Finite Group Databasebeta

← #10765307

Subgroups — #10765307

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
1C3×#2972C211yesmaximal, characteristic, complemented
2#82308C211yesmaximal, characteristic
3#79290C211yesmaximal, characteristic, complemented
4#71599C211yesmaximal, characteristic, complemented
5#70844C212yesmaximal, complemented
6#70841C212yesmaximal, complemented
7#70840C212yesmaximal, complemented
8C8×#213C212yesmaximal, complemented
9#70761C211yesmaximal, characteristic, complemented
10#70747C211yesmaximal, characteristic, complemented
11#70743C211yesmaximal, characteristic, complemented
12#17922—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14#5477C2211yescharacteristic
15C3×C8×D4—22no—
16C3×C8×D4C2212yes—
17C3×C8×D4C2212yes—
18C3×#431C2211yescharacteristic
19C3×#431C2211yescharacteristic
20C2×C3×C22⋊C8—22no—
21#5217C412yes—
22#5015C412yes—
23C4×#213C2212yes—
24C4×#213C2212yes—
25#5007C412yes—
26#4967C412yes—
27#4926C2211yescharacteristic
28#4925C2211yescharacteristic
29C4×#208C2211yescharacteristic
30#4836—22no—
31#4727—22no—
32#4688C2212yescomplemented
33C8×D12C2212yescomplemented
34D4⋊(C3⋊C8)C2212yes—
35D4⋊(C3⋊C8)C2212yes—
36#4483C2212yescomplemented
37#4483C2212yescomplemented
38#4480C2212yescomplemented
39#4479C2212yescomplemented
40#4455C2211yescharacteristic, complemented
41#4454C2211yescharacteristic, complemented
42#4454C2211yescharacteristic, complemented
43C8×C3⋊C8C2211yescharacteristic, complemented
44#3100—31no—
45#2981—31no—
46#2972D311yescharacteristic, complemented
47#1586—32no—
48#1581—32no—
49#1580—32no—
50C8×D8—32no—
51#1572—31no—
52#1490—31no—
53#1484—31no—
54#1453—31no—
55C22×C3×D4C2×C411yescharacteristic
56C23×C3×C4—21no—
57C2×#235—21no—
58C22×C3×C8—22no—
59C22×C3×C8—22no—
60C3×C4×D4—21no—
61C3×C4×D4—21no—
62C3×C4×D4C2×C412yes—
63C3×C4×D4C2312yes—
64C2×C3×C4⋊C4C2×C411yescharacteristic
65C2×C3×C22⋊C4—21no—
66C2×C3×C42C2311yescharacteristic
67C2×#213C2×C412yes—
68C2×#213C2×C412yes—
69#930—21no—
70#930C2×C412yes—
71C2×#208—21no—
72C2×ΓL(1,𝔽25)—22no—
73C2×D3×C8—22no—
74#875—21no—
75#874C2×C411yescharacteristic
76C4×D12—21no—
77C4×D12C2312yes—
78#869C2×C411yescharacteristic
79C3×C4⋊C8D412yes—
80C3×C4⋊C8C2312yes—
81C3×C22⋊C8—21no—
82C3×C22⋊C8—22no—
83C3×C22⋊C8—22no—
84C3×C22⋊C8D412yes—
85C3×C4×C8—21no—
86C3×C4×C8C2311yescharacteristic
87C3×C4×C8C2311yescharacteristic
88#832C2×C412yes—
89#832C2×C412yes—
90#830—21no—
91#830C2×C412yes—
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)C2312yes—
100#803—21no—
101#803—21no—
102C4×C3⋊C8C2311yescharacteristic
103C4×C3⋊C8C2311yescharacteristic
104C4×C3⋊C8C2311yescharacteristic
105C4×C3⋊C8C2311yescharacteristic
106#532—31no—
107C2×C4×D4C2×D311yescharacteristic
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—22no—
117C8×D4—32no—
118C8×D4C2×D312yes—
119C8×D4C2×D312yes—
120#432—31no—
121#431—31no—
122#431C2×D311yescharacteristic
123#431C2×D311yescharacteristic
