Finite Group Databasebeta

← #10801200

Subgroups — #10801200

Order 768.

1328 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#87166C211yesmaximal, characteristic, complemented
2#82936C211yesmaximal, characteristic
3#82916C211yesmaximal, characteristic, complemented
4#78626C212yesmaximal, complemented
5C3×#1891C211yesmaximal, characteristic, complemented
6#72026C211yesmaximal, characteristic, complemented
7#70143C211yesmaximal, characteristic, complemented
8#70133C211yesmaximal, characteristic, complemented
9#69992C212yesmaximal, complemented
10#69034C212yesmaximal, complemented
11#69014C212yesmaximal, complemented
12#21483—31nomaximal, Sylow
13#5964C2211yescharacteristic
14C3×#568C2211yescharacteristic
15C3×#525C2211yescharacteristic
16C2×#948C2211yescharacteristic
17#5791C2212yes—
18C23×C3⋊C8C2211yescharacteristic, complemented
19#5750—24no—
20D3×#131—24no—
21#5745—22no—
22#5745C2212yes—
23C2×#907—22no—
24C2×#907C2212yes—
25C3×#413C2211yescharacteristic
26C3×#407C2212yescomplemented
27#5240C2211yescharacteristic
28#5234C2211yescharacteristic
29#5207C2211yescharacteristic, complemented
30#5132C2212yes—
31#5129C2212yes—
32#5122C2212yescomplemented
33#5119C2212yescomplemented
34#5053C414yes—
35#5032C414yes—
36#5011—22no—
37D4×C3⋊C8—22no—
38#5006C2212yes—
39#4999C2211yescharacteristic, complemented
40#4997C2211yescharacteristic, complemented
41#4730—24no—
42#67⋊C8—24no—
43C3×#344C2212yescomplemented
44#4559C2212yescomplemented
45#4556C2212yescomplemented
46#3573—31no—
47#3025—31no—
48#2929—32no—
49#2929—31no—
50#1891—32no—
51#1891D311yescharacteristic, complemented
52#1868—32no—
53#1868—31no—
54#1791—31no—
55#1757—31no—
56#1757—32no—
57C2×C3×#41C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59#1010—22no—
60#1009—22no—
61D3×#41—24no—
62#1007—21no—
63#1004—21no—
64#1004C2×C414yes—
65C2×#235—22no—
66C2×#235C2312yes—
67C3×#131—24no—
68C2×C3×M16—22no—
69C2×C3×M16C2312yes—
70C2×C3×M16C2312yes—
71C3×#123C2×C412yes—
72C3×C22⋊Q8—21no—
73C3×#121—21no—
74C3×#120C2×C412yes—
75C3×C4×D4—22no—
76C3×#117C2311yescharacteristic
77#953C2×C414yes—
78#948—22no—
79#948—22no—
80#930—22no—
81#929C2×C414yes—
82C4×#67—24no—
83#926C2312yes—
84#924—22no—
85C2×#208D412yescomplemented
86C2×#208C2311yescharacteristic
87C2×#208C2311yescharacteristic
88C22×C3⋊C8—22no—
89C22×C3⋊C8—22no—
90C22×C3⋊C8—21no—
91C22×C3⋊C8—21no—
92C22×C3⋊C8D412yescomplemented
93C22×C3⋊C8C2311yescharacteristic
94C22×C3⋊C8C2311yescharacteristic
95#907—24no—
96#907—24no—
97#907—22no—
98#907—24no—
99#907—22no—
100#907—22no—
101#907—22no—
102D3×M16—28no—
103D3×M16—24no—
104#901—28no—
105#901—24no—
106C2×ΓL(1,𝔽25)—24no—
107C2×ΓL(1,𝔽25)D414yescomplemented
108C2×D3×C8—24no—
109C2×D3×C8D414yescomplemented
110C3×#101—21no—
111C3×#100—21no—
112C3×C22⋊C8—24no—
