Finite Group Databasebeta

← #10819911

Subgroups — #10819911

Order 768.

880 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83910C211yesmaximal, characteristic, complemented
2#83904C211yesmaximal, characteristic
3#78990C212yesmaximal, complemented
4#78580C211yesmaximal, characteristic, complemented
5C3×#1976C211yesmaximal, characteristic, complemented
6#70660C211yesmaximal, characteristic, complemented
7#70658C211yesmaximal, characteristic, complemented
8#70417C212yesmaximal, complemented
9#70045C212yesmaximal, complemented
10#70042C212yesmaximal, complemented
11#69890C211yesmaximal, characteristic, complemented
12#23197—31nomaximal, Sylow
13C2×C3×#135C2211yescharacteristic
14#5832C2212yes—
15C2×#950C2212yes—
16#5819C2212yes—
17#5745C2212yes—
18C2×#904C2211yescharacteristic
19C3×#427C2211yescharacteristic
20C3×#421C2212yes—
21C3×#410C2212yes—
22C3×#405C2211yescharacteristic
23#5236C2212yes—
24#5197—22no—
25#5196—22no—
26#5190C412yescomplemented
27#5176C412yescomplemented
28#5167C412yescomplemented
29#5154C412yescomplemented
30C2×#822C2211yescharacteristic
31C2×#819C2211yescharacteristic
32#5101C2211yescharacteristic
33C2×#815C2211yescharacteristic
34#5006C2212yes—
35#5005C2212yes—
36#5005C2212yes—
37#4998C2211yescharacteristic
38#4997C2212yes—
39#4824—24no—
40#4823—24no—
41#4717—22no—
42#4700—22no—
43#4555C2211yescharacteristic
44#4532C2211yescharacteristic
45#4532C2212yes—
46#3150—31no—
47#2957—32no—
48#2957—31no—
49#2957—31no—
50#1976—31no—
51#1976—32no—
52#1976D311yescharacteristic, complemented
53#1884—32no—
54#1884—32no—
55#1773—31no—
56#1773—31no—
57C2×C3×#41C2312yes—
58C2×#235C2312yes—
59C3×#135—21no—
60C3×#135—21no—
61C3×#135—21no—
62C3×#135—21no—
63C2×C3×Q16C2×C411yescharacteristic
64C2×C3×SD16C2×C412yes—
65C2×C3×D8C2×C411yescharacteristic
66C2×C3×M16C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×#117C2312yes—
69C3×#117C2311yescharacteristic
70#950—22no—
71#950—22no—
72#950—22no—
73#950—22no—
74#948—24no—
75#948—24no—
76#947C2×C412yes—
77#944C2×C412yes—
78C2×#216C2×C412yes—
79C2×#215C2×C412yes—
80#940C2×C412yes—
81#939—42no—
82#936C2×C412yes—
83C2×#214C2×C412yes—
84C2×#213C2×C412yes—
85C4×#67—42no—
86#926C2311yescharacteristic
87C2×C4×Dic3C2311yescharacteristic
88C2×#208—42no—
89C2×#208C2311yescharacteristic
90C2×#208C2312yes—
91C22×C3⋊C8C2311yescharacteristic
92#907—24no—
93D3×M16—42no—
94C2×Dic12C2×C411yescharacteristic
95#904—21no—
96#904—21no—
97#904—21no—
98#904—21no—
99C2×D24C2×C411yescharacteristic
100C2×#204C2×C412yes—
101#901—24no—
102C3×#107—21no—
103C3×#106—21no—
104C3×#104—24no—
105C3×#99—42no—
106C3×#97—21no—
107C3×C4×C8—21no—
108#837—24no—
109#822D414yescomplemented
110#819D412yescomplemented
111#819D412yescomplemented
112#815—21no—
113#815C2×C412yes—
114#815C2×C412yes—
115C8×Dic3—21no—
116#808—21no—
117#807—21no—
118#806—42no—
119#805—24no—
120#805—24no—
121#803—22no—
122C4×C3⋊C8—22no—
123C2×#135—32no—
124C2×#135—31no—
125C2×#135C2×D311yescharacteristic
126#568—32no—
127#568—32no—
128#532—32no—
129#497—32no—
130#497—32no—
131#496—32no—
132#496—32no—
133#444—62no—
134#444—64no—
135#444—62no—
136#443—62no—
137#443—64no—
138#443—62no—
139C2×#108—31no—
140C2×#108—31no—
141#427—31no—
142#427C2×D311yescharacteristic
143#421—32no—
144#421—32no—
145#421—32no—
146#421C2×D312yes—
147#410—32no—
148#410C2×D312yes—
149#405—31no—
150#405—32no—
151#405C2×D311yescharacteristic
152C2×#97—31no—
153#339—31no—
154#339—32no—
155#339—31no—
156C3×#41—22no—
157C3×#41—22no—
158C3×#41—22no—
159C3×#41—22no—
160C2×C3×Q8C22×C412yes—
161C2×C3×D4—22no—
