Finite Group Databasebeta

← #10764124

Subgroups — #10764124

Order 768.

1112 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86214C211yesmaximal, characteristic
2#79340C212yesmaximal, complemented
3C3×#1558C211yesmaximal, characteristic, complemented
4C2×#4546C211yesmaximal, characteristic, complemented
5#70816C212yesmaximal, complemented
6#68489C211yesmaximal, characteristic, complemented
7#68487C211yesmaximal, characteristic, complemented
8#68485C212yesmaximal, complemented
9#68483C212yesmaximal, complemented
10#68482C211yesmaximal, characteristic, complemented
11D3×#333C211yesmaximal, characteristic, complemented
12#17779—31nomaximal, Sylow
13C2×#1007C412yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15#5816C412yes—
16#5727—22no—
17#5726—22no—
18#5725C2211yescharacteristic
19#5725C2211yescharacteristic
20#5724C412yes—
21#5724C412yes—
22D3×C4⋊Q8C2211yescharacteristic
23D3×C4⋊Q8C2211yescharacteristic
24#5507—22no—
25#5507—22no—
26#5477C2211yescharacteristic
27C4×#235C2211yescharacteristic
28C3×#432C2212yes—
29C2×C4⋊(C3⋊C8)C2212yes—
30#4837C2212yes—
31#4835C2212yes—
32#4833C2212yes—
33D3×C4⋊C8C2212yes—
34#4730—22no—
35#67⋊C8—22no—
36#4706—22no—
37#4701—22no—
38C3×#333C2211yescharacteristic
39C3×#333C2211yescharacteristic
40C3×#333C2212yes—
41#4546C2211yescharacteristic
42#4546C2212yes—
43#4546C2211yescharacteristic
44#4485C2212yes—
45#4485C2212yes—
46#4481C2212yes—
47#4481C2212yes—
48#3499—31no—
49#2990—32no—
50#1560—31no—
51#1558—31no—
52#1558D311yescharacteristic, complemented
53#1551—32no—
54#1551—32no—
55#1548—32no—
56#1543—31no—
57C2×#333—31no—
58C2×#333—31no—
59C22×C3×Q8C2×C412yes—
60#1007—48no—
61C2×#239—22no—
62C2×#239C2×C412yes—
63C2×#239C2×C412yes—
64C2×D3×Q8—22no—
65C2×D3×Q8C2×C412yes—
66C2×D3×Q8C2×C412yes—
67C2×#235—22no—
68C2×#235—22no—
69C2×#235C2×C411yescharacteristic
70C2×#235C2×C412yes—
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8C2×C412yes—
73C3×C4⋊Q8C2311yescharacteristic
74C3×C4⋊Q8C2311yescharacteristic
75C2×C3×C4⋊C4—22no—
76C2×C3×C42C2311yescharacteristic
77#947C2×C412yes—
78#947C2×C412yes—
79#946—24no—
80#944C2×C412yes—
81#944C2×C412yes—
82#930—22no—
83#929—24no—
84C4×#67D412yes—
85C4×#67D412yes—
86#926C2×C411yescharacteristic
87C22×C3⋊C8—22no—
88C2×ΓL(1,𝔽25)—22no—
89C2×D3×C8—22no—
90#896—24no—
91#896—24no—
92#895—22no—
93#895—22no—
94#894—24no—
95#894—24no—
96#893—22no—
97#893—22no—
98D3×C4⋊C4—22no—
99D3×C4⋊C4—22no—
100#888—22no—
101#888—22no—
102#875C2×C412yes—
103#874C2311yescharacteristic
104C4×D12—22no—
105C4×D12D412yes—
106#872C2×C411yescharacteristic
107#872C2×C411yescharacteristic
