Finite Group Databasebeta

← #10883543

Subgroups — #10883543

Order 768.

806 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79256C211yesmaximal, characteristic, complemented
2#79180C211yesmaximal, characteristic, complemented
3#78580C211yesmaximal, characteristic, complemented
4#78553C211yesmaximal, characteristic, complemented
5C3×#1833C211yesmaximal, characteristic, complemented
6#70711C211yesmaximal, characteristic, complemented
7#70665C211yesmaximal, characteristic, complemented
8#69036C212yesmaximal, complemented
9#69035C212yesmaximal, complemented
10#68703C212yesmaximal, complemented
11#68700C212yesmaximal, complemented
12#28180—31nomaximal, Sylow
13#5748C2212yes—
14#5745C2212yes—
15C2×#904C2211yescharacteristic
16C2×#901C2211yescharacteristic
17C22×D3×C8C2211yescharacteristic
18#5524C2211yescharacteristic
19#5524C2212yes—
20C2×C3×#108C2211yescharacteristic
21C3×#427C2211yescharacteristic
22C3×#405C2211yescharacteristic
23#5132C2212yes—
24#5129C2211yescharacteristic
25#5126C2211yescharacteristic
26#5126C2212yes—
27C2×#819C2211yescharacteristic
28#5005C2212yes—
29#5005C2212yes—
30#5002C2212yes—
31#4999C2211yescharacteristic
32#4898—22no—
33#4897—21no—
34#4896—21no—
35#4887—21no—
36#4883—21no—
37#4864—22no—
38#4717—22no—
39#4706—22no—
40#4700—22no—
41C8×D12—22no—
42C3×#344C2212yes—
43C3×H(3,ℤ/4)C2212yes—
44#4558C2212yes—
45#4533C2212yes—
46#3067—31no—
47#3050—31no—
48#2957—31no—
49#2929—31no—
50#2080—32no—
51#2010—32no—
52#1891—32no—
53#1881—32no—
54#1833D311yescharacteristic, complemented
55#1814—31no—
56#1810—31no—
57C2×#235C2312yes—
58C2×#235C2311yescharacteristic
59C22×C4×D3C2311yescharacteristic
60C2×C3×M16C2312yes—
61C2×C3×M16C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×#117C2312yes—
64C3×#117C2311yescharacteristic
65C2×#208C2311yescharacteristic
66C2×#208C2312yes—
67C22×C3⋊C8C2311yescharacteristic
68#909—22no—
69#909—22no—
70#908—22no—
71#908—22no—
72#907—42no—
73D3×M16—42no—
74C2×Dic12D411yescharacteristic, complemented
75#904—21no—
76#904—21no—
77#904—21no—
78#904—21no—
79#904—42no—
80C2×D24D411yescharacteristic, complemented
81C2×#204D412yescomplemented
82#901—21no—
83#901—21no—
84#901—24no—
85#901—21no—
86#901—21no—
87C2×ΓL(1,𝔽25)—21no—
88C2×D3×C8—21no—
89C2×D3×C8—21no—
90C2×D3×C8—21no—
91C2×D3×C8—21no—
92C2×D3×C8—21no—
93C2×D3×C8—21no—
94C2×D3×C8—21no—
95#897—21no—
96#897—22no—
97#895—21no—
98#895—22no—
99#886—21no—
100#886—22no—
101#882—21no—
102#882—22no—
103C4×D12—22no—
104C4×D12—42no—
105C3×#108—21no—
106C3×#108—21no—
107C3×#107—21no—
108C3×#106—21no—
109C3×#97D412yescomplemented
110C3×C4×C8D412yescomplemented
111#824—22no—
112#823—22no—
113(C2×D3)⋊C8—22no—
114(C2×D3)⋊C8—22no—
115(C2×D3)⋊C8—42no—
116#819—21no—
117#819—21no—
118#810—22no—
119#809—22no—
120#805—24no—
121#805—42no—
122C4⋊(C3⋊C8)—22no—
