Finite Group Databasebeta

← #10765310

Subgroups — #10765310

Order 768.

678 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83298C211yesmaximal, characteristic
2#83280C211yesmaximal, characteristic, complemented
3#79286C211yesmaximal, characteristic, complemented
4C3×#1484C211yesmaximal, characteristic, complemented
5#70747C211yesmaximal, characteristic, complemented
6#69855C211yesmaximal, characteristic, complemented
7#69848C211yesmaximal, characteristic, complemented
8#68158C212yesmaximal, complemented
9#68153C212yesmaximal, complemented
10#68151C212yesmaximal, complemented
11#68148C212yesmaximal, complemented
12#17922—31nomaximal, Sylow
13C3×#532C2211yescharacteristic
14C2×C4×D12C2211yescharacteristic
15C3×#431C2211yescharacteristic
16C3×C4×M16C2211yescharacteristic
17#5234—22no—
18#5087C412yes—
19#5072C412yes—
20#5036C412yes—
21#5033C412yes—
22C4×#213C2212yes—
23C4×#213C2212yes—
24#5012C2212yescomplemented
25#5011—22no—
26D4×C3⋊C8C2212yescomplemented
27#4926C2211yescharacteristic
28#4925C2211yescharacteristic
29#4925C2211yescharacteristic
30#4834C2212yes—
31#4834C2212yes—
32D3×C22⋊C8—22no—
33C8×D12—22no—
34C3×D4⋊C8C2212yescomplemented
35C3×D4⋊C8C2212yescomplemented
36#4480C2212yescomplemented
37#4479C2212yescomplemented
38#4459C2212yes—
39#4459C2212yes—
40#4455C2211yescharacteristic, complemented
41#4454C2211yescharacteristic, complemented
42#4454C2211yescharacteristic, complemented
43C8×C3⋊C8C2211yescharacteristic, complemented
44#3100—31no—
45#2981—31no—
46#2972—31no—
47#1586—32no—
48#1581—32no—
49#1580—32no—
50C8×D8—32no—
51#1572—31no—
52#1490—31no—
53#1484D311yescharacteristic, complemented
54#1453—31no—
55C2×C3×#41—21no—
56C22×D12C2×C411yescharacteristic
57C22×C4×D3—21no—
58C2×C3×M16—21no—
59C3×C4⋊Q8C2×C411yescharacteristic
60C3×#127C2×C411yescharacteristic
61C3×#124—21no—
62C3×#121—21no—
63C3×#121C2×C412yes—
64C3×C4×D4—21no—
65C3×C4×D4C2312yes—
66C2×C3×C42C2311yescharacteristic
67C2×#213C2×C412yes—
68C2×#213C2×C412yes—
69C2×(C2×D3)⋊C4—21no—
70C2×C4⋊Dic3C2×C411yescharacteristic
71C2×#208—22no—
72C22×C3⋊C8—22no—
73C2×D3×C8—22no—
74C2×D3×C8—22no—
75C4×D12—21no—
76C4×D12—21no—
77C4×D12C2×C412yes—
78C4×D12C2312yes—
79C3×C4⋊C8C2312yes—
80C3×C22⋊C8—21no—
81C3×#97—21no—
82C3×C4×C8—21no—
83C3×C4×C8C2311yescharacteristic
84C3×C4×C8C2311yescharacteristic
85#832C2×C412yes—
86#832C2×C412yes—
87#830—21no—
88#830—22no—
89#830—22no—
90#830C2×C412yes—
91#830D412yes—
92(C2×D3)⋊C8—22no—
93(C2×D3)⋊C8—22no—
94#809C2×C412yes—
95#809C2×C412yes—
96#807C2×C412yes—
97#807C2×C412yes—
98C4⋊(C3⋊C8)C2×C412yes—
99C4⋊(C3⋊C8)D412yes—
100C4⋊(C3⋊C8)C2312yes—
101#803—21no—
102C4×C3⋊C8C2311yescharacteristic
103C4×C3⋊C8C2311yescharacteristic
104C4×C3⋊C8C2311yescharacteristic
105C4×C3⋊C8C2311yescharacteristic
106#532C2×D311yescharacteristic
107C2×C4×D4—31no—
108#480—32no—
109#474—32no—
110#459—32no—
111#447—32no—
112C4×D8—32no—
113C4×D8—32no—
114#436—32no—
115#435—62no—
116C8×D4—32no—
117C8×D4—32no—
118C8×D4—32no—
119C8×D4—62no—
120#432—31no—
121#431—31no—
122#431—31no—
123#431C2×D311yescharacteristic
124#408—62no—
125C2×C22⋊C8—62no—
