Finite Group Databasebeta

← #10763238

Subgroups — #10763238

Order 768.

1230 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#532C211yesmaximal, characteristic
2#78517C212yesmaximal, complemented
3C3×#1525C211yesmaximal, characteristic, complemented
4#71797C211yesmaximal, characteristic, complemented
5#69859C212yesmaximal, complemented
6#68349C214yesmaximal, complemented
7#68347C214yesmaximal, complemented
8#17663—31nomaximal, Sylow
9C2×D3×#41C412yes—
10C3×#532C2211yescharacteristic
11#5832C412yes—
12#5725C411yescharacteristic
13D3×C4⋊Q8C411yescharacteristic
14#5716C411yescharacteristic
15D3×#127C411yescharacteristic
16D3×#124C2212yes—
17#5668C2212yes—
18D3×#121—22no—
19#5596—22no—
20#5548—22no—
21C4×D3×D4—22no—
22#5477C2211yescharacteristic
23C2×C42×D3C2211yescharacteristic
24C3×C4×M16C2212yescomplemented
25#5126—22no—
26#5118—22no—
27C4×#208C2212yescomplemented
28#4705C2214yes—
29#4705C2214yescomplemented
30#4685—22no—
31#4685—22no—
32C3×#329C2214yescomplemented
33#4540C2214yescomplemented
34#4464C2218yescomplemented
35C2×#532—31no—
36#2920—32no—
37#1536—34no—
38#1533—34no—
39#1525—32no—
40#1525—31no—
41#1525D311yescharacteristic, complemented
42C2×C3×#41C2×C412yes—
43D3×#41—48no—
44C2×#239C2×C412yes—
45C2×D3×Q8C2×C412yes—
46C2×#237—22no—
47C2×#237C2×C412yes—
48C2×D3×D4—22no—
49C2×D3×D4—22no—
50C2×D3×D4C2×C412yes—
51C2×#235—22no—
52C2×#235C2×C412yes—
53C22×C4×D3—22no—
54C22×C4×D3C2×C411yescharacteristic
55C22×C4×D3D412yes—
56C2×C3×M16—22no—
57C3×C4⋊Q8C2×C411yescharacteristic
58C3×#127C2×C411yescharacteristic
59C3×#124C2312yes—
60C3×#121—22no—
61C3×C4×D4—22no—
62C2×C3×C42C2311yescharacteristic
63#947C2×C412yes—
64#944C2×C412yes—
65#940—22no—
66#940C2×C412yes—
67#939—24no—
68#938—22no—
69#936C2×C412yes—
70D4×Dic3—22no—
71#930—22no—
72C4×#67—24no—
73C2×C4×Dic3C2×C411yescharacteristic
74C2×C4×Dic3D412yes—
75C2×#208—22no—
76#895—22no—
77#893—22no—
78D3×C4⋊C4—22no—
79#888—22no—
80#885—24no—
81#884—24no—
82#67⋊C4—24no—
83D3×C22⋊C4—24no—
84#875C2312yes—
85#874C2×C411yescharacteristic
86C4×D12—22no—
87C42×D3C2×C411yescharacteristic
88C42×D3C2×C411yescharacteristic
89C42×D3D412yes—
90C42×D3D412yes—
91C42×D3C2311yescharacteristic
92C42×D3C2311yescharacteristic
93#869C2×C411yescharacteristic
94C3×#97C2314yes—
95C3×C4×C8—22no—
96(C2×D3)⋊C8—24no—
97(C2×D3)⋊C8—24no—
98Dic3⋊C8—24no—
99Dic3⋊C8—24no—
100#803C2314yes—
101C4×C3⋊C8—22no—
102C22×#41—32no—
103#532—32no—
104#532—31no—
105#532—31no—
106#532—32no—
107#532—31no—
108#532—62no—
109#532—62no—