162C2×C3×D4C22×C412yes—
163C22×C3×C4—22no—
164C22×C3×C4C2411yescharacteristic
165C2×#67—22no—
166C2×#67—42no—
167C22×Dic3—21no—
168#235—22no—
169#235—22no—
170#235—22no—
171#235—22no—
172C2×D12C22×C412yes—
173C2×C4×D3—22no—
174C2×Dic6C22×C412yes—
175C3×Q16—21no—
176C3×Q16—21no—
177C3×SD16—22no—
178C3×SD16—22no—
179C3×D8—21no—
180C3×D8—21no—
181C3×M16—21no—
182C3×M16C2×D412yes—
183C2×C3×C8—21no—
184C2×C3×C8C22×C411yescharacteristic
185C2×C3×C8C22×C411yescharacteristic
186C2×C3×C8C2×D411yescharacteristic
187C2×C3×C8C2×D411yescharacteristic
188C2×C3×C8#4111yescharacteristic
189C2×C3×C8#4111yescharacteristic
190C3×C4⋊C4—21no—
191C3×C4⋊C4—22no—
192C3×C22⋊C4—22no—
193C3×C22⋊C4—21no—
194C3×C42—22no—
195C3×C42—21no—
196#217—21no—
197#217—42no—
198#216—22no—
199#216—22no—
200#215—22no—
201#215—22no—
202#214—22no—
203#214—22no—
204#213—22no—
205#213—22no—
206(C2×D3)⋊C4—42no—
207C4⋊Dic3—21no—
208Dic3⋊C4—42no—
209C4×Dic3—21no—
210C4×Dic3—21no—
211C4×Dic3C22×C412yes—
212#208—22no—
213#208—21no—
214#208—44no—
215#208C2×D412yes—
216#208C2×D414yes—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—44no—
220C2×C3⋊C8C22×C412yes—
221C2×C3⋊C8C2×D412yes—
222C2×C3⋊C8#4111yescharacteristic
223C2×C3⋊C8#4111yescharacteristic
224Dic12—21no—
225Dic12—21no—
226D24—21no—
227D24—21no—
228#204—22no—
229#204—22no—
230ΓL(1,𝔽25)—44no—
231D3×C8—44no—
232C2×#41—32no—
233C2×#41C22×D312yes—
234#135—21no—
235#135—21no—
236#135—21no—
237#135—21no—
238#135—62no—
239#135—62no—
240#135—62no—
241#135—62no—
242#135—61no—
243#135—61no—
244#135—61no—
245#135—61no—
246C2×Q16—32no—
247C2×Q16—31no—
248C2×Q16C4×D311yescharacteristic
249C2×SD16—32no—
250C2×SD16—32no—
251C2×SD16—32no—
252C2×SD16C4×D312yes—
253C2×D8—32no—
254C2×D8—31no—
255C2×D8C4×D311yescharacteristic
256#131—64no—
257#131—64no—
258#131—64no—
259#131—64no—
260C2×M16—31no—
261C2×M16—32no—
262C2×M16—122no—
263C2×M16—122no—
264C2×M16C22×D311yescharacteristic
265C22×C8—31no—
266C22×C8C22×D311yescharacteristic
267C4⋊Q8—32no—
268#127—32no—
269#124—32no—
270#124—32no—
271#121—122no—
272C4×D4—122no—
273#117—31no—
274#117C22×D312yes—
275#117C22×D311yescharacteristic
276C2×C42—31no—
277#108—34no—
278#108—32no—
279#108—32no—
280#107—21no—
281#107—61no—
282#106—21no—
283#106—61no—
284#104—24no—
285#104—64no—
286#104—64no—
287#104—64no—
288#99—42no—
289#99—122no—
290#97—21no—
291#97—32no—
292#97—32no—
293#97—62no—
294#97—61no—
295C4×C8—21no—
296C4×C8—62no—
297C4×C8—61no—
298C23×C3—22no—
299C23×C3H(5,𝔽2)11yescharacteristic
300C22×D3—22no—
301C3×Q8—22no—
302C3×Q8—22no—
303C3×D4—22no—
304C3×D4—22no—
305C3×D4—42no—
306C3×D4—42no—
307C2×C3×C4—22no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4C23×C411yescharacteristic
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4H(5,𝔽2)11yescharacteristic
318#67—42no—
319#67—42no—
320#67—84no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—41no—
324C2×Dic3—81no—
325D12—22no—
326D12—22no—
327C4×D3—42no—
328C4×D3—42no—
329Dic6—22no—
330Dic6—22no—
331C3×C8—22no—
332C3×C8C4×D411yescharacteristic
333C3×C8C4×D411yescharacteristic
334C3×C8C4×D411yescharacteristic
335C3×C8C4×D411yescharacteristic
336C3⋊C8—24no—
337C3⋊C8—22no—
338C3⋊C8C4×D412yes—
339C3⋊C8C4×D412yes—
340#41—22no—
341#41—22no—
342#41—22no—
343#41—22no—
344#41—62no—
345#41—62no—
346#41—62no—
347#41—62no—
348C2×Q8—32no—
349C2×Q8C2×C4×D312yes—
350C2×D4—22no—
351C2×D4—32no—
352C2×D4—62no—
353C2×D4—122no—
354C2×D4C2×C4×D312yes—
355C22×C4—22no—
356C22×C4—61no—
357C22×C4—62no—
358C22×C4C23×D311yescharacteristic
359Q16—21no—
360Q16—21no—
361Q16—62no—
362Q16—62no—
