Finite Group Databasebeta

← #10823166

Subgroups — #10823166

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×#919C211yesmaximal, characteristic
2C2×#4839C211yesmaximal, characteristic, complemented
3C2×#4803C212yesmaximal, complemented
4#76482C212yesmaximal, complemented
5C3×#2056C211yesmaximal, characteristic, complemented
6#72068C211yesmaximal, characteristic, complemented
7#70481C211yesmaximal, characteristic, complemented
8#70280C212yesmaximal, complemented
9#69680C212yesmaximal, complemented
10D3×#340C211yesmaximal, characteristic, complemented
11#68862C211yesmaximal, characteristic, complemented
12#23543—31nomaximal, Sylow
13C22×#239C2212yes—
14C22×C3×Q16C2211yescharacteristic
15C22×#215C2212yes—
16C2×#919—28no—
17C2×#919—24no—
18C22×D24C2211yescharacteristic
19C22×ΓL(1,𝔽25)C2211yescharacteristic
20C22×D3×C8C2211yescharacteristic
21C2×D3×C4⋊C4C2212yes—
22C2×#872C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2212yes—
25C3×#391C2212yes—
26#5229C2212yes—
27C2×#835C2212yes—
28C2×#821C2212yes—
29C2×Dic3⋊C8C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4988C2212yes—
32C2×#809C2212yes—
33#4939C2212yes—
34C2×C4⋊(C3⋊C8)C2211yescharacteristic
35#4839—24no—
36#4803—28no—
37#4667—22no—
38#4667—22no—
39#4665—22no—
40#4665—22no—
41C3×#340C2211yescharacteristic
42#4551C2211yescharacteristic
43#4529C2212yes—
44C22×#135—31no—
45C2×#424—31no—
46C2×#417—32no—
47#2311—32no—
48#2056—31no—
49#2056D311yescharacteristic, complemented
50#2055—32no—
51#2055—32no—
52#2054—31no—
53#2054—31no—
54C2×#340—31no—
55C22×C3×Q8C2312yes—
56C2×#239—24no—
57C2×#239—24no—
58C2×#239—24no—
59C2×#239—24no—
60C2×#239—24no—
61C2×#239D418yes—
62C22×D12—22no—
63C22×D12C2312yes—
64C22×C4×D3—22no—
65C22×C4×D3—22no—
66C22×C4×D3C2311yescharacteristic
67C2×C3×Q16—24no—
68C2×C3×Q16—22no—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C42C2311yescharacteristic
73C2×#215—24no—
74C2×#215—24no—
75C2×#215—24no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—22no—
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#919—232no—
90C2×D24—24no—
91C2×D24—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×#103—24no—
102C3×#95—22no—
103C3×#95—22no—
104#835—24no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107#821—24no—
108Dic3⋊C8—22no—
109Dic3⋊C8—22no—
110#809—24no—
111C4⋊(C3⋊C8)—22no—
112C22×#41—32no—
113C2×#135—64no—
114C2×#135—68no—
115C22×Q16C2×D311yescharacteristic
116C22×SD16—32no—
117C22×D8—31no—
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×#103C2×D312yes—
130C2×#102—32no—
131C2×#102—32no—
132#391—32no—
133#391C2×D312yes—
134#390—32no—
135#390—32no—
136#386—62no—
137#386—62no—
138#380—62no—
139#380—62no—
140#340—31no—
141#340—32no—
142#340C2×D311yescharacteristic
143C23×D3—22no—
144C23×D3C2×D411yescharacteristic
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8C2×D414yes—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2411yescharacteristic
156C22×Dic3—22no—
157C22×Dic3C2×D411yescharacteristic
158C22×Dic3C2×D411yescharacteristic
159C22×Dic3C2×D411yescharacteristic
160#239—216no—
161#239—216no—
162#239—48no—
163#239—48no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—22no—
168C2×D12—28no—
169C2×D12—22no—
170C2×D12—44no—
171C2×D12—44no—
172C2×D12C2×D414yes—
173C2×C4×D3—28no—
174C2×C4×D3—44no—
175C2×C4×D3—44no—
176C2×C4×D3—44no—
177C2×C4×D3—82no—
178C2×C4×D3—82no—
179C2×C4×D3C2×D414yes—
180C2×C4×D3C2×D412yes—
181C2×C4×D3#4112yes—
182C2×C4×D3#4114yes—
183C3×Q16—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×C42—21no—
191C3×C42—21no—
192#215—216no—
193(C2×D3)⋊C4—22no—
194(C2×D3)⋊C4—22no—
195(C2×D3)⋊C4—82no—
196(C2×D3)⋊C4—82no—
197(C2×D3)⋊C4—82no—
198(C2×D3)⋊C4—82no—
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—
210D24—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×Q8C22×D312yes—
220C22×D4—32no—
221C22×D4—62no—
222C23×C4—31no—
223C23×C4—62no—
224C23×C4—62no—
225#135—632no—
226C2×Q16—24no—
227C2×Q16—22no—
228C2×SD16—64no—
229C2×SD16—64no—
230C2×SD16—64no—
231C2×D8—64no—
232C2×D8—62no—
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—31no—
251C2×C22⋊C4—31no—
252C2×C22⋊C4—62no—
253C2×C22⋊C4—62no—
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—24no—
263#103—64no—
264#102—64no—
265#102—64no—
266#95—22no—
267#95—22no—
268#95—62no—
269#95—62no—
270C23×C3C22×D411yescharacteristic
271C22×D3—21no—
