Finite Group Databasebeta

← #10952014

Subgroups — #10952014

Order 768.

622 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D8×#67C211yesmaximal, characteristic
2#80753C211yesmaximal, characteristic, complemented
3C3×#2187C211yesmaximal, characteristic, complemented
4C2×#4562C211yesmaximal, characteristic, complemented
5#71487C211yesmaximal, characteristic, complemented
6#71485C211yesmaximal, characteristic, complemented
7#71475C211yesmaximal, characteristic, complemented
8Dic3×SD32C211yesmaximal, characteristic, complemented
9#71082C211yesmaximal, characteristic, complemented
10#69200C211yesmaximal, characteristic, complemented
11D3×#357C211yesmaximal, characteristic, complemented
12#69183C211yesmaximal, characteristic, complemented
13#69179C211yesmaximal, characteristic, complemented
14#67⋊C16C211yesmaximal, characteristic, complemented
15#69096C211yesmaximal, characteristic, complemented
16#33862—31nomaximal, Sylow
17C22×C3×D8C2211yescharacteristic
18D4×#67—21no—
19C2×D3×D8C2211yescharacteristic
20#5635—21no—
21C2×C3×SD32C2211yescharacteristic
22C2×C3×SD32C2211yescharacteristic, complemented
23C3×#467C2211yescharacteristic, complemented
24#5217—21no—
25#5188C2211yescharacteristic, complemented
26Dic3×Q16C2211yescharacteristic, complemented
27C2×#828C2211yescharacteristic, complemented
28#5157C2211yescharacteristic
29#5156—21no—
30#5151C2211yescharacteristic
31#5150—21no—
32Dic3×D8C2211yescharacteristic
33#5115C2211yescharacteristic
34C8×#67C2211yescharacteristic
35C2×#818C2211yescharacteristic, complemented
36C22×C3⋊C16C2211yescharacteristic, complemented
37#5044—21no—
38C2×D3×C16C2211yescharacteristic, complemented
39D3×#107C2211yescharacteristic, complemented
40#4874C2211yescharacteristic, complemented
41#4811—21no—
42#4793—21no—
43C3×#357C2211yescharacteristic
44C3×#357C2211yescharacteristic, complemented
45C3×C22⋊C16C2211yescharacteristic, complemented
46#4564C2211yescharacteristic, complemented
47#4562C2211yescharacteristic
48#4562C2211yescharacteristic
49#4562C2211yescharacteristic, complemented
50#4562C2211yescharacteristic, complemented
51#4509C2211yescharacteristic, complemented
52#4509C2211yescharacteristic, complemented
53(C2×D3)⋊C16C2211yescharacteristic, complemented
54#4505C2211yescharacteristic, complemented
55#4505C2211yescharacteristic, complemented
56Dic3⋊C16C2211yescharacteristic, complemented
57Dic3×C16C2211yescharacteristic, complemented
58#4491C2211yescharacteristic, complemented
59#4490C2211yescharacteristic, complemented
60#3338—31no—
61D4×D8—31no—
62#2240—31no—
63#2233—31no—
64#2233—31no—
65#2226—31no—
66#2217—31no—
67#2217—31no—
68#2207—31no—
69#2187—31no—
70#2187D311yescharacteristic, complemented
71C4×SD32—31no—
72D4×C16—31no—
73C2×#357—31no—
74C2×#357—31no—
75C22×C3×D4—21no—
76C22×#67—41no—
77C2×D3×D4—21no—
78C2×C3×Q16C2311yescharacteristic
79C2×C3×D8—22no—
80C2×C3×D8—21no—
81C2×C3×D8—41no—
82C2×C3×D8C2311yescharacteristic
83C2×C3×D8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C22⋊Q8—21no—
86#953—41no—
87#946—21no—
88Q8×Dic3—21no—
89#940—21no—
90#939—41no—
91#938—21no—
92#937—41no—
93D4×Dic3—21no—
94C2×#213—21no—
95C2×#213—21no—
96#930—21no—
97C4×#67D411yescharacteristic
98C2×C4⋊Dic3—21no—
99D3×D8—22no—
100D3×D8—41no—
101C2×D24C2311yescharacteristic
102C2×D3×C8C2311yescharacteristic
103#897—41no—
104D3×C4⋊C4—21no—
105#889—41no—
106#888—21no—
107#67⋊C4—41no—
108#878—41no—
109C3×SD32—22no—
110C3×SD32—41no—
111C2×C3×C16C2311yescharacteristic
112C2×C3×C16C2311yescharacteristic
113C3×#107C2311yescharacteristic
114C3×#103—21no—
115#835—21no—
116#835—21no—
117#832—21no—
118#832—21no—
119#830D411yescharacteristic
120#828—41no—
121#821—21no—
122(C2×D3)⋊C8D411yescharacteristic
123#818—21no—
124#818C2311yescharacteristic
125#818C2311yescharacteristic
126#818C2311yescharacteristic
127Dic3⋊C8D411yescharacteristic