363Q16—61no—
364Q16—61no—
365SD16—22no—
366SD16—22no—
367SD16—62no—
368SD16—62no—
369SD16—62no—
370SD16—62no—
371SD16—62no—
372SD16—62no—
373D8—21no—
374D8—21no—
375D8—62no—
376D8—62no—
377D8—61no—
378D8—61no—
379M16—21no—
380M16—34no—
381M16—32no—
382M16—62no—
383M16—61no—
384M16—124no—
385M16—124no—
386M16D3×D412yes—
387C2×C8—21no—
388C2×C8—32no—
389C2×C8—32no—
390C2×C8—31no—
391C2×C8—31no—
392C2×C8—61no—
393C2×C8—62no—
394C2×C8—124no—
395C2×C8—124no—
396C2×C8C2×C4×D311yescharacteristic
397C2×C8C2×C4×D311yescharacteristic
398C2×C8D3×D411yescharacteristic
399C2×C8D3×D411yescharacteristic
400C2×C8#23711yescharacteristic
401C2×C8#23711yescharacteristic
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—61no—
405C4⋊C4—122no—
406C22⋊C4—21no—
407C22⋊C4—22no—
408C22⋊C4—61no—
409C22⋊C4—122no—
410C22⋊C4—122no—
411C42—21no—
412C42—22no—
413C42—32no—
414C42—61no—
415C42—61no—
416C22×C3—41no—
417C22×C3—42no—
418C22×C3—82no—
419C22×C3#51811yescharacteristic
420C22×C3#55011yescharacteristic
421C22×C3#55011yescharacteristic
422C2×D3—42no—
423C2×D3—82no—
424C3×C4—42no—
425C3×C4—81no—
426C3×C4—82no—
427C3×C4C2×C4×D411yescharacteristic
428C3×C4C2×C4×D411yescharacteristic
429C3×C4D4211yescharacteristic
430C3×C4#55911yescharacteristic
431Dic3—42no—
432Dic3—42no—
433Dic3—81no—
434C23—22no—
435C23—62no—
436C23#100411yescharacteristic
437Q8—22no—
438Q8—22no—
439Q8—62no—
440Q8—62no—
441D4—22no—
442D4—22no—
443D4—42no—
444D4—42no—
445D4—62no—
446D4—62no—
447D4—122no—
448D4—122no—
449D4—244no—
450C2×C4—22no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—121no—
460C2×C4—241no—
461C2×C4C22×C4×D311yescharacteristic
462C2×C4C2×D3×D411yescharacteristic
463C2×C4C2×D3×D411yescharacteristic
464C2×C4C2×#23711yescharacteristic
465C2×C4D3×#4111yescharacteristic
466C2×C4#100911yescharacteristic
467C8—22no—
468C8—32no—
469C8—32no—
470C8—64no—
471C8—62no—
472C8#67⋊C411yescharacteristic
473C8#67⋊C411yescharacteristic
474C8#67⋊C411yescharacteristic
475C8#67⋊C411yescharacteristic
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—82no—
480C2×C3#236611yescharacteristic
481D3—82no—
482C22—41no—
483C22—42no—
484C22—82no—
485C22—122no—
486C22—242no—
487C22#554511yescharacteristic
488C22#559911yescharacteristic
489C22#571811yescharacteristic
490C4—42no—
491C4—82no—
492C4—81no—
493C4—122no—
494C4—122no—
495C4—241no—
496C4C2×#67⋊C411yescharacteristic
497C4C4×D3×D411yescharacteristic
498C4#560911yescharacteristic
499C4#568111yescharacteristic
500C3#2319711yescharacteristic, complemented, Sylow
501C2—21no—
502C2—21no—
503C2—21no—
504C2—82no—
505C2—242no—
506C2#7630611yescharacteristic
subgroupG/H
centerC4#5609
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1976C2
socleC2×C3#2366
prime radical#83904C2
cosocleC22×C3×C4C24
hypercenter#1976D3
hypocenterC3#23197
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1976

Complements

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

NH ≅ G/Ntype
1. #83910C2semidirect
3. #78990C2semidirect
4. #78580C2semidirect
5. C3×#1976C2semidirect
6. #70660C2semidirect
7. #70658C2semidirect
8. #70417C2semidirect
9. #70045C2semidirect
10. #70042C2semidirect
11. #69890C2semidirect
26. #5190C4semidirect
27. #5176C4semidirect
28. #5167C4semidirect
29. #5154C4semidirect
52. #1976D3semidirect
109. #822D4semidirect
110. #819D4semidirect
111. #819D4semidirect
500. C3#23197semidirect

Maximal subgroups

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

subgroupindex
#839102
#839042
#789902
#785802
C3×#19762
#706602
#706582
#704172
#700452
#700422
#698902
#231973