Finite Group Databasebeta

← #10830139

Subgroups — #10830139

Order 768.

1132 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×#5585C211yesmaximal, characteristic
2C2×#5030C211yesmaximal, characteristic, complemented
3C2×#4848C211yesmaximal, characteristic
4#74191C211yesmaximal, characteristic, complemented
5C3×#1805C211yesmaximal, characteristic, complemented
6#70473C211yesmaximal, characteristic, complemented
7#70278C212yesmaximal, complemented
8#70245C212yesmaximal, complemented
9#69553C211yesmaximal, characteristic
10#68896C212yesmaximal, complemented
11#68895C212yesmaximal, complemented
12#24202—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#947C2212yes—
15C22×#216C2211yescharacteristic
16C22×#204C2212yes—
17#5585—24no—
18#5585—24no—
19C2×#894—22no—
20C2×#893C2212yes—
21C2×#875—21no—
22C2×#874C2211yescharacteristic
23C2×C4×D12—21no—
24C2×C4×Dic6C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2212yes—
28C3×#382C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#821C2212yes—
31#5030—24no—
32#4984C2212yes—
33#4973C2212yes—
34C2×#810C2211yescharacteristic
35C2×#809C2212yes—
36C2×#809C2212yes—
37C2×#808C2211yescharacteristic
38C2×#803C2211yescharacteristic
39#4848—24no—
40#4659—21no—
41#4656—21no—
42#4653—22no—
43C3×#340C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#559—31no—
47C2×#460—31no—
48C2×#441—31no—
49#2581—31no—
50#2042—32no—
51#2015—32no—
52#1985—31no—
53#1961—31no—
54#1877—32no—
55#1864—32no—
56#1805D311yescharacteristic, complemented
57C22×C3×Q8C2311yescharacteristic
58C22×D12—21no—
59C22×D12—21no—
60C22×D12C2312yes—
61C22×C4×D3—22no—
62C22×Dic6D411yescharacteristic
63C22×Dic6C2311yescharacteristic
64C22×C3×C8C2312yes—
65C3×C4×Q8—22no—
66C3×C4×Q8D414yes—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C42—21no—
71C2×C3×C42C2311yescharacteristic
72#947—24no—
73#947—24no—
74C2×#216—22no—
75C2×#216—22no—
76C2×#216—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3D411yescharacteristic
82C2×C4⋊Dic3C2311yescharacteristic
83C2×Dic3⋊C4—21no—
84C2×Dic3⋊C4D412yes—
85C2×C4×Dic3—21no—
86C2×C4×Dic3C2312yes—
87C22×C3⋊C8C2311yescharacteristic
88C22×C3⋊C8C2311yescharacteristic
89C2×#204—24no—
90C2×#204—24no—
91C2×#204—42no—
92#894—216no—
93#893—24no—
94#893—24no—
95#875—28no—
96#874—22no—
97#874D414yes—
98C4×D12—28no—
99C4×Dic6—22no—
100C4×Dic6—22no—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C3×C4⋊C8—22no—
104C3×#103—42no—
105C3×#95—21no—
106C3×#95—21no—
107#835—22no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111#821—24no—
112#810—22no—
113#809—24no—
114#809—42no—
115#808—22no—
116#803—22no—
117C22×Q16—31no—
118C22×SD16—32no—
119#559—64no—
120#559—64no—
121C2×#127—31no—
122C2×#124—32no—
123C2×#124—61no—
124C2×#123—62no—
125C2×C4×Q8—31no—
126C2×C4×Q8—31no—
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4—32no—
129C2×C4×D4—61no—
130#460—64no—
131#441—64no—
132C2×#106—31no—
133C2×C4⋊C8C2×D311yescharacteristic
134C2×#103—31no—
135C2×#103—31no—
136C2×#103C2×D312yes—
137C2×#102—32no—
138C2×#102—32no—
139C2×#102—32no—
140C2×#97—31no—
141#400—61no—
142#395—61no—
143#391—32no—
144#389—32no—
145#387—62no—
146#382C2×D311yescharacteristic