123#575—32no—
124C2×#135—31no—
125#568—32no—
126C2×#131—31no—
127C23×C8—31no—
128#525—31no—
129#525—32no—
130#473—61no—
131#472—61no—
132#471—62no—
133#467—61no—
134#466—61no—
135#465—62no—
136#444—62no—
137#443—62no—
138#435—62no—
139C8×D4—62no—
140C2×#108—31no—
141C2×#108C2×D311yescharacteristic
142#427C2×D311yescharacteristic
143#424—31no—
144#421—32no—
145#421—32no—
146#416—32no—
147#413—32no—
148#408—31no—
149#407—31no—
150#407—32no—
151#405C2×D311yescharacteristic
152#344—32no—
153#344C2×D312yes—
154#341—32no—
155H(3,ℤ/4)C2×D312yes—
156C23×D3#4111yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C2×#67—21no—
159C2×#67—22no—
160C22×Dic3#4111yescharacteristic
161#235—22no—
162#235—21no—
163#235—21no—
164#235—22no—
165#235—22no—
166#235—21no—
167#235—21no—
168#235—22no—
169C2×D12—22no—
170C2×D12—42no—
171C2×D12C2×D412yes—
172C2×D12#4111yescharacteristic
173C2×C4×D3—21no—
174C2×C4×D3—21no—
175C2×C4×D3—21no—
176C2×C4×D3—21no—
177C2×C4×D3—21no—
178C2×C4×D3—21no—
179C2×C4×D3—21no—
180C2×C4×D3—22no—
181C2×Dic6C2×D412yes—
182C2×Dic6#4111yescharacteristic
183C3×M16—21no—
184C3×M16—21no—
185C3×M16—22no—
186C3×M16—22no—
187C2×C3×C8—22no—
188C2×C3×C8C2×D412yes—
189C2×C3×C8C2×D411yescharacteristic
190C2×C3×C8C2×D411yescharacteristic
191C2×C3×C8C2×D411yescharacteristic
192C2×C3×C8C2×D411yescharacteristic
193C2×C3×C8#4111yescharacteristic
194C2×C3×C8#4111yescharacteristic
195C3×C4⋊C4—21no—
196C3×C4⋊C4—22no—
197C3×C22⋊C4—21no—
198C3×C22⋊C4—22no—
199C3×C42—22no—
200C3×C42C2×D412yes—
201(C2×D3)⋊C4—22no—
202(C2×D3)⋊C4—22no—
203(C2×D3)⋊C4—42no—
204(C2×D3)⋊C4—42no—
205C4⋊Dic3—22no—
206C4⋊Dic3—42no—
207#208—22no—
208#208—21no—
209#208—21no—
210#208—22no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—22no—
216Dic12—21no—
217Dic12—21no—
218Dic12—42no—
219D24—21no—
220D24—21no—
221D24—42no—
222#204—22no—
223#204—22no—
224#204—42no—
225#204—42no—
226ΓL(1,𝔽25)—44no—
227ΓL(1,𝔽25)—41no—
228ΓL(1,𝔽25)—41no—
229D3×C8—44no—
230D3×C8—41no—
231D3×C8—41no—
232D3×C8—41no—
233D3×C8—41no—
234D3×C8—41no—
235D3×C8—41no—
236C2×#41—31no—
237C2×#41—32no—
238C23×C4—31no—
239#137—62no—
240#137—62no—
241AGL(1,ℤ/8)—62no—
242AGL(1,ℤ/8)—62no—
243#135—61no—
244#135—61no—
245#135—61no—
246#135—61no—
247#135—122no—
248C2×Q16—31no—
249C2×SD16—32no—
250C2×D8—31no—
251#131—64no—
252#131—61no—
253#131—61no—
254#131—61no—
255#131—61no—
256#131—122no—
257C2×M16—31no—
258C2×M16—32no—
259C2×M16—61no—
260C2×M16—122no—
261C2×M16C22×D312yes—
262C2×M16C22×D311yescharacteristic
263C22×C8—31no—
264C22×C8—61no—
265C22×C8—61no—
266C22×C8—61no—
267C22×C8—61no—
268C22×C8—61no—
269C22×C8—61no—
270C22×C8—61no—
271C22×C8C22×D311yescharacteristic
272#123—61no—
273#123—62no—
274C22⋊Q8—61no—
275C22⋊Q8—62no—
276#121—61no—
277#121—62no—
278#120—61no—
279#120—62no—
280C4×D4—62no—
281C4×D4—122no—
282#117C22×D312yes—
