Finite Group Databasebeta

← #10819906

Subgroups — #10819906

Order 768.

884 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#83910C211yesmaximal, characteristic, complemented
2#83903C211yesmaximal, characteristic
3#78990C212yesmaximal, complemented
4#78580C211yesmaximal, characteristic, complemented
5C3×#1976C211yesmaximal, characteristic, complemented
6#70659C211yesmaximal, characteristic, complemented
7#70658C211yesmaximal, characteristic, complemented
8#70417C212yesmaximal, complemented
9#70045C212yesmaximal, complemented
10#70042C212yesmaximal, complemented
11#69889C211yesmaximal, characteristic, complemented
12#23196—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#5191C411yescharacteristic, complemented
27#5183C411yescharacteristic, complemented
28#5175C412yescomplemented
29#5168C412yescomplemented
30#5155C411yescharacteristic, complemented
31#5151C411yescharacteristic, complemented
32C2×#822C2211yescharacteristic
33C2×#819C2211yescharacteristic
34#5101C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#5006C2212yes—
37#5005C2212yes—
38#5005C2212yes—
39#4998C2211yescharacteristic
40#4997C2212yes—
41#4824—24no—
42#4823—24no—
43#4717—22no—
44#4700—22no—
45#4555C2211yescharacteristic
46#4532C2211yescharacteristic
47#4532C2212yes—
48#3147—31no—
49#2957—32no—
50#2957—31no—
51#2957—31no—
52#1976—31no—
53#1976—32no—
54#1976D311yescharacteristic, complemented
55#1884—32no—
56#1884—32no—
57#1772—31no—
58#1772—31no—
59C2×C3×#41C2312yes—
60C2×#235C2312yes—
61C3×#135—21no—
62C3×#135—21no—
63C3×#135—21no—
64C3×#135—21no—
65C2×C3×Q16C2×C411yescharacteristic
66C2×C3×SD16C2×C412yes—
67C2×C3×D8C2×C411yescharacteristic
68C2×C3×M16C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C3×#117C2312yes—
71C3×#117C2311yescharacteristic
72#950—22no—
73#950—22no—
74#950—22no—
75#950—22no—
76#948—24no—
77#948—24no—
78#947C2×C412yes—
79#944C2×C412yes—
80C2×#216C2×C412yes—
81C2×#215C2×C412yes—
82#940C2×C412yes—
83#939—42no—
84#936C2×C412yes—
85C2×#214C2×C412yes—
86C2×#213C2×C412yes—
87C4×#67—42no—
88#926C2311yescharacteristic
89C2×C4×Dic3C2311yescharacteristic
90C2×#208—42no—
91C2×#208C2311yescharacteristic
92C2×#208C2312yes—
93C22×C3⋊C8C2311yescharacteristic
94#907—24no—
95D3×M16—42no—
96C2×Dic12C2×C411yescharacteristic
97#904—21no—
98#904—21no—
99#904—21no—
100#904—21no—
101C2×D24C2×C411yescharacteristic
102C2×#204C2×C412yes—
103#901—24no—
104C3×#107—21no—
105C3×#106—21no—
106C3×#104—24no—
107C3×#99—42no—
108C3×#97—21no—
109C3×C4×C8—21no—
110#837—24no—
111#822D412yescomplemented
112#822D412yescomplemented
113#819D412yescomplemented
114#819D412yescomplemented
115#815—21no—
116C8×Dic3—21no—
117C8×Dic3C2×C411yescharacteristic
118C8×Dic3C2×C411yescharacteristic
119C8×Dic3C2×C411yescharacteristic
