Finite Group Databasebeta

← #10691592

Subgroups — #10691592

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
1C3×D4×M16C211yesmaximal, characteristic, complemented
2C2×#5015C211yesmaximal, characteristic
3#71598C211yesmaximal, characteristic, complemented
4#70502C211yesmaximal, characteristic, complemented
5#69876C211yesmaximal, characteristic, complemented
6#67711C211yesmaximal, characteristic, complemented
7#67668C211yesmaximal, characteristic, complemented
8#9253—31nomaximal, Sylow
9C22×C3×M16—21no—
10C2×C3×C4×D4C2211yescharacteristic
11C22×#213C412yes—
12C2×#874C2211yescharacteristic
13C3×#436—21no—
14C3×#435—22no—
15C3×C8×D4—21no—
16C3×#423C2211yescharacteristic
17C3×#407—21no—
18C3×C4×M16C2211yescharacteristic
19C2×#823—21no—
20C2×#823—21no—
21#5015—21no—
22#5015—21no—
23#5015—21no—
24#5015—21no—
25C2×#809C412yes—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29D4⋊(C3⋊C8)—21no—
30D4⋊(C3⋊C8)—21no—
31#4485—22no—
32C2×#459—31no—
33D4×M16D311yescharacteristic, complemented
34#1965—31no—
35#1545—31no—
36#1486—31no—
37#1292—31no—
38#1283—31no—
39C22×C3×D4C2×C411yescharacteristic
40C23×C3×C4—21no—
41C22×D12—21no—
42C22×D12—21no—
43C22×D12C2×C412yes—
44C2×C3×M16—22no—
45C2×C3×M16—21no—
46C2×C3×M16—42no—
47C2×C3×M16—42no—
48C2×C3×M16D411yescharacteristic
49C2×C3×M16D411yescharacteristic
50C22×C3×C8—22no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4D411yescharacteristic
54C3×C4×D4D411yescharacteristic
55C3×C4×D4D411yescharacteristic
56C3×C4×D4D411yescharacteristic
57C2×C3×C4⋊C4C2×C411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#213—22no—
61C2×#213—22no—
62C2×#213—44no—
63C2×#208—21no—
64C22×C3⋊C8C2×C412yes—
65#874—21no—
66#874—21no—
67#874—21no—
68#874—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8D412yes—
71C3×C22⋊C8—22no—
72C3×C22⋊C8—22no—
73C3×#97—21no—
74C3×C4×C8—21no—
75#830—21no—
76#823—42no—
77#823—42no—
78#809—22no—
79#809—22no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)—21no—
85C4×C3⋊C8—21no—
86C22×D8—32no—
87C22×M16—21no—
88C2×#127—31no—
89C2×C4×D4C2×D311yescharacteristic
90#459—61no—
91#459—61no—
92#459—61no—
93#459—61no—
94#436—21no—
95#435—22no—
96C8×D4—21no—
97#431—31no—
98#423—31no—
99#423C2×D311yescharacteristic
100C2×C4⋊C8—31no—
101C2×#102—32no—
102C2×#100—61no—
103C2×#100—61no—
104#407—21no—
105C4×M16C2×D311yescharacteristic
106#331—62no—
107D4⋊C8—61no—
108D4⋊C8—61no—
109C24×C3—21no—
110C23×D3—22no—
111C2×C3×D4—22no—
112C2×C3×D4—21no—
113C2×C3×D4—41no—
114C2×C3×D4C22⋊C411yescharacteristic
115C2×C3×D4C22⋊C411yescharacteristic
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4C22×C411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C2×D12—22no—
129C2×D12—22no—
130C2×D12—42no—
131C2×D12—42no—
132C2×D12—41no—
133C2×D12—41no—
134C2×D12—41no—
135C2×D12—41no—
136C2×D12—41no—
137C2×D12—41no—
138C2×D12—41no—
139C2×D12—41no—
140C2×D12—82no—
141C3×M16—21no—
142C3×M16—21no—
143C3×M16—21no—
144C3×M16—44no—
145C3×M16—41no—
146C3×M16—41no—
147C3×M16D812yescomplemented
148C2×C3×C8—21no—
149C2×C3×C8—21no—
150C2×C3×C8—22no—
151C2×C3×C8—44no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4C22⋊C411yescharacteristic
154C3×C4⋊C4C22⋊C411yescharacteristic
155C3×C22⋊C4—21no—
156C3×C22⋊C4—21no—
157C3×C22⋊C4—21no—
158C3×C22⋊C4—21no—
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42#4111yescharacteristic
163#213—44no—
164#213—44no—
165#208—41no—
166#208—41no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—42no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171C22×D4—32no—
172C22×D4—61no—
173C22×D4—61no—
174C22×D4C4×D311yescharacteristic
175C23×C4—21no—
176C2×D8—62no—
177C2×D8—62no—
178C2×D8—124no—
179C2×M16—22no—
180C2×M16—21no—
181C2×M16—42no—
182C2×M16—42no—
183C2×M16—61no—
184C2×M16#6711yescharacteristic
185C2×M16#6711yescharacteristic
186C22×C8—22no—
187C22×C8—32no—
188#127—61no—
189#127—61no—
190#127—61no—
191#127—61no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4D1211yescharacteristic
195C4×D4D1211yescharacteristic
196C4×D4#6711yescharacteristic
197C4×D4#6711yescharacteristic
198C2×C4⋊C4C4×D311yescharacteristic
199C2×C22⋊C4—21no—
200C2×C42C22×D311yescharacteristic
201C4⋊C8—21no—
202C4⋊C8—61no—
203C4⋊C8—61no—
204C4⋊C8—61no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207C4⋊C8#6712yes—
208#102—62no—
209#102—62no—
210#100—122no—
211#100—122no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—61no—
215#97—21no—
216C4×C8—21no—
217C4×C8—61no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—42no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—81no—
224C23×C3C4×D411yescharacteristic
225C22×D3—42no—
226C22×D3—42no—
227C22×D3—42no—
228C22×D3—82no—
229C22×D3—82no—
230C3×D4—21no—
