Finite Group Databasebeta

← #10877203

Subgroups — #10877203

Order 768.

970 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
1C2×#4975C211yesmaximal, characteristic, complemented
2#73865C211yesmaximal, characteristic
3C3×#1853C211yesmaximal, characteristic, complemented
4#71560C211yesmaximal, characteristic
5#70587C212yesmaximal, complemented
6#70563C212yesmaximal, complemented
7#69932C211yesmaximal, characteristic
8#69908C211yesmaximal, characteristic, complemented
9#69505C211yesmaximal, characteristic, complemented
10#68892C212yesmaximal, complemented
11#68811C212yesmaximal, complemented
12#27678—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C2×#930C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×Dic3⋊C4—21no—
17C22×Dic3⋊C4—21no—
18C22×C4×Dic3C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×#67⋊C4—21no—
21C2×C3×#106C2212yes—
22C2×C3×C22⋊C8C2211yescharacteristic
23C3×#380C2211yescharacteristic
24C2×#830C2211yescharacteristic
25C2×#821C2212yes—
26C2×Dic3⋊C8C2212yes—
27#4988C2211yescharacteristic
28#4975—21no—
29#4975—21no—
30#4975C2212yes—
31#4975C2212yes—
32Dic3×C4⋊C4C2211yescharacteristic
33#4972C2211yescharacteristic
34C2×#809C2212yes—
35C2×#808C2212yes—
36#4958—21no—
37#4669—21no—
38#4666—21no—
39C3×#340C2212yes—
40#4551C2212yes—
41#4550C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44C4×#121—31no—
45C2×#384—31no—
46#2100—32no—
47#2094—32no—
48#1948—32no—
49#1874—31no—
50#1874—31no—
51#1853—31no—
52#1853D311yescharacteristic, complemented
53#1802—32no—
54#1797—31no—
55C23×C3×C4—21no—
56C23×C3×C4C2311yescharacteristic
57C22×#67—21no—
58C23×Dic3—21no—
59C23×Dic3D411yescharacteristic
60C23×Dic3D411yescharacteristic
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C22×C3×C8C2312yes—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—42no—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—21no—
70#930—22no—
71#930Q814yes—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—21no—
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—42no—
78C2×C4⋊Dic3C2312yes—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×Dic3⋊C4—21no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—44no—
86C2×Dic3⋊C4—44no—
87C2×C4×Dic3—24no—
88C2×C4×Dic3—21no—
89C2×C4×Dic3D411yescharacteristic
90C2×C4×Dic3D411yescharacteristic
91C2×C4×Dic3C2311yescharacteristic
92C2×C4×Dic3C2311yescharacteristic
93C22×C3⋊C8C2312yes—
94#893—24no—
95#67⋊C4—44no—
96C3×#106—42no—
97C3×C22⋊C8—22no—
98C3×#95—21no—
99C3⋊#95—21no—
100C3⋊#95—21no—
101#830—22no—
102#821—42no—
103Dic3⋊C8—24no—
104#809—42no—
105#808—42no—
106C2×#121—31no—
107C2×C4×D4—31no—
108C2×C4×D4—61no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4—61no—
111C22×C4⋊C4—61no—
112C22×C4⋊C4C2×D311yescharacteristic
113C22×C42—31no—
114C2×#106—32no—
115C2×#106C2×D312yes—
116C2×C4⋊C8—32no—
117C2×#102—32no—
118C2×#102—32no—
119C2×C22⋊C8—31no—
120C2×C22⋊C8C2×D311yescharacteristic
121#390—31no—
122#389—31no—
123#388—61no—
124#386—61no—
125#384—32no—
126#384—32no—
127#384—61no—
128#384—61no—
129#380C2×D311yescharacteristic
130C4×C4⋊C4—31no—
131C4×C22⋊C4—61no—
132#340—32no—
133#340—32no—
134#340—32no—
135#340C2×D312yes—
136#336—32no—
137C24×C3C2×D411yescharacteristic
138C23×D3—21no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4—42no—
145C22×C3×C4—42no—
146C22×C3×C4—42no—
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×Q812yes—
149C22×C3×C4#4112yes—
150C22×C3×C4C2411yescharacteristic
151C2×#67—24no—
152C2×#67—42no—
153C2×#67—42no—
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—21no—