110#532C2×D311yescharacteristic
111C2×C4⋊Q8—31no—
112C2×#127—31no—
113C2×#124—32no—
114C2×#121—62no—
115C2×C4×D4—62no—
116C22×C42—31no—
117#432—34no—
118#432—34no—
119#431—62no—
120#431—62no—
121#423—62no—
122#407—62no—
123C4×M16—32no—
124C4×M16C2×D312yescomplemented
125#329—34no—
126#329—38no—
127#329C2×D314yescomplemented
128C23×D3—22no—
129C23×D3C22⋊C411yescharacteristic
130C3×#41—44no—
131C2×C3×Q8C22×C412yes—
132C2×C3×D4—22no—
133C2×C3×D4—22no—
134C2×C3×D4C22×C412yes—
135C22×C3×C4—22no—
136C22×C3×C4C22×C411yescharacteristic
137C22×C3×C4C2×D412yes—
138C2×#67—22no—
139C2×#67—22no—
140C2×#67—24no—
141C2×#67—22no—
142C22×Dic3—22no—
143C22×Dic3C22⋊C411yescharacteristic
144C22×Dic3C22⋊C412yes—
145#239—44no—
146D3×Q8—44no—
147#237—44no—
148#237—48no—
149D3×D4—44no—
150D3×D4—48no—
151D3×D4—48no—
152#235—48no—
153#235—44no—
154C2×D12—22no—
155C2×D12—22no—
156C2×D12C22×C412yes—
157C2×C4×D3—24no—
158C2×C4×D3—22no—
159C2×C4×D3—22no—
160C2×C4×D3—22no—
161C2×C4×D3—44no—
162C2×C4×D3—48no—
163C2×C4×D3C22⋊C412yes—
164C2×C4×D3C22⋊C412yes—
165C2×C4×D3C22⋊C411yescharacteristic
166C2×C4×D3C22⋊C411yescharacteristic
167C2×C4×D3C22×C411yescharacteristic
168C2×C4×D3C22×C411yescharacteristic
169C2×C4×D3C2×D412yes—
170C2×C4×D3C2×D412yes—
171C2×Dic6—22no—
172C2×Dic6C22×C412yes—
173C3×M16—44no—
174C2×C3×C8—24no—
175C3×C4⋊C4—22no—
176C3×C22⋊C4—24no—
177C3×C42C22×C411yescharacteristic
178C3×C42C2×D412yes—
179C3×C42C2411yescharacteristic
180#217—24no—
181(C2×D3)⋊C4—24no—
182(C2×D3)⋊C4—24no—
183C4⋊Dic3—22no—
184Dic3⋊C4—24no—
185C4×Dic3C22⋊C412yes—
186C4×Dic3C22⋊C412yes—
187C4×Dic3C22⋊C412yes—
188C4×Dic3C22×C411yescharacteristic
189C4×Dic3C22×C411yescharacteristic
190C4×Dic3C2×D412yes—
191C4×Dic3C2×D412yes—
192#208—44no—
193C2×C3⋊C8—24no—
194C2×#41—32no—
195C2×#41—32no—
196C2×#41—32no—
197C2×#41—62no—
198C2×#41—62no—
199C2×#41—128no—
200C2×#41C4×D312yes—
201C22×Q8—32no—
202C22×D4—32no—
203C22×D4—62no—
204C22×D4—62no—
205C23×C4—31no—
206C23×C4—32no—
207C23×C4—62no—
208C2×M16—22no—
209C2×M16—62no—
210C4⋊Q8—32no—
211C4⋊Q8—31no—
212C4⋊Q8—62no—
213C4⋊Q8C4×D311yescharacteristic
214#127—32no—
215#127—31no—
216#127—62no—
217#127C4×D311yescharacteristic
218#124—32no—
219#124—32no—
220#124—32no—
221#124—64no—
222#124C22×D312yes—
223#121—22no—
224#121—64no—
225#121—62no—
226#121—62no—
227#121—64no—
228#121—62no—
229C4×D4—22no—
230C4×D4—64no—
231C4×D4—62no—
232C4×D4—64no—
233C4×D4—62no—
234C4×D4—62no—
235C2×C4⋊C4—62no—
236C2×C22⋊C4—64no—
237C2×C42—31no—
238C2×C42—32no—
239C2×C42—31no—
240C2×C42—31no—