231C3×D4—21no—
232C3×D4—21no—
233C3×D4—42no—
234C3×D4#10012yescomplemented
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4C2×C22⋊C411yescharacteristic
251C2×C3×C4C4×D411yescharacteristic
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12311yescharacteristic
256C2×C3×C4C2×D811yescharacteristic
257C2×C3×C4#13511yescharacteristic
258C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
259C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
260D12—42no—
261D12—42no—
262D12—82no—
263D12—82no—
264D12—82no—
265D12—81no—
266D12—81no—
267D12—81no—
268D12—81no—
269C3×C8—22no—
270C3×C8—42no—
271C3×C8—41no—
272C3⋊C8—42no—
273C3⋊C8—81no—
274C3⋊C8—81no—
275C24—21no—
276C24—62no—
277C2×D4—22no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—62no—
281C2×D4—62no—
282C2×D4—122no—
283C2×D4—122no—
284C2×D4—121no—
285C2×D4—121no—
286C2×D4—121no—
287C2×D4—121no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4—121no—
291C2×D4—121no—
292C2×D4—242no—
293C2×D4(C2×D3)⋊C411yescharacteristic
294C2×D4(C2×D3)⋊C411yescharacteristic
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4C2×C4×D311yescharacteristic
305C22×C4#23511yescharacteristic
306C22×C4C2×#6711yescharacteristic
307D8—124no—
308D8—124no—
309M16—21no—
310M16—21no—
311M16—21no—
312M16—44no—
313M16—41no—
314M16—41no—
315M16—121no—
316M16—121no—
317M16#21312yescomplemented
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—44no—
322C2×C8—62no—
323C2×C8—122no—
324C2×C8—121no—
325C2×C8—121no—
326C4⋊C4—21no—
327C4⋊C4(C2×D3)⋊C411yescharacteristic
328C4⋊C4(C2×D3)⋊C411yescharacteristic
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—21no—
333C42C2×D1211yescharacteristic
334C42#23511yescharacteristic
335C42C2×#6711yescharacteristic
336C42C2×#6711yescharacteristic
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—22no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—82no—
346C22×C3—82no—
347C22×C3#38611yescharacteristic
348C22×C3C4×D811yescharacteristic
349C22×C3#44211yescharacteristic
350C2×D3—82no—
351C2×D3—82no—
352C2×D3—82no—
353C2×D3—162no—
354C2×D3—162no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4C2×#10011yescharacteristic
361C3×C4#41311yescharacteristic
362C3×C4#44711yescharacteristic
363C3×C4#44911yescharacteristic
364C3×C4#45911yescharacteristic
365C3×C4#46311yescharacteristic
366C23—21no—
367C23—21no—
368C23—42no—
369C23—41no—
370C23—41no—
371C23—81no—
372C23—122no—
373C23—122no—
374C23—122no—
375C23—242no—
376C23—242no—
377C23C4×#6711yescharacteristic
378D4—21no—
379D4—21no—
380D4—21no—
381D4—42no—
382D4—122no—
383D4—122no—
384D4—242no—
385D4—242no—
386D4—242no—
387D4—241no—
388D4—241no—
389D4—241no—
390D4—241no—
391D4#82312yescomplemented
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4C2×(C2×D3)⋊C411yescharacteristic
408C2×C4C4×#6711yescharacteristic
409C2×C4#92911yescharacteristic
410C2×C4#93011yescharacteristic
411C2×C4#93011yescharacteristic
412C2×C4C2×#21311yescharacteristic
413C2×C4#93211yescharacteristic
414C2×C4#94911yescharacteristic
415C2×C4#95011yescharacteristic
416C2×C4#95311yescharacteristic
417C8—22no—
418C8—42no—
419C8—41no—
420C8—122no—
421C8—241no—
422C8—241no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—42no—
426C2×C3—81no—
427C2×C3#187911yescharacteristic
428C2×C3#190011yescharacteristic
429C2×C3#190311yescharacteristic
430D3—162no—
431C22—21no—
432C22—21no—
433C22—22no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—82no—
440C22—82no—
441C22—242no—
442C22—242no—
443C22—242no—
444C22—482no—
445C22—482no—
446C22C4×#21311yescharacteristic
447C22#501411yescharacteristic
448C22#521311yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—41no—
453C4—41no—
454C4#501511yescharacteristic
455C4#501611yescharacteristic
456C4C2×#82311yescharacteristic
457C4#513211yescharacteristic
458C4#521711yescharacteristic
459C4#523811yescharacteristic
460C3#925311yescharacteristic, complemented, Sylow
461C2—21no—
462C2—21no—
463C2—42no—
464C2—81no—
465C2—482no—
466C2#7161311yescharacteristic
467C2#7200211yescharacteristic
468C2#7202311yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×D4×M16C2
socleC22×C3#386
prime radicalC2×#5015C2
cosocleC2×C3×C42C23
hypercenterD4×M16D3
hypocenterC3#9253
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
1. C3×D4×M16C2semidirect
3. #71598C2semidirect
4. #70502C2semidirect
5. #69876C2semidirect
6. #67711C2semidirect
7. #67668C2semidirect
33. D4×M16D3semidirect
147. C3×M16D8semidirect
234. C3×D4#100semidirect
317. M16#213semidirect
391. D4#823semidirect
460. C3#9253semidirect

Maximal subgroups

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

subgroupindex
C3×D4×M162
C2×#50152
#715982
#705022
#698762
#677112
#676682
#92533