120C8×Dic3C2×C411yescharacteristic
121#808—21no—
122#807—21no—
123#806—42no—
124#805—24no—
125#805—24no—
126#803—22no—
127C4×C3⋊C8—22no—
128C2×#135—32no—
129C2×#135—31no—
130C2×#135C2×D311yescharacteristic
131#568—32no—
132#568—32no—
133#532—32no—
134#495—31no—
135#495—32no—
136#495—31no—
137#494—31no—
138#493—31no—
139#492—32no—
140#444—62no—
141#444—64no—
142#444—62no—
143#443—62no—
144#443—64no—
145#443—62no—
146C2×#108—31no—
147C2×#108—31no—
148#427—31no—
149#427C2×D311yescharacteristic
150#421—32no—
151#421—32no—
152#421—32no—
153#421C2×D312yes—
154#410—32no—
155#410C2×D312yes—
156#405—31no—
157#405—32no—
158#405C2×D311yescharacteristic
159C2×C4×C8—31no—
160#339—31no—
161#339—32no—
162#339—31no—
163C3×#41—22no—
164C3×#41—22no—
165C3×#41—22no—
166C3×#41—22no—
167C2×C3×Q8C22×C412yes—
168C2×C3×D4—22no—
169C2×C3×D4C22×C412yes—
170C22×C3×C4—22no—
171C22×C3×C4C2411yescharacteristic
172C2×#67—22no—
173C2×#67—42no—
174C22×Dic3—21no—
175#235—22no—
176#235—22no—
177#235—22no—
178#235—22no—
179C2×D12C22×C412yes—
180C2×C4×D3—22no—
181C2×Dic6C22×C412yes—
182C3×Q16—21no—
183C3×Q16—21no—
184C3×SD16—22no—
185C3×SD16—22no—
186C3×D8—21no—
187C3×D8—21no—
188C3×M16—21no—
189C3×M16C2×D412yes—
190C2×C3×C8—21no—
191C2×C3×C8C22×C411yescharacteristic
192C2×C3×C8C22×C411yescharacteristic
193C2×C3×C8C2×D411yescharacteristic
194C2×C3×C8C2×D411yescharacteristic
195C2×C3×C8#4111yescharacteristic
196C2×C3×C8#4111yescharacteristic
197C3×C4⋊C4—21no—
198C3×C4⋊C4—22no—
199C3×C22⋊C4—22no—
200C3×C22⋊C4—21no—
201C3×C42—22no—
202C3×C42—21no—
203#217—21no—
204#217—42no—
205#216—22no—
206#216—22no—
207#215—22no—
208#215—22no—
209#214—22no—
210#214—22no—
211#213—22no—
212#213—22no—
213(C2×D3)⋊C4—42no—
214C4⋊Dic3—21no—
215Dic3⋊C4—42no—
216C4×Dic3—21no—
217C4×Dic3—21no—
218C4×Dic3C22×C411yescharacteristic
219C4×Dic3C22×C411yescharacteristic
220#208—22no—
221#208—21no—
222#208—44no—
223#208C2×D412yes—
224#208C2×D414yes—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—44no—
228C2×C3⋊C8C22×C411yescharacteristic
229C2×C3⋊C8C22×C411yescharacteristic
230C2×C3⋊C8C2×D411yescharacteristic
231C2×C3⋊C8C2×D411yescharacteristic
232C2×C3⋊C8#4111yescharacteristic
233C2×C3⋊C8#4111yescharacteristic
234Dic12—21no—
235Dic12—21no—
236D24—21no—
237D24—21no—
238#204—22no—
239#204—22no—
240ΓL(1,𝔽25)—44no—
241D3×C8—44no—
242C2×#41—32no—
243C2×#41C22×D312yes—
244#135—21no—
245#135—21no—
246#135—21no—
247#135—21no—
248#135—62no—
249#135—62no—
250#135—62no—
251#135—62no—
252#135—61no—
253#135—61no—
254#135—61no—
255#135—61no—
256C2×Q16—32no—
257C2×Q16—31no—
258C2×Q16C4×D311yescharacteristic
259C2×SD16—32no—
260C2×SD16—32no—
261C2×SD16—32no—
262C2×SD16C4×D312yes—
263C2×D8—32no—
264C2×D8—31no—
265C2×D8C4×D311yescharacteristic
266#131—64no—
267#131—64no—
268#131—64no—