113#834C2×C414yes—
114#830—22no—
115#830—22no—
116#824—22no—
117#823—22no—
118(C2×D3)⋊C8—24no—
119#815—22no—
120Dic3⋊C8—24no—
121C8×Dic3—22no—
122#806C2×C418yes—
123C4⋊(C3⋊C8)—22no—
124#803—21no—
125C4×C3⋊C8—21no—
126#585—31no—
127#568—32no—
128#568—62no—
129#568—64no—
130#568C2×D311yescharacteristic
131C2×#131—31no—
132C2×#131—32no—
133C2×#131—62no—
134C2×#131—64no—
135C23×C8—31no—
136#525—32no—
137#525C2×D311yescharacteristic
138#458—34no—
139B(4,𝔽2)—34no—
140#435—64no—
141#435—62no—
142C8×D4—64no—
143C8×D4—62no—
144#424—32no—
145#424—31no—
146#413—32no—
147#413C2×D311yescharacteristic
148#410—31no—
149#410—32no—
150#408—31no—
151#408—32no—
152#407—31no—
153#407C2×D312yescomplemented
154#405—32no—
155#405—31no—
156#344—32no—
157#344—32no—
158#344C2×D312yescomplemented
159C24×C3C22×C411yescharacteristic
160C3×#41—22no—
161C3×#41—22no—
162C2×C3×Q8#4111yescharacteristic
163C2×C3×D4—22no—
164C2×C3×D4C22×C412yes—
165C2×C3×D4#4111yescharacteristic
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4C22×C411yescharacteristic
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2411yescharacteristic
173C2×#67—24no—
174C2×#67—24no—
175C2×#67C22×C414yes—
176#239—24no—
177D3×Q8—24no—
178#237—24no—
179#237—28no—
180D3×D4—24no—
181D3×D4—28no—
182#235—24no—
183#235—24no—
184#235—24no—
185#235—22no—
186#235—22no—
187#235—22no—
188#235—22no—
189C2×D12—22no—
190C2×D12#4112yes—
191C2×C4×D3—24no—
192C2×C4×D3C2×D414yes—
193C2×Dic6—22no—
194C2×Dic6#4112yes—
195C3×M16—24no—
196C3×M16—22no—
197C3×M16—22no—
198C3×M16—22no—
199C3×M16—22no—
200C2×C3×C8—24no—
201C2×C3×C8—22no—
202C2×C3×C8C2×D414yes—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—21no—
205C3×C22⋊C4—22no—
206C3×C22⋊C4—22no—
207C3×C22⋊C4C22×C412yes—
208C3×C42—21no—
209#217—24no—
210#217C22×C414yes—
211(C2×D3)⋊C4—24no—
212C4⋊Dic3—22no—
213Dic3⋊C4—24no—
214C4×Dic3—22no—
215#208—22no—
216#208—24no—
217#208—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—41no—
224C2×C3⋊C8—42no—
225C2×C3⋊C8—41no—
226C2×C3⋊C8C2×D412yes—
227C2×C3⋊C8C2×D411yescharacteristic
228C2×C3⋊C8C2×D411yescharacteristic
229C2×C3⋊C8C2×D412yes—
230C2×C3⋊C8C2×D412yes—
231ΓL(1,𝔽25)—28no—
232ΓL(1,𝔽25)—24no—
233ΓL(1,𝔽25)—24no—
234D3×C8—28no—
235D3×C8—24no—
236D3×C8—24no—
2372−1+4—62no—
2382−1+4—61no—
239H(5,𝔽2)—34no—
240H(5,𝔽2)—62no—
241H(5,𝔽2)—61no—
242C2×#41—32no—
243C2×#41—64no—
244C2×#41—62no—
245C2×#41C22×D311yescharacteristic
246C23×C4C22×D311yescharacteristic
247#131—24no—
248#131—68no—
249#131—62no—
250#131—62no—
251#131—62no—
252#131—62no—
253#131—64no—
254#131—62no—
255#131—62no—
256#131—64no—
257#131—64no—
258#131—64no—
259C2×M16—22no—
260C2×M16—32no—
261C2×M16—31no—
262C2×M16—34no—
263C2×M16—31no—
264C2×M16—64no—
265C2×M16—64no—
266C2×M16—68no—
267C2×M16C22×D312yes—
268C2×M16C22×D312yes—
