Finite Group Databasebeta

← #10846316

Subgroups — #10846316

Order 768.

970 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#494C211yesmaximal, characteristic, complemented
2C2×#5188C211yesmaximal, characteristic
3C2×#5188C211yesmaximal, characteristic
4C2×Dic3×Q16C211yesmaximal, characteristic, complemented
5#83254C212yesmaximal
6#72071C212yesmaximal
7#72066C211yesmaximal, characteristic, complemented
8#72065C212yesmaximal, complemented
9#71781C212yesmaximal, complemented
10C8×(C2×D3)⋊C4C211yesmaximal, characteristic
11#69770C211yesmaximal, characteristic, complemented
12#25415—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C22×C3×Q16C2211yescharacteristic
16C2×C3×C4⋊Q8C2212yes—
17C2×#946C2212yes—
18C2×#946C2212yes—
19C2×Q8×Dic3C2212yes—
20C22×D3×C8C2211yescharacteristic
21C3×#494—24no—
22C2×C3×C4×C8C2211yescharacteristic
23#5229C2212yes—
24C2×#835C2212yes—
25C2×#835C2212yes—
26C2×#835C2212yes—
27C2×#835C2212yes—
28#5188—24no—
29#5188—24no—
30Dic3×Q16—24no—
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×#818C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35#4994—22no—
36#4990—22no—
37#4982—22no—
38#4976—22no—
39C4×(C2×D3)⋊C4C2211yescharacteristic
40#4928C2212yes—
41#4551C2212yes—
42#4550C2211yescharacteristic
43#4524C2211yescharacteristic
44C2×#494D311yescharacteristic, complemented
45C2×#467—31no—
46C2×#467—31no—
47C2×C4×Q16—31no—
48#2429—32no—
49#2043—32no—
50#2001—32no—
51#1969—31no—
52#1942—31no—
53#1878—32no—
54C8×C22⋊C4—31no—
55C22×C3×Q8C2312yes—
56C22×C3×Q8C2312yes—
57C22×C4×D3C2311yescharacteristic
58C2×C3×Q16—24no—
59C2×C3×Q16—22no—
60C2×C3×Q16—22no—
61C2×C3×Q16—22no—
62C2×C3×Q16—22no—
63C2×C3×Q16—24no—
64C2×C3×Q16—22no—
65C2×C3×Q16—42no—
66C2×C3×Q16D414yes—
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C3×C4⋊Q8—24no—
70C3×C4⋊Q8—42no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C42C2311yescharacteristic
74#946—24no—
75#946—24no—
76#946—42no—
77#946—42no—
78Q8×Dic3—24no—
79Q8×Dic3—42no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4D411yescharacteristic
83C2×(C2×D3)⋊C4D411yescharacteristic
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3C2312yes—
86C2×C4⋊Dic3C2312yes—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3C2311yescharacteristic
92C22×C3⋊C8C2311yescharacteristic
93C22×C3⋊C8C2311yescharacteristic
94C2×D3×C8—22no—
95C2×D3×C8—22no—
96C2×D3×C8—22no—
97C3×C4×C8—22no—
98#835—24no—
99#835—24no—
100#835—24no—
101#835—42no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—22no—
105C3⋊#95D411yescharacteristic
106C3⋊#95D411yescharacteristic
107(C2×D3)⋊C8—22no—
108#818—22no—
109#818D414yes—
110C8×Dic3—22no—
111C22×Q16C2×D311yescharacteristic
112C22×Q16C2×D312yes—
113C22×Q16C2×D311yescharacteristic
114C23×C8—31no—
115C2×C4⋊Q8C2×D312yes—
116C2×C22⋊Q8—32no—
117C2×C22⋊Q8—32no—
118C2×C4×Q8—32no—
119#494—24no—
120#467—64no—
121#467—64no—
122C4×Q16—64no—
123C2×#107—31no—
124C2×#107—31no—
125C2×#103—32no—
126C2×#103—32no—
127C2×#103—32no—
128C2×#103—32no—
129C2×C22⋊C8—31no—
130C2×C4×C8—31no—
131C2×C4×C8C2×D311yescharacteristic
132#398—62no—
133#396—62no—
134#391—32no—
135#389—32no—
136#389—62no—
137#388—62no—
138C4×C22⋊C4—31no—
139#340—32no—
140#336—31no—
141#336—31no—
142C23×D3C2×D411yescharacteristic
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8C2×D414yes—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2411yescharacteristic
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3C2×D411yescharacteristic
