Finite Group Databasebeta

← #10876127

Subgroups — #10876127

Order 768.

912 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5229C211yesmaximal, characteristic
2#73864C211yesmaximal, characteristic
3C3×#1787C211yesmaximal, characteristic
4#72132C211yesmaximal, characteristic
5#70250C212yesmaximal, complemented
6#70241C212yesmaximal, complemented
7#70233C212yesmaximal, complemented
8#70223C212yesmaximal, complemented
9#69928C211yesmaximal, characteristic
10#69910C211yesmaximal, characteristic
11#69505C211yesmaximal, characteristic
12#27583—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×C4×Dic3C2211yescharacteristic
16C2×#877—21no—
17C2×C3×#103C2212yes—
18C2×C3×#103C2212yes—
19C2×C3×C22⋊C8C2211yescharacteristic
20C3×#380C2211yescharacteristic
21#5229—21no—
22#5229—21no—
23#5229C2212yes—
24#5229C2212yes—
25C2×#835C2212yes—
26C2×#835C2212yes—
27#5212C2211yescharacteristic
28C2×C3⋊#95—21no—
29C2×C3⋊#95—21no—
30C2×#830C2211yescharacteristic
31C2×#817C2212yes—
32#4975C2211yescharacteristic
33#4975C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36C2×#808C2212yes—
37#4942—21no—
38#4650—21no—
39#4649—21no—
40#4551C2212yes—
41#4550C2212yes—
42#4550C2212yes—
43#4529C2212yes—
44C2×#391—31no—
45#2285—31no—
46#2096—32no—
47#2072—32no—
48#1927—32no—
49#1927—32no—
50#1853—31no—
51#1853—31no—
52#1817—31no—
53#1787—31no—
54#1787D311yescharacteristic
55C22×C3×Q8—24no—
56C22×C3×Q8—21no—
57C22×C3×Q8—22no—
58C22×C3×Q8—22no—
59C22×C3×Q8—42no—
60C22×C3×Q8C2312yes—
61C23×C3×C4—21no—
62C23×C3×C4C2311yescharacteristic
63C23×Dic3—21no—
64C23×Dic3D411yescharacteristic
65C23×Dic3D411yescharacteristic
66C22×C3×C8C2312yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—21no—
70C2×#217—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3—42no—
73C2×C4⋊Dic3C2311yescharacteristic
74C2×C4⋊Dic3C2311yescharacteristic
75C2×C4⋊Dic3C2312yes—
76C2×Dic3⋊C4—21no—
77C2×Dic3⋊C4—21no—
78C2×Dic3⋊C4—21no—
79C2×Dic3⋊C4—21no—
80C2×C4×Dic3—22no—
81C2×C4×Dic3—22no—
82C2×C4×Dic3—21no—
83C2×C4×Dic3—21no—
84C2×C4×Dic3D411yescharacteristic
85C2×C4×Dic3D411yescharacteristic
86C2×C4×Dic3C2311yescharacteristic
87C2×C4×Dic3C2311yescharacteristic
88C22×C3⋊C8C2312yes—
89#877—44no—
90C3×#103—24no—
91C3×#103—42no—
92C3×C22⋊C8—22no—
93C3×#95—21no—
94#835—24no—
95#835—42no—
96C3⋊#95—21no—
97C3⋊#95—21no—
98C3⋊#95—21no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102C3⋊#95—22no—
103#830—22no—
104#817—42no—
105#808—42no—
106C23×Q8C2×D311yescharacteristic
107C2×#117—61no—
108C22×C4⋊C4—31no—
109C22×C42—31no—
110C2×#106—32no—
111C2×#106—32no—
112C2×#103—32no—
113C2×#103—32no—
114C2×#103C2×D312yes—
115C2×#103C2×D312yes—
116C2×C22⋊C8—31no—
117C2×C22⋊C8C2×D311yescharacteristic
118#391—32no—
119#391—32no—
120#391—61no—
121#391—61no—
122#384—31no—
123#384—31no—
124#382—61no—
125#380—31no—
126#380—61no—
127#380C2×D311yescharacteristic
128C4×C4⋊C4—31no—
129C4×C4⋊C4—31no—
130C4×C4⋊C4—61no—
131C2×#95—61no—
132C2×#95—61no—
133#340—32no—
134#340—32no—
135#336—32no—
136#336—32no—
137C24×C3C2×D411yescharacteristic
138C2×C3×Q8—24no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8—48no—
145C2×C3×Q8—48no—
146C2×C3×Q8—42no—
147C2×C3×Q8—42no—
148C2×C3×Q8#4114yes—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—24no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—42no—