269#131—64no—
270C2×M16—31no—
271C2×M16—32no—
272C2×M16—122no—
273C2×M16—122no—
274C2×M16C22×D311yescharacteristic
275C22×C8—31no—
276C22×C8C22×D311yescharacteristic
277C4⋊Q8—32no—
278#127—32no—
279#124—32no—
280#124—32no—
281#121—122no—
282C4×D4—122no—
283#117—31no—
284#117C22×D312yes—
285#117C22×D311yescharacteristic
286C2×C42—31no—
287#108—32no—
288#108—32no—
289#108—32no—
290#108—32no—
291#107—21no—
292#107—61no—
293#106—21no—
294#106—61no—
295#104—24no—
296#104—64no—
297#104—64no—
298#104—64no—
299#99—42no—
300#99—122no—
301#97—21no—
302#97—62no—
303#97—61no—
304C4×C8—21no—
305C4×C8—31no—
306C4×C8—31no—
307C4×C8—31no—
308C4×C8—31no—
309C4×C8—62no—
310C4×C8—61no—
311C23×C3—22no—
312C23×C3H(5,𝔽2)11yescharacteristic
313C22×D3—22no—
314C3×Q8—22no—
315C3×Q8—22no—
316C3×D4—22no—
317C3×D4—22no—
318C3×D4—42no—
319C3×D4—42no—
320C2×C3×C4—22no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4C23×C411yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C2×#4111yescharacteristic
329C2×C3×C4C2×#4111yescharacteristic
330C2×C3×C4H(5,𝔽2)11yescharacteristic
331#67—42no—
332#67—42no—
333#67—84no—
334C2×Dic3—21no—
335C2×Dic3—21no—
336C2×Dic3—22no—
337C2×Dic3—42no—
338C2×Dic3—41no—
339D12—22no—
340D12—22no—
341C4×D3—42no—
342C4×D3—42no—
343Dic6—22no—
344Dic6—22no—
345C3×C8—22no—
346C3×C8C4×D411yescharacteristic
347C3×C8C4×D411yescharacteristic
348C3×C8C4×D411yescharacteristic
349C3×C8C4×D411yescharacteristic
350C3⋊C8—24no—
351C3⋊C8—22no—
352C3⋊C8C4×D411yescharacteristic
353C3⋊C8C4×D411yescharacteristic
354C3⋊C8C4×D411yescharacteristic
355C3⋊C8C4×D411yescharacteristic
356#41—22no—
357#41—22no—
358#41—22no—
359#41—22no—
360#41—62no—
361#41—62no—
362#41—62no—
363#41—62no—
364C2×Q8—32no—
365C2×Q8C2×C4×D312yes—
366C2×D4—22no—
367C2×D4—32no—
368C2×D4—62no—
369C2×D4—122no—
370C2×D4C2×C4×D312yes—
371C22×C4—22no—
372C22×C4—61no—
373C22×C4—62no—
374C22×C4C23×D311yescharacteristic
375Q16—21no—
376Q16—21no—
377Q16—62no—
378Q16—62no—
379Q16—61no—
380Q16—61no—
381SD16—22no—
382SD16—22no—
383SD16—62no—
384SD16—62no—
385SD16—62no—
386SD16—62no—
387SD16—62no—
388SD16—62no—
389D8—21no—
390D8—21no—
391D8—62no—
392D8—62no—
393D8—61no—
394D8—61no—
395M16—21no—
396M16—34no—
397M16—32no—
398M16—62no—
399M16—61no—
400M16—124no—
401M16—124no—
402M16D3×D412yes—
403C2×C8—21no—
404C2×C8—31no—
405C2×C8—31no—
406C2×C8—31no—
407C2×C8—31no—
408C2×C8—31no—
409C2×C8—31no—
410C2×C8—61no—
411C2×C8—62no—
412C2×C8—124no—
413C2×C8—124no—
414C2×C8C2×C4×D311yescharacteristic
415C2×C8C2×C4×D311yescharacteristic
416C2×C8D3×D411yescharacteristic
417C2×C8D3×D411yescharacteristic
418C2×C8#23711yescharacteristic
419C2×C8#23711yescharacteristic
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—61no—
423C4⋊C4—122no—
424C22⋊C4—21no—
425C22⋊C4—22no—
