Finite Group Databasebeta

← #10765309

Subgroups — #10765309

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#79297C211yesmaximal, characteristic
2#79290C211yesmaximal, characteristic, complemented
3#79286C211yesmaximal, characteristic, complemented
4C3×#1453C211yesmaximal, characteristic, complemented
5#70751C211yesmaximal, characteristic, complemented
6#69855C211yesmaximal, characteristic, complemented
7#69853C211yesmaximal, characteristic, complemented
8#68081C212yesmaximal, complemented
9#68078C212yesmaximal, complemented
10#68077C212yesmaximal, complemented
11C8×D24C212yesmaximal, complemented
12#17922—31nomaximal, Sylow
13#5477C2211yescharacteristic
14C2×C4×D12C2211yescharacteristic
15C3×#431C2211yescharacteristic
16C3×#431C2211yescharacteristic
17C3×C4×M16C2211yescharacteristic
18#4926C2211yescharacteristic
19#4925C2211yescharacteristic
20#4846C412yes—
21#4843C412yes—
22#4836—22no—
23#4834—22no—
24#4736C412yes—
25#4732C412yes—
26#4727—22no—
27D3×C22⋊C8—22no—
28C4×D24C2212yes—
29C4×D24C2212yes—
30#4688C2212yescomplemented
31C8×D12C2212yes—
32C8×D12C2212yes—
33C8×D12C2212yescomplemented
34C3×#322C2211yescharacteristic, complemented
35C3×#322C2211yescharacteristic, complemented
36C3×#322C2211yescharacteristic, complemented
37C3×C82C2211yescharacteristic, complemented
38#4459C2212yes—
39#4459C2212yes—
40#4459C2212yescomplemented
41#4459C2212yescomplemented
42#4458C2212yescomplemented
43#4457C2212yescomplemented
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#1484—31no—
54#1453D311yescharacteristic, complemented
55C2×#235—21no—
56C22×D12C2×C411yescharacteristic
57C22×C4×D3—21no—
58C2×C3×M16—21no—
59C2×C3×C42C2311yescharacteristic
60#930—21no—
61#930C2×C412yes—
62C2×(C2×D3)⋊C4—21no—
63C2×C4⋊Dic3C2×C411yescharacteristic
64C2×D24C2×C412yes—
65C2×D24C2×C412yes—
66C2×ΓL(1,𝔽25)—22no—
67C2×D3×C8—22no—
68C2×D3×C8—22no—
69C2×D3×C8—22no—
70#875—21no—
71#874C2×C411yescharacteristic
72C4×D12—21no—
73C4×D12—21no—
74C4×D12—21no—
75C4×D12C2×C412yes—
76C4×D12C2312yes—
77C4×D12C2312yes—
78#869C2×C411yescharacteristic
79C3×C4⋊C8C2×C412yes—
80C3×C4⋊C8D412yes—
81C3×C22⋊C8C2×C412yes—
82C3×C22⋊C8D412yes—
83C3×#97—21no—
84C3×C4×C8C2311yescharacteristic
85C3×C4×C8C2311yescharacteristic
86C3×C4×C8C2311yescharacteristic
87C3×C4×C8C2311yescharacteristic
88C3×C4×C8C2311yescharacteristic
89C3×C4×C8C2311yescharacteristic
90#830—21no—
91#830—21no—
92#821C2×C412yes—
93#821C2×C412yes—
94#821C2×C412yes—
95#821C2×C412yes—
96(C2×D3)⋊C8—22no—
97(C2×D3)⋊C8—22no—
98(C2×D3)⋊C8—22no—
99(C2×D3)⋊C8—22no—
100#818C2×C412yes—
101#818C2×C412yes—
102C4⋊(C3⋊C8)C2312yes—
103C4⋊(C3⋊C8)C2312yes—
104#803—21no—
105C4×C3⋊C8—21no—
106#532—31no—
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#431C2×D311yescharacteristic
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⋊C8—32no—
132D4⋊C8—32no—
