Finite Group Databasebeta

← #10823160

Subgroups — #10823160

Order 768.

1458 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#912C211yesmaximal, characteristic
2C2×#4839C211yesmaximal, characteristic, complemented
3C2×#4771C212yesmaximal, complemented
4#76294C212yesmaximal, complemented
5C3×#2054C211yesmaximal, characteristic, complemented
6#72040C211yesmaximal, characteristic, complemented
7#70487C211yesmaximal, characteristic, complemented
8#70217C212yesmaximal, complemented
9#69651C212yesmaximal, complemented
10D3×#340C211yesmaximal, characteristic, complemented
11#68864C211yesmaximal, characteristic, complemented
12#23543—31nomaximal, Sylow
13C22×#237C2212yes—
14C22×C3×D8C2211yescharacteristic
15C22×#214C2212yes—
16C2×#912—28no—
17C2×#912—24no—
18C22×Dic12C2211yescharacteristic
19C22×ΓL(1,𝔽25)C2211yescharacteristic
20C22×D3×C8C2211yescharacteristic
21C2×D3×C4⋊C4C2212yes—
22C2×#872C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#102C2212yes—
25C3×#390C2212yes—
26#5221C2212yes—
27C2×#832C2212yes—
28C2×#816C2212yes—
29C2×Dic3⋊C8C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4973C2212yes—
32C2×#810C2212yes—
33#4956—22no—
34#4956—22no—
35#4942—22no—
36#4942—22no—
37#4929C2212yes—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4839—24no—
40#4771—28no—
41C3×#340C2211yescharacteristic
42#4551C2211yescharacteristic
43#4529C2212yes—
44C22×#135—31no—
45C2×#424—31no—
46C2×#417—32no—
47#2311—32no—
48#2056—31no—
49#2056—31no—
50#2055—32no—
51#2055—32no—
52#2054—31no—
53#2054D311yescharacteristic, complemented
54C2×#340—31no—
55C22×C3×D4C2312yes—
56C22×#67—22no—
57C23×Dic3—22no—
58C2×#237—24no—
59C2×#237—24no—
60C2×#237—24no—
61C2×#237—24no—
62C2×#237—24no—
63C2×#237D418yes—
64C22×C4×D3—22no—
65C22×C4×D3C2311yescharacteristic
66C22×Dic6C2312yes—
67C2×C3×D8—24no—
68C2×C3×D8—22no—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C42C2311yescharacteristic
75C2×#217—22no—
76C2×#217—22no—
77C2×#214—24no—
78C2×#214—24no—
79C2×#214—24no—
80C2×(C2×D3)⋊C4D411yescharacteristic
81C2×(C2×D3)⋊C4D411yescharacteristic
82C2×C4⋊Dic3C2312yes—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4D411yescharacteristic
85C2×Dic3⋊C4D411yescharacteristic
86C2×C4×Dic3C2311yescharacteristic
87C22×C3⋊C8C2311yescharacteristic
88C22×C3⋊C8C2311yescharacteristic
89#912—232no—
90C2×Dic12—24no—
91C2×Dic12—22no—
92C2×ΓL(1,𝔽25)—24no—
93C2×ΓL(1,𝔽25)—22no—
94C2×D3×C8—24no—
95C2×D3×C8D414yes—
96D3×C4⋊C4—24no—
97D3×C4⋊C4—24no—
98#872—22no—
99#872—22no—
100C3×C4⋊C8—22no—
101C3×#102—24no—
102#832—24no—
103C3⋊#95—22no—
104C3⋊#95—22no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107#816—24no—
108Dic3⋊C8—22no—
109Dic3⋊C8—22no—
110#810—24no—
111C4⋊(C3⋊C8)—22no—
112C22×#41—32no—
113C2×#135—64no—
114C2×#135—68no—
115C22×Q16—31no—
116C22×SD16—32no—
117C22×D8C2×D311yescharacteristic
118C22×M16—31no—
119C23×C8—31no—
120C2×#117—31no—
121C22×C4⋊C4—32no—
122#424—64no—
123C2×C4⋊C8—31no—
124C2×C4⋊C8—31no—
125C2×C4⋊C8—31no—
126C2×C4⋊C8C2×D311yescharacteristic
127#417—68no—
128C2×#103—32no—
129C2×#103—32no—
130C2×#102—32no—
131C2×#102C2×D312yes—
132#391—32no—
133#391—32no—
134#390—32no—
135#390C2×D312yes—
136#386—62no—
137#386—62no—
138#380—62no—
139#380—62no—
140#340—31no—
141#340—32no—
142#340C2×D311yescharacteristic
143C24×C3—22no—
144C23×D3C2×D411yescharacteristic
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4C2×D414yes—
151C22×C3×C4—22no—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2411yescharacteristic