108C42×D3C2311yescharacteristic
109C42×D3C2311yescharacteristic
110#869C2311yescharacteristic
111C4×Dic6D412yes—
112C3×C4⋊C8C2312yes—
113C3×C4⋊C8C2312yes—
114C3×C22⋊C8—22no—
115C3×#97—22no—
116(C2×D3)⋊C8—22no—
117(C2×D3)⋊C8—22no—
118#815—22no—
119Dic3⋊C8—22no—
120Dic3⋊C8—22no—
121C8×Dic3—22no—
122C4⋊(C3⋊C8)—22no—
123C4⋊(C3⋊C8)C2312yes—
124C4⋊(C3⋊C8)C2312yes—
125C2×2−1+4—32no—
126D4×Q8—62no—
127#547—62no—
128#547—62no—
129#547—62no—
130#536—32no—
131#536—32no—
132#536—32no—
133#532—31no—
134#532—31no—
135#532—31no—
136C2×C4⋊Q8—31no—
137C2×C4⋊Q8—31no—
138C2×C4⋊Q8C2×D311yescharacteristic
139C4×#41—31no—
140#446—62no—
141#435—62no—
142#435—62no—
143C8×D4—62no—
144#432—32no—
145#432C2×D312yes—
146#431—32no—
147#423—32no—
148C2×C4⋊C8—32no—
149C2×C4⋊C8—32no—
150#333—31no—
151#333—32no—
152#333—31no—
153#333—32no—
154#333—32no—
155#333C2×D311yescharacteristic
156#333C2×D311yescharacteristic
157#333C2×D312yes—
158#331—32no—
159#331—32no—
160C2×C3×Q8—22no—
161C2×C3×Q8—44no—
162C2×C3×Q8C22×C412yes—
163C2×C3×Q8C22×C412yes—
164C22×C3×C4—22no—
165C22×C3×C4C22×C411yescharacteristic
166C22×C3×C4C2×D412yes—
167C2×#67—22no—
168C2×#67C22⋊C411yescharacteristic
169C2×#67C22⋊C411yescharacteristic
170#239—44no—
171#239—44no—
172#239—48no—
173D3×Q8—48no—
174D3×Q8—44no—
175D3×Q8—44no—
176#235—24no—
177#235—48no—
178#235—48no—
179#235—44no—
180C2×D12—22no—
181C2×D12—22no—
182C2×D12—22no—
183C2×D12C22⋊C411yescharacteristic
184C2×D12C22×C412yes—
185C2×C4×D3—22no—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3C22×C411yescharacteristic
191C2×C4×D3C22×C411yescharacteristic
192C2×C4×D3C2×D412yes—
193C2×C4×D3C2×D412yes—
194C2×Dic6—22no—
195C2×Dic6—22no—
196C2×Dic6C22⋊C411yescharacteristic
197C2×Dic6C22×C412yes—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C42C22×C411yescharacteristic
204C3×C42C2×D412yes—
205C3×C42C2411yescharacteristic
206#217C22⋊C412yes—
207(C2×D3)⋊C4—24no—
208(C2×D3)⋊C4—24no—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4C22⋊C412yes—
211(C2×D3)⋊C4C22⋊C412yes—
212C4⋊Dic3—22no—
213C4⋊Dic3C22⋊C412yes—
214Dic3⋊C4—24no—
215Dic3⋊C4—24no—
216Dic3⋊C4C22⋊C412yes—
217Dic3⋊C4C22⋊C412yes—
218C4×Dic3C22×C411yescharacteristic
219C4×Dic3C22×C411yescharacteristic
220C4×Dic3C2×D412yes—
221C4×Dic3C2×D412yes—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—82no—
225ΓL(1,𝔽25)—82no—
226D3×C8—82no—
2272−1+4—128no—
228C2×#41—31no—
229C2×#41—32no—
230C2×#41—32no—
231C2×#41—32no—
232C2×#41—62no—
233C2×#41—62no—
234C2×#41—62no—
235C22×Q8—32no—
236C22×Q8—32no—
237C22×Q8—62no—
238C22×Q8C4×D312yes—
239C2×M16—62no—
240C22×C8—62no—