241C2×C42—32no—
242C2×C42—32no—
243C2×C42—31no—
244C2×C42—31no—
245C2×C42C22×D311yescharacteristic
246C4⋊C8—64no—
247C4⋊C8—64no—
248C22⋊C8—64no—
249C22⋊C8—64no—
250#97—34no—
251#97C22×D314yes—
252C4×C8—22no—
253C4×C8—62no—
254C23×C3—22no—
255C23×C3C2×C22⋊C411yescharacteristic
256C22×D3—22no—
257C22×D3—22no—
258C22×D3—42no—
259C22×D3—48no—
260C22×D3—44no—
261C22×D3C2×C22⋊C411yescharacteristic
262C22×D3C2×C22⋊C411yescharacteristic
263C3×Q8—42no—
264C3×D4—42no—
265C3×D4—44no—
266C3×D4—44no—
267C2×C3×C4—22no—
268C2×C3×C4—24no—
269C2×C3×C4—22no—
270C2×C3×C4—44no—
271C2×C3×C4C2×C22⋊C412yes—
272C2×C3×C4C2×C22⋊C411yescharacteristic
273C2×C3×C4C23×C411yescharacteristic
274C2×C3×C4C22×D412yes—
275#67—44no—
276#67—48no—
277#67—44no—
278#67—44no—
279C2×Dic3—22no—
280C2×Dic3—22no—
281C2×Dic3—42no—
282C2×Dic3—44no—
283C2×Dic3C2×C22⋊C412yes—
284C2×Dic3C2×C22⋊C412yes—
285C2×Dic3C2×C22⋊C411yescharacteristic
286C2×Dic3C2×C22⋊C411yescharacteristic
287D12—42no—
288D12—44no—
289D12—44no—
290C4×D3—22no—
291C4×D3—22no—
292C4×D3—24no—
293C4×D3—24no—
294C4×D3—22no—
295C4×D3—42no—
296C4×D3—42no—
297C4×D3—44no—
298C4×D3—44no—
299C4×D3—44no—
300C4×D3#10014yes—
301Dic6—44no—
302Dic6—42no—
303C3×C8—44no—
304C3⋊C8—44no—
305C24—31no—
306C24—62no—
307#41—44no—
308#41—124no—
309#41—128no—
310#41—128no—
311#41—124no—
312#41—124no—
313C2×Q8—32no—
314C2×Q8—62no—
315C2×Q8—124no—
316C2×Q8C2×C4×D312yes—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—32no—
320C2×D4—62no—
321C2×D4—64no—
322C2×D4—62no—
323C2×D4—62no—
324C2×D4—62no—
325C2×D4—62no—
326C2×D4—124no—
327C2×D4—128no—
328C2×D4—128no—
329C2×D4C2×C4×D312yes—
330C22×C4—22no—
331C22×C4—31no—
332C22×C4—31no—
333C22×C4—32no—
334C22×C4—32no—
335C22×C4—32no—
336C22×C4—31no—
337C22×C4—31no—
338C22×C4—31no—
339C22×C4—32no—
340C22×C4—32no—
341C22×C4—64no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4—62no—
345C22×C4—62no—
346C22×C4—124no—
347C22×C4—128no—
348C22×C4C2×C4×D311yescharacteristic
349C22×C4D3×D412yes—
350M16—44no—
351M16—124no—
352C2×C8—24no—
353C2×C8—64no—
354C4⋊C4—22no—
355C4⋊C4—64no—
356C4⋊C4—62no—
357C22⋊C4—24no—
358C22⋊C4—64no—
359C22⋊C4—64no—
360C22⋊C4—64no—
361C42—32no—
362C42—32no—
363C42—32no—
364C42—31no—
365C42—31no—
366C42—32no—
367C42—32no—
368C42C2×C4×D311yescharacteristic
369C42D3×D412yes—
370C42C23×D311yescharacteristic
371C22×C3—21no—
372C22×C3—22no—
373C22×C3—42no—
374C22×C3—44no—
375C22×C3C22×C22⋊C411yescharacteristic
376C2×D3—22no—
377C2×D3—21no—
378C2×D3—22no—
379C2×D3—44no—
380C2×D3—42no—