155C2×#67—28no—
156C2×#67—44no—
157C2×#67—44no—
158C22×Dic3—22no—
159C22×Dic3—28no—
160C22×Dic3—22no—
161C22×Dic3—44no—
162C22×Dic3—44no—
163C22×Dic3C2×D411yescharacteristic
164C22×Dic3C2×D411yescharacteristic
165C22×Dic3C2×D411yescharacteristic
166#237—216no—
167#237—216no—
168#237—48no—
169#237—48no—
170C2×C4×D3—44no—
171C2×C4×D3—82no—
172C2×C4×D3—82no—
173C2×C4×D3C2×D414yes—
174C2×C4×D3C2×D412yes—
175C2×C4×D3#4112yes—
176C2×C4×D3#4114yes—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6C2×D414yes—
183C3×D8—28no—
184C2×C3×C8—22no—
185C2×C3×C8—21no—
186C2×C3×C8—22no—
187C2×C3×C8C2×D412yes—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C22⋊C4—82no—
191C3×C22⋊C4—82no—
192C3×C42—21no—
193C3×C42—21no—
194#217—82no—
195#217—82no—
196#214—216no—
197(C2×D3)⋊C4—22no—
198(C2×D3)⋊C4—22no—
199C4⋊Dic3—22no—
200C4⋊Dic3—22no—
201Dic3⋊C4—22no—
202Dic3⋊C4—22no—
203Dic3⋊C4—44no—
204C4×Dic3—21no—
205C4×Dic3—21no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8C2×D412yes—
210Dic12—28no—
211ΓL(1,𝔽25)—44no—
212D3×C8—28no—
213C2×#41—38no—
214C2×#41—64no—
215C2×#41—64no—
216C2×#41—64no—
217C2×#41—64no—
218C2×#41—64no—
219C22×Q8—32no—
220C22×D4—62no—
221C22×D4C22×D312yes—
222C23×C4—31no—
223C23×C4—62no—
224C23×C4—62no—
225#135—632no—
226C2×Q16—64no—
227C2×Q16—62no—
228C2×SD16—64no—
229C2×SD16—64no—
230C2×SD16—64no—
231C2×D8—24no—
232C2×D8—22no—
233C2×M16—64no—
234C2×M16—62no—
235C22×C8—31no—
236C22×C8—34no—
237C22×C8—31no—
238C22×C8—64no—
239C22×C8C22×D311yescharacteristic
240C22×C8C22×D311yescharacteristic
241#117—62no—
242#117—62no—
243C2×C4⋊C4—31no—
244C2×C4⋊C4—31no—
245C2×C4⋊C4—32no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4C22×D312yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—31no—
253C2×C22⋊C4—31no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—62no—
256C2×C42—31no—
257C2×C42C22×D311yescharacteristic
258C4⋊C8—22no—
259C4⋊C8—62no—
260C4⋊C8—62no—
261C4⋊C8—62no—
262#103—64no—
263#103—64no—
264#102—24no—
265#102—64no—
266#95—62no—
267#95—62no—
268#95—62no—
269#95—62no—
270C23×C3—24no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3C22×D411yescharacteristic
278C22×D3—21no—
279C22×D3—22no—
280C22×D3—21no—
281C22×D3#12112yes—
282C22×D3#12114yes—
283C3×D4—24no—
284C3×D4—24no—
285C3×D4—42no—
286C3×D4—42no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—22no—
291C2×C3×C4—21no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4C22×D412yes—
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C4C2×#4112yes—
299#67—416no—
300C2×Dic3—21no—
301C2×Dic3—21no—
302C2×Dic3—24no—
303C2×Dic3—21no—
304C2×Dic3—22no—
305C2×Dic3—21no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—416no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3#12112yes—
313C2×Dic3#12114yes—
314C4×D3—88no—
315C4×D3#12014yes—
316C4×D3#12318yes—
317C4×D3#12414yes—
318Dic6—24no—
319Dic6—24no—
320Dic6—42no—
321Dic6—42no—
322C3×C8—22no—
323C3×C8—41no—
324C3⋊C8—22no—
325C3⋊C8—41no—
326C24—22no—
327C24—31no—
328#41—616no—
329#41—616no—
330#41—128no—
331#41—128no—
332C2×Q8—34no—
333C2×Q8—62no—
334C2×Q8—62no—
335C2×Q8—62no—
336C2×Q8—62no—
337C2×Q8—62no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—68no—
344C2×D4—124no—
345C2×D4—124no—
346C2×D4D3×D414yes—
347C22×C4—22no—
348C22×C4—31no—
349C22×C4—34no—
350C22×C4—32no—
351C22×C4—34no—
352C22×C4—32no—
353C22×C4—31no—
354C22×C4—31no—
355C22×C4—68no—
356C22×C4—62no—
357C22×C4—62no—
358C22×C4—124no—
359C22×C4—124no—
360C22×C4—124no—
361C22×C4—242no—
362C22×C4—242no—