126C4×M16C2×D311yescharacteristic
127#335—32no—
128#334—32no—
129D4⋊C8—32no—
130D4⋊C8—32no—
131D4⋊C8C2×D312yescomplemented
132D4⋊C8C2×D312yescomplemented
133#322—31no—
134#322—31no—
135#322—31no—
136C82—31no—
137C23×D3—21no—
138C3×#41—82no—
139C2×C3×Q8—21no—
140C2×C3×D4—22no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4C22×C412yes—
144C22×C3×C4—22no—
145C22×C3×C4C22×C411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C22×Dic3—21no—
149C2×D12—21no—
150C2×D12—24no—
151C2×D12C22×C412yes—
152C2×C4×D3—22no—
153C2×C4×D3—22no—
154C2×C4×D3—82no—
155C3×M16—42no—
156C2×C3×C8—21no—
157C2×C3×C8—22no—
158C2×C3×C8—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4C22×C412yes—
161C3×C22⋊C4—22no—
162C3×C22⋊C4—22no—
163C3×C42C22×C411yescharacteristic
164C3×C42C2×D411yescharacteristic
165C3×C42#4111yescharacteristic
166C3×C42C2411yescharacteristic
167#213—24no—
168#213—24no—
169(C2×D3)⋊C4—22no—
170(C2×D3)⋊C4—22no—
171C4⋊Dic3—21no—
172C4⋊Dic3C22×C412yes—
173#208—82no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—82no—
176C2×C3⋊C8C22×C412yes—
177C2×C3⋊C8C22×C412yes—
178C2×C3⋊C8C2×D412yes—
179C2×C3⋊C8C2×D412yes—
180D3×C8—82no—
181D3×C8—82no—
182C2×#41—21no—
183C22×D4—31no—
184C23×C4—61no—
185C2×D8—32no—
186C2×D8—32no—
187C2×M16—21no—
188C2×M16—62no—
189C22×C8—62no—
190C22×C8—62no—
191C22×C8—62no—
192C4⋊Q8C4×D311yescharacteristic
193#127C4×D311yescharacteristic
194#124—21no—
195#121—21no—
196#121C4×D312yes—
197C4×D4—21no—
198C4×D4—32no—
199C4×D4—32no—
200C4×D4—61no—
201C4×D4—61no—
202C4×D4C22×D312yes—
203C2×C4⋊C4—31no—
204C2×C22⋊C4—61no—
205C2×C42C22×D311yescharacteristic
206#107—32no—
207#107—32no—
208C4⋊C8—32no—
209C4⋊C8—32no—
210C4⋊C8—32no—
211C4⋊C8C22×D312yes—
212#102—32no—
213#102—32no—
214#102—32no—
215#102—32no—
216C22⋊C8—21no—
217C22⋊C8—32no—
218C22⋊C8—32no—
219C22⋊C8—62no—
220C22⋊C8—62no—
221C22⋊C8—62no—
222C22⋊C8—62no—
223C22⋊C8—61no—
224#97—21no—
225#97—61no—
226C4×C8—21no—
227C4×C8—31no—
228C4×C8—31no—
229C4×C8—31no—
230C4×C8—31no—
231C4×C8C22×D311yescharacteristic
232C4×C8C22×D311yescharacteristic
233C23×C3—22no—
234C23×C3C4×D411yescharacteristic
235C22×D3—22no—
236C22×D3—24no—
237C22×D3—81no—
238C22×D3—81no—
239C3×Q8—81no—
240C3×D4—22no—
241C3×D4—22no—
242C3×D4—81no—
243C3×D4—81no—
244C3×D4—82no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4C4×D411yescharacteristic
252C2×C3×C4C4×D411yescharacteristic
253C2×C3×C4C4×D411yescharacteristic
254C2×C3×C4C23×C411yescharacteristic
255C2×C3×C4C22×D411yescharacteristic
256C2×C3×C4C2×#4111yescharacteristic
257C2×Dic3—22no—
258C2×Dic3—81no—
259D12—24no—
260D12—22no—
261D12#13114yes—
262C4×D3—42no—
263C4×D3—42no—
264C4×D3—82no—
265C3×C8—22no—
266C3×C8—41no—
267C3×C8—42no—
268C3⋊C8—22no—
269C3⋊C8—22no—
270C3⋊C8—22no—
271C3⋊C8—22no—
272C3⋊C8—42no—
273C24—61no—
274#41—82no—
275C2×Q8—21no—
276C2×D4—22no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—32no—
280C2×D4—64no—
281C2×D4—61no—
282C2×D4C2×C4×D312yes—
283C22×C4—22no—
284C22×C4—61no—
285C22×C4—62no—
286C22×C4—62no—
287C22×C4—242no—
288C22×C4C2×C4×D311yescharacteristic
289C22×C4D3×D411yescharacteristic