156C22×C3×C4—42no—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4#4112yes—
159C22×C3×C4#4112yes—
160C22×C3×C4C2411yescharacteristic
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3—41no—
169C22×Dic3—42no—
170C22×Dic3—42no—
171C22×Dic3—42no—
172C22×Dic3SD1614yes—
173C22×Dic3C2×D411yescharacteristic
174C22×Dic3C2×D411yescharacteristic
175C22×Dic3C2×D411yescharacteristic
176C22×Dic3C2×D411yescharacteristic
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C22⋊C4—42no—
183#217—42no—
184C4⋊Dic3—22no—
185C4⋊Dic3—22no—
186C4⋊Dic3—22no—
187C4⋊Dic3—22no—
188C4⋊Dic3—42no—
189Dic3⋊C4—42no—
190Dic3⋊C4—42no—
191Dic3⋊C4—42no—
192Dic3⋊C4—42no—
193C4×Dic3—22no—
194C4×Dic3—22no—
195C4×Dic3—21no—
196C4×Dic3—21no—
197C4×Dic3—21no—
198C4×Dic3—21no—
199C4×Dic3—42no—
200C4×Dic3—42no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C22×Q8—24no—
205C22×Q8—21no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—42no—
209C22×Q8C22×D312yes—
210C23×C4—21no—
211C23×C4—31no—
212C23×C4—31no—
213C23×C4—61no—
214C23×C4C22×D311yescharacteristic
215C22×C8—32no—
216C22×C8C22×D312yes—
217#117—124no—
218C2×C4⋊C4—32no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—31no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4—61no—
226C2×C4⋊C4—122no—
227C2×C4⋊C4C22×D311yescharacteristic
228C2×C4⋊C4C22×D311yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—61no—
231C2×C42—31no—
232C2×C42—31no—
233C2×C42—31no—
234C2×C42—31no—
235C2×C42—62no—
236C2×C42—62no—
237C2×C42—61no—
238C2×C42—61no—
239#106—122no—
240#106—122no—
241#103—24no—
242#103—42no—
243#103—64no—
244#103—122no—
245C22⋊C8—22no—
246C22⋊C8—62no—
247#95—21no—
248#95—62no—
249#95—62no—
250#95—62no—
251#95—62no—
252#95—61no—
253#95—61no—
254#95—61no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3—21no—
259C23×C3#12312yes—
260C23×C3C2×SD1612yes—
261C23×C3#13712yes—
262C23×C3C22×D411yescharacteristic
263C3×Q8—24no—
264C3×Q8—24no—
265C3×Q8—48no—
266C3×Q8—42no—
267C3×Q8—42no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—48no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4#12312yes—
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C4C2×#4112yes—
286C2×C3×C4H(5,𝔽2)12yes—
287C2×Dic3—21no—
288C2×Dic3—21no—
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—28no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—44no—
298C2×Dic3—41no—
299C2×Dic3—42no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—41no—
304C2×Dic3—42no—
305C2×Dic3—41no—
306C2×Dic3—42no—
307C2×Dic3—84no—
308C2×Dic3C2×SD1612yes—
309C2×Dic3C2×SD1612yes—
310C2×Dic3#13712yes—
311C3×C8—42no—
312C3⋊C8—42no—
313C24D3×D411yescharacteristic
314C2×Q8—24no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—48no—
321C2×Q8—48no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8#23714yes—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—34no—
332C22×C4—31no—
333C22×C4—31no—
334C22×C4—31no—
335C22×C4—31no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—62no—
339C22×C4—62no—
340C22×C4—62no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—61no—
344C22×C4—61no—
345C22×C4—121no—
346C22×C4—122no—
347C22×C4—122no—
348C22×C4—122no—
349C22×C4#23512yes—
350C22×C4D3×D411yescharacteristic