363C22×C4D3×D411yescharacteristic
364C22×C4D3×D411yescharacteristic
365C22×C4C23×D311yescharacteristic
366Q16—68no—
367SD16—616no—
368D8—28no—
369M16—124no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—21no—
373C2×C8—32no—
374C2×C8—68no—
375C2×C8—62no—
376C2×C8—62no—
377C2×C8—61no—
378C2×C8D3×D412yes—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—62no—
382C4⋊C4—62no—
383C4⋊C4—62no—
384C4⋊C4—62no—
385C4⋊C4—124no—
386C22⋊C4—62no—
387C22⋊C4—62no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—242no—
391C22⋊C4—242no—
392C42—21no—
393C42—21no—
394C42—61no—
395C42—61no—
396C22×C3—44no—
397C22×C3—44no—
398C22×C3—44no—
399C22×C3—82no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3—82no—
403C22×C3C2×#12112yes—
404C22×C3C2×#12111yescharacteristic
405C22×C3#57611yescharacteristic
406C22×C3#57712yes—
407C22×C3#57811yescharacteristic
408C2×D3—24no—
409C2×D3—24no—
410C2×D3—22no—
411C2×D3—22no—
412C2×D3#39414yes—
413C3×C4—41no—
414C3×C4—41no—
415C3×C4—82no—
416C3×C4C2×#12011yescharacteristic
417C3×C4C2×#12312yes—
418C3×C4C2×#12411yescharacteristic
419Dic3—44no—
420Dic3—41no—
421Dic3—41no—
422Dic3—82no—
423Dic3#39414yes—
424C23—24no—
425C23—32no—
426C23—34no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—61no—
434C23—61no—
435C23—62no—
436C23C2×D3×D411yescharacteristic
437Q8—64no—
438Q8—64no—
439Q8—122no—
440Q8—122no—
441D4—24no—
442D4—24no—
443D4—42no—
444D4—42no—
445D4—1216no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—32no—
452C2×C4—34no—
453C2×C4—38no—
454C2×C4—34no—
455C2×C4—34no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4—62no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—64no—
465C2×C4—1216no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—248no—
472C2×C4C2×D3×D412yes—
473C2×C4C2×D3×D411yescharacteristic
474C2×C4D3×#4111yescharacteristic
475C2×C4D3×#4112yes—
476C8—22no—
477C8—41no—
478C8—62no—
479C8—121no—
480C2×C3—84no—
481C2×C3C2×#39411yescharacteristic
482C2×C3#304512yes—
483C2×C3#304612yes—
484C2×C3#306011yescharacteristic
485C2×C3#306111yescharacteristic
486D3—24no—
487C22—34no—
488C22—44no—
489C22—44no—
490C22—44no—
491C22—62no—
492C22—64no—
493C22—64no—
494C22—62no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22D3×#12111yescharacteristic
500C22D3×#12112yes—
501C22#575711yescharacteristic
502C22#577111yescharacteristic
503C22#577512yes—
504C4—34no—
505C4—41no—
506C4—41no—
507C4—82no—
508C4—124no—
509C4—121no—
510C4—121no—
511C4—242no—
512C4D3×#12011yescharacteristic
513C4D3×#12312yes—
514C4D3×#12411yescharacteristic
515C3#2354311yescharacteristic, complemented, Sylow
516C2—64no—
517C2—84no—
518C2D3×#39411yescharacteristic
519C2#8033512yes—
520C2#8053912yes—
521C2#8071711yescharacteristic
522C2#8086611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2054C2
socleC23×C3C22×D4
prime radicalC22×#912C2
hypercenter#2054D3
hypocenterC3#23543
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2054

Complements

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

NH ≅ G/Ntype
2. C2×#4839C2semidirect
3. C2×#4771C2semidirect
4. #76294C2semidirect
5. C3×#2054C2semidirect
6. #72040C2semidirect
7. #70487C2semidirect
8. #70217C2semidirect
9. #69651C2semidirect
10. D3×#340C2semidirect
11. #68864C2semidirect
53. #2054D3semidirect
515. C3#23543semidirect

Maximal subgroups

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

subgroupindex
C22×#9122
C2×#48392
C2×#47712
#762942
C3×#20542
#720402
#704872
#702172
#696512
D3×#3402
#688642
#235433