128C8×Dic3C2311yescharacteristic
129C2×C3⋊C16—41no—
130C2×C3⋊C16C2311yescharacteristic
131C2×C3⋊C16C2311yescharacteristic
132#807—21no—
133#807—21no—
134D3×C16—41no—
135C22×D8—31no—
136C22×D8C2×D311yescharacteristic
137#554—61no—
138D42—61no—
139C2×SD32—31no—
140C2×SD32C2×D311yescharacteristic
141C2×SD32C2×D311yescharacteristic, complemented
142C22×C16—31no—
143C22×C16—31no—
144#500—31no—
145#493—31no—
146#484—61no—
147#484—61no—
148#477—61no—
149#467—31no—
150#467C2×D311yescharacteristic, complemented
151#466—31no—
152#466—31no—
153#459—61no—
154#447—61no—
155#447—61no—
156C4×Q16—31no—
157C4×D8—31no—
158C8×D4—31no—
159C2×#107—31no—
160C2×#107—31no—
161#367—31no—
162#367—31no—
163#367—31no—
164C4⋊C16—31no—
165#358—31no—
166#357—31no—
167#357—31no—
168#357—31no—
169#357—31no—
170#357—31no—
171#357—31no—
172#357—31no—
173#357C2×D311yescharacteristic
174#357C2×D311yescharacteristic, complemented
175C22⋊C16—31no—
176C22⋊C16C2×D311yescharacteristic, complemented
177C4×C16—31no—
178C24×C3—41no—
179C23×D3—41no—
180C2×C3×Q8—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—42no—
184C2×C3×D4—41no—
185C2×C3×D4—81no—
186C22×C3×C4C2×D411yescharacteristic
187C2×#67—41no—
188C2×#67—41no—
189C2×#67—41no—
190C2×#67—41no—
191C2×#67—41no—
192C2×#67—84no—
193C2×#67D811yescharacteristic
194C22×Dic3—41no—
195C22×Dic3—41no—
196D3×D4—42no—
197D3×D4—81no—
198C2×D12—21no—
199C2×C4×D3—41no—
200C2×C4×D3C2×D411yescharacteristic
201C2×Dic6—41no—
202C3×Q16—21no—
203C3×Q16—21no—
204C3×D8—21no—
205C3×D8—21no—
206C3×D8—21no—
207C3×D8—42no—
208C3×D8#4112yes—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8C2×D411yescharacteristic
212C2×C3×C8C2411yescharacteristic
213C3×C4⋊C4—21no—
214C3×C4⋊C4—41no—
215C3×C22⋊C4—41no—
216#217—41no—
217#217—41no—
218#217D811yescharacteristic
219#213—42no—
220#213—81no—
221(C2×D3)⋊C4—41no—
222(C2×D3)⋊C4—41no—
223(C2×D3)⋊C4D811yescharacteristic
224C4⋊Dic3—21no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—41no—
228C4⋊Dic3—41no—
229Dic3⋊C4—41no—
230Dic3⋊C4—41no—
231Dic3⋊C4—41no—
232Dic3⋊C4D811yescharacteristic
233C4×Dic3—41no—
234C4×Dic3C2×D411yescharacteristic
235C2×C3⋊C8C2×D411yescharacteristic
236C2×C3⋊C8C2×D411yescharacteristic
237D24—21no—
238D24—21no—
239D3×C8—21no—
240D3×C8—21no—
241C3×C16—21no—
242C3×C16—21no—
243C3⋊C16—21no—
244C3⋊C16—21no—
245C22×D4—21no—
246C22×D4—61no—
247C22×D4—121no—
248C2×Q16C22×D311yescharacteristic
249C2×D8—22no—
250C2×D8—21no—
251C2×D8—31no—
252C2×D8—41no—
253C2×D8—61no—
254C2×D8—61no—
255C2×D8—62no—
256C2×D8—121no—
257C2×D8C22×D311yescharacteristic
258C2×D8C22×D311yescharacteristic
259C22×C8—31no—
260C22×C8C22×D311yescharacteristic
261C4⋊Q8—61no—
262#127—61no—
263#125—121no—
264C22⋊Q8—21no—
265C22⋊Q8—61no—
266C22⋊Q8—121no—
267C22⋊Q8—121no—
268#121—61no—
269#121—61no—
270#121—121no—
271#120—121no—
272#120—121no—
273C4×Q8—61no—
274C4×D4—31no—
275C4×D4—61no—
276C4×D4—121no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279SD32—22no—
280SD32—41no—
281SD32—121no—
282C2×C16—31no—
283C2×C16—31no—
284C2×C16—121no—
285C2×C16—121no—
286C2×C16C22×D311yescharacteristic
287C2×C16C22×D311yescharacteristic
288#107—31no—
289#107—31no—
290#107—31no—
291#107—61no—
292#107—61no—
293#107—61no—
294#107C22×D311yescharacteristic
295C4⋊C8—31no—
296#103—21no—
297#103—61no—
298#103—61no—
299#102—61no—
300#102—61no—
301#102—61no—
302C22⋊C8—31no—
303C22⋊C8—31no—
304C4×C8—31no—
305C23×C3—41no—
306C23×C3—41no—
307C23×C3—82no—
308C23×C3—82no—
309C23×C3—161no—
310C23×C3C2×D811yescharacteristic
311C22×D3—41no—
312C22×D3—82no—
313C22×D3—82no—
314C22×D3—161no—
315C22×D3C2×D811yescharacteristic
316C3×Q8—41no—
317C3×Q8—41no—
318C3×D4—22no—
319C3×D4—41no—
320C3×D4—41no—
321C3×D4—41no—
322C3×D4—82no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4C2×D811yescharacteristic
328C2×C3×C4C22×D411yescharacteristic
329#67—82no—
330#67—82no—