351C22×C4#23712yes—
352C22×C4C23×D311yescharacteristic
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—62no—
357C2×C8—62no—
358C2×C8—62no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—62no—
362C4⋊C4—62no—
363C4⋊C4—62no—
364C4⋊C4—62no—
365C4⋊C4—122no—
366C4⋊C4—122no—
367C4⋊C4—122no—
368C4⋊C4—122no—
369C4⋊C4—122no—
370C22⋊C4—42no—
371C22⋊C4—122no—
372C42—62no—
373C42—62no—
374C42—61no—
375C42—61no—
376C42—61no—
377C42—61no—
378C42—122no—
379C42—122no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3#48314yes—
387C22×C3C2×#12311yescharacteristic
388C22×C3#52511yescharacteristic
389C22×C3#53511yescharacteristic
390C22×C3C22×SD1611yescharacteristic
391C22×C3C2×#13711yescharacteristic
392C22×C3C2×#13711yescharacteristic
393C22×C3#57511yescharacteristic
394C3×C4—22no—
395C3×C4—44no—
396C3×C4—81no—
397C3×C4—81no—
398C3×C4#55112yes—
399C3×C4#55112yes—
400Dic3—22no—
401Dic3—22no—
402Dic3—22no—
403Dic3—41no—
404Dic3—81no—
405Dic3—81no—
406Dic3—82no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23#88312yes—
412C23D3×SD1612yes—
413C23#91812yes—
414C23C2×D3×D411yescharacteristic
415Q8—24no—
416Q8—24no—
417Q8—48no—
418Q8—42no—
419Q8—42no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—32no—
426C2×C4—32no—
427C2×C4—32no—
428C2×C4—48no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—61no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—68no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—124no—
448C2×C4—121no—
449C2×C4—121no—
450C2×C4—122no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—122no—
454C2×C4—121no—
455C2×C4—122no—
456C2×C4—121no—
457C2×C4—244no—
458C2×C4#88312yes—
459C2×C4C2×#23511yescharacteristic
460C2×C4C2×#23712yes—
461C2×C4C2×#23711yescharacteristic
462C2×C4#100412yes—
463C8—42no—
464C8—122no—
465C2×C3—24no—
466C2×C3#248211yescharacteristic
467C2×C3C2×#48311yescharacteristic
468C2×C3#309511yescharacteristic
469C2×C3#310311yescharacteristic
470C2×C3#311311yescharacteristic
471C2×C3#320611yescharacteristic
472C2×C3#321511yescharacteristic
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22#480514yes—
480C22C2×#88311yescharacteristic
481C22#558611yescharacteristic
482C22#559211yescharacteristic
483C22C2×D3×SD1611yescharacteristic
484C22C2×#91811yescharacteristic
485C22D3×#13711yescharacteristic
486C22#577711yescharacteristic
487C4—22no—
488C4—44no—
489C4—62no—
490C4—62no—
491C4—62no—
492C4—81no—
493C4—81no—
494C4—121no—
495C4—242no—
496C4—241no—
497C4—241no—
498C4#549512yes—
499C4#558712yes—
500C3#2758311yescharacteristic, complemented, Sylow
501C2—24no—
502C2#7383211yescharacteristic
503C2C2×#480511yescharacteristic
504C2#7896511yescharacteristic
505C2#8002811yescharacteristic
506C2#8003411yescharacteristic
507C2#8017411yescharacteristic
508C2#8020211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1787C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1787D3
hypocenterC3#27583
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
5. #70250C2semidirect
6. #70241C2semidirect
7. #70233C2semidirect
8. #70223C2semidirect
500. C3#27583semidirect

Maximal subgroups

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

subgroupindex
C2×#52292
#738642
C3×#17872
#721322
#702502
#702412
#702332
#702232
#699282
#699102
#695052
#275833