Finite Group Databasebeta

← #10944142

Subgroups — #10944142

Order 768.

1372 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×#5516C211yesmaximal, characteristic, complemented
2#82821C211yesmaximal, characteristic
3C2×#5050C211yesmaximal, characteristic, complemented
4C2×#4798C212yesmaximal, complemented
5C2×#4738C211yesmaximal, characteristic
6C3×#1873C211yesmaximal, characteristic, complemented
7#71586C211yesmaximal, characteristic
8#70281C212yesmaximal, complemented
9#69909C211yesmaximal, characteristic, complemented
10#68909C212yesmaximal, complemented
11#68904C212yesmaximal, complemented
12#32918—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C2×#947C2211yescharacteristic
17C22×#216C2212yes—
18C2×#930C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×#204C2212yes—
21#5516—28no—
22C2×#887C2212yes—
23C2×#887C2212yes—
24C2×#877—22no—
25C2×C4×Dic6—22no—
26C2×C3×#103C2212yes—
27C2×C3×C22⋊C8C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#830C2211yescharacteristic
30C2×#816C2212yes—
31C2×Dic3⋊C8C2212yes—
32#5050—24no—
33#4994—22no—
34#4988C2211yescharacteristic
35#4975C2211yescharacteristic
36C2×#810C2212yes—
37C2×#809C2212yes—
38#4959—22no—
39#4798—28no—
40#4738—24no—
41C3×#340C2212yes—
42#4550C2212yes—
43#4529C2212yes—
44C2×#519—31no—
45C2×#464—32no—
46C2×#451—31no—
47C2×#448—31no—
48#2620—31no—
49#1880—32no—
50#1878—32no—
51#1875—31no—
52#1873D311yescharacteristic, complemented
53#1803—32no—
54#1798—31no—
55C22×C3×Q8C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C22×#67—21no—
58C23×Dic3—21no—
59C22×D12C2311yescharacteristic
60C22×Dic6—21no—
61C22×Dic6—22no—
62C22×Dic6—24no—
63C22×Dic6—24no—
64C22×Dic6D414yes—
65C22×Dic6C2312yes—
66C22×C3×C8C2312yes—
67C3×C22⋊Q8—22no—
68C3×C22⋊Q8—22no—
69C3×#117—24no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C42—21no—
76C2×#217D412yes—
77#947—24no—
78C2×#216—24no—
79C2×#216—24no—
80C2×#216—24no—
81#930—22no—
82#930—22no—
83C2×(C2×D3)⋊C4—21no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86#926—24no—
87C2×C4⋊Dic3D412yes—
88C2×C4⋊Dic3C2311yescharacteristic
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—22no—
92C2×C4×Dic3—22no—
93C2×C4×Dic3—22no—
94C2×C4×Dic3C2311yescharacteristic
95C2×C4×Dic3C2311yescharacteristic
96C22×C3⋊C8C2312yes—
97C2×#204—24no—
98C2×#204—24no—
99C2×#204—24no—
100#887—24no—
101#887—24no—
102#887—24no—
103#887D418yescomplemented
104#877—216no—
105C4×Dic6—216no—
106C3×#103—24no—
107C3×C22⋊C8—22no—
108C3⋊#95—22no—
109#830—22no—
110#816—24no—
111Dic3⋊C8—24no—
112#810—24no—
113#809—24no—
114C23×Q8—31no—
115C22×Q16—32no—
116C22×SD16—32no—
117C2×#124—31no—
118C2×C22⋊Q8C2×D311yescharacteristic
119C2×#121—31no—
120#519—68no—
121C2×C4×Q8—32no—
122C2×C4×Q8—32no—
123C2×C4×Q8—62no—
124C2×#117—31no—
125C2×#117—62no—
126C2×#117C2×D311yescharacteristic
127#464—68no—
128#451—64no—
129#448—64no—
130C2×C4⋊C8—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103C2×D312yes—
134C2×#102—32no—
135C2×C22⋊C8—31no—
136C2×C22⋊C8C2×D311yescharacteristic
137#396—62no—
138#394—62no—
139#391—31no—
140#390—31no—
141#384—31no—
142#340—32no—
143#340C2×D312yes—
144#336—32no—
145C24×C3C2×D411yescharacteristic
146C23×D3—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—21no—
