Finite Group Databasebeta

← #10802128

Subgroups — #10802128

Order 768.

876 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79256C211yesmaximal, characteristic
2#78579C212yesmaximal, complemented
3C2×#4700C211yesmaximal, characteristic, complemented
4C4×#901C211yesmaximal, characteristic, complemented
5C3×#1772C211yesmaximal, characteristic, complemented
6#70707C211yesmaximal, characteristic, complemented
7#70665C211yesmaximal, characteristic, complemented
8#70658C211yesmaximal, characteristic, complemented
9#69898C212yesmaximal, complemented
10#68778C212yesmaximal, complemented
11#68750C212yesmaximal, complemented
12#21558—31nomaximal, Sylow
13#5748C2212yes—
14C2×#904C2211yescharacteristic
15C2×#901C2212yes—
16C2×#901C2211yescharacteristic
17C2×#901C2211yescharacteristic
18C4×#235C2211yescharacteristic
19C2×C3×#108C2211yescharacteristic
20C3×#427C2211yescharacteristic
21C3×#405C2212yes—
22C2×C3×C4×C8C2211yescharacteristic
23#5132C2212yes—
24#5125C2212yes—
25C2×#819C2211yescharacteristic
26#5101C2211yescharacteristic
27#5101C2211yescharacteristic
28#5006C2212yes—
29#5005C2212yes—
30#5005C2212yes—
31C2×#805C2212yes—
32C4×#208C2211yescharacteristic
33#4898C414yescomplemented
34#4897C412yescomplemented
35#4896C412yescomplemented
36#4717—24no—
37#4700—24no—
38#4700—21no—
39#4700—21no—
40#4700—21no—
41#4700—21no—
42#4689—21no—
43#4689—21no—
44C4×ΓL(1,𝔽25)—21no—
45C4×D3×C8—21no—
46C3×#339C2212yes—
47#4559C2212yes—
48#4533C2212yes—
49#3067—31no—
50#2955—32no—
51C2×#443—31no—
52C4×#131—31no—
53#1976—31no—
54#1953—31no—
55#1932—32no—
56#1906—32no—
57#1814—31no—
58#1772D311yescharacteristic, complemented
59#1757—32no—
60C2×#235C2312yes—
61C2×#235C2311yescharacteristic
62C2×C3×M16C2312yes—
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C3×#117C2312yes—
66C2×C3×C42C2311yescharacteristic
67C4×#67—22no—
68#926C2311yescharacteristic
69C2×#208C2312yes—
70C2×#208C2311yescharacteristic
71C2×#208C2311yescharacteristic
72#909—22no—
73#909C2×C414yes—
74#908—22no—
75#908C2×C414yes—
76C2×Dic12C2×C411yescharacteristic, complemented
77#904—24no—
78#904—21no—
79#904—21no—
80#904—21no—
81#904—21no—
82C2×D24C2×C411yescharacteristic, complemented
83C2×#204C2×C412yescomplemented
84#901—21no—
85#901—21no—
86#901—21no—
87#901—21no—
88#901—22no—
89#901—22no—
90#901—22no—
91#901—22no—
92#901C2×C412yes—
93#901C2×C412yes—
94#901D412yes—
95#901D412yes—
96C2×ΓL(1,𝔽25)—21no—
97C2×ΓL(1,𝔽25)—21no—
98C2×ΓL(1,𝔽25)—22no—
99C2×D3×C8—21no—
100C2×D3×C8—21no—
101C2×D3×C8—22no—
102C4×D12—21no—
103#872—21no—
104C42×D3—21no—
105C4×Dic6—21no—
106C3×#108C2×C412yescomplemented
107C3×#108C2×C412yescomplemented
108C3×#107—21no—
109C3×#106—21no—
110C3×#97—22no—
111C3×C4×C8—22no—
112C3×C4×C8D411yescharacteristic
113C3×C4×C8D411yescharacteristic
114C3×C4×C8D411yescharacteristic
115C3×C4×C8D411yescharacteristic
116#824C2×C414yescomplemented
117#823C2×C414yescomplemented
118#819—21no—
119#819—21no—
120#819—24no—
121#815—21no—
122#815—21no—
