Finite Group Databasebeta

← #10691595

Subgroups — #10691595

Order 768.

612 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5072C211yesmaximal, characteristic
2M16×D12C211yesmaximal, characteristic, complemented
3C3×#1965C211yesmaximal, characteristic, complemented
4#71759C211yesmaximal, characteristic, complemented
5#70496C211yesmaximal, characteristic, complemented
6#67711C211yesmaximal, characteristic, complemented
7#67668C211yesmaximal, characteristic, complemented
8#9253—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C22×#213C412yes—
11C2×D3×M16—21no—
12C2×C4×D12C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×C3×#100—21no—
15C2×C3×#100—21no—
16C3×C4×M16C2211yescharacteristic
17C2×#832C412yes—
18#5126—21no—
19#5072—21no—
20#5072—21no—
21#5072—21no—
22#5072—21no—
23#4925C2211yescharacteristic
24#4924C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26#4706—22no—
27#4688—21no—
28C8×D12—21no—
29#4544—22no—
30#4483—21no—
31#4483—21no—
32C2×#459—31no—
33D4×M16—31no—
34#1965D311yescharacteristic, complemented
35#1545—31no—
36#1486—31no—
37#1292—31no—
38#1283—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4C2×C412yes—
42C22×D12C2×C411yescharacteristic
43C22×C4×D3—21no—
44C2×C3×M16—21no—
45C2×C3×M16D411yescharacteristic
46C2×C3×M16D411yescharacteristic
47C3×#127—21no—
48C3×#127—21no—
49C3×#127—21no—
50C3×#127—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#213—22no—
53C2×#213—22no—
54C2×#213—44no—
55C2×(C2×D3)⋊C4—21no—
56C2×C4⋊Dic3C2×C411yescharacteristic
57C2×#208—21no—
58C22×C3⋊C8C2×C412yes—
59D3×M16—42no—
60D3×M16—42no—
61C2×ΓL(1,𝔽25)—22no—
62C2×D3×C8—22no—
63C4×D12—21no—
64C4×D12—21no—
65C4×D12D411yescharacteristic
66C4×D12D411yescharacteristic
67C4×D12D411yescharacteristic
68C4×D12D411yescharacteristic
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×#100—42no—
72C3×#100—42no—
73C3×#97—21no—
74C3×C4×C8—21no—
75#832—22no—
76#832—22no—
77#830—21no—
78(C2×D3)⋊C8—22no—
79(C2×D3)⋊C8—22no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)D412yes—
85C4×C3⋊C8—21no—
86C22×D8—32no—
87C22×M16—61no—
88C2×#127C2×D311yescharacteristic
89C2×C4×D4—31no—
90#459—61no—
91#459—61no—
92#459—61no—
93#459—61no—
94#436—61no—
95#435—62no—
96C8×D4—61no—
97#431—31no—
98#423—31no—
99#423C2×D311yescharacteristic
100C2×C4⋊C8—31no—
101C2×#102—32no—
102C2×#100—21no—
103C2×#100—21no—
104#407—61no—
105C4×M16C2×D311yescharacteristic
106#331—62no—
107D4⋊C8—61no—
108D4⋊C8—61no—
109C24×C3—22no—
110C23×D3—21no—
111C2×C3×D4—22no—
112C2×C3×D4—22no—
113C2×C3×D4—42no—
114C2×C3×D4—42no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—82no—
124C22×C3×C4C22×C411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C22×Dic3—21no—
128C2×D12—22no—
129C2×D12—21no—
130C2×D12—41no—
131C2×D12C22⋊C411yescharacteristic
132C2×D12C22⋊C411yescharacteristic
133C2×C4×D3—21no—
134C2×C4×D3—21no—
135C2×C4×D3—21no—
136C2×C4×D3—21no—
137C2×C4×D3—41no—
138C2×C4×D3—41no—
139C2×C4×D3—41no—
140C2×C4×D3—41no—
141C3×M16—21no—
142C3×M16—21no—
143C3×M16—21no—
144C3×M16—41no—
145C3×M16—41no—
146C3×M16D812yescomplemented
147C2×C3×C8—21no—
148C2×C3×C8—21no—
149C2×C3×C8—41no—
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42#4111yescharacteristic