160C22×Dic3—22no—
161C22×Dic3—42no—
162C22×Dic3—42no—
163C22×Dic3—42no—
164C22×Dic3SD1614yes—
165C22×Dic3C2×D411yescharacteristic
166C22×Dic3C2×D411yescharacteristic
167C22×Dic3C2×D411yescharacteristic
168C22×Dic3C2×D411yescharacteristic
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—21no—
172C2×D12C2×Q812yes—
173C2×C4×D3—42no—
174C2×C4×D3—42no—
175C2×C4×D3—42no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—42no—
184C3×C22⋊C4—42no—
185(C2×D3)⋊C4—24no—
186(C2×D3)⋊C4—42no—
187(C2×D3)⋊C4—42no—
188(C2×D3)⋊C4—42no—
189C4⋊Dic3—21no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192C4⋊Dic3—42no—
193C4⋊Dic3C2×Q812yes—
194Dic3⋊C4—44no—
195Dic3⋊C4—44no—
196Dic3⋊C4—44no—
197Dic3⋊C4—42no—
198Dic3⋊C4—44no—
199C4×Dic3—22no—
200C4×Dic3—22no—
201C4×Dic3—22no—
202C4×Dic3—42no—
203C4×Dic3#4114yes—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C22×D4—31no—
208C22×D4—61no—
209C23×C4—21no—
210C23×C4—31no—
211C23×C4—31no—
212C23×C4—61no—
213C23×C4—61no—
214C23×C4C22×D311yescharacteristic
215C22×C8—32no—
216C22×C8C22×D312yes—
217#121—34no—
218#121—62no—
219C4×D4—64no—
220C4×D4—124no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—31no—
223C2×C4⋊C4—32no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—62no—
226C2×C4⋊C4—62no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4—61no—
231C2×C4⋊C4—122no—
232C2×C4⋊C4—124no—
233C2×C4⋊C4—124no—
234C2×C4⋊C4C22×D312yes—
235C2×C4⋊C4C22×D311yescharacteristic
236C2×C4⋊C4C22×D311yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—61no—
239C2×C22⋊C4—61no—
240C2×C22⋊C4—61no—
241C2×C22⋊C4—61no—
242C2×C42—31no—
243C2×C42—31no—
244C2×C42—31no—
245C2×C42—31no—
246C2×C42—64no—
247C2×C42—61no—
248#106—42no—
249#106—122no—
250C4⋊C8—64no—
251#102—122no—
252#102—122no—
253C22⋊C8—22no—
254C22⋊C8—62no—
255#95—21no—
256#95—61no—
257#95—61no—
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3C22⋊Q812yes—
263C23×C3C2×SD1612yes—
264C23×C3AGL(1,ℤ/8)12yes—
265C23×C3C22×D411yescharacteristic
266C22×D3—22no—
267C22×D3—41no—
268C22×D3—41no—
269C22×D3—41no—
270C22×D3—41no—
271C22×D3—41no—
272C22×D3—41no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—41no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—84no—
287C2×C3×C4C22⋊Q812yes—
288C2×C3×C4C22×Q811yescharacteristic
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C4C2×#4112yes—
291C2×C3×C4H(5,𝔽2)12yes—
292#67—48no—
293C2×Dic3—28no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—21no—
298C2×Dic3—21no—
299C2×Dic3—21no—
300C2×Dic3—28no—
301C2×Dic3—22no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—84no—
308C2×Dic3#12312yes—
309C2×Dic3#12312yes—
310C2×Dic3C2×SD1612yes—
311C2×Dic3C2×SD1612yes—
312C2×Dic3AGL(1,ℤ/8)12yes—
313C2×Dic3#13712yes—
314D12—24no—
315D12—42no—
316C4×D3—84no—
317C3×C8—42no—
318C3⋊C8—42no—
319C24—61no—
320C24D3×D411yescharacteristic
321C2×D4—32no—
322C2×D4—64no—
323C2×D4—61no—
324C2×D4—62no—
325C2×D4—62no—
326C2×D4—122no—
327C2×D4—122no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—31no—
334C22×C4—34no—
335C22×C4—31no—
336C22×C4—31no—
337C22×C4—31no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4—62no—
345C22×C4—62no—
346C22×C4—61no—
347C22×C4—62no—
348C22×C4—122no—
349C22×C4—122no—
350C22×C4—122no—
351C22×C4—122no—
352C22×C4—122no—
353C22×C4—122no—
354C22×C4#23512yes—
355C22×C4D3×D411yescharacteristic
356C22×C4D3×Q812yes—
357C22×C4C23×D311yescharacteristic
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—62no—
362C2×C8—62no—
363C2×C8—62no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—32no—