123C8×Dic3—21no—
124C8×Dic3—21no—
125#806—24no—
126#805—24no—
127#805—22no—
128#805—22no—
129#805—24no—
130#803—21no—
131C4×C3⋊C8—21no—
132#575—32no—
133C2×#135—31no—
134C2×#131—31no—
135C2×#131—31no—
136C2×#131—32no—
137C4×#41—31no—
138#473—32no—
139#472—32no—
140#471—34no—
141#444—64no—
142#443—64no—
143#443—61no—
144#443—61no—
145#443—61no—
146#443—61no—
147#430—32no—
148C2×#108—31no—
149C2×#108C2×D311yescharacteristic
150#427C2×D311yescharacteristic
151#421—32no—
152#421—32no—
153C2×#104—32no—
154#413—32no—
155#410—32no—
156#405—31no—
157#405—31no—
158#405—61no—
159#405—61no—
160#405C2×D312yes—
161C4×M16—31no—
162C4×M16—61no—
163C2×C4×C8—61no—
164C2×C4×C8C2×D311yescharacteristic
165#344—32no—
166#344—32no—
167#339C2×D312yes—
168C22×C3×C4—21no—
169C22×C3×C4C2411yescharacteristic
170C2×#67—21no—
171C2×#67—22no—
172#235—21no—
173#235—21no—
174#235—22no—
175#235—22no—
176#235—22no—
177#235—22no—
178#235C22×C412yes—
179#235C2×D412yes—
180C2×D12C22×C412yes—
181C2×D12#4111yescharacteristic
182C2×C4×D3—21no—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×Dic6C22×C412yes—
186C2×Dic6#4111yescharacteristic
187C3×M16—22no—
188C3×M16C22×C414yes—
189C2×C3×C8—22no—
190C2×C3×C8C22×C411yescharacteristic
191C2×C3×C8C22×C411yescharacteristic
192C2×C3×C8C2×D411yescharacteristic
193C2×C3×C8C2×D411yescharacteristic
194C2×C3×C8C2×D411yescharacteristic
195C2×C3×C8C2×D411yescharacteristic
196C2×C3×C8C2×D411yescharacteristic
197C2×C3×C8C2×D411yescharacteristic
198C2×C3×C8#4111yescharacteristic
199C2×C3×C8#4111yescharacteristic
200C2×C3×C8#4111yescharacteristic
201C2×C3×C8#4111yescharacteristic
202C3×C4⋊C4—22no—
203C3×C22⋊C4—22no—
204C3×C42—21no—
205C3×C42—22no—
206C3×C42C2×D411yescharacteristic
207C3×C42C2×D411yescharacteristic
208#217#4112yes—
209(C2×D3)⋊C4—22no—
210C4⋊Dic3—21no—
211Dic3⋊C4—22no—
212C4×Dic3—22no—
213C4×Dic3—21no—
214#208—21no—
215#208—21no—
216#208—22no—
217#208—22no—
218#208C22×C412yes—
219#208C2×D412yes—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—22no—
223Dic12—24no—
224Dic12—21no—
225Dic12—21no—
226D24—24no—
227D24—21no—
228D24—21no—
229#204—24no—
230#204—24no—
231#204—22no—
232#204—22no—
233ΓL(1,𝔽25)—22no—
234ΓL(1,𝔽25)—22no—
235ΓL(1,𝔽25)—22no—
236ΓL(1,𝔽25)—22no—
237ΓL(1,𝔽25)—42no—
238ΓL(1,𝔽25)—42no—
239D3×C8—22no—
240D3×C8—22no—
241D3×C8—22no—
242D3×C8—22no—
243D3×C8—42no—
244D3×C8—42no—
245C2×#41—31no—
246C2×#41—32no—
247#137—34no—
248#137—62no—
249AGL(1,ℤ/8)—34no—
250AGL(1,ℤ/8)—62no—
251#135—64no—
252#135—61no—
253#135—61no—
254#135—61no—
255#135—61no—
256C2×Q16—31no—
257C2×SD16—32no—
258C2×D8—31no—
259#131—32no—
260#131—32no—
261#131—32no—
262#131—32no—
263#131—61no—
264#131—61no—
265#131—61no—
266#131—61no—
267#131—62no—
268#131—62no—
269#131—62no—
270#131—62no—
271C2×M16—31no—
272C2×M16—31no—
273C2×M16—32no—
274C2×M16—61no—
275C2×M16—61no—
276C2×M16—62no—
277C2×M16C22×D312yes—
278C22×C8—61no—
279C22×C8—61no—
280C22×C8—62no—
281C22×C8C22×D311yescharacteristic
282C22×C8C22×D311yescharacteristic