149C2×C3×Q8—22no—
150C2×C3×Q8—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D412yes—
158C22×C3×C4#4112yes—
159C22×C3×C4C2411yescharacteristic
160C2×#67—42no—
161C2×#67—82no—
162C22×Dic3—24no—
163C22×Dic3—24no—
164C22×Dic3—24no—
165C22×Dic3—22no—
166C22×Dic3—21no—
167C22×Dic3C2×D412yes—
168C22×Dic3C2×D412yes—
169C2×D12—22no—
170C2×D12—21no—
171C2×D12—22no—
172C2×D12—21no—
173C2×Dic6—24no—
174C2×Dic6—24no—
175C2×Dic6—216no—
176C2×Dic6—24no—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—216no—
181C2×Dic6C2×D414yes—
182C2×Dic6C2×D414yes—
183C2×Dic6#4114yes—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4C2×D414yes—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—42no—
194C3×C42—24no—
195#217—24no—
196#216—48no—
197(C2×D3)⋊C4—42no—
198(C2×D3)⋊C4—44no—
199(C2×D3)⋊C4—82no—
200C4⋊Dic3—24no—
201C4⋊Dic3—21no—
202C4⋊Dic3—21no—
203Dic3⋊C4—28no—
204Dic3⋊C4—28no—
205Dic3⋊C4—82no—
206C4×Dic3—28no—
207C4×Dic3—22no—
208C4×Dic3—28no—
209C4×Dic3C2×D414yes—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213#204—48no—
214C22×Q8—34no—
215C22×Q8—32no—
216C22×Q8—64no—
217C22×Q8—64no—
218C22×Q8—62no—
219C22×Q8—61no—
220C22×Q8C22×D311yescharacteristic
221C22×D4—31no—
222C22×D4—61no—
223C23×C4—61no—
224C23×C4C22×D311yescharacteristic
225C2×Q16—64no—
226C2×Q16—64no—
227C2×Q16—64no—
228C2×SD16—64no—
229C2×SD16—64no—
230C2×SD16—64no—
231C22×C8—32no—
232C22×C8C22×D312yes—
233#124—64no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236#121—62no—
237#121—62no—
238C4×Q8—38no—
239C4×Q8—616no—
240C4×Q8—64no—
241C4×Q8—64no—
242C4×Q8—64no—
243#117—24no—
244#117—64no—
245#117—616no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4C22×D311yescharacteristic
253C2×C4⋊C4C22×D312yes—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—32no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—61no—
259C2×C22⋊C4—62no—
260C2×C42—21no—
261C2×C42—31no—
262C2×C42—31no—
263C2×C42—62no—
264C2×C42—62no—
265C4⋊C8—64no—
266#103—24no—
267#103—64no—
268#103—64no—
269#102—64no—
270C22⋊C8—22no—
271C22⋊C8—62no—
272#95—62no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3#12112yes—
278C23×C3C22×D411yescharacteristic
279C22×D3—41no—
280C22×D3—41no—
281C22×D3—41no—
282C22×D3—42no—
283C22×D3—42no—
284C3×Q8—42no—
285C3×Q8—42no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4#12112yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×D412yes—
301C2×C3×C4C2×#4111yescharacteristic
302C2×C3×C4C2×#4112yes—
303#67—84no—
304C2×Dic3—216no—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—41no—
314C2×Dic3—42no—
315C2×Dic3#12014yes—
316D12—42no—
317D12—42no—
318Dic6—216no—
319Dic6—24no—
320Dic6—24no—
321Dic6—24no—
322Dic6#12118yes—
323C3×C8—42no—
324C3⋊C8—42no—
325C24—61no—
326C24D3×D411yescharacteristic
327C2×Q8—21no—
328C2×Q8—22no—
329C2×Q8—21no—
330C2×Q8—22no—
331C2×Q8—34no—
332C2×Q8—34no—
333C2×Q8—34no—
334C2×Q8—616no—
335C2×Q8—64no—
336C2×Q8—64no—
337C2×Q8—616no—
338C2×Q8—64no—
339C2×Q8—62no—
340C2×Q8—62no—
341C2×Q8—62no—
342C2×D4—61no—
343C2×D4—62no—
344C2×D4—61no—
345C2×D4—62no—
346C2×D4—122no—
347C2×D4—242no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—32no—
354C22×C4—32no—
355C22×C4—64no—
356C22×C4—64no—