147#340C2×D312yes—
148#336—32no—
149#336—32no—
150C23×D3—22no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8C2×D412yes—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C22×Dic3—21no—
164C22×Dic3—21no—
165C22×Dic3C2×D412yes—
166C22×Dic3C2×D412yes—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—24no—
172C2×D12—22no—
173C2×D12—24no—
174C2×D12—44no—
175C2×D12—82no—
176C2×D12C2×D414yes—
177C2×C4×D3—28no—
178C2×C4×D3—84no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—22no—
184C2×Dic6—21no—
185C2×Dic6—42no—
186C2×Dic6#4112yes—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4C2×D412yes—
195C3×C42—24no—
196C3×C42C2×D412yes—
197C3×C42C2×D412yes—
198#216—28no—
199(C2×D3)⋊C4—28no—
200(C2×D3)⋊C4—28no—
201(C2×D3)⋊C4—28no—
202C4⋊Dic3—22no—
203C4⋊Dic3—21no—
204C4⋊Dic3—21no—
205C4⋊Dic3—44no—
206Dic3⋊C4—24no—
207Dic3⋊C4—42no—
208C4×Dic3—22no—
209C4×Dic3—22no—
210C4×Dic3—42no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8#4112yes—
217#204—48no—
218C22×Q8—31no—
219C22×Q8—31no—
220C22×Q8C22×D311yescharacteristic
221C22×D4—32no—
222C22×D4—61no—
223C22×D4—61no—
224C23×C4—62no—
225C2×Q16—62no—
226C2×Q16—62no—
227C2×Q16—62no—
228C2×SD16—64no—
229C2×SD16—64no—
230C2×SD16—122no—
231C22×C8—31no—
232C22×C8—31no—
233C22×C8C22×D312yes—
234#127—34no—
235#127—62no—
236#124—68no—
237#124—64no—
238#124—64no—
239#123—616no—
240C4×Q8—22no—
241C4×Q8—62no—
242C4×Q8—62no—
243C4×Q8—62no—
244C4×Q8—62no—
245C4×Q8D1214yes—
246C4×D4—68no—
247C4×D4—64no—
248C4×D4—64no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4C22×D312yes—
256C2×C4⋊C4C22×D311yescharacteristic
257C2×C22⋊C4—62no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C42—21no—
261C2×C42—32no—
262C2×C42—61no—
263C2×C42C22×D311yescharacteristic
264#106—62no—
265C4⋊C8—22no—
266#103—42no—
267#103—62no—
268#103—62no—
269#102—64no—
270#102—64no—
271#102—122no—
272#97—62no—
273#95—21no—
274#95—21no—
275#95—62no—
276#95—62no—
277#95—62no—
278C23×C3C22×D411yescharacteristic
279C22×D3—24no—
280C22×D3—42no—
281C22×D3—42no—
282C22×D3—44no—
283C22×D3—44no—
284C3×Q8—22no—
285C3×Q8—22no—
286C3×Q8—41no—
287C3×Q8—41no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4#12012yes—
299C2×C3×C4#12012yes—
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C2×#4111yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C4H(5,𝔽2)11yescharacteristic
305C2×C3×C42−1+411yescharacteristic
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—24no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—42no—
316D12—24no—
317D12—24no—
318D12—48no—
319D12—42no—
320D12—42no—
321D12—84no—
322C4×D3—88no—
323Dic6—22no—
324Dic6—22no—
325Dic6—44no—
326Dic6—41no—
327Dic6—41no—
328C3×C8—42no—
329C3⋊C8—22no—
330C3⋊C8—41no—
331C24—62no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—32no—
338C2×Q8—62no—
339C2×Q8—61no—
340C2×Q8—61no—
341C2×Q8—61no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—122no—
345C2×Q8C2×D1212yes—
346C2×D4—34no—
347C2×D4—64no—
348C2×D4—64no—
349C2×D4—62no—
350C2×D4—62no—
351C2×D4—62no—
352C2×D4—62no—
353C2×D4—62no—
354C2×D4—124no—
355C2×D4—242no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—32no—
360C22×C4—32no—
361C22×C4—68no—