154#213—44no—
155#213—44no—
156(C2×D3)⋊C4—21no—
157(C2×D3)⋊C4—21no—
158(C2×D3)⋊C4—21no—
159(C2×D3)⋊C4—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3C22⋊C411yescharacteristic
162C4⋊Dic3C22⋊C411yescharacteristic
163#208—41no—
164#208—41no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—42no—
168C2×C3⋊C8—41no—
169ΓL(1,𝔽25)—44no—
170D3×C8—44no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—31no—
174C22×D4C4×D312yes—
175C23×C4—61no—
176C2×D8—62no—
177C2×D8—62no—
178C2×D8—124no—
179C2×M16—21no—
180C2×M16—62no—
181C2×M16—61no—
182C2×M16—122no—
183C2×M16—122no—
184C2×M16#6711yescharacteristic
185C2×M16#6711yescharacteristic
186C22×C8—32no—
187C22×C8—62no—
188#127—21no—
189#127—21no—
190#127—21no—
191#127—21no—
192C4×D4—31no—
193C4×D4—31no—
194C4×D4—31no—
195C4×D4—31no—
196C4×D4—61no—
197C4×D4—61no—
198C2×C4⋊C4—31no—
199C2×C22⋊C4—61no—
200C2×C42C22×D311yescharacteristic
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—32no—
204C4⋊C8—61no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208#102—62no—
209#102—62no—
210#100—42no—
211#100—42no—
212C22⋊C8—62no—
213C22⋊C8—62no—
214C22⋊C8—61no—
215#97—21no—
216C4×C8—21no—
217C4×C8—61no—
218C23×C3—42no—
219C23×C3—42no—
220C23×C3—42no—
221C23×C3—82no—
222C23×C3—82no—
223C23×C3C4×D411yescharacteristic
224C22×D3—21no—
225C22×D3—21no—
226C22×D3—42no—
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—81no—
230C3×D4—42no—
231C3×D4—42no—
232C3×D4—82no—
233C3×D4—82no—
234C3×D4—82no—
235C3×D4—81no—
236C3×D4—81no—
237C3×D4—81no—
238C3×D4—81no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—41no—
242C2×C3×C4C2×C22⋊C411yescharacteristic
243C2×C3×C4C4×D411yescharacteristic
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12311yescharacteristic
248C2×C3×C4C2×D811yescharacteristic
249C2×C3×C4#13511yescharacteristic
250C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
251C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
252C2×Dic3—21no—
253C2×Dic3—21no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256D12—21no—
257D12—21no—
258D12—21no—
259D12—42no—
260D12#10012yescomplemented
261C4×D3—41no—
262C4×D3—41no—
263C4×D3—41no—
264C4×D3—41no—
265C4×D3—41no—
266C4×D3—41no—
267C4×D3—41no—
268C4×D3—41no—
269C3×C8—22no—
270C3×C8—41no—
271C3×C8—81no—
272C3⋊C8—42no—
273C3⋊C8—42no—
274C3⋊C8—81no—
275C24—22no—
276C24—61no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—31no—
280C2×D4—31no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—62no—
292C2×D4—61no—
293C2×D4—82no—
294C2×D4—121no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—121no—
301C22×C4—121no—
302C22×C4—121no—
303C22×C4—121no—
304C22×C4C2×C4×D311yescharacteristic
305C22×C4#23511yescharacteristic
306C22×C4C2×#6711yescharacteristic
307D8—124no—
308D8—124no—
309M16—21no—
310M16—21no—
311M16—21no—
312M16—41no—
313M16—41no—
314M16—124no—
315M16—121no—
316M16—121no—
317M16#21312yescomplemented
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—41no—
321C2×C8—62no—
322C2×C8—62no—
323C2×C8—124no—
324C2×C8—122no—
325C2×C8—121no—
326C4⋊C4—31no—
327C4⋊C4—31no—
328C4⋊C4—61no—
329C22⋊C4—61no—
330C22⋊C4—61no—
331C22⋊C4—61no—
332C22⋊C4—61no—
333C42D3×D411yescharacteristic