272C22×D3—22no—
273C22×D3—21no—
274C22×D3—24no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—42no—
279C22×D3—42no—
280C22×D3—42no—
281C22×D3#12112yes—
282C22×D3#12114yes—
283C3×Q8—24no—
284C3×Q8—24no—
285C3×Q8—42no—
286C3×Q8—42no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—24no—
290C2×C3×C4—21no—
291C2×C3×C4—22no—
292C2×C3×C4—21no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4C22×D412yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C2×#4111yescharacteristic
301C2×C3×C4C2×#4112yes—
302C2×Dic3—21no—
303C2×Dic3—21no—
304C2×Dic3—21no—
305C2×Dic3—22no—
306C2×Dic3—21no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3#12112yes—
311C2×Dic3#12114yes—
312D12—24no—
313D12—24no—
314D12—416no—
315D12—42no—
316D12—42no—
317C4×D3—416no—
318C4×D3—88no—
319C4×D3#12014yes—
320C4×D3#12318yes—
321C4×D3#12414yes—
322C3×C8—22no—
323C3×C8—41no—
324C3⋊C8—22no—
325C3⋊C8—41no—
326C24—31no—
327C24—62no—
328#41—616no—
329#41—616no—
330#41—128no—
331#41—128no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8D3×D414yes—
338C2×D4—34no—
339C2×D4—68no—
340C2×D4—62no—
341C2×D4—62no—
342C2×D4—62no—
343C2×D4—62no—
344C2×D4—62no—
345C2×D4—124no—
346C2×D4—124no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—31no—
350C22×C4—34no—
351C22×C4—32no—
352C22×C4—34no—
353C22×C4—32no—
354C22×C4—31no—
355C22×C4—31no—
356C22×C4—68no—
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—28no—
367SD16—616no—
368D8—68no—
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—242no—
389C22⋊C4—242no—
390C22⋊C4—242no—
391C22⋊C4—242no—
392C42—21no—
393C42—21no—
394C42—61no—
395C42—61no—
396C22×C3C2×#12112yes—
397C22×C3C2×#12111yescharacteristic
398C22×C3#57611yescharacteristic
399C22×C3#57712yes—
400C22×C3#57811yescharacteristic
401C2×D3—24no—
402C2×D3—24no—
403C2×D3—22no—
404C2×D3—22no—
405C2×D3—44no—
406C2×D3—44no—
407C2×D3—44no—
408C2×D3—82no—
409C2×D3—82no—
410C2×D3—82no—
411C2×D3—82no—
412C2×D3#39414yes—
413C3×C4—44no—
414C3×C4—41no—
415C3×C4—41no—
416C3×C4—82no—
417C3×C4C2×#12011yescharacteristic
418C3×C4C2×#12312yes—
419C3×C4C2×#12411yescharacteristic
420Dic3—41no—
421Dic3—41no—
422Dic3—82no—
423Dic3#39414yes—
424C23—32no—
425C23—34no—
426C23—61no—
427C23—61no—
428C23—62no—
429C23—64no—
430C23—122no—
431C23—122no—
432C23—122no—
433C23—122no—
434C23—122no—
435C23—122no—
436C23C2×D3×D411yescharacteristic
437Q8—24no—
438Q8—24no—
439Q8—42no—
440Q8—42no—
441D4—64no—
442D4—64no—
443D4—1216no—
444D4—122no—
445D4—122no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—32no—
453C2×C4—34no—
454C2×C4—38no—
455C2×C4—34no—
456C2×C4—34no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—62no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—1216no—
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×C3C2×#39411yescharacteristic
481C2×C3#304512yes—
482C2×C3#304612yes—
483C2×C3#306011yescharacteristic
484C2×C3#306111yescharacteristic
485D3—24no—
486D3—84no—
487C22—34no—
488C22—62no—
489C22—64no—
490C22—62no—
491C22—64no—
492C22—124no—
493C22—124no—
494C22—124no—
495C22—242no—
496C22—242no—
497C22—242no—
498C22—242no—
499C22D3×#12111yescharacteristic
500C22D3×#12112yes—
501C22#576611yescharacteristic
502C22#576911yescharacteristic
503C22#577812yes—
504C4—34no—
505C4—41no—
506C4—44no—
507C4—41no—
508C4—82no—
509C4—121no—
510C4—121no—
511C4—242no—
512C4D3×#12011yescharacteristic
513C4D3×#12312yes—
514C4D3×#12411yescharacteristic
515C3#2354311yescharacteristic, complemented, Sylow
516C2—64no—
517C2—244no—
518C2D3×#39411yescharacteristic
519C2#8047912yes—
520C2#8060212yes—
521C2#8076111yescharacteristic
522C2#8090611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2056C2
socleC23×C3C22×D4
prime radicalC22×#919C2
hypercenter#2056D3
hypocenterC3#23543
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2056

Complements

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

NH ≅ G/Ntype
2. C2×#4839C2semidirect
3. C2×#4803C2semidirect
4. #76482C2semidirect
5. C3×#2056C2semidirect
6. #72068C2semidirect
7. #70481C2semidirect
8. #70280C2semidirect
9. #69680C2semidirect
10. D3×#340C2semidirect
11. #68862C2semidirect
49. #2056D3semidirect
515. C3#23543semidirect

Maximal subgroups

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

subgroupindex
C22×#9192
C2×#48392
C2×#48032
#764822
C3×#20562
#720682
#704812
#702802
#696802
D3×#3402
#688622
#235433