163C22×Dic3C2×D411yescharacteristic
164C22×Dic3C2×D411yescharacteristic
165C2×C4×D3—21no—
166C2×C4×D3—21no—
167C2×C4×D3—21no—
168C2×C4×D3—21no—
169C2×C4×D3—21no—
170C2×C4×D3—21no—
171C3×Q16—28no—
172C3×Q16—48no—
173C3×Q16—44no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8C2×D412yes—
177C2×C3×C8C2×D412yes—
178C2×C3×C8#4112yes—
179C2×C3×C8#4112yes—
180C3×C4⋊C4—44no—
181C3×C4⋊C4—44no—
182C3×C42—21no—
183C3×C42—21no—
184(C2×D3)⋊C4—22no—
185(C2×D3)⋊C4—44no—
186(C2×D3)⋊C4—44no—
187(C2×D3)⋊C4Q1614yescomplemented
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191C4⋊Dic3—44no—
192C4⋊Dic3C2×D414yes—
193Dic3⋊C4—44no—
194Dic3⋊C4—44no—
195Dic3⋊C4—44no—
196C4×Dic3—21no—
197C4×Dic3—21no—
198C4×Dic3—44no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—21no—
205D3×C8—44no—
206C22×Q8C22×D312yes—
207C22×Q8C22×D312yes—
208C23×C4—31no—
209C2×Q16—24no—
210C2×Q16—22no—
211C2×Q16—22no—
212C2×Q16—22no—
213C2×Q16—24no—
214C2×Q16—22no—
215C2×Q16—22no—
216C2×Q16—42no—
217C2×Q16#6714yes—
218C22×C8—31no—
219C22×C8—31no—
220C22×C8—62no—
221C22×C8—62no—
222C22×C8—62no—
223C22×C8C22×D311yescharacteristic
224C22×C8C22×D311yescharacteristic
225C4⋊Q8—24no—
226C4⋊Q8—42no—
227C22⋊Q8—64no—
228C22⋊Q8—64no—
229C22⋊Q8—122no—
230C22⋊Q8—122no—
231C4×Q8—64no—
232C4×Q8—122no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—32no—
236C2×C4⋊C4—32no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—62no—
241C2×C22⋊C4—31no—
242C2×C22⋊C4—31no—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—62no—
245C2×C42—31no—
246C2×C42—62no—
247C2×C42C22×D311yescharacteristic
248#107—34no—
249#107—62no—
250#103—64no—
251#103—64no—
252#103—64no—
253#103—122no—
254C22⋊C8—62no—
255C4×C8—22no—
256C4×C8—62no—
257#95—31no—
258#95—31no—
259#95—62no—
260#95—62no—
261#95—62no—
262C23×C3C22×D411yescharacteristic
263C22×D3—21no—
264C22×D3—21no—
265C22×D3—21no—
266C22×D3—21no—
267C22×D3—21no—
268C22×D3—21no—
269C22×D3C2×Q1612yes—
270C3×Q8—24no—
271C3×Q8—24no—
272C3×Q8—42no—
273C3×Q8—42no—
274C3×Q8—42no—
275C3×Q8—42no—
276C3×Q8—42no—
277C3×Q8—42no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—24no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4C2×Q1612yes—
289C2×C3×C4#13512yes—
290C2×C3×C4C22×D411yescharacteristic
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4C2×#4111yescharacteristic
293C2×C3×C4C2×#4111yescharacteristic
294C2×C3×C4H(5,𝔽2)11yescharacteristic
295C2×C3×C4H(5,𝔽2)11yescharacteristic
296C2×Dic3—21no—
297C2×Dic3—21no—
298C2×Dic3—21no—
299C2×Dic3—21no—
300C2×Dic3—24no—
301C2×Dic3—21no—
302C2×Dic3—21no—
303C2×Dic3—21no—
304C2×Dic3—21no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3C2×Q1612yes—
311C4×D3—41no—
312C4×D3—41no—
313C4×D3—41no—
314C4×D3—41no—
315C3×C8—22no—
316C3×C8#12114yes—
317C3⋊C8—41no—
318C3⋊C8—41no—
319C24—31no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8C2×#6714yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—31no—
335C22×C4—31no—
336C22×C4—31no—
337C22×C4—62no—
338C22×C4—61no—
339C22×C4—61no—
340C22×C4—61no—
341C22×C4—61no—
342C22×C4—61no—
343C22×C4—61no—
344C22×C4—62no—
345C22×C4D3×D411yescharacteristic
346C22×C4D3×D411yescharacteristic
347C22×C4C23×D311yescharacteristic
348Q16—28no—
349Q16—48no—
350Q16—44no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—61no—
354C2×C8—61no—
355C2×C8—61no—
356C2×C8—61no—
357C2×C8—61no—
358C2×C8—61no—
359C2×C8—124no—
360C2×C8D3×D412yes—
361C2×C8#23712yes—
362C2×C8#23712yes—
363C2×C8C2×#6712yes—