331#67—82no—
332#67—82no—
333#67—161no—
334#67SD3214yescomplemented
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—82no—
339C2×Dic3—161no—
340C2×Dic3C2×D811yescharacteristic
341C2×Dic3C2×D811yescharacteristic
342C2×Dic3C2×D811yescharacteristic
343D12—41no—
344D12—41no—
345C4×D3—21no—
346C4×D3—21no—
347C4×D3—161no—
348Dic6—161no—
349C3×C8—21no—
350C3×C8C2×#4111yescharacteristic
351C3×C8H(5,𝔽2)11yescharacteristic
352C3⋊C8—21no—
353C3⋊C8—21no—
354C24—41no—
355C24—121no—
356C2×Q8—21no—
357C2×Q8—121no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—31no—
361C2×D4—42no—
362C2×D4—41no—
363C2×D4—61no—
364C2×D4—81no—
365C2×D4—121no—
366C2×D4—121no—
367C2×D4—121no—
368C2×D4—122no—
369C2×D4—121no—
370C2×D4—121no—
371C2×D4—241no—
372C2×D4—244no—
373C22×C4—31no—
374C22×C4—121no—
375C22×C4—121no—
376C22×C4—121no—
377C22×C4D3×D411yescharacteristic
378Q16—21no—
379Q16—21no—
380D8—21no—
381D8—21no—
382D8—21no—
383D8—42no—
384D8—61no—
385D8—61no—
386D8—122no—
387D8—241no—
388D8#23712yes—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—31no—
392C2×C8—31no—
393C2×C8—61no—
394C2×C8—61no—
395C2×C8D3×D411yescharacteristic
396C2×C8C23×D311yescharacteristic
397C4⋊C4—21no—
398C4⋊C4—31no—
399C4⋊C4—41no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—121no—
404C4⋊C4—121no—
405C4⋊C4—121no—
406C4⋊C4—121no—
407C4⋊C4—121no—
408C22⋊C4—31no—
409C22⋊C4—31no—
410C22⋊C4—41no—
411C22⋊C4—121no—
412C22⋊C4—121no—
413C22⋊C4—121no—
414C22⋊C4—121no—
415C42—31no—
416C42—121no—
417C16—21no—
418C16—21no—
419C16—61no—
420C16—61no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—42no—
424C22×C3—81no—
425C22×C3—81no—
426C22×C3—81no—
427C22×C3—81no—
428C22×C3—81no—
429C22×C3—164no—
430C22×C3C2×SD3212yes—
431C22×C3C22×D811yescharacteristic
432C2×D3—21no—
433C2×D3—21no—
434C2×D3—81no—
435C2×D3—81no—
436C2×D3—81no—
437C2×D3—164no—
438C2×D3C2×SD3212yes—
439C3×C4—21no—
440C3×C4—81no—
441C3×C4—81no—
442C3×C4#54611yescharacteristic
443C3×C4#57711yescharacteristic
444Dic3—21no—
445Dic3—21no—
446Dic3—81no—
447Dic3—81no—
448Dic3—81no—
449Dic3C2×SD3212yes—
450C23—31no—
451C23—41no—
452C23—41no—
453C23—82no—
454C23—82no—
455C23—121no—
456C23—161no—
457C23—242no—
458C23—242no—
459C23—481no—
460C23D3×D811yescharacteristic
461Q8—41no—
462Q8—41no—
463Q8—481no—
464D4—22no—
465D4—34no—
466D4—41no—
467D4—41no—
468D4—41no—
469D4—82no—
470D4—121no—
471D4—121no—
472D4—242no—
473D4—242no—
474D4—242no—
475D4—242no—
476D4—481no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—31no—
480C2×C4—31no—
481C2×C4—31no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—61no—
485C2×C4—61no—
486C2×C4—121no—
487C2×C4—121no—
488C2×C4—121no—
489C2×C4—242no—
490C2×C4—481no—
491C2×C4—481no—
492C2×C4D3×D811yescharacteristic
493C2×C4C2×D3×D411yescharacteristic
494C8—21no—
495C8—61no—
496C8—61no—
497C8C2×#23711yescharacteristic
498C8#100411yescharacteristic
499C2×C3—22no—
500C2×C3—82no—
501C2×C3—161no—
502C2×C3#329411yescharacteristic
503C2×C3C22×SD3211yescharacteristic
504C2×C3#341511yescharacteristic
505D3—22no—
506D3—161no—
507C22—21no—
508C22—21no—
509C22—32no—
510C22—42no—
511C22—61no—
512C22—61no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—164no—
519C22—241no—
520C22—241no—
521C22—241no—
522C22—484no—
523C22D3×SD3212yes—
524C22C2×D3×D811yescharacteristic
525C4—21no—
526C4—32no—
527C4—61no—
528C4—61no—
529C4—81no—
530C4—81no—
531C4—241no—
532C4—241no—
533C4—241no—
534C4#559111yescharacteristic
535C4#577511yescharacteristic
536C3#3386211yescharacteristic, complemented, Sylow
537C2—22no—
538C2—62no—
539C2—82no—
540C2—161no—
541C2—481no—
542C2#8001411yescharacteristic
543C2C2×D3×SD3211yescharacteristic
544C2#8203611yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2187C2
socleC22×C3C22×D8
prime radicalD8×#67C2
hypercenter#2187D3
hypocenterC3#33862
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2187

