Finite Group Databasebeta

← #10763239

Subgroups — #10763239

Order 768.

1230 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#532C211yesmaximal, characteristic
2#79331C212yesmaximal, complemented
3C3×#1536C211yesmaximal, characteristic, complemented
4#71799C211yesmaximal, characteristic, complemented
5#70811C212yesmaximal, complemented
6#68347C214yesmaximal, complemented
7#68342C214yesmaximal, 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×#432C2212yescomplemented
25#4926C2212yescomplemented
26#4837—22no—
27#4833—22no—
28#4726—22no—
29#4718—22no—
30#4707C2214yescomplemented
31#4705C2214yes—
32C3×#329C2214yescomplemented
33#4540C2214yescomplemented
34#4464C2218yescomplemented
35C2×#532—31no—
36#2920—32no—
37#1536—32no—
38#1536—31no—
39#1536D311yescharacteristic, complemented
40#1533—34no—
41#1525—34no—
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—
56C3×C4⋊Q8C2×C411yescharacteristic
57C3×#127C2×C411yescharacteristic
58C3×#124C2312yes—
59C3×#121—22no—
60C3×C4×D4—22no—
61C2×C3×C42C2311yescharacteristic
62#947C2×C412yes—
63#944C2×C412yes—
64#940—22no—
65#940C2×C412yes—
66#939—24no—
67#938—22no—
68#936C2×C412yes—
69D4×Dic3—22no—
70#930—22no—
71C4×#67—24no—
72C2×C4×Dic3C2×C411yescharacteristic
73C2×C4×Dic3D412yes—
74C2×ΓL(1,𝔽25)—24no—
75#895—22no—
76#893—22no—
77D3×C4⋊C4—22no—
78#888—22no—
79#885—24no—
80#884—24no—
81#67⋊C4—24no—
82D3×C22⋊C4—24no—
83#875C2312yes—
84#874C2×C411yescharacteristic
85C4×D12—22no—
86C42×D3C2×C411yescharacteristic
87C42×D3C2×C411yescharacteristic
88C42×D3D412yes—
89C42×D3D412yes—
90C42×D3C2311yescharacteristic
91C42×D3C2311yescharacteristic
92#869C2×C411yescharacteristic
93C3×C4⋊C8—22no—
94C3×C22⋊C8—22no—
95C3×#97C2314yes—
96#830—22no—
97(C2×D3)⋊C8—24no—
98Dic3⋊C8—24no—
99C8×Dic3—24no—
100C4⋊(C3⋊C8)—22no—
101#803C2314yes—
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—32no—
118#432—34no—
119#432C2×D312yescomplemented
120#431—62no—
121#431—62no—
122#423—62no—
123#407—62no—
124C4×M16—34no—
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⋊C412yes—
144C22×Dic3C22⋊C411yescharacteristic
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⋊C411yescharacteristic
165C2×C4×D3C22⋊C412yes—
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—
173C2×C3×C8—24no—
174C3×C4⋊C4—22no—
175C3×C22⋊C4—24no—
176C3×C42C22×C411yescharacteristic
177C3×C42C2×D412yes—
178C3×C42C2411yescharacteristic
179#217—24no—
180(C2×D3)⋊C4—24no—
181(C2×D3)⋊C4—24no—
182C4⋊Dic3—22no—
183Dic3⋊C4—24no—
184C4×Dic3C22⋊C412yes—
185C4×Dic3C22⋊C412yes—
186C4×Dic3C22⋊C412yes—
187C4×Dic3C22×C411yescharacteristic
188C4×Dic3C22×C411yescharacteristic
189C4×Dic3C2×D412yes—
190C4×Dic3C2×D412yes—
191C2×C3⋊C8—24no—
192ΓL(1,𝔽25)—48no—
193C2×#41—32no—
194C2×#41—32no—
195C2×#41—32no—
196C2×#41—62no—
197C2×#41—62no—
198C2×#41—128no—
199C2×#41C4×D312yes—