357C22×C4—64no—
358C22×C4—62no—
359C22×C4—61no—
360C22×C4#23512yes—
361C22×C4D3×D411yescharacteristic
362C22×C4D3×D412yes—
363C22×C4C23×D311yescharacteristic
364Q16—128no—
365SD16—128no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—62no—
370C2×C8—62no—
371C2×C8—62no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—42no—
376C4⋊C4—64no—
377C4⋊C4—68no—
378C4⋊C4—68no—
379C4⋊C4—61no—
380C4⋊C4—61no—
381C4⋊C4—242no—
382C4⋊C4D3×D414yes—
383C22⋊C4—24no—
384C22⋊C4—42no—
385C22⋊C4—64no—
386C22⋊C4—124no—
387C22⋊C4—122no—
388C22⋊C4—242no—
389C42—24no—
390C42—34no—
391C42—68no—
392C42—68no—
393C42—62no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3C2×#12011yescharacteristic
400C22×C3C2×#12111yescharacteristic
401C22×C3#52511yescharacteristic
402C22×C3#57712yes—
403C22×C3#57812yes—
404C2×D3—81no—
405C2×D3—82no—
406C2×D3—82no—
407C2×D3—81no—
408C2×D3—81no—
409C3×C4—22no—
410C3×C4—44no—
411C3×C4—81no—
412C3×C4—81no—
413C3×C4C2×#12112yes—
414C3×C4#52512yes—
415Dic3—28no—
416Dic3—24no—
417Dic3—42no—
418Dic3—81no—
419C23—21no—
420C23—22no—
421C23—22no—
422C23—21no—
423C23—121no—
424C23—121no—
425C23—122no—
426C23—122no—
427C23—121no—
428C23#88412yes—
429C23C2×D3×D411yescharacteristic
430Q8—38no—
431Q8—42no—
432Q8—42no—
433Q8—616no—
434Q8—64no—
435Q8—64no—
436Q8—64no—
437D4—122no—
438D4—122no—
439D4—244no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—34no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—616no—
454C2×C4—64no—
455C2×C4—64no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—64no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—121no—
463C2×C4—122no—
464C2×C4#88412yes—
465C2×C4C2×#23511yescharacteristic
466C2×C4C2×D3×D412yes—
467C2×C4C2×D3×D411yescharacteristic
468C2×C4D3×#4112yes—
469C8—42no—
470C8—122no—
471C2×C3—42no—
472C2×C3#284711yescharacteristic
473C2×C3#301312yes—
474C2×C3#304311yescharacteristic
475C2×C3#304411yescharacteristic
476C2×C3#312112yes—
477D3—161no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—42no—
482C22—42no—
483C22—241no—
484C22—242no—
485C22—242no—
486C22—241no—
487C22—241no—
488C22C2×#88411yescharacteristic
489C22D3×#12011yescharacteristic
490C22#565611yescharacteristic
491C22#577012yes—
492C22#577912yes—
493C4—22no—
494C4—44no—
495C4—64no—
496C4—68no—
497C4—81no—
498C4—81no—
499C4—122no—
500C4—241no—
501C4#554712yes—
502C4D3×#12112yes—
503C3#3291811yescharacteristic, complemented, Sylow
504C2—42no—
505C2—481no—
506C2#7493511yescharacteristic
507C2#7886411yescharacteristic
508C2#7897711yescharacteristic
509C2#8023212yes—
510C2#8149712yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1873C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1873D3
hypocenterC3#32918
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
1. C2×#5516C2semidirect
3. C2×#5050C2semidirect
4. C2×#4798C2semidirect
6. C3×#1873C2semidirect
8. #70281C2semidirect
9. #69909C2semidirect
10. #68909C2semidirect
11. #68904C2semidirect
52. #1873D3semidirect
103. #887D4semidirect
503. C3#32918semidirect

Maximal subgroups

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

subgroupindex
C2×#55162
#828212
C2×#50502
C2×#47982
C2×#47382
C3×#18732
#715862
#702812
#699092
#689092
#689042
#329183