Finite Group Databasebeta

← #10798409

Subgroups — #10798409

Order 768.

800 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78519C211yesmaximal, characteristic, complemented
2C4×#907C211yesmaximal, characteristic, complemented
3C4×#904C211yesmaximal, characteristic
4C3×C4×#108C211yesmaximal, characteristic, complemented
5#70722C212yesmaximal, complemented
6#70699C212yesmaximal, complemented
7C4×#825C211yesmaximal, characteristic, complemented
8C4×#825C211yesmaximal, characteristic, complemented
9#69750C211yesmaximal, characteristic, complemented
10#68946C212yesmaximal
11#68925C212yesmaximal, complemented
12#21137—31nomaximal, Sylow
13C2×#907C2212yes—
14C2×#904C2211yescharacteristic
15C4×#235C2211yescharacteristic
16C4×#235C2211yescharacteristic
17C2×C3×#108C2212yes—
18C3×C4×M16C2211yescharacteristic
19C3×C4×M16C2211yescharacteristic
20C2×C3×C4×C8C2211yescharacteristic
21C2×#825C2212yes—
22C2×#825C2212yes—
23#5129C2212yes—
24#5122C2212yes—
25#5119C2212yes—
26#5113C2211yescharacteristic
27#5113C2211yescharacteristic
28#5106C2211yescharacteristic
29C42×Dic3C2211yescharacteristic
30C2×C4×C3⋊C8C2211yescharacteristic
31#4709—22no—
32#4708—22no—
33C4×Dic12C411yescharacteristic, complemented
34C4×Dic12C411yescharacteristic, complemented
35C4×D24C411yescharacteristic, complemented
36C4×D24C411yescharacteristic, complemented
37C4×#204C411yescharacteristic, complemented
38C4×#204C411yescharacteristic, complemented
39C4×#204C411yescharacteristic, complemented
40C4×#204C411yescharacteristic, complemented
41C4×ΓL(1,𝔽25)—21no—
42#4685—21no—
43#4685—21no—
44C4×D3×C8—21no—
45C3×#341C2212yes—
46#4555C2212yes—
47#4555C2212yes—
48#4550C2212yes—
49#4550C2212yes—
50C4×#135—31no—
51#2922—31no—
52C4×#131—31no—
53#1929—32no—
54#1922—32no—
55#1876—32no—
56#1866—32no—
57C4×#108D311yescharacteristic, complemented
58C4×#104—31no—
59C4×#104—31no—
60#1744—31no—
61C2×#235C2311yescharacteristic
62C2×#235C2311yescharacteristic
63C2×C3×M16C2312yes—
64C2×C3×M16C2312yes—
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C2×C3×C42C2311yescharacteristic
68C4×#67—21no—
69C4×#67—21no—
70C4×#67—21no—
71C4×#67—21no—
72#926C2311yescharacteristic
73#926C2311yescharacteristic
74C2×C4×Dic3—21no—
75C2×C4×Dic3—21no—
76C2×C4×Dic3C2312yes—
77C22×C3⋊C8C2312yes—
78#907—24no—
79#907—24no—
80C2×Dic12C2×C411yescharacteristic
81#904—24no—
82C2×D24C2×C411yescharacteristic
83C2×#204C2×C411yescharacteristic
84C2×#204C2×C411yescharacteristic
85C2×ΓL(1,𝔽25)—22no—
86C2×D3×C8—22no—
87C4×D12C2×C411yescharacteristic
88C4×D12C2×C411yescharacteristic
89C4×D12C2×C411yescharacteristic
90C4×D12C2×C411yescharacteristic
91#872—21no—
92#872—21no—
93C42×D3—21no—
94C42×D3—21no—
95C4×Dic6C2×C411yescharacteristic
96C4×Dic6C2×C411yescharacteristic
97C4×Dic6C2×C411yescharacteristic
98C4×Dic6C2×C411yescharacteristic
99C3×#108—24no—
100C3×#97—21no—
101C3×#97—21no—
102C3×C4×C8—21no—
103C3×C4×C8—21no—
104C3×C4×C8C2×C411yescharacteristic
105C3×C4×C8C2×C411yescharacteristic
106C3×C4×C8C2×C411yescharacteristic
