Finite Group Databasebeta

← #10846307

Subgroups — #10846307

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×C4×Q16C211yesmaximal, characteristic, complemented
2C2×#5188C211yesmaximal, characteristic
3C2×#5183C211yesmaximal, characteristic, complemented
4#82496C212yesmaximal
5C2×#4887C211yesmaximal, characteristic
6#71632C212yesmaximal, complemented
7#70612C211yesmaximal, characteristic, complemented
8#70273C212yesmaximal
9#70246C212yesmaximal, complemented
10C8×(C2×D3)⋊C4C211yesmaximal, characteristic
11#69775C211yesmaximal, characteristic, complemented
12#25415—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C2×C3×C4×Q8C2212yes—
15C2×#946C2212yes—
16C2×#944C2212yes—
17C22×#216C2212yes—
18C22×Dic12C2211yescharacteristic
19C22×D3×C8C2211yescharacteristic
20C2×#897C2212yes—
21C3×C4×Q16—24no—
22C2×C3×#107C2211yescharacteristic
23C2×C3×#103C2212yes—
24C2×C3×C4×C8C2211yescharacteristic
25C2×#835C2212yes—
26#5188—24no—
27#5183—24no—
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#816C2212yes—
31C2×C8×Dic3C2211yescharacteristic
32#4994—22no—
33#4984C2212yes—
34#4981—22no—
35C2×#810C2212yes—
36#4940—22no—
37C4×(C2×D3)⋊C4C2211yescharacteristic
38#4929C2212yes—
39#4928—22no—
40#4887—24no—
41#4550C2211yescharacteristic
42#4529C2212yes—
43#4524C2211yescharacteristic
44C2×#494—31no—
45C2×#467—31no—
46C2×#467—31no—
47C2×C4×Q16D311yescharacteristic, complemented
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×C4×D3C2311yescharacteristic
57C22×Dic6C2312yes—
58C2×C3×Q16—22no—
59C2×C3×Q16—22no—
60C2×C3×Q16D414yes—
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×C4×Q8—24no—
64C3×C4×Q8—42no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C42—22no—
68C2×C3×C42C2311yescharacteristic
69#946—24no—
70#946—42no—
71#944—24no—
72#944—42no—
73C2×#216—24no—
74C2×#216—24no—
75C2×#216—42no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4D411yescharacteristic
79C2×(C2×D3)⋊C4D411yescharacteristic
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3C2312yes—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×C4×Dic3C2311yescharacteristic
87C22×C3⋊C8C2311yescharacteristic
88C22×C3⋊C8C2311yescharacteristic
89C2×Dic12—22no—
90C2×Dic12—22no—
91C2×Dic12—22no—
92C2×D3×C8—22no—
93C2×D3×C8—22no—
94C2×D3×C8—22no—
95#897—24no—
96#897—42no—
97C3×#107D414yes—
98C3×#103—24no—
99C3×C4×C8—22no—
100#835—24no—
101C3⋊#95—22no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95D411yescharacteristic
105C3⋊#95D411yescharacteristic
106(C2×D3)⋊C8—22no—
107#818—22no—
108#816—42no—
109C8×Dic3—22no—
110#810—24no—
111C22×Q16—32no—
112C22×Q16—31no—
113C22×Q16C2×D311yescharacteristic
114C23×C8—31no—
115C2×C4⋊Q8—32no—
116C2×C22⋊Q8—32no—
117C2×C22⋊Q8—32no—
118C2×C4×Q8C2×D312yes—
119#494—64no—
120#467—64no—
121#467—64no—
122C4×Q16—24no—
123C2×#107—31no—
124C2×#107C2×D311yescharacteristic
125C2×#103—32no—
126C2×#103—32no—
127C2×#103—32no—
128C2×#103C2×D312yes—
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×Q8C2×D414yes—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2411yescharacteristic
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3C2×D411yescharacteristic
157C22×Dic3C2×D411yescharacteristic
158C22×Dic3C2×D411yescharacteristic
159C2×C4×D3—21no—
160C2×C4×D3—21no—
161C2×C4×D3—21no—
162C2×C4×D3—21no—
163C2×C4×D3—21no—
164C2×C4×D3—21no—
165C2×Dic6—22no—
166C2×Dic6—22no—
167C2×Dic6—22no—
168C2×Dic6—22no—
169C2×Dic6—22no—
170C2×Dic6—22no—
171C3×Q16—28no—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8C2×D412yes—
176C2×C3×C8C2×D412yes—
177C2×C3×C8#4112yes—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—44no—
180C3×C4⋊C4C2×D414yes—
181C3×C42—21no—
182C3×C42—21no—
183C3×C42—44no—
184#216—48no—
185(C2×D3)⋊C4—22no—
186(C2×D3)⋊C4—44no—
187(C2×D3)⋊C4—44no—
188(C2×D3)⋊C4Q1614yescomplemented
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191C4⋊Dic3—44no—
192C4⋊Dic3—44no—
193Dic3⋊C4—44no—
194Dic3⋊C4—44no—
195Dic3⋊C4—44no—
196C4×Dic3—21no—
197C4×Dic3—21no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8#4112yes—
204Dic12—44no—
205D3×C8—44no—
206C22×Q8—32no—
207C22×Q8C22×D312yes—
208C23×C4—31no—
209C2×Q16—22no—
210C2×Q16—22no—
211C2×Q16—64no—
212C2×Q16—64no—
213C2×Q16—62no—
214C2×Q16—62no—
215C2×Q16—62no—
216C2×Q16—122no—
217C2×Q16D1214yes—
218C22×C8—31no—
219C22×C8—31no—
220C22×C8—62no—
221C22×C8—62no—
222C22×C8—62no—
223C22×C8C22×D311yescharacteristic
