Finite Group Databasebeta

← #10691590

Subgroups — #10691590

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×#5015C211yesmaximal, characteristic
2D4×#208C211yesmaximal, characteristic, complemented
3C3×#1486C211yesmaximal, characteristic, complemented
4#70774C211yesmaximal, characteristic, complemented
5#70521C211yesmaximal, characteristic, complemented
6#67744C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#9253—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C22×#213C412yes—
11C22×#208—21no—
12C2×#874C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15#5207—21no—
16C2×#823—21no—
17C2×#823—21no—
18#5015—21no—
19#5015—21no—
20#5015—21no—
21#5015—21no—
22#5012—21no—
23#5011—22no—
24D4×C3⋊C8—21no—
25C2×#809C412yes—
26#4924C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28C4×#208C2211yescharacteristic
29C3×D4⋊C8—21no—
30C3×D4⋊C8—21no—
31#4485—22no—
32C2×#459—31no—
33D4×M16—31no—
34#1965—31no—
35#1545—31no—
36#1486D311yescharacteristic, complemented
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—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4D411yescharacteristic
48C3×C4×D4D411yescharacteristic
49C3×C4×D4D411yescharacteristic
50C3×C4×D4D411yescharacteristic
51C2×C3×C4⋊C4C2×C411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C42C2311yescharacteristic
54C2×#213—22no—
55C2×#213—22no—
56C2×#213—44no—
57C2×#208—22no—
58C2×#208—21no—
59C2×#208—42no—
60C2×#208—42no—
61C2×#208D411yescharacteristic
62C2×#208D411yescharacteristic
63C22×C3⋊C8—22no—
64C22×C3⋊C8C2×C412yes—
65#874—21no—
66#874—21no—
67#874—21no—
68#874—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C4×C8—21no—
74#830—22no—
75#830—22no—
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)D412yes—
84#803—21no—
85C4×C3⋊C8—21no—
86C22×D8—32no—
87C22×M16—61no—
88C2×#127—31no—
89C2×C4×D4C2×D311yescharacteristic
90#459—61no—
91#459—61no—
92#459—61no—
93#459—61no—
94#436—61no—
95#435—62no—
96C8×D4—61no—
97#431C2×D311yescharacteristic
98#423—31no—
99#423C2×D311yescharacteristic
100C2×C4⋊C8—31no—
101C2×#102—32no—
102C2×#100—61no—
103C2×#100—61no—
104#407—61no—
105C4×M16—31no—
106#331—62no—
107D4⋊C8—21no—
108D4⋊C8—21no—
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—41no—
142C3×M16—41no—
143C2×C3×C8—41no—
144C2×C3×C8—41no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4C22⋊C411yescharacteristic
147C3×C4⋊C4C22⋊C411yescharacteristic
148C3×C22⋊C4—21no—
149C3×C22⋊C4—21no—
150C3×C22⋊C4—21no—
151C3×C22⋊C4—21no—
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42#4111yescharacteristic
156#213—44no—
157#213—44no—
158#208—21no—
159#208—21no—
160#208—21no—
161#208—44no—
162#208—41no—
163#208—41no—
164#208D812yescomplemented
165C2×C3⋊C8—21no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—21no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—44no—
170C2×C3⋊C8—42no—
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—21no—
180C2×M16—31no—
181C2×M16—31no—
182C2×M16—62no—
183C2×M16—61no—
184C2×M16—122no—
185C2×M16—122no—
186C22×C8—32no—
187C22×C8—62no—
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—21no—
203C4⋊C8—21no—
204C4⋊C8—32no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208#102—62no—
209#102—62no—
210#100—122no—
211#100—122no—
212C22⋊C8—21no—
213C22⋊C8—62no—
214C22⋊C8—62no—
215#97—61no—
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—81no—
270C3×C8—81no—
271C3⋊C8—22no—
272C3⋊C8—42no—
273C3⋊C8—42no—
274C3⋊C8—41no—
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×C4D3×D411yescharacteristic
306C22×C4#23711yescharacteristic
307D8—124no—
308D8—124no—
309M16—32no—
310M16—41no—
311M16—41no—
312M16—61no—
313M16—61no—
314M16—61no—
315M16—124no—
316M16—121no—
317M16—121no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—61no—
321C2×C8—61no—
322C2×C8—62no—
323C2×C8—62no—
324C2×C8—124no—
325C2×C8—122no—
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
334C42D3×D411yescharacteristic
335C42#23711yescharacteristic
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—
377C23#67⋊C411yescharacteristic
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×C4#67⋊C411yescharacteristic
408C2×C4#88211yescharacteristic
409C2×C4#88611yescharacteristic
410C2×C4D3×D811yescharacteristic
411C2×C4#91111yescharacteristic
412C2×C4#91411yescharacteristic
413C2×C4#91611yescharacteristic
414C2×C4C2×(C2×D3)⋊C411yescharacteristic
415C2×C4#93911yescharacteristic
416C2×C4#93911yescharacteristic
417C8—62no—
418C8—81no—
419C8—81no—
420C8—122no—
421C8—122no—
422C8—121no—
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—
446C22#475611yescharacteristic
447C22#478511yescharacteristic
448C22#495811yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—41no—
453C4—41no—
454C4#477311yescharacteristic
455C4#478111yescharacteristic
456C4C2×#82311yescharacteristic
457C4#513211yescharacteristic
458C4#515011yescharacteristic
459C4#516511yescharacteristic
460C3#925311yescharacteristic, complemented, Sylow
461C2—21no—
462C2—21no—
463C2—42no—
464C2—81no—
465C2—482no—
466C2#6998511yescharacteristic
467C2#7017211yescharacteristic
468C2#7040011yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1486C2
socleC22×C3#386
prime radicalC2×#5015C2
cosocleC2×C3×C42C23
hypercenter#1486D3
hypocenterC3#9253
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
2. D4×#208C2semidirect
3. C3×#1486C2semidirect
4. #70774C2semidirect
5. #70521C2semidirect
6. #67744C2semidirect
7. #67711C2semidirect
36. #1486D3semidirect
164. #208D8semidirect
234. C3×D4#100semidirect
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
C2×#50152
D4×#2082
C3×#14862
#707742
#705212
#677442
#677112
#92533