200C22×Q8—32no—
201C22×D4—32no—
202C22×D4—62no—
203C22×D4—62no—
204C23×C4—31no—
205C23×C4—32no—
206C23×C4—62no—
207C2×M16—64no—
208C4⋊Q8—32no—
209C4⋊Q8—31no—
210C4⋊Q8—62no—
211C4⋊Q8C4×D311yescharacteristic
212#127—32no—
213#127—31no—
214#127—62no—
215#127C4×D311yescharacteristic
216#124—32no—
217#124—32no—
218#124—32no—
219#124—64no—
220#124C22×D312yes—
221#121—22no—
222#121—64no—
223#121—62no—
224#121—62no—
225#121—64no—
226#121—62no—
227C4×D4—22no—
228C4×D4—64no—
229C4×D4—62no—
230C4×D4—64no—
231C4×D4—62no—
232C4×D4—62no—
233C2×C4⋊C4—62no—
234C2×C22⋊C4—64no—
235C2×C42—31no—
236C2×C42—32no—
237C2×C42—31no—
238C2×C42—31no—
239C2×C42—32no—
240C2×C42—32no—
241C2×C42—31no—
242C2×C42—31no—
243C2×C42C22×D311yescharacteristic
244C4⋊C8—22no—
245C4⋊C8—64no—
246C4⋊C8—62no—
247C22⋊C8—22no—
248C22⋊C8—64no—
249C22⋊C8—62no—
250#97—34no—
251#97C22×D314yes—
252C4×C8—64no—
253C23×C3—22no—
254C23×C3C2×C22⋊C411yescharacteristic
255C22×D3—22no—
256C22×D3—22no—
257C22×D3—42no—
258C22×D3—48no—
259C22×D3—44no—
260C22×D3C2×C22⋊C411yescharacteristic
261C22×D3C2×C22⋊C411yescharacteristic
262C3×Q8—42no—
263C3×D4—42no—
264C3×D4—44no—
265C3×D4—44no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—42no—
269C2×C3×C4—44no—
270C2×C3×C4C2×C22⋊C412yes—
271C2×C3×C4C2×C22⋊C411yescharacteristic
272C2×C3×C4C23×C411yescharacteristic
273C2×C3×C4C22×D412yes—
274#67—44no—
275#67—44no—
276#67—44no—
277#67—48no—
278C2×Dic3—22no—
279C2×Dic3—24no—
280C2×Dic3—22no—
281C2×Dic3—44no—
282C2×Dic3C2×C22⋊C412yes—
283C2×Dic3C2×C22⋊C411yescharacteristic
284C2×Dic3C2×C22⋊C411yescharacteristic
285C2×Dic3C2×C22⋊C412yes—
286D12—42no—
287D12—44no—
288D12—44no—
289C4×D3—22no—
290C4×D3—22no—
291C4×D3—22no—
292C4×D3—24no—
293C4×D3—24no—
294C4×D3—42no—
295C4×D3—42no—
296C4×D3—44no—
297C4×D3—44no—
298C4×D3—44no—
299C4×D3#10014yes—
300Dic6—44no—
301Dic6—42no—
302C3×C8—44no—
303C3⋊C8—44no—
304C24—31no—
305C24—62no—
306#41—44no—
307#41—124no—
308#41—128no—
309#41—128no—
310#41—124no—
311#41—124no—
312C2×Q8—32no—
313C2×Q8—62no—
314C2×Q8—124no—
315C2×Q8C2×C4×D312yes—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—32no—
319C2×D4—62no—
320C2×D4—64no—
321C2×D4—62no—
322C2×D4—62no—
323C2×D4—62no—
324C2×D4—62no—
325C2×D4—124no—
326C2×D4—128no—
327C2×D4—128no—
328C2×D4C2×C4×D312yes—
329C22×C4—22no—
330C22×C4—31no—
331C22×C4—32no—
332C22×C4—31no—
333C22×C4—32no—
334C22×C4—31no—
335C22×C4—31no—
336C22×C4—31no—
337C22×C4—32no—
338C22×C4—32no—
339C22×C4—32no—
340C22×C4—64no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4—62no—
345C22×C4—128no—
346C22×C4—124no—
347C22×C4C2×C4×D311yescharacteristic
348C22×C4D3×D412yes—
349M16—128no—
350C2×C8—24no—
351C2×C8—64no—
352C4⋊C4—22no—
353C4⋊C4—64no—
354C4⋊C4—62no—
355C22⋊C4—24no—
356C22⋊C4—64no—
357C22⋊C4—64no—
358C22⋊C4—64no—
359C42—32no—
360C42—32no—
361C42—32no—
362C42—31no—