Complements

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

NH ≅ G/Ntype
2. #80753C2semidirect
3. C3×#2187C2semidirect
4. C2×#4562C2semidirect
5. #71487C2semidirect
6. #71485C2semidirect
7. #71475C2semidirect
8. Dic3×SD32C2semidirect
9. #71082C2semidirect
10. #69200C2semidirect
11. D3×#357C2semidirect
12. #69183C2semidirect
13. #69179C2semidirect
14. #67⋊C16C2semidirect
15. #69096C2semidirect
22. C2×C3×SD32C22semidirect
23. C3×#467C22semidirect
25. #5188C22semidirect
26. Dic3×Q16C22semidirect
27. C2×#828C22semidirect
35. C2×#818C22semidirect
36. C22×C3⋊C16C22semidirect
38. C2×D3×C16C22semidirect
39. D3×#107C22semidirect
40. #4874C22semidirect
44. C3×#357C22semidirect
45. C3×C22⋊C16C22semidirect
46. #4564C22semidirect
49. #4562C22semidirect
50. #4562C22semidirect
51. #4509C22semidirect
52. #4509C22semidirect
53. (C2×D3)⋊C16C22semidirect
54. #4505C22semidirect
55. #4505C22semidirect
56. Dic3⋊C16C22semidirect
57. Dic3×C16C22semidirect
58. #4491C22semidirect
59. #4490C22semidirect
70. #2187D3semidirect
141. C2×SD32C2×D3semidirect
150. #467C2×D3semidirect
174. #357C2×D3semidirect
176. C22⋊C16C2×D3semidirect
334. #67SD32semidirect
536. C3#33862semidirect

Maximal subgroups

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

subgroupindex
D8×#672
#807532
C3×#21872
C2×#45622
#714872
#714852
#714752
Dic3×SD322
#710822
#692002
D3×#3572
#691832
#691792
#67⋊C162
#690962
#338623