283#117C22×D311yescharacteristic
284#108—21no—
285#108—21no—
286#108—61no—
287#108—61no—
288#107—21no—
289#106—21no—
290C4⋊C8—62no—
291#104—64no—
292#104—122no—
293#103—62no—
294#102—62no—
295#101—62no—
296#100—62no—
297C22⋊C8—62no—
298C22⋊C8—62no—
299C22⋊C8—122no—
300#97D1212yescomplemented
301C4×C8D1212yescomplemented
302C23×C3C2×#4111yescharacteristic
303C22×D3—21no—
304C22×D3—21no—
305C22×D3—21no—
306C22×D3—21no—
307C22×D3—22no—
308C22×D3—42no—
309C2×C3×C4—22no—
310C2×C3×C4—42no—
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4H(5,𝔽2)11yescharacteristic
316C2×C3×C4H(5,𝔽2)11yescharacteristic
317#67—42no—
318#67—41no—
319#67—41no—
320#67—42no—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—22no—
325C2×Dic3—21no—
326D12—22no—
327D12—21no—
328D12—22no—
329D12—21no—
330D12—44no—
331D12—84no—
332C4×D3—41no—
333C4×D3—41no—
334C4×D3—42no—
335C4×D3—41no—
336C4×D3—41no—
337C4×D3—41no—
338C4×D3—41no—
339C4×D3—42no—
340Dic6—22no—
341Dic6—21no—
342Dic6—22no—
343Dic6—21no—
344C3×C8—22no—
345C3×C8—42no—
346C3×C8#12111yescharacteristic
347C3×C8#12111yescharacteristic
348C3×C8#12111yescharacteristic
349C3×C8#12111yescharacteristic
350C3⋊C8—41no—
351C3⋊C8—41no—
352C3⋊C8—42no—
353C24—31no—
354#41—62no—
355#41—61no—
356#41—61no—
357#41—62no—
358#41—62no—
359#41—61no—
360#41—62no—
361#41—61no—
362C2×Q8—32no—
363C2×Q8—31no—
364C2×D4—32no—
365C2×D4—31no—
366C2×D4—62no—
367C2×D4—62no—
368C2×D4—61no—
369C2×D4—122no—
370C22×C4—31no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4—61no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—62no—
379C22×C4C23×D311yescharacteristic
380Q16—61no—
381Q16—61no—
382Q16—122no—
383SD16—62no—
384SD16—62no—
385SD16—122no—
386SD16—122no—
387D8—61no—
388D8—61no—
389D8—122no—
390M16—21no—
391M16—21no—
392M16—22no—
393M16—22no—
394M16—62no—
395M16—61no—
396M16—61no—
397M16—62no—
398M16—124no—
399M16—121no—
400M16—121no—
401C2×C8—22no—
402C2×C8—61no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—61no—
406C2×C8—62no—
407C2×C8—124no—
408C2×C8—121no—
409C2×C8—121no—
410C2×C8—121no—
411C2×C8—121no—
412C2×C8—121no—
413C2×C8—121no—
414C2×C8C2×D1212yes—
415C2×C8C2×D1211yescharacteristic
416C2×C8C2×D1211yescharacteristic
417C2×C8D3×D411yescharacteristic
418C2×C8D3×D411yescharacteristic
419C2×C8#23911yescharacteristic
420C2×C8#23911yescharacteristic
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—62no—
424C4⋊C4—122no—
425C22⋊C4—21no—
426C22⋊C4—22no—
427C22⋊C4—62no—
428C22⋊C4—62no—
429C22⋊C4—122no—
430C22⋊C4—122no—
431C42—22no—
432C42C2×D1212yes—
433C22×C3—41no—
434C22×C3#52511yescharacteristic
435C22×C3#54611yescharacteristic
436C22×C3#55311yescharacteristic
437C2×D3—22no—
438C2×D3—22no—
439C2×D3—22no—
440C2×D3—41no—
441C2×D3—41no—
442C2×D3—41no—
443C2×D3—41no—
444C2×D3—42no—
445C2×D3—41no—
446C2×D3—42no—