283C4×Q8—61no—
284C4×D4—61no—
285C4×D4—62no—
286#117—31no—
287#117—61no—
288#117C22×D312yes—
289C2×C42—61no—
290C2×C42C22×D311yescharacteristic
291#108—64no—
292#108—61no—
293#108—61no—
294#108C4×D312yescomplemented
295#108C4×D312yescomplemented
296#107—21no—
297#106—21no—
298#104—64no—
299#104—64no—
300#104—62no—
301#104—62no—
302#101—34no—
303#100—34no—
304#99—64no—
305#97—22no—
306#97—61no—
307#97—61no—
308#97—61no—
309C4×C8—22no—
310C4×C8—61no—
311C4×C8—61no—
312C4×C8—61no—
313C4×C8D1211yescharacteristic
314C4×C8D1211yescharacteristic
315C4×C8D1211yescharacteristic
316C4×C8D1211yescharacteristic
317C23×C3C2×#4111yescharacteristic
318C22×D3—21no—
319C22×D3—22no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4C23×C411yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C2×#4111yescharacteristic
328C2×C3×C4C2×#4111yescharacteristic
329C2×C3×C4C2×#4111yescharacteristic
330#67—22no—
331#67—22no—
332#67—42no—
333#67—42no—
334C2×Dic3—21no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337D12—22no—
338D12—22no—
339D12—21no—
340D12C4×D412yes—
341C4×D3—22no—
342C4×D3—22no—
343C4×D3—44no—
344C4×D3—42no—
345C4×D3—42no—
346Dic6—21no—
347Dic6—22no—
348Dic6—22no—
349Dic6C4×D412yes—
350C3×C8—24no—
351C3×C8C4×D411yescharacteristic
352C3×C8C4×D411yescharacteristic
353C3×C8C4×D411yescharacteristic
354C3×C8C4×D411yescharacteristic
355C3×C8#12111yescharacteristic
356C3×C8#12111yescharacteristic
357C3×C8#12111yescharacteristic
358C3×C8#12111yescharacteristic
359C3⋊C8—22no—
360C3⋊C8—22no—
361C3⋊C8—42no—
362#41—32no—
363#41—32no—
364#41—61no—
365#41—61no—
366#41—62no—
367#41—62no—
368#41—62no—
369#41—62no—
370C2×Q8—32no—
371C2×Q8—31no—
372C2×D4—32no—
373C2×D4—31no—
374C2×D4—61no—
375C2×D4—62no—
376C22×C4—21no—
377C22×C4—62no—
378C22×C4—61no—
379C22×C4—62no—
380C22×C4C23×D311yescharacteristic
381Q16—64no—
382Q16—61no—
383Q16—61no—
384SD16—64no—
385SD16—64no—
386SD16—62no—
387SD16—62no—
388D8—64no—
389D8—61no—
390D8—61no—
391M16—22no—
392M16—32no—
393M16—32no—
394M16—61no—
395M16—61no—
396M16—62no—
397M16—62no—
398M16—62no—
399M16—62no—
400M16—62no—
401M16—62no—
402M16—122no—
403M16—122no—
404M16C2×C4×D314yes—
405C2×C8—22no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—62no—
409C2×C8—62no—
410C2×C8—62no—
411C2×C8—62no—
412C2×C8—62no—
413C2×C8—122no—
414C2×C8—122no—
415C2×C8C2×C4×D311yescharacteristic
416C2×C8C2×C4×D311yescharacteristic
417C2×C8C2×D1211yescharacteristic
418C2×C8C2×D1211yescharacteristic
419C2×C8C2×D1211yescharacteristic
420C2×C8C2×D1211yescharacteristic
421C2×C8#23511yescharacteristic
422C2×C8#23511yescharacteristic
423C2×C8D3×D411yescharacteristic
424C2×C8D3×D411yescharacteristic
425C2×C8#23911yescharacteristic
426C2×C8#23911yescharacteristic
427C4⋊C4—22no—
428C4⋊C4—62no—
429C4⋊C4—61no—
430C22⋊C4—22no—
431C22⋊C4—32no—
432C22⋊C4—62no—
433C42—21no—
434C42—22no—
435C42—62no—
436C42—61no—
437C42C2×D1211yescharacteristic
438C42C2×D1211yescharacteristic
439C22×C3—41no—
440C22×C3C4×#4111yescharacteristic