290C22×C4#23711yescharacteristic
291D8—64no—
292D8—64no—
293M16—42no—
294M16—242no—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—21no—
298C2×C8—32no—
299C2×C8—32no—
300C2×C8—32no—
301C2×C8—32no—
302C2×C8—62no—
303C2×C8—242no—
304C2×C8—242no—
305C2×C8—242no—
306C4⋊C4—21no—
307C4⋊C4—32no—
308C4⋊C4—61no—
309C4⋊C4C2×C4×D312yes—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—62no—
313C22⋊C4—62no—
314C42C2×C4×D311yescharacteristic
315C42D3×D411yescharacteristic
316C42#23711yescharacteristic
317C42C23×D311yescharacteristic
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—42no—
322C22×C3—42no—
323C22×C3—42no—
324C22×C3C2×C4×D411yescharacteristic
325C2×D3—24no—
326C2×D3—42no—
327C2×D3—42no—
328C2×D3—84no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—42no—
334C3×C4C2×#13112yes—
335C3×C4#57612yes—
336Dic3—42no—
337C23—22no—
338C23—64no—
339C23—62no—
340C23—241no—
341C23—241no—
342C23#67⋊C411yescharacteristic
343Q8—81no—
344D4—22no—
345D4—22no—
346D4—34no—
347D4—64no—
348D4—62no—
349D4—81no—
350D4—81no—
351D4—82no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—62no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—242no—
362C2×C4—241no—
363C2×C4#67⋊C411yescharacteristic
364C2×C4#67⋊C411yescharacteristic
365C2×C4#67⋊C411yescharacteristic
366C2×C4C22×C4×D311yescharacteristic
367C2×C4C2×D3×D411yescharacteristic
368C2×C4C2×#23711yescharacteristic
369C8—22no—
370C8—41no—
371C8—42no—
372C8—62no—
373C8—62no—
374C8—62no—
375C8—62no—
376C8—122no—
377C2×C3—41no—
378C2×C3—82no—
379C2×C3#292911yescharacteristic
380C2×C3#294811yescharacteristic
381C2×C3#295611yescharacteristic
382D3—44no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—64no—
390C22—122no—
391C22—122no—
392C22—244no—
393C22C2×#67⋊C411yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—122no—
400C4D3×#13112yes—
401C4#577412yes—
402C3#1792211yescharacteristic, complemented, Sylow
403C2—41no—
404C2—82no—
405C2—124no—
406C2#7865511yescharacteristic
407C2#7891511yescharacteristic
408C2#7898011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1484C2
socleC22×C3C2×C4×D4
prime radical#83298C2
cosocleC3×C42C24
hypercenter#1484D3
hypocenterC3#17922
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1484

Complements

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

NH ≅ G/Ntype
2. #83280C2semidirect
3. #79286C2semidirect
4. C3×#1484C2semidirect
5. #70747C2semidirect
6. #69855C2semidirect
7. #69848C2semidirect
8. #68158C2semidirect
9. #68153C2semidirect
10. #68151C2semidirect
11. #68148C2semidirect
24. #5012C22semidirect
26. D4×C3⋊C8C22semidirect
34. C3×D4⋊C8C22semidirect
35. C3×D4⋊C8C22semidirect
36. #4480C22semidirect
37. #4479C22semidirect
40. #4455C22semidirect
41. #4454C22semidirect
42. #4454C22semidirect
43. C8×C3⋊C8C22semidirect
53. #1484D3semidirect
131. D4⋊C8C2×D3semidirect
132. D4⋊C8C2×D3semidirect
402. C3#17922semidirect

Maximal subgroups

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

subgroupindex
#832982
#832802
#792862
C3×#14842
#707472
#698552
#698482
#681582
#681532
#681512
#681482
#179223