364C4⋊C4—34no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—62no—
368C4⋊C4—62no—
369C4⋊C4—62no—
370C4⋊C4—124no—
371C4⋊C4—124no—
372C4⋊C4—124no—
373C4⋊C4—124no—
374C22⋊C4—34no—
375C22⋊C4—62no—
376C22⋊C4—124no—
377C22⋊C4—124no—
378C42—21no—
379C42—21no—
380C42—61no—
381C42—61no—
382C42—124no—
383C22×C3#53211yescharacteristic
384C22×C3D4211yescharacteristic
385C22×C3D4211yescharacteristic
386C22×C3C22×Q1611yescharacteristic
387C22×C3C2×#13511yescharacteristic
388C22×C3#57811yescharacteristic
389C22×C3#57811yescharacteristic
390C2×D3—22no—
391C2×D3—22no—
392C2×D3—22no—
393C2×D3—41no—
394C2×D3—41no—
395C2×D3—41no—
396C2×D3—41no—
397C3×C4—22no—
398C3×C4—22no—
399C3×C4—44no—
400C3×C4—82no—
401C3×C4C2×#12111yescharacteristic
402C3×C4#54011yescharacteristic
403C3×C4#54611yescharacteristic
404C3×C4#55011yescharacteristic
405Dic3—22no—
406Dic3—41no—
407Dic3—41no—
408Dic3—44no—
409Dic3—82no—
410C23—32no—
411C23—61no—
412C23—61no—
413C23—61no—
414C23—61no—
415C23—61no—
416C23—61no—
417C23C2×D3×D411yescharacteristic
418Q8—24no—
419Q8—24no—
420Q8—42no—
421Q8—42no—
422Q8—42no—
423Q8—42no—
424Q8—42no—
425Q8—42no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—24no—
431C2×C4—32no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—61no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—64no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—121no—
449C2×C4—121no—
450C2×C4—121no—
451C2×C4—121no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4D3×Q1612yes—
456C2×C4#91912yes—
457C2×C4C2×D3×D411yescharacteristic
458C2×C4C2×#23711yescharacteristic
459C2×C4C2×#23711yescharacteristic
460C2×C4#100411yescharacteristic
461C2×C4#100411yescharacteristic
462C2×C4C22×#6711yescharacteristic
463C8—22no—
464C8—121no—
465C8—121no—
466C8#93914yes—
467C2×C3#242811yescharacteristic
468C2×C3#317011yescharacteristic
469C2×C3#317711yescharacteristic
470C2×C3#328311yescharacteristic
471C2×C3#328511yescharacteristic
472C2×C3D4×Q1611yescharacteristic
473C2×C3D4×Q1611yescharacteristic
474D3—42no—
475C22—62no—
476C22—62no—
477C22—62no—
478C22—121no—
479C22—121no—
480C22—121no—
481C22—121no—
482C22#571611yescharacteristic
483C22#571711yescharacteristic
484C22C2×D3×Q1611yescharacteristic
485C22C2×#91911yescharacteristic
486C22#576611yescharacteristic
487C22#576611yescharacteristic
488C22D4×#6711yescharacteristic
489C4—22no—
490C4—22no—
491C4—44no—
492C4—62no—
493C4—82no—
494C4—121no—
495C4—124no—
496C4—121no—
497C4—242no—
498C4#571811yescharacteristic
499C4#571911yescharacteristic
500C4C2×#93911yescharacteristic
501C4#580611yescharacteristic
502C3#2541511yescharacteristic, complemented, Sylow
503C2—122no—
504C2#8179211yescharacteristic
505C2#8179311yescharacteristic
506C2#8180611yescharacteristic
507C2#8180711yescharacteristic
508C2#8318611yescharacteristic
509C2#8388911yescharacteristic
510C2Q16×#6711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#494C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC2×#494D3
hypocenterC3#25415
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#494

Complements

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

NH ≅ G/Ntype
1. C2×C3×#494C2semidirect
4. C2×Dic3×Q16C2semidirect
7. #72066C2semidirect
8. #72065C2semidirect
9. #71781C2semidirect
11. #69770C2semidirect
44. C2×#494D3semidirect
187. (C2×D3)⋊C4Q16semidirect
502. C3#25415semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4942
C2×#51882
C2×Dic3×Q162
#832542
#720712
#720662
#720652
#717812
C8×(C2×D3)⋊C42
#697702
#254153