362C22×C4—61no—
363C22×C4—61no—
364C22×C4—244no—
365C22×C4D3×D411yescharacteristic
366C22×C4D3×D411yescharacteristic
367C22×C4C23×D311yescharacteristic
368Q16—68no—
369SD16—128no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—32no—
374C2×C8—61no—
375C2×C8—61no—
376C2×C8—61no—
377C2×C8—61no—
378C2×C8—61no—
379C4⋊C4—24no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—62no—
384C4⋊C4—64no—
385C4⋊C4—61no—
386C4⋊C4—61no—
387C4⋊C4—124no—
388C4⋊C4—122no—
389C4⋊C4C2×D1212yes—
390C22⋊C4—68no—
391C22⋊C4—68no—
392C22⋊C4—68no—
393C42—24no—
394C42—62no—
395C42—62no—
396C42—122no—
397C42C2×D1212yes—
398C42D3×D412yes—
399C22×C3C2×#12011yescharacteristic
400C22×C3#54811yescharacteristic
401C22×C3#55011yescharacteristic
402C22×C3#57512yes—
403C22×C3#57712yes—
404C2×D3—44no—
405C2×D3—44no—
406C2×D3—44no—
407C2×D3—84no—
408C2×D3—84no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—42no—
412C3×C4—42no—
413C3×C4—82no—
414C3×C4#52511yescharacteristic
415C3×C4#53711yescharacteristic
416C3×C4#54611yescharacteristic
417C3×C4#54711yescharacteristic
418Dic3—42no—
419Dic3—42no—
420Dic3—42no—
421Dic3—81no—
422C23—64no—
423C23—122no—
424C23—124no—
425C23—124no—
426C23—122no—
427C23C2×D3×D411yescharacteristic
428Q8—22no—
429Q8—22no—
430Q8—41no—
431Q8—41no—
432Q8—62no—
433Q8—62no—
434Q8—124no—
435Q8—121no—
436Q8—121no—
437D4—64no—
438D4—64no—
439D4—128no—
440D4—122no—
441D4—122no—
442D4—244no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—64no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—122no—
463C2×C4—248no—
464C2×C4#88212yes—
465C2×C4#88212yes—
466C2×C4C22×D1211yescharacteristic
467C2×C4C2×D3×D411yescharacteristic
468C2×C4#100411yescharacteristic
469C2×C4#100711yescharacteristic
470C2×C4D3×#4111yescharacteristic
471C2×C4D3×#4111yescharacteristic
472C8—42no—
473C8—62no—
474C8—121no—
475C2×C3#240711yescharacteristic
476C2×C3#300511yescharacteristic
477C2×C3#300911yescharacteristic
478C2×C3#337112yes—
479C2×C3#340112yes—
480D3—84no—
481C22—124no—
482C22—124no—
483C22—124no—
484C22—244no—
485C22—244no—
486C22C2×#88211yescharacteristic
487C22#559911yescharacteristic
488C22#563311yescharacteristic
489C22#577012yes—
490C22#577712yes—
491C4—22no—
492C4—22no—
493C4—42no—
494C4—42no—
495C4—82no—
496C4—122no—
497C4—122no—
498C4—122no—
499C4—241no—
500C4#551011yescharacteristic
501C4#552411yescharacteristic
502C4#561211yescharacteristic
503C4#563111yescharacteristic
504C3#2420211yescharacteristic, complemented, Sylow
505C2—244no—
506C2#7401511yescharacteristic
507C2#7883511yescharacteristic
508C2#7895911yescharacteristic
509C2#8059312yes—
510C2#8121812yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1805C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#1805D3
hypocenterC3#24202
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
2. C2×#5030C2semidirect
4. #74191C2semidirect
5. C3×#1805C2semidirect
6. #70473C2semidirect
7. #70278C2semidirect
8. #70245C2semidirect
10. #68896C2semidirect
11. #68895C2semidirect
56. #1805D3semidirect
504. C3#24202semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
C2×#50302
C2×#48482
#741912
C3×#18052
#704732
#702782
#702452
#695532
#688962
#688952
#242023