269C22×C8—31no—
270C22×C8—32no—
271C22×C8—34no—
272C22×C8—31no—
273C22×C8—64no—
274C22×C8—62no—
275C22×C8—61no—
276C22×C8—62no—
277C22×C8—61no—
278#123—34no—
279#123C4×D312yes—
280C22⋊Q8—21no—
281C22⋊Q8—62no—
282#121—21no—
283#121—62no—
284#120—34no—
285#120C4×D312yes—
286C4×D4—22no—
287C4×D4—64no—
288#117—32no—
289#117C22×D311yescharacteristic
290C4⋊C8—62no—
291C4⋊C8—64no—
292#101—21no—
293#101—62no—
294#100—21no—
295#100—62no—
296#99—34no—
297#99—38no—
298C22⋊C8—24no—
299C22⋊C8—64no—
300C22⋊C8—62no—
301C22⋊C8—62no—
302#97—61no—
303#97—62no—
304C4×C8—61no—
305C4×C8—62no—
306C23×C3—22no—
307C23×C3—21no—
308C23×C3—42no—
309C23×C3C4×D412yes—
310C23×C3C23×C411yescharacteristic
311C22×D3—24no—
312C22×D3C4×D414yes—
313C3×Q8—22no—
314C3×D4—24no—
315C3×D4—22no—
316C3×D4—44no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—42no—
325C2×C3×C4C4×D412yes—
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D412yes—
328C2×C3×C4C2×#4111yescharacteristic
329C2×C3×C4C2×#4112yes—
330#67—28no—
331#67—24no—
332#67—24no—
333#67—48no—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3C4×D414yes—
337D12—24no—
338D12—22no—
339D12—22no—
340C4×D3—28no—
341C4×D3—24no—
342C4×D3—24no—
343Dic6—24no—
344Dic6—22no—
345Dic6—22no—
346C3×C8—24no—
347C3×C8—42no—
348C3⋊C8—22no—
349C3⋊C8—41no—
350C3⋊C8—41no—
351C3⋊C8#12011yescharacteristic
352C3⋊C8#12012yes—
353C3⋊C8#12011yescharacteristic
354C24C2×C4×D311yescharacteristic
355#41—22no—
356#41—22no—
357#41—64no—
358#41—64no—
359#41—68no—
360#41—64no—
361#41—64no—
362#41—64no—
363#41—62no—
364#41—62no—
365#41—62no—
366#41—62no—
367C2×Q8—32no—
368C2×Q8—62no—
369C2×Q8—64no—
370C2×Q8#23711yescharacteristic
371C2×D4—22no—
372C2×D4—34no—
373C2×D4—32no—
374C2×D4—62no—
375C2×D4—68no—
376C2×D4—64no—
377C2×D4—64no—
378C2×D4—64no—
379C2×D4C2×C4×D312yes—
380C2×D4#23711yescharacteristic
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—34no—
386C22×C4—64no—
387C22×C4C2×C4×D311yescharacteristic
388C22×C4D3×D412yes—
389C22×C4C23×D311yescharacteristic
390M16—24no—
391M16—22no—
392M16—22no—
393M16—22no—
394M16—22no—
395M16—64no—
396M16—64no—
397M16—68no—
398M16—64no—
399M16—62no—
400M16—62no—
401C2×C8—24no—
402C2×C8—22no—
403C2×C8—31no—
404C2×C8—32no—
405C2×C8—31no—
406C2×C8—32no—
407C2×C8—32no—
408C2×C8—64no—
409C2×C8—64no—
410C2×C8—68no—
411C2×C8—62no—
412C2×C8—62no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—62no—
416C2×C8—121no—
417C2×C8—122no—
418C2×C8—121no—
419C2×C8D3×D414yes—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—64no—
423C4⋊C4—62no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—34no—
427C22⋊C4—64no—
428C22⋊C4—64no—
429C22⋊C4C2×C4×D312yes—
430C42—21no—
431C42—62no—
432C22×C3—24no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—41no—
436C22×C3—42no—
437C22×C3—42no—