224C22×C8C22×D311yescharacteristic
225C4⋊Q8—64no—
226C4⋊Q8—122no—
227C22⋊Q8—64no—
228C22⋊Q8—64no—
229C22⋊Q8—122no—
230C22⋊Q8—122no—
231C4×Q8—24no—
232C4×Q8—42no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—32no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4C22×D312yes—
241C2×C22⋊C4—31no—
242C2×C22⋊C4—31no—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—62no—
245C2×C42—22no—
246C2×C42—31no—
247C2×C42C22×D311yescharacteristic
248#107—62no—
249#107#6714yes—
250#103—24no—
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—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4C2×Q1612yes—
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C4H(5,𝔽2)11yescharacteristic
291C2×C3×C4H(5,𝔽2)11yescharacteristic
292C2×Dic3—21no—
293C2×Dic3—21no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—21no—
298C2×Dic3—21no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3C2×Q1612yes—
306C2×Dic3#13512yes—
307C4×D3—41no—
308C4×D3—41no—
309C4×D3—41no—
310C4×D3—41no—
311Dic6—42no—
312Dic6—42no—
313Dic6—42no—
314Dic6—42no—
315C3×C8—41no—
316C3×C8#12114yes—
317C3⋊C8—22no—
318C3⋊C8—41no—
319C24—31no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—62no—
326C2×Q8—62no—
327C2×Q8—62no—
328C2×Q8—62no—
329C2×Q8—62no—
330C2×Q8—62no—
331C2×Q8C2×D1214yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—31no—
335C22×C4—31no—
336C22×C4—31no—
337C22×C4—61no—
338C22×C4—61no—
339C22×C4—61no—
340C22×C4—61no—
341C22×C4—61no—
342C22×C4—61no—
343C22×C4—62no—
344C22×C4—62no—
345C22×C4D3×D411yescharacteristic
346C22×C4D3×D411yescharacteristic
347C22×C4C23×D311yescharacteristic
348Q16—28no—
349Q16—128no—
350Q16—124no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—32no—
355C2×C8—61no—
356C2×C8—61no—
357C2×C8—61no—
358C2×C8—61no—
359C2×C8—61no—
360C2×C8—124no—
361C2×C8C2×D1212yes—
362C2×C8#23512yes—
363C2×C8C2×#6712yes—
364C4⋊C4—22no—
365C4⋊C4—44no—
366C4⋊C4—62no—
367C4⋊C4—62no—
368C4⋊C4—124no—
369C4⋊C4—124no—
370C4⋊C4—124no—
371C4⋊C4—124no—
372C4⋊C4—124no—
373C4⋊C4C2×#6714yes—
374C22⋊C4—34no—
375C22⋊C4—62no—
376C22⋊C4—124no—
377C22⋊C4—124no—
378C42—21no—
379C42—21no—
380C42—44no—
381C42—61no—
382C42—61no—
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—44no—
399C3×C4—41no—
400C3×C4—44no—
401C3×C4C2×#12111yescharacteristic
402C3×C4#54011yescharacteristic
403C3×C4#54611yescharacteristic
404C3×C4#55011yescharacteristic
405Dic3—22no—
406Dic3—22no—
407Dic3—41no—
408Dic3—82no—
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—122no—
423Q8—122no—
424Q8—122no—
425Q8—122no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—32no—
434C2×C4—32no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—121no—
449C2×C4—121no—
450C2×C4—122no—
451C2×C4—122no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4D3×Q1612yes—
457C2×C4C22×D1211yescharacteristic
458C2×C4C2×#23511yescharacteristic
459C2×C4#100411yescharacteristic
460C2×C4D3×#4111yescharacteristic
461C2×C4#100911yescharacteristic
462C2×C4C22×#6711yescharacteristic
463C8—41no—
464C8—62no—
465C8—121no—
466C8#93014yes—
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#554811yescharacteristic
483C22D4×D1211yescharacteristic
484C22C2×D3×Q1611yescharacteristic
485C22#576611yescharacteristic
486C22D3×#13511yescharacteristic
487C22#577111yescharacteristic
488C22D4×#6711yescharacteristic
489C4—22no—
490C4—41no—
491C4—44no—
492C4—44no—
493C4—62no—
494C4—62no—
495C4—121no—
496C4—242no—
497C4—242no—
498C4#555411yescharacteristic
499C4#556211yescharacteristic
500C4C2×#93011yescharacteristic
501C4#583411yescharacteristic
502C3#2541511yescharacteristic, complemented, Sylow
503C2—122no—
504C2#7972511yescharacteristic
505C2#7972811yescharacteristic
506C2#7973211yescharacteristic
507C2Q16×D1211yescharacteristic
508C2#8240911yescharacteristic
509C2Q16×#6711yescharacteristic
510C2#8392611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×Q16C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC2×C4×Q16D3
hypocenterC3#25415
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×Q16C2semidirect
3. C2×#5183C2semidirect
6. #71632C2semidirect
7. #70612C2semidirect
9. #70246C2semidirect
11. #69775C2semidirect
47. C2×C4×Q16D3semidirect
188. (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×C4×Q162
C2×#51882
C2×#51832
#824962
C2×#48872
#716322
#706122
#702732
#702462
C8×(C2×D3)⋊C42
#697752
#254153