107C3×C4×C8C2×C411yescharacteristic
108#825—24no—
109#825—24no—
110#821C2×C411yescharacteristic
111#821C2×C411yescharacteristic
112#821C2×C411yescharacteristic
113#821C2×C411yescharacteristic
114(C2×D3)⋊C8—22no—
115(C2×D3)⋊C8—22no—
116#818C2×C411yescharacteristic
117#818C2×C411yescharacteristic
118#817C2×C411yescharacteristic
119#817C2×C411yescharacteristic
120#816C2×C411yescharacteristic
121#816C2×C411yescharacteristic
122#816C2×C411yescharacteristic
123#816C2×C411yescharacteristic
124#815—22no—
125Dic3⋊C8—22no—
126Dic3⋊C8—22no—
127C8×Dic3—22no—
128C4×C3⋊C8D411yescharacteristic
129C4×C3⋊C8D411yescharacteristic
130C4×C3⋊C8D412yes—
131C2×#135—31no—
132C2×#131—32no—
133C4×#41—31no—
134C4×#41—31no—
135C4×Q16—31no—
136C4×Q16—31no—
137C4×SD16—31no—
138C4×SD16—31no—
139C4×SD16—31no—
140C4×SD16—31no—
141C4×D8—31no—
142C4×D8—31no—
143#433—62no—
144#431—61no—
145#431—61no—
146C2×#108C2×D312yes—
147#427—31no—
148#424—32no—
149C2×#104—32no—
150C2×#104—32no—
151#416—31no—
152#416—31no—
153#408—32no—
154#405—32no—
155#405—62no—
156C4×M16—61no—
157C4×M16C2×D311yescharacteristic
158C4×M16C2×D311yescharacteristic
159C2×C4×C8—31no—
160C2×C4×C8—61no—
161C2×C4×C8C2×D311yescharacteristic
162C43—31no—
163#341C2×D312yes—
164#339—32no—
165#339—32no—
166#336—32no—
167#336—32no—
168C22×C3×C4#4111yescharacteristic
169C22×C3×C4#4111yescharacteristic
170C22×C3×C4C2411yescharacteristic
171C2×#67—21no—
172C2×#67—21no—
173C22×Dic3—22no—
174#235—22no—
175#235—22no—
176#235—22no—
177#235#4114yes—
178C2×D12C22×C411yescharacteristic
179C2×D12C22×C411yescharacteristic
180C2×C4×D3—21no—
181C2×C4×D3—21no—
182C2×C4×D3—21no—
183C2×C4×D3—21no—
184C2×C4×D3—21no—
185C2×C4×D3—21no—
186C2×Dic6C22×C411yescharacteristic
187C2×Dic6C22×C411yescharacteristic
188C3×M16—22no—
189C3×M16—22no—
190C3×M16—22no—
191C3×M16#4114yes—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8C22×C411yescharacteristic
197C2×C3×C8C22×C411yescharacteristic
198C2×C3×C8C22×C411yescharacteristic
199C2×C3×C8C22×C411yescharacteristic
200C3×C42C22×C411yescharacteristic
201C3×C42C22×C411yescharacteristic
202C3×C42C2×D411yescharacteristic
203C3×C42C2×D411yescharacteristic
204#217—21no—
205#217—21no—
206(C2×D3)⋊C4—21no—
207(C2×D3)⋊C4—21no—
208(C2×D3)⋊C4—21no—
209(C2×D3)⋊C4—21no—
210C4⋊Dic3C22×C411yescharacteristic
211C4⋊Dic3C22×C411yescharacteristic
212C4⋊Dic3C22×C411yescharacteristic
213C4⋊Dic3C22×C411yescharacteristic
214Dic3⋊C4—21no—
215Dic3⋊C4—21no—
216Dic3⋊C4—21no—
217Dic3⋊C4—21no—
218C4×Dic3—24no—
219C4×Dic3—21no—
220C4×Dic3—21no—
221C4×Dic3—21no—
222C4×Dic3—21no—
223C4×Dic3—21no—
224C4×Dic3—21no—
225C4×Dic3—22no—
226C4×Dic3—22no—
227C4×Dic3—42no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8C2×D412yes—
231C2×C3⋊C8C2×D412yes—
232Dic12—22no—
233D24—22no—
234#204—22no—
235#204—22no—
236ΓL(1,𝔽25)—28no—
237D3×C8—28no—
238C2×#41—31no—
239C2×#41—31no—
240#135—64no—
241C2×Q16—31no—
242C2×SD16—31no—
243C2×SD16—31no—
244C2×D8—31no—
245#131—64no—
246#131—64no—
247C2×M16—62no—
248C2×M16C22×D312yes—