363C42—31no—
364C42—32no—
365C42—32no—
366C42C2×C4×D311yescharacteristic
367C42D3×D412yes—
368C42C23×D311yescharacteristic
369C22×C3—21no—
370C22×C3—22no—
371C22×C3—42no—
372C22×C3—44no—
373C22×C3C22×C22⋊C411yescharacteristic
374C2×D3—22no—
375C2×D3—21no—
376C2×D3—22no—
377C2×D3—44no—
378C2×D3—42no—
379C2×D3—44no—
380C2×D3—88no—
381C2×D3C2×#10012yes—
382C3×C4—21no—
383C3×C4—22no—
384C3×C4—22no—
385C3×C4—42no—
386C3×C4C2×#10012yes—
387Dic3—21no—
388Dic3—22no—
389Dic3—42no—
390Dic3C2×#10012yes—
391Dic3#42114yes—
392C23—22no—
393C23—31no—
394C23—31no—
395C23—62no—
396C23—62no—
397C23—122no—
398C23—124no—
399C23—128no—
400C23D3×C22⋊C411yescharacteristic
401Q8—42no—
402Q8—122no—
403Q8—124no—
404D4—42no—
405D4—44no—
406D4—44no—
407D4—122no—
408D4—124no—
409D4—128no—
410D4—124no—
411D4—124no—
412D4—124no—
413D4—124no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—32no—
417C2×C4—31no—
418C2×C4—34no—
419C2×C4—32no—
420C2×C4—31no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—64no—
424C2×C4—62no—
425C2×C4—64no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—64no—
429C2×C4—62no—
430C2×C4—62no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—124no—
434C2×C4—124no—
435C2×C4—124no—
436C2×C4—124no—
437C2×C4D3×C22⋊C412yes—
438C2×C4D3×C22⋊C411yescharacteristic
439C2×C4C22×C4×D311yescharacteristic
440C2×C4C2×D3×D412yes—
441C8—44no—
442C8—124no—
443C2×C3—41no—
444C2×C3—82no—
445C2×C3C22×#10011yescharacteristic
446C2×C3C2×#42112yes—
447D3—22no—
448D3—41no—
449D3—82no—
450C22—21no—
451C22—22no—
452C22—32no—
453C22—42no—
454C22—44no—
455C22—62no—
456C22—61no—
457C22—62no—
458C22—124no—
459C22—122no—
460C22—124no—
461C22—248no—
462C22C2×D3×C22⋊C411yescharacteristic
463C4—21no—
464C4—22no—
465C4—22no—
466C4—32no—
467C4—34no—
468C4—42no—
469C4—61no—
470C4—62no—
471C4—122no—
472C4D3×#10012yes—
473C3#1766311yescharacteristic, complemented, Sylow
474C2—41no—
475C2—62no—
476C2—82no—
477C2—121no—
478C2—242no—
479C2C2×D3×#10011yescharacteristic
480C2D3×#42112yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1536C2
socleC22×C3C22×C22⋊C4
prime radicalD3×#532C2
cosocleC3×C42C24
hypercenter#1536D3
hypocenterC3#17663
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1536

Complements

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

NH ≅ G/Ntype
2. #79331C2semidirect
3. C3×#1536C2semidirect
4. #71799C2semidirect
5. #70811C2semidirect
6. #68347C2semidirect
7. #68342C2semidirect
24. C3×#432C22semidirect
25. #4926C22semidirect
30. #4707C22semidirect
32. C3×#329C22semidirect
33. #4540C22semidirect
34. #4464C22semidirect
39. #1536D3semidirect
119. #432C2×D3semidirect
127. #329C2×D3semidirect
473. C3#17663semidirect

Maximal subgroups

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

subgroupindex
D3×#5322
#793312
C3×#15362
#717992
#708112
#683472
#683422
#176633