334C42D3×D411yescharacteristic
335C42D3×D411yescharacteristic
336C42#23711yescharacteristic
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—82no—
340C22×C3—82no—
341C22×C3—82no—
342C22×C3—162no—
343C22×C3—162no—
344C22×C3#38611yescharacteristic
345C22×C3C4×D811yescharacteristic
346C22×C3#44211yescharacteristic
347C2×D3—22no—
348C2×D3—41no—
349C2×D3—41no—
350C2×D3—41no—
351C2×D3—41no—
352C2×D3—41no—
353C2×D3—82no—
354C2×D3—82no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4C2×#10011yescharacteristic
359C3×C4#41311yescharacteristic
360C3×C4#44711yescharacteristic
361C3×C4#44911yescharacteristic
362C3×C4#45911yescharacteristic
363C3×C4#46311yescharacteristic
364Dic3—41no—
365Dic3—41no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—61no—
370C23—61no—
371C23—82no—
372C23—82no—
373C23—122no—
374C23—121no—
375C23—121no—
376C23—241no—
377C23C4×#6711yescharacteristic
378D4—32no—
379D4—42no—
380D4—42no—
381D4—61no—
382D4—61no—
383D4—61no—
384D4—82no—
385D4—82no—
386D4—82no—
387D4—81no—
388D4—81no—
389D4—81no—
390D4—81no—
391D4—122no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—41no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4D3×C22⋊C411yescharacteristic
408C2×C4#67⋊C411yescharacteristic
409C2×C4#88311yescharacteristic
410C2×C4#88411yescharacteristic
411C2×C4C2×#21311yescharacteristic
412C2×C4#93211yescharacteristic
413C2×C4#93711yescharacteristic
414C2×C4#93811yescharacteristic
415C2×C4#94911yescharacteristic
416C2×C4#95011yescharacteristic
417C8—22no—
418C8—41no—
419C8—81no—
420C8—122no—
421C8—122no—
422C8—241no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—162no—
426C2×C3#187911yescharacteristic
427C2×C3#190011yescharacteristic
428C2×C3#190311yescharacteristic
429D3—42no—
430D3—81no—
431C22—21no—
432C22—21no—
433C22—62no—
434C22—82no—
435C22—82no—
436C22—82no—
437C22—121no—
438C22—121no—
439C22—121no—
440C22—121no—
441C22—121no—
442C22—162no—
443C22—162no—
444C22—242no—
445C22—242no—
446C22#495611yescharacteristic
447C22C4×#21311yescharacteristic
448C22#501411yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—121no—
453C4—121no—
454C4D3×#10011yescharacteristic
455C4#474511yescharacteristic
456C4#503611yescharacteristic
457C4#503711yescharacteristic
458C4#505711yescharacteristic
459C4#507211yescharacteristic
460C3#925311yescharacteristic, complemented, Sylow
461C2—21no—
462C2—21no—
463C2—122no—
464C2—162no—
465C2—241no—
466C2#6965711yescharacteristic
467C2#7008111yescharacteristic
468C2#7008711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1965C2
socleC22×C3#386
prime radicalC2×#5072C2
cosocleC2×C3×C42C23
hypercenter#1965D3
hypocenterC3#9253
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1965

Complements

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

NH ≅ G/Ntype
2. M16×D12C2semidirect
3. C3×#1965C2semidirect
4. #71759C2semidirect
5. #70496C2semidirect
6. #67711C2semidirect
7. #67668C2semidirect
34. #1965D3semidirect
146. C3×M16D8semidirect
260. D12#100semidirect
317. M16#213semidirect
460. C3#9253semidirect

Maximal subgroups

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

subgroupindex
C2×#50722
M16×D122
C3×#19652
#717592
#704962
#677112
#676682
#92533