249C2×M16C22×D312yes—
250C22×C8—32no—
251C22×C8—62no—
252C22×C8C22×D311yescharacteristic
253C22×C8C22×D311yescharacteristic
254C4×Q8—31no—
255C4×Q8—31no—
256C4×Q8—31no—
257C4×Q8—31no—
258C4×D4—31no—
259C4×D4—31no—
260C4×D4—31no—
261C4×D4—31no—
262C4×D4—61no—
263C4×D4—61no—
264C4×D4—61no—
265C4×D4—61no—
266#117—31no—
267#117—31no—
268#117—61no—
269#117—61no—
270C2×C42—32no—
271C2×C42—61no—
272C2×C42—61no—
273C2×C42—61no—
274C2×C42—61no—
275C2×C42C22×D311yescharacteristic
276#108—24no—
277#107—31no—
278#107—31no—
279#106—31no—
280#106—31no—
281C4⋊C8—62no—
282C4⋊C8—62no—
283#104—64no—
284#104—64no—
285#103—31no—
286#103—31no—
287#103—31no—
288#103—31no—
289#102—31no—
290#102—31no—
291#102—31no—
292#102—31no—
293C22⋊C8—62no—
294C22⋊C8—62no—
295#97—21no—
296#97—21no—
297#97—62no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—32no—
301C4×C8—31no—
302C4×C8—31no—
303C4×C8—62no—
304C4×C8C4×D311yescharacteristic
305C4×C8C4×D311yescharacteristic
306C4×C8C4×D311yescharacteristic
307C4×C8C4×D311yescharacteristic
308C23×C3C2×#4111yescharacteristic
309C22×D3—21no—
310C22×D3—21no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4C4×D411yescharacteristic
314C2×C3×C4C4×D411yescharacteristic
315C2×C3×C4C4×D411yescharacteristic
316C2×C3×C4C4×D411yescharacteristic
317C2×C3×C4C23×C411yescharacteristic
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C2×#4112yes—
320C2×C3×C4C2×#4112yes—
321#67—24no—
322#67—42no—
323C2×Dic3—21no—
324C2×Dic3—24no—
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—21no—
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332D12—21no—
333D12—21no—
334D12—21no—
335D12#11712yes—
336C4×D3—24no—
337C4×D3—42no—
338C4×D3—42no—
339C4×D3—42no—
340C4×D3—42no—
341C4×D3—42no—
342Dic6—21no—
343Dic6—21no—
344Dic6—21no—
345Dic6#11712yes—
346C3×C8—21no—
347C3×C8—21no—
348C3×C8—21no—
349C3×C8—21no—
350C3×C8—24no—
351C3×C8—42no—
352C3⋊C8—22no—
353C3⋊C8—22no—
354#41—34no—
355#41—62no—
356#41—62no—
357#41—62no—
358C2×Q8—31no—
359C2×Q8—31no—
360C2×D4—31no—
361C2×D4—31no—
362C2×D4—61no—
363C2×D4—61no—
364C22×C4—61no—
365C22×C4—62no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4—61no—
369C22×C4—61no—
370C22×C4—61no—
371C22×C4#23911yescharacteristic
372C22×C4#23911yescharacteristic
373C22×C4C23×D311yescharacteristic
374Q16—62no—
375SD16—62no—
376SD16—62no—
377D8—62no—
378M16—22no—
379M16—22no—
380M16—22no—
381M16—68no—
382M16#23514yes—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—32no—
388C2×C8—32no—
389C2×C8—68no—
390C2×C8—62no—
391C2×C8—62no—
392C2×C8C2×C4×D311yescharacteristic
393C2×C8C2×C4×D311yescharacteristic
394C2×C8C2×C4×D311yescharacteristic
395C2×C8C2×C4×D311yescharacteristic
396C4⋊C4—31no—
397C4⋊C4—31no—
398C4⋊C4—31no—
399C4⋊C4—31no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—61no—
404C22⋊C4—61no—
405C22⋊C4—61no—
406C22⋊C4—61no—
407C22⋊C4—61no—
408C22⋊C4—61no—
409C22⋊C4—61no—
410C42—64no—
411C42—61no—
412C42—61no—
413C42—61no—
414C42—61no—
415C42—61no—
416C42—61no—
417C42—62no—
418C42—62no—
419C42—122no—