124#408—62no—
125C2×C22⋊C8—22no—
126C4×M16—31no—
127#335—32no—
128#334—32no—
129D4⋊C8—32no—
130D4⋊C8—32no—
131D4⋊C8—32no—
132D4⋊C8—32no—
133#322—31no—
134#322—31no—
135#322—31no—
136C82—31no—
137C24×C3—21no—
138C2×C3×D4—24no—
139C2×C3×D4—21no—
140C2×C3×D4C22×C412yes—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—82no—
145C22×C3×C4C22×C411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C2×#67—22no—
149#235—82no—
150C2×D12—21no—
151C2×D12—21no—
152C2×D12C22×C412yes—
153C2×C4×D3—22no—
154C2×Dic6—21no—
155C2×C3×C8—22no—
156C2×C3×C8—82no—
157C2×C3×C8—82no—
158C2×C3×C8C2×D412yes—
159C2×C3×C8C2×D412yes—
160C3×C4⋊C4—21no—
161C3×C4⋊C4C22×C412yes—
162C3×C22⋊C4—22no—
163C3×C22⋊C4—22no—
164C3×C42C22×C411yescharacteristic
165C3×C42C2×D411yescharacteristic
166C3×C42#4111yescharacteristic
167C3×C42C2411yescharacteristic
168#213—24no—
169#213—24no—
170(C2×D3)⋊C4—22no—
171(C2×D3)⋊C4—22no—
172C4⋊Dic3—21no—
173C4⋊Dic3C22×C412yes—
174#208—42no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—21no—
178C2×C3⋊C8C22×C412yes—
179C2×C3⋊C8C22×C412yes—
180ΓL(1,𝔽25)—82no—
181D3×C8—82no—
182C2×#41—61no—
183C22×D4C4×D311yescharacteristic
184C23×C4—21no—
185C2×D8—32no—
186C2×D8—32no—
187C2×M16—61no—
188C2×M16—62no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—62no—
192C4⋊Q8—31no—
193#127—31no—
194#124—61no—
195#121—32no—
196#121—61no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—32no—
200C4×D4—61no—
201C4×D4C4×D312yes—
202C4×D4C22×D312yes—
203C2×C4⋊C4C4×D311yescharacteristic
204C2×C22⋊C4—21no—
205C2×C42C22×D311yescharacteristic
206#107—32no—
207#107—32no—
208C4⋊C8—32no—
209C4⋊C8—32no—
210C4⋊C8#6712yes—
211C4⋊C8C22×D312yes—
212#102—32no—
213#102—32no—
214#102—32no—
215#102—32no—
216C22⋊C8—21no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8—32no—
220C22⋊C8—61no—
221C22⋊C8—62no—
222C22⋊C8—62no—
223C22⋊C8#6712yes—
224#97—61no—
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—24no—
234C23×C3—22no—
235C23×C3—81no—
236C23×C3—81no—
237C23×C3C4×D411yescharacteristic
238C22×D3—22no—
239C3×D4—24no—
240C3×D4—22no—
241C3×D4#13114yes—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—41no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—82no—
249C2×C3×C4—81no—
250C2×C3×C4C4×D411yescharacteristic
251C2×C3×C4C4×D411yescharacteristic
252C2×C3×C4C4×D411yescharacteristic
253C2×C3×C4C23×C411yescharacteristic
254C2×C3×C4C22×D411yescharacteristic
255C2×C3×C4C2×#4111yescharacteristic
256#67—82no—
257C2×Dic3—22no—
258D12—22no—
259D12—22no—
260D12—81no—
261D12—81no—
262C4×D3—42no—
263C4×D3—42no—
264Dic6—81no—
265C3×C8—22no—
266C3×C8—22no—
267C3×C8—42no—
268C3⋊C8—22no—
269C3⋊C8—22no—
270C3⋊C8—22no—
271C3⋊C8—41no—
272C3⋊C8—42no—
273C24—21no—
274#41—242no—
275C2×Q8—61no—
276C2×D4—24no—
277C2×D4—21no—
278C2×D4—32no—
279C2×D4—62no—
280C2×D4—61no—
281C2×D4—61no—
282C2×D4C2×C4×D312yes—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—62no—
287C22×C4—82no—