426C22⋊C4—61no—
427C22⋊C4—122no—
428C22⋊C4—122no—
429C42—21no—
430C42—22no—
431C42—31no—
432C42—31no—
433C42—61no—
434C42—61no—
435C22×C3—41no—
436C22×C3—42no—
437C22×C3—82no—
438C22×C3#51811yescharacteristic
439C22×C3#55011yescharacteristic
440C22×C3#55011yescharacteristic
441C2×D3—42no—
442C2×D3—82no—
443C3×C4—42no—
444C3×C4—81no—
445C3×C4—82no—
446C3×C4C2×C4×D411yescharacteristic
447C3×C4C2×C4×D411yescharacteristic
448C3×C4D4211yescharacteristic
449C3×C4#55911yescharacteristic
450Dic3—22no—
451Dic3—41no—
452Dic3—42no—
453Dic3—81no—
454C23—22no—
455C23—62no—
456C23#100411yescharacteristic
457Q8—22no—
458Q8—22no—
459Q8—62no—
460Q8—62no—
461D4—22no—
462D4—22no—
463D4—42no—
464D4—42no—
465D4—62no—
466D4—62no—
467D4—122no—
468D4—122no—
469D4—244no—
470C2×C4—22no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—62no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—121no—
482C2×C4C22×C4×D311yescharacteristic
483C2×C4C2×D3×D411yescharacteristic
484C2×C4C2×D3×D411yescharacteristic
485C2×C4C2×#23711yescharacteristic
486C2×C4D3×#4111yescharacteristic
487C2×C4#100911yescharacteristic
488C8—22no—
489C8—31no—
490C8—31no—
491C8—31no—
492C8—31no—
493C8—64no—
494C8—62no—
495C8#67⋊C411yescharacteristic
496C8#67⋊C411yescharacteristic
497C8#67⋊C411yescharacteristic
498C8#67⋊C411yescharacteristic
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—82no—
503C2×C3#236611yescharacteristic
504D3—82no—
505C22—41no—
506C22—42no—
507C22—82no—
508C22—122no—
509C22—242no—
510C22#554511yescharacteristic
511C22#559911yescharacteristic
512C22#571811yescharacteristic
513C4—42no—
514C4—62no—
515C4—82no—
516C4—81no—
517C4—121no—
518C4—122no—
519C4—241no—
520C4C2×#67⋊C411yescharacteristic
521C4C4×D3×D411yescharacteristic
522C4#560911yescharacteristic
523C4#568111yescharacteristic
524C3#2319611yescharacteristic, complemented, Sylow
525C2—21no—
526C2—21no—
527C2—21no—
528C2—82no—
529C2—242no—
530C2#7630611yescharacteristic
subgroupG/H
centerC4#5609
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1976C2
socleC2×C3#2366
prime radical#83903C2
cosocleC22×C3×C4C24
hypercenter#1976D3
hypocenterC3#23196
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. #70659C2semidirect
7. #70658C2semidirect
8. #70417C2semidirect
9. #70045C2semidirect
10. #70042C2semidirect
11. #69889C2semidirect
26. #5191C4semidirect
27. #5183C4semidirect
28. #5175C4semidirect
29. #5168C4semidirect
30. #5155C4semidirect
31. #5151C4semidirect
54. #1976D3semidirect
111. #822D4semidirect
112. #822D4semidirect
113. #819D4semidirect
114. #819D4semidirect
524. C3#23196semidirect

Maximal subgroups

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

subgroupindex
#839102
#839032
#789902
#785802
C3×#19762
#706592
#706582
#704172
#700452
#700422
#698892
#231963