381C2×D3—44no—
382C2×D3—88no—
383C2×D3C2×#10012yes—
384C3×C4—21no—
385C3×C4—22no—
386C3×C4—42no—
387C3×C4C2×#10012yes—
388C3×C4#42114yes—
389Dic3—21no—
390Dic3—22no—
391Dic3—22no—
392Dic3—42no—
393Dic3C2×#10012yes—
394C23—22no—
395C23—31no—
396C23—31no—
397C23—62no—
398C23—62no—
399C23—122no—
400C23—124no—
401C23—128no—
402C23D3×C22⋊C411yescharacteristic
403Q8—42no—
404Q8—122no—
405Q8—124no—
406D4—42no—
407D4—44no—
408D4—44no—
409D4—122no—
410D4—124no—
411D4—124no—
412D4—128no—
413D4—124no—
414D4—124no—
415D4—124no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—31no—
420C2×C4—34no—
421C2×C4—32no—
422C2×C4—32no—
423C2×C4—31no—
424C2×C4—44no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—64no—
430C2×C4—64no—
431C2×C4—62no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—124no—
435C2×C4—124no—
436C2×C4—124no—
437C2×C4—122no—
438C2×C4—124no—
439C2×C4D3×C22⋊C412yes—
440C2×C4D3×C22⋊C411yescharacteristic
441C2×C4C22×C4×D311yescharacteristic
442C2×C4C2×D3×D412yes—
443C8—44no—
444C8—124no—
445C2×C3—41no—
446C2×C3—82no—
447C2×C3C22×#10011yescharacteristic
448C2×C3C2×#42112yes—
449D3—22no—
450D3—41no—
451D3—82no—
452C22—21no—
453C22—22no—
454C22—32no—
455C22—42no—
456C22—44no—
457C22—62no—
458C22—61no—
459C22—62no—
460C22—124no—
461C22—122no—
462C22—124no—
463C22—248no—
464C22C2×D3×C22⋊C411yescharacteristic
465C4—21no—
466C4—22no—
467C4—32no—
468C4—42no—
469C4—61no—
470C4—62no—
471C4—62no—
472C4—122no—
473C4D3×#10012yes—
474C4#482114yes—
475C3#1766311yescharacteristic, complemented, Sylow
476C2—41no—
477C2—62no—
478C2—82no—
479C2—121no—
480C2—242no—
481C2C2×D3×#10011yescharacteristic
482C2C2×#482112yes—
subgroupG/H
centerC2×C4D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1525C2
socleC22×C3C22×C22⋊C4
prime radicalD3×#532C2
cosocleC3×C42C24
hypercenter#1525D3
hypocenterC3#17663
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1525

Complements

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

NH ≅ G/Ntype
2. #78517C2semidirect
3. C3×#1525C2semidirect
4. #71797C2semidirect
5. #69859C2semidirect
6. #68349C2semidirect
7. #68347C2semidirect
24. C3×C4×M16C22semidirect
27. C4×#208C22semidirect
29. #4705C22semidirect
32. C3×#329C22semidirect
33. #4540C22semidirect
34. #4464C22semidirect
41. #1525D3semidirect
124. C4×M16C2×D3semidirect
127. #329C2×D3semidirect
475. C3#17663semidirect

Maximal subgroups

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

subgroupindex
D3×#5322
#785172
C3×#15252
#717972
#698592
#683492
#683472
#176633