288C22×C4C2×C4×D311yescharacteristic
289C22×C4#23511yescharacteristic
290C22×C4C2×#6711yescharacteristic
291D8—64no—
292D8—64no—
293M16—122no—
294M16—242no—
295C2×C8—22no—
296C2×C8—32no—
297C2×C8—32no—
298C2×C8—61no—
299C2×C8—61no—
300C2×C8—62no—
301C2×C8—82no—
302C2×C8—82no—
303C2×C8—242no—
304C2×C8C2×#6712yes—
305C2×C8C2×#6712yes—
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
315C42#23511yescharacteristic
316C42C2×#6711yescharacteristic
317C42C23×D311yescharacteristic
318C22×C3—21no—
319C22×C3—24no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—42no—
323C22×C3—42no—
324C22×C3—84no—
325C22×C3C2×C4×D411yescharacteristic
326C2×D3—42no—
327C2×D3—42no—
328C2×D3—42no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—42no—
334C3×C4C2×#13112yes—
335C3×C4#57612yes—
336Dic3—42no—
337C23—24no—
338C23—22no—
339C23—62no—
340C23—81no—
341C23—81no—
342C23C4×#6711yescharacteristic
343Q8—241no—
344D4—24no—
345D4—22no—
346D4—62no—
347D4—62no—
348D4—241no—
349D4—241no—
350D4—242no—
351D4#90114yes—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—62no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4C4×#6711yescharacteristic
364C2×C4C4×#6711yescharacteristic
365C2×C4C4×#6711yescharacteristic
366C2×C4C22×C4×D311yescharacteristic
367C2×C4C2×#23511yescharacteristic
368C2×C4C22×#6711yescharacteristic
369C8—22no—
370C8—22no—
371C8—42no—
372C8—62no—
373C8—62no—
374C8—62no—
375C8—121no—
376C8—122no—
377C2×C3—41no—
378C2×C3—44no—
379C2×C3#292911yescharacteristic
380C2×C3#294811yescharacteristic
381C2×C3#295611yescharacteristic
382D3—82no—
383C22—21no—
384C22—24no—
385C22—21no—
386C22—21no—
387C22—42no—
388C22—42no—
389C22—84no—
390C22—122no—
391C22—122no—
392C22—122no—
393C22C2×C4×#6711yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—122no—
400C4C2×#90112yes—
401C4#583512yes—
402C3#1792211yescharacteristic, complemented, Sylow
403C2—41no—
404C2—44no—
405C2—242no—
406C2#8253811yescharacteristic
407C2#8373411yescharacteristic
408C2#8381111yescharacteristic
subgroupG/H
centerC2×C4C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#2972C2
socleC22×C3C2×C4×D4
prime radical#82308C2
cosocleC3×C42C24
hypercenter#2972D3
hypocenterC3#17922
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. C3×#2972C2semidirect
3. #79290C2semidirect
4. #71599C2semidirect
5. #70844C2semidirect
6. #70841C2semidirect
7. #70840C2semidirect
8. C8×#213C2semidirect
9. #70761C2semidirect
10. #70747C2semidirect
11. #70743C2semidirect
32. #4688C22semidirect
33. C8×D12C22semidirect
36. #4483C22semidirect
37. #4483C22semidirect
38. #4480C22semidirect
39. #4479C22semidirect
40. #4455C22semidirect
41. #4454C22semidirect
42. #4454C22semidirect
43. C8×C3⋊C8C22semidirect
46. #2972D3semidirect
402. C3#17922semidirect

Maximal subgroups

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

subgroupindex
C3×#29722
#823082
#792902
#715992
#708442
#708412
#708402
C8×#2132
#707612
#707472
#707432
#179223