241C22×C8—62no—
242C4⋊Q8—22no—
243C4⋊Q8—32no—
244C4⋊Q8—32no—
245C4⋊Q8—31no—
246C4⋊Q8—62no—
247C4⋊Q8—62no—
248C4⋊Q8C4×D312yes—
249C4⋊Q8C22×D311yescharacteristic
250C4⋊Q8C22×D311yescharacteristic
251#127—31no—
252#124—32no—
253#124—32no—
254#124—32no—
255#123—64no—
256#123—64no—
257#123—64no—
258C22⋊Q8—64no—
259C22⋊Q8—64no—
260C22⋊Q8—64no—
261#121—62no—
262#121—62no—
263#121—62no—
264C4×Q8—32no—
265C4×D4—32no—
266C4×D4—32no—
267C4×D4—32no—
268C4×D4—62no—
269C4×D4—62no—
270C4×D4—62no—
271#117—31no—
272#117—31no—
273#117—31no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4—62no—
277C2×C42—31no—
278C2×C42—31no—
279C2×C42C22×D311yescharacteristic
280C4⋊C8—32no—
281C4⋊C8—32no—
282C4⋊C8—62no—
283C4⋊C8—62no—
284C4⋊C8—62no—
285C4⋊C8C22×D312yes—
286C4⋊C8C22×D312yes—
287C22⋊C8—22no—
288C22⋊C8—62no—
289C22⋊C8—62no—
290#97—22no—
291#97—62no—
292C4×C8—62no—
293C23×C3C2×C22⋊C411yescharacteristic
294C22×D3—22no—
295C22×D3C2×C22⋊C411yescharacteristic
296C22×D3C2×C22⋊C411yescharacteristic
297C3×Q8—42no—
298C3×Q8—42no—
299C3×Q8—44no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—44no—
306C2×C3×C4C2×C22⋊C412yes—
307C2×C3×C4C2×C22⋊C411yescharacteristic
308C2×C3×C4C23×C411yescharacteristic
309C2×C3×C4C22×D412yes—
310#67—22no—
311#67—44no—
312#67#10014yes—
313C2×Dic3—22no—
314C2×Dic3C2×C22⋊C412yes—
315C2×Dic3C2×C22⋊C412yes—
316C2×Dic3C2×C22⋊C411yescharacteristic
317C2×Dic3C2×C22⋊C411yescharacteristic
318D12—22no—
319D12—42no—
320D12—44no—
321D12—44no—
322D12—42no—
323D12—42no—
324C4×D3—22no—
325C4×D3—22no—
326C4×D3—22no—
327C4×D3—22no—
328C4×D3—22no—
329C4×D3—22no—
330C4×D3—44no—
331C4×D3—44no—
332C4×D3—44no—
333C4×D3—44no—
334C4×D3—44no—
335Dic6—22no—
336Dic6—44no—
337Dic6—44no—
338Dic6—42no—
339C3×C8—42no—
340C3×C8—42no—
341C3⋊C8—42no—
342C3⋊C8—42no—
343#41—64no—
344#41—124no—
345#41—128no—
346#41—128no—
347#41—124no—
348#41—124no—
349#41—128no—
350C2×Q8—22no—
351C2×Q8—31no—
352C2×Q8—32no—
353C2×Q8—44no—
354C2×Q8—62no—
355C2×Q8—62no—
356C2×Q8—124no—
357C2×Q8—124no—
358C2×Q8—128no—
359C2×Q8C2×C4×D312yes—
360C2×Q8C2×C4×D312yes—
361C2×D4—31no—
362C2×D4—31no—
363C2×D4—31no—
364C2×D4—32no—
365C2×D4—62no—
366C2×D4—62no—
367C2×D4—62no—
368C2×D4—62no—
369C22×C4—22no—
370C22×C4—31no—
371C22×C4—31no—
372C22×C4—32no—
373C22×C4—32no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4C2×C4×D311yescharacteristic
380C22×C4D3×D412yes—
381M16—242no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—62no—
385C2×C8—62no—
386C2×C8—242no—
387C2×C8—242no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—32no—
392C4⋊C4—32no—
393C4⋊C4—32no—
394C4⋊C4—64no—