420C42C2×C4×D311yescharacteristic
421C42C2×C4×D311yescharacteristic
422C42D3×D411yescharacteristic
423C42D3×D411yescharacteristic
424C22×C3—21no—
425C22×C3—21no—
426C22×C3C2×C4×D411yescharacteristic
427C22×C3#52912yes—
428C2×D3—22no—
429C2×D3—41no—
430C2×D3—41no—
431C2×D3—41no—
432C2×D3—41no—
433C2×D3—41no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4#44312yes—
437C3×C4#44412yes—
438C3×C4C2×#11711yescharacteristic
439C3×C4C4×#4111yescharacteristic
440C3×C4#52512yes—
441Dic3—22no—
442Dic3—44no—
443Dic3—41no—
444Dic3—41no—
445Dic3—41no—
446Dic3—41no—
447Dic3—41no—
448C23—61no—
449C23—61no—
450C23C2×#23911yescharacteristic
451Q8—32no—
452Q8—61no—
453Q8—61no—
454Q8—61no—
455D4—32no—
456D4—64no—
457D4—61no—
458D4—61no—
459D4—61no—
460D4—122no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—61no—
464C2×C4—64no—
465C2×C4—64no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4—122no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4#89311yescharacteristic
479C2×C4#89311yescharacteristic
480C2×C4#89311yescharacteristic
481C2×C4#89311yescharacteristic
482C2×C4C22×C4×D311yescharacteristic
483C2×C4C2×#23512yes—
484C2×C4C2×D3×D411yescharacteristic
485C2×C4D3×#4112yes—
486C8—21no—
487C8—21no—
488C8—21no—
489C8—21no—
490C8—24no—
491C8—42no—
492C8—62no—
493C8—62no—
494C2×C3—22no—
495C2×C3#229011yescharacteristic
496C2×C3C2×#44311yescharacteristic
497C2×C3C2×#44411yescharacteristic
498D3—42no—
499D3—81no—
500C22—21no—
501C22—21no—
502C22—62no—
503C22—121no—
504C22—121no—
505C22—121no—
506C22—121no—
507C22—121no—
508C22C2×#89311yescharacteristic
509C22#558012yes—
510C4—21no—
511C4—21no—
512C4—62no—
513C4—124no—
514C4—121no—
515C4—121no—
516C4—121no—
517C4—121no—
518C4—121no—
519C4#489412yes—
520C4#489512yes—
521C4C4×#23511yescharacteristic
522C4D3×#11711yescharacteristic
523C4#554712yes—
524C3#2113711yescharacteristic, complemented, Sylow
525C2—22no—
526C2—122no—
527C2—241no—
528C2#7375411yescharacteristic
529C2C2×#489511yescharacteristic
530C2C2×#489411yescharacteristic
subgroupG/H
centerC42D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#108C2
socleC22×C3C2×C4×D4
prime radicalC2×#904C22
cosocleC22×C3×C4C24
hypercenterC4×#108D3
hypocenterC3#21137
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C42 ◁ C2×C42 ◁ C4×#108

Complements

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

NH ≅ G/Ntype
1. #78519C2semidirect
2. C4×#907C2semidirect
4. C3×C4×#108C2semidirect
5. #70722C2semidirect
6. #70699C2semidirect
7. C4×#825C2semidirect
8. C4×#825C2semidirect
9. #69750C2semidirect
11. #68925C2semidirect
33. C4×Dic12C4semidirect
34. C4×Dic12C4semidirect
35. C4×D24C4semidirect
36. C4×D24C4semidirect
37. C4×#204C4semidirect
38. C4×#204C4semidirect
39. C4×#204C4semidirect
40. C4×#204C4semidirect
57. C4×#108D3semidirect
524. C3#21137semidirect

Maximal subgroups

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

subgroupindex
#785192
C4×#9072
C4×#9042
C3×C4×#1082
#707222
#706992
C4×#8252
#697502
#689462
#689252
#211373