369C4⋊C4—42no—
370C4⋊C4—61no—
371C4⋊C4—62no—
372C4⋊C4—62no—
373C4⋊C4—122no—
374C4⋊C4—124no—
375C4⋊C4—122no—
376C4⋊C4—124no—
377C4⋊C4—124no—
378C4⋊C4—124no—
379C22⋊C4—42no—
380C22⋊C4—64no—
381C22⋊C4—122no—
382C22⋊C4—122no—
383C22⋊C4—122no—
384C42—34no—
385C42—62no—
386C42—62no—
387C42—62no—
388C42—122no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3#47614yes—
396C22×C3C2×C22⋊Q811yescharacteristic
397C22×C3C2×#12311yescharacteristic
398C22×C3#53511yescharacteristic
399C22×C3C22×SD1611yescharacteristic
400C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
401C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
402C22×C3C2×#13711yescharacteristic
403C2×D3—42no—
404C2×D3—42no—
405C2×D3—42no—
406C2×D3—81no—
407C2×D3—81no—
408C2×D3—81no—
409C2×D3—81no—
410C3×C4—22no—
411C3×C4—81no—
412C3×C4—81no—
413C3×C4—82no—
414C3×C4#55312yes—
415C3×C4#55412yes—
416Dic3—22no—
417Dic3—22no—
418Dic3—22no—
419Dic3—42no—
420Dic3—82no—
421Dic3#48114yes—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—62no—
427C23—121no—
428C23—121no—
429C23—121no—
430C23—121no—
431C23—121no—
432C23—121no—
433C23#89612yes—
434C23D3×SD1612yes—
435C23#91412yes—
436C23C2×D3×D411yescharacteristic
437D4—64no—
438D4—128no—
439D4—122no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—32no—
445C2×C4—32no—
446C2×C4—32no—
447C2×C4—32no—
448C2×C4—32no—
449C2×C4—32no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—68no—
460C2×C4—68no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—62no—
468C2×C4—84no—
469C2×C4—121no—
470C2×C4—121no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4—244no—
475C2×C4—244no—
476C2×C4#89612yes—
477C2×C4C2×#23511yescharacteristic
478C2×C4#100412yes—
479C2×C4C2×D3×Q811yescharacteristic
480C2×C4D3×#4112yes—
481C8—42no—
482C8—122no—
483C2×C3—24no—
484C2×C3#248711yescharacteristic
485C2×C3C2×#47611yescharacteristic
486C2×C3#307911yescharacteristic
487C2×C3C2×#48111yescharacteristic
488C2×C3#309211yescharacteristic
489C2×C3#320611yescharacteristic
490C2×C3#321711yescharacteristic
491D3—82no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—122no—
499C22—122no—
500C22—122no—
501C22—241no—
502C22—241no—
503C22—241no—
504C22—241no—
505C22#487014yes—
506C22C2×#89611yescharacteristic
507C22#559211yescharacteristic
508C22D3×#12311yescharacteristic
509C22C2×D3×SD1611yescharacteristic
510C22C2×#91411yescharacteristic
511C22D3×AGL(1,ℤ/8)11yescharacteristic
512C22D3×#13711yescharacteristic
513C4—22no—
514C4—34no—
515C4—62no—
516C4—62no—
517C4—62no—
518C4—82no—
519C4—81no—
520C4—81no—
521C4—122no—
522C4—242no—
523C4#554012yes—
524C4#563612yes—
525C3#2767811yescharacteristic, complemented, Sylow
526C2—24no—
527C2—242no—
528C2#7445711yescharacteristic
529C2C2×#487011yescharacteristic
530C2#7924011yescharacteristic
531C2#8003411yescharacteristic
532C2#8014111yescharacteristic
533C2D3×#48111yescharacteristic
534C2#8142711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1853C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1853D3
hypocenterC3#27678
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
1. C2×#4975C2semidirect
3. C3×#1853C2semidirect
5. #70587C2semidirect
6. #70563C2semidirect
8. #69908C2semidirect
9. #69505C2semidirect
10. #68892C2semidirect
11. #68811C2semidirect
52. #1853D3semidirect
525. C3#27678semidirect

Maximal subgroups

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

subgroupindex
C2×#49752
#738652
C3×#18532
#715602
#705872
#705632
#699322
#699082
#695052
#688922
#688112
#276783