447C2×D3—82no—
448C3×C4—42no—
449C3×C4—82no—
450C3×C4C2×#12111yescharacteristic
451C3×C4#53511yescharacteristic
452C3×C4#54611yescharacteristic
453C3×C4#55311yescharacteristic
454Dic3—42no—
455Dic3—41no—
456Dic3—41no—
457C23—61no—
458C23—61no—
459C23—61no—
460C23—61no—
461C23—62no—
462C23—122no—
463C23D3×#4111yescharacteristic
464Q8—61no—
465Q8—62no—
466Q8—61no—
467Q8—62no—
468D4—61no—
469D4—62no—
470D4—61no—
471D4—62no—
472D4—124no—
473D4—121no—
474D4—121no—
475D4—122no—
476D4—122no—
477D4—244no—
478C2×C4—22no—
479C2×C4—42no—
480C2×C4—61no—
481C2×C4—62no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—61no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4—122no—
488C2×C4—121no—
489C2×C4—121no—
490C2×C4—122no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4C22×D1211yescharacteristic
494C2×C4C2×D3×D411yescharacteristic
495C2×C4C2×#23911yescharacteristic
496C2×C4D3×#4111yescharacteristic
497C2×C4#100911yescharacteristic
498C2×C4#100911yescharacteristic
499C8—22no—
500C8—42no—
501C8—121no—
502C8—121no—
503C8—122no—
504C8#89511yescharacteristic
505C8#89511yescharacteristic
506C8#89511yescharacteristic
507C8#89511yescharacteristic
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3#253511yescharacteristic
512D3—42no—
513D3—81no—
514D3—82no—
515C22—41no—
516C22—62no—
517C22—62no—
518C22—62no—
519C22—121no—
520C22—121no—
521C22—121no—
522C22—122no—
523C22—121no—
524C22—121no—
525C22—122no—
526C22—242no—
527C22#552411yescharacteristic
528C22#567611yescharacteristic
529C22#569511yescharacteristic
530C4—42no—
531C4—82no—
532C4—121no—
533C4—122no—
534C4—121no—
535C4C2×#89511yescharacteristic
536C4#552511yescharacteristic
537C4#567611yescharacteristic
538C4#569511yescharacteristic
539C3#2818011yescharacteristic, complemented, Sylow
540C2—21no—
541C2—21no—
542C2—21no—
543C2—122no—
544C2—241no—
545C2—242no—
546C2#7423911yescharacteristic
subgroupG/H
centerC4#5525
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1833C2
socleC2×C3#2535
hypercenter#1833D3
hypocenterC3#28180
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1833

Complements

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

NH ≅ G/Ntype
1. #79256C2semidirect
2. #79180C2semidirect
3. #78580C2semidirect
4. #78553C2semidirect
5. C3×#1833C2semidirect
6. #70711C2semidirect
7. #70665C2semidirect
8. #69036C2semidirect
9. #69035C2semidirect
10. #68703C2semidirect
11. #68700C2semidirect
54. #1833D3semidirect
74. C2×Dic12D4semidirect
80. C2×D24D4semidirect
81. C2×#204D4semidirect
109. C3×#97D4semidirect
110. C3×C4×C8D4semidirect
300. #97D12semidirect
301. C4×C8D12semidirect
539. C3#28180semidirect

Maximal subgroups

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

subgroupindex
#792562
#791802
#785802
#785532
C3×#18332
#707112
#706652
#690362
#690352
#687032
#687002
#281803