438C22×C3—41no—
439C22×C3—41no—
440C22×C3C2×C4×D411yescharacteristic
441C22×C3C2×C4×D412yes—
442C2×D3—24no—
443C2×D3—44no—
444C2×D3—84no—
445C3×C4—22no—
446C3×C4—42no—
447C3×C4—41no—
448C3×C4C2×#12011yescharacteristic
449C3×C4#52511yescharacteristic
450C3×C4#52512yes—
451Dic3—24no—
452Dic3—44no—
453C23—21no—
454C23—22no—
455C23—34no—
456C23—42no—
457C23—64no—
458C23#67⋊C412yes—
459C23C22×C4×D311yescharacteristic
460Q8—22no—
461Q8—64no—
462Q8—62no—
463Q8—62no—
464D4—24no—
465D4—22no—
466D4—44no—
467D4—64no—
468D4—68no—
469D4—64no—
470D4—64no—
471D4—62no—
472D4—62no—
473D4—128no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—34no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—68no—
484C2×C4—64no—
485C2×C4—64no—
486C2×C4—64no—
487C2×C4—64no—
488C2×C4#67⋊C412yes—
489C2×C4C2×D3×D411yescharacteristic
490C2×C4C2×D3×D412yes—
491C2×C4C2×#23711yescharacteristic
492C2×C4D3×#4112yes—
493C8—24no—
494C8—31no—
495C8—32no—
496C8—31no—
497C8—42no—
498C8—62no—
499C8—121no—
500C8—121no—
501C2×C3—22no—
502C2×C3—21no—
503C2×C3—42no—
504C2×C3—42no—
505C2×C3C4×#12011yescharacteristic
506D3—44no—
507C22—22no—
508C22—24no—
509C22—22no—
510C22—41no—
511C22—42no—
512C22—41no—
513C22—42no—
514C22—41no—
515C22—64no—
516C22—124no—
517C22—244no—
518C22C2×#67⋊C411yescharacteristic
519C22C4×D3×D412yes—
520C4—22no—
521C4—42no—
522C4—41no—
523C4—64no—
524C4—124no—
525C4D3×#12011yescharacteristic
526C4#560612yes—
527C4#565611yescharacteristic
528C3#2148311yescharacteristic, complemented, Sylow
529C2—21no—
530C2—22no—
531C2—42no—
532C2—42no—
533C2—124no—
534C2#7465911yescharacteristic
subgroupG/H
centerC4D3×#120
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1891C2
socleC2×C3C4×#120
prime radical#82936C2
cosocleC22×C3×C4C24
hypercenter#1891D3
hypocenterC3#21483
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1891

Complements

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

NH ≅ G/Ntype
1. #87166C2semidirect
3. #82916C2semidirect
4. #78626C2semidirect
5. C3×#1891C2semidirect
6. #72026C2semidirect
7. #70143C2semidirect
8. #70133C2semidirect
9. #69992C2semidirect
10. #69034C2semidirect
11. #69014C2semidirect
18. C23×C3⋊C8C22semidirect
26. C3×#407C22semidirect
29. #5207C22semidirect
32. #5122C22semidirect
33. #5119C22semidirect
39. #4999C22semidirect
40. #4997C22semidirect
43. C3×#344C22semidirect
44. #4559C22semidirect
45. #4556C22semidirect
51. #1891D3semidirect
85. C2×#208D4semidirect
92. C22×C3⋊C8D4semidirect
107. C2×ΓL(1,𝔽25)D4semidirect
109. C2×D3×C8D4semidirect
153. #407C2×D3semidirect
158. #344C2×D3semidirect
528. C3#21483semidirect

Maximal subgroups

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

subgroupindex
#871662
#829362
#829162
#786262
C3×#18912
#720262
#701432
#701332
#699922
#690342
#690142
#214833