213C22×C8—32no—
214C22×C8C22×D311yescharacteristic
215C22×C8C22×D311yescharacteristic
216C4⋊Q8—62no—
217C4⋊Q8—62no—
218#121—64no—
219#121—122no—
220#121—124no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—62no—
224C4×D4—62no—
225C2×C4⋊C4—31no—
226C2×C4⋊C4—32no—
227C2×C4⋊C4—32no—
228C2×C4⋊C4—61no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4—61no—
231C2×C4⋊C4C22×D311yescharacteristic
232C2×C4⋊C4C22×D311yescharacteristic
233C2×C4⋊C4C22×D311yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—32no—
236C2×C22⋊C4—62no—
237C2×C22⋊C4—62no—
238C2×C22⋊C4—62no—
239C2×C22⋊C4—62no—
240C2×C42—21no—
241C2×C42C22×D311yescharacteristic
242#107—122no—
243#106—64no—
244C4⋊C8—34no—
245#102—22no—
246#102—22no—
247#102—64no—
248C22⋊C8—64no—
249C4×C8—22no—
250#95—21no—
251#95—62no—
252#95—62no—
253#95—62no—
254C23×C3—22no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3C22×D411yescharacteristic
262C22×D3—22no—
263C22×D3—22no—
264C22×D3—22no—
265C22×D3—22no—
266C22×D3—22no—
267C22×D3—22no—
268C22×D3—22no—
269C3×D4—22no—
270C3×D4—22no—
271C3×D4—41no—
272C3×D4—41no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—84no—
285C2×C3×C4#12112yes—
286C2×C3×C4#12112yes—
287C2×C3×C4AGL(1,ℤ/8)12yes—
288C2×C3×C4#13712yes—
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—42no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305D12—44no—
306C4×D3—42no—
307C4×D3—42no—
308C4×D3—42no—
309C4×D3—42no—
310Dic6—84no—
311C3×C8—22no—
312C3×C8—41no—
313C3⋊C8—24no—
314C24—21no—
315C24—32no—
316C2×Q8—122no—
317C2×Q8—241no—
318C2×Q8—241no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—62no—
325C2×D4—62no—
326C2×D4—62no—
327C2×D4—124no—
328C2×D4—124no—
329C2×D4—242no—
330C2×D4#23512yes—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—32no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—62no—
339C22×C4—62no—
340C22×C4—62no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4—62no—
345C22×C4—62no—
346C22×C4D3×D411yescharacteristic
347C22×C4D3×D411yescharacteristic
348C22×C4C23×D311yescharacteristic
349M16—128no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—34no—
356C2×C8—62no—
357C2×C8—62no—
358C2×C8#23512yes—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—62no—
366C4⋊C4—62no—
367C4⋊C4—62no—
368C4⋊C4—62no—
369C4⋊C4—62no—
370C4⋊C4—122no—
371C4⋊C4—122no—
372C4⋊C4—241no—
373C22⋊C4—42no—
374C22⋊C4—64no—
375C22⋊C4—124no—
376C22⋊C4—124no—
377C22⋊C4—242no—
378C22⋊C4—242no—
379C42—42no—
380C42#23512yes—
381C42D3×D412yes—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3C2×#12111yescharacteristic
390C22×C3#54611yescharacteristic
391C22×C3#54711yescharacteristic
392C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
393C22×C3C2×#13711yescharacteristic
394C22×C3#57611yescharacteristic
395C22×C3#57711yescharacteristic
396C2×D3—42no—
397C2×D3—42no—
398C2×D3—42no—
399C2×D3—42no—
400C2×D3—42no—
401C2×D3—42no—
402C2×D3—42no—
403C3×C4—22no—
404C3×C4—81no—
405C3×C4—81no—
406C3×C4—81no—
407C3×C4#46814yes—
408C3×C4#52911yescharacteristic
409C3×C4#54211yescharacteristic
410C3×C4#54611yescharacteristic
411C3×C4#54811yescharacteristic
412Dic3—42no—
413Dic3—82no—
414Dic3—82no—
415C23—22no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—62no—
423C23—62no—
424C23—62no—
425C23—62no—
426C23—62no—
427C23—62no—
428C23—62no—
429C23C2×D3×D411yescharacteristic
430Q8—244no—
431D4—22no—
432D4—22no—
433D4—41no—
434D4—41no—
435D4—124no—
436D4—248no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—84no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4#88412yes—
463C2×C4#88412yes—
464C2×C4#91112yes—
465C2×C4#91512yes—
466C2×C4C2×#23511yescharacteristic
467C2×C4C2×#23511yescharacteristic
468C2×C4C2×#23511yescharacteristic
469C2×C4C2×D3×D411yescharacteristic
470C2×C4#100911yescharacteristic
471C2×C4#101011yescharacteristic
472C8—22no—
473C8—41no—
474C8—64no—
475C2×C3—82no—
476C2×C3#242711yescharacteristic
477C2×C3C2×#46811yescharacteristic
478C2×C3#306211yescharacteristic
479C2×C3#332511yescharacteristic
480C2×C3#333111yescharacteristic
481C2×C3#335411yescharacteristic
482C2×C3#335711yescharacteristic
483D3—82no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22—122no—
492C22—122no—
493C22—122no—
494C22—122no—
495C22—122no—
496C22—122no—
497C22—122no—
498C22C2×#88411yescharacteristic
499C22#555011yescharacteristic
500C22#555111yescharacteristic
501C22C2×#91111yescharacteristic
502C22C2×#91511yescharacteristic
503C22#576911yescharacteristic
504C22#577011yescharacteristic
505C4—22no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4—122no—
510C4—242no—
511C4—242no—
512C4#478014yes—
513C4#547911yescharacteristic
514C4#553411yescharacteristic
515C4#555011yescharacteristic
516C4#555211yescharacteristic
517C3#2534611yescharacteristic, complemented, Sylow
518C2—82no—
519C2—242no—
520C2#7348611yescharacteristic
521C2C2×#478011yescharacteristic
522C2#7884011yescharacteristic
523C2#7961211yescharacteristic
524C2#7961711yescharacteristic
525C2#7968611yescharacteristic
526C2#7969311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#102C2
socleC23×C3C22×D4
prime radical#82417C2
hypercenterC4×#102D3
hypocenterC3#25346
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. C2×#5008C2semidirect
3. C2×#4688C2semidirect
4. #73807C2semidirect
5. C3×C4×#102C2semidirect
6. #70206C2semidirect
7. #70203C2semidirect
8. #69777C2semidirect
9. #69466C2semidirect
10. #68870C2semidirect
11. #68833C2semidirect
54. C4×#102D3semidirect
104. C4⋊(C3⋊C8)D4semidirect
517. C3#25346semidirect

Maximal subgroups

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

subgroupindex
#824172
C2×#50082
C2×#46882
#738072
C3×C4×#1022
#702062
#702032
#697772
#694662
#688702
#688332
#253463