395C4⋊C4—64no—
396C4⋊C4—62no—
397C22⋊C4—32no—
398C22⋊C4—32no—
399C22⋊C4—32no—
400C22⋊C4—64no—
401C22⋊C4—64no—
402C22⋊C4—64no—
403C42—31no—
404C42—31no—
405C42—32no—
406C42—32no—
407C42C2×C4×D311yescharacteristic
408C42D3×D412yes—
409C42C23×D311yescharacteristic
410C22×C3—22no—
411C22×C3C2×#10012yes—
412C22×C3C22×C22⋊C411yescharacteristic
413C2×D3—22no—
414C2×D3—21no—
415C2×D3—22no—
416C2×D3—42no—
417C2×D3—44no—
418C2×D3C2×#10012yes—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—22no—
422C3×C4—42no—
423C3×C4—42no—
424C3×C4#57712yes—
425C3×C4#57812yes—
426Dic3—21no—
427Dic3—22no—
428Dic3—22no—
429Dic3—42no—
430Dic3C2×#10012yes—
431C23—31no—
432C23—31no—
433C23—62no—
434C23D3×C22⋊C411yescharacteristic
435Q8—42no—
436Q8—42no—
437Q8—44no—
438Q8—62no—
439Q8—122no—
440Q8—124no—
441Q8—124no—
442D4—34no—
443D4—62no—
444D4—62no—
445D4—122no—
446D4—124no—
447D4—124no—
448D4—124no—
449D4—122no—
450D4—122no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—31no—
457C2×C4—32no—
458C2×C4—32no—
459C2×C4—31no—
460C2×C4—44no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—124no—
469C2×C4—124no—
470C2×C4—124no—
471C2×C4—124no—
472C2×C4—124no—
473C2×C4D3×C22⋊C412yes—
474C2×C4D3×C22⋊C411yescharacteristic
475C2×C4C22×C4×D311yescharacteristic
476C2×C4C2×D3×D412yes—
477C8—42no—
478C8—42no—
479C8—122no—
480C8—122no—
481C2×C3—22no—
482C2×C3C22×#10011yescharacteristic
483C2×C3#289812yes—
484D3—22no—
485D3—41no—
486D3—82no—
487C22—22no—
488C22—32no—
489C22—62no—
490C22—61no—
491C22—62no—
492C22—122no—
493C22—124no—
494C22D3×#10012yes—
495C22C2×D3×C22⋊C411yescharacteristic
496C4—21no—
497C4—21no—
498C4—22no—
499C4—32no—
500C4—42no—
501C4—42no—
502C4—61no—
503C4—62no—
504C4—62no—
505C4—122no—
506C4#577812yes—
507C4#577912yes—
508C3#1777911yescharacteristic, complemented, Sylow
509C2—22no—
510C2—62no—
511C2—121no—
512C2—242no—
513C2C2×D3×#10011yescharacteristic
514C2#7895712yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1558C2
socleC22×C3C22×C22⋊C4
prime radical#86214C2
cosocleC3×C42C24
hypercenter#1558D3
hypocenterC3#17779
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
2. #79340C2semidirect
3. C3×#1558C2semidirect
4. C2×#4546C2semidirect
5. #70816C2semidirect
6. #68489C2semidirect
7. #68487C2semidirect
8. #68485C2semidirect
9. #68483C2semidirect
10. #68482C2semidirect
11. D3×#333C2semidirect
52. #1558D3semidirect
508. C3#17779semidirect

Maximal subgroups

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

subgroupindex
#862142
#793402
C3×#15582
C2×#45462
#708162
#684892
#684872
#684852
#684832
#684822
D3×#3332
#177793