441C22×C3#52511yescharacteristic
442C22×C3#53211yescharacteristic
443C2×D3—22no—
444C2×D3—42no—
445C2×D3—42no—
446C2×D3—82no—
447C3×C4—22no—
448C3×C4—41no—
449C3×C4—82no—
450C3×C4C2×C4×D411yescharacteristic
451C3×C4C2×C4×D411yescharacteristic
452C3×C4C2×#12111yescharacteristic
453C3×C4#52911yescharacteristic
454Dic3—22no—
455Dic3—42no—
456Dic3—42no—
457C23—61no—
458C23—62no—
459C23D3×#4111yescharacteristic
460Q8—32no—
461Q8—61no—
462Q8—62no—
463Q8—62no—
464D4—32no—
465D4—62no—
466D4—62no—
467D4—62no—
468D4—62no—
469D4—61no—
470D4—122no—
471D4—122no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—61no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—124no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4C22×C4×D311yescharacteristic
485C2×C4C22×D1211yescharacteristic
486C2×C4C2×#23511yescharacteristic
487C2×C4C2×D3×D411yescharacteristic
488C2×C4C2×#23911yescharacteristic
489C2×C4D3×#4111yescharacteristic
490C8—24no—
491C8—62no—
492C8—62no—
493C8—122no—
494C8C4×D1211yescharacteristic
495C8C4×D1211yescharacteristic
496C8C4×D1211yescharacteristic
497C8C4×D1211yescharacteristic
498C8#89511yescharacteristic
499C8#89511yescharacteristic
500C8#89511yescharacteristic
501C8#89511yescharacteristic
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3C4×#12111yescharacteristic
506D3—42no—
507D3—82no—
508C22—41no—
509C22—62no—
510C22—122no—
511C22—122no—
512C22—242no—
513C22#552411yescharacteristic
514C22#554811yescharacteristic
515C22C4×#23911yescharacteristic
516C4—22no—
517C4—41no—
518C4—62no—
519C4—82no—
520C4—122no—
521C4—122no—
522C4C2×C4×D1211yescharacteristic
523C4C2×#89511yescharacteristic
524C4C4×D3×D411yescharacteristic
525C4#558011yescharacteristic
526C3#2155811yescharacteristic, complemented, Sylow
527C2—21no—
528C2—21no—
529C2—21no—
530C2—122no—
531C2—242no—
532C2C4×#89511yescharacteristic
subgroupG/H
centerC8#895
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1772C2
socleC2×C3C4×#121
prime radical#79256C2
cosocleC22×C3×C4C24
hypercenter#1772D3
hypocenterC3#21558
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C8 ◁ C22×C8 ◁ #1772

Complements

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

NH ≅ G/Ntype
2. #78579C2semidirect
3. C2×#4700C2semidirect
4. C4×#901C2semidirect
5. C3×#1772C2semidirect
6. #70707C2semidirect
7. #70665C2semidirect
8. #70658C2semidirect
9. #69898C2semidirect
10. #68778C2semidirect
11. #68750C2semidirect
33. #4898C4semidirect
34. #4897C4semidirect
35. #4896C4semidirect
58. #1772D3semidirect
76. C2×Dic12C2×C4semidirect
82. C2×D24C2×C4semidirect
83. C2×#204C2×C4semidirect
106. C3×#108C2×C4semidirect
107. C3×#108C2×C4semidirect
116. #824C2×C4semidirect
117. #823C2×C4semidirect
294. #108C4×D3semidirect
295. #108C4×D3semidirect
526. C3#21558semidirect

Maximal subgroups

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

subgroupindex
#792562
#785792
C2×#47002
C4×#9012
C3×#17722
#707072
#706652
#706582
#698982
#687782
#687502
#215583