133#322C2×D311yescharacteristic, complemented
134#322C2×D311yescharacteristic, complemented
135#322C2×D311yescharacteristic, complemented
136C82C2×D311yescharacteristic, complemented
137C23×D3—21no—
138C22×C3×C4C22×C411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C2×#67—22no—
142C22×Dic3—21no—
143#235—82no—
144C2×D12—21no—
145C2×D12—24no—
146C2×D12—21no—
147C2×D12—21no—
148C2×D12C22×C412yes—
149C2×D12C22×C412yes—
150C2×C4×D3—22no—
151C2×C4×D3—22no—
152C2×C4×D3—22no—
153C2×C4×D3—82no—
154C2×Dic6—21no—
155C3×M16—42no—
156C2×C3×C8—21no—
157C2×C3×C8—21no—
158C2×C3×C8C22×C412yes—
159C2×C3×C8C22×C412yes—
160C2×C3×C8C2×D412yes—
161C2×C3×C8C2×D412yes—
162C3×C42C22×C411yescharacteristic
163C3×C42C2×D411yescharacteristic
164C3×C42#4111yescharacteristic
165C3×C42C2411yescharacteristic
166(C2×D3)⋊C4—22no—
167(C2×D3)⋊C4—22no—
168(C2×D3)⋊C4—22no—
169(C2×D3)⋊C4—22no—
170C4⋊Dic3—21no—
171C4⋊Dic3—21no—
172C4⋊Dic3C22×C412yes—
173C4⋊Dic3C22×C412yes—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—22no—
176D24—24no—
177D24—24no—
178ΓL(1,𝔽25)—82no—
179D3×C8—82no—
180D3×C8—82no—
181D3×C8—82no—
182C2×#41—61no—
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⋊Q8—31no—
193#127—31no—
194#124—61no—
195#121—32no—
196#121—61no—
197C4×D4—32no—
198C4×D4—32no—
199C4×D4—32no—
200C4×D4—61no—
201C4×D4—61no—
202C4×D4—61no—
203C2×C4⋊C4—31no—
204C2×C22⋊C4—61no—
205C2×C42C22×D311yescharacteristic
206#107—32no—
207#107—32no—
208C4⋊C8—32no—
209C4⋊C8—32no—
210C4⋊C8C4×D312yes—
211C4⋊C8D1212yes—
212#102—32no—
213#102—32no—
214#102—32no—
215#102—32no—
216C22⋊C8—61no—
217C22⋊C8—61no—
218C22⋊C8—62no—
219C22⋊C8—62no—
220C22⋊C8—62no—
221C22⋊C8—62no—
222C22⋊C8C4×D312yes—
223C22⋊C8D1212yes—
224#97—21no—
225#97—61no—
226C4×C8—61no—
227C4×C8C22×D311yescharacteristic
228C4×C8C22×D311yescharacteristic
229C4×C8C22×D311yescharacteristic
230C4×C8C22×D311yescharacteristic
231C4×C8C22×D311yescharacteristic
232C4×C8C22×D311yescharacteristic
233C23×C3C4×D411yescharacteristic
234C22×D3—22no—
235C22×D3—24no—
236C22×D3—22no—
237C22×D3—81no—
238C22×D3—81no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—41no—
242C2×C3×C4C4×D411yescharacteristic
243C2×C3×C4C4×D411yescharacteristic
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4C23×C411yescharacteristic
246C2×C3×C4C22×D411yescharacteristic
247C2×C3×C4C2×#4111yescharacteristic
248#67—82no—
249C2×Dic3—22no—
250C2×Dic3—22no—
251C2×Dic3—81no—
252D12—24no—
253D12—22no—
254D12—22no—
255D12—22no—
256D12—81no—
257D12—81no—
258D12#13114yes—
259C4×D3—42no—
260C4×D3—42no—
261C4×D3—42no—
262C4×D3—42no—
263C4×D3—82no—
264Dic6—81no—
265C3×C8—22no—
266C3×C8—22no—
267C3×C8—22no—
268C3×C8—22no—
269C3×C8—22no—
270C3×C8—41no—
271C3⋊C8—42no—
272C3⋊C8—42no—
273C24—61no—
274#41—242no—
275C2×Q8—61no—
276C2×D4—32no—
277C2×D4—32no—
278C2×D4—64no—
279C2×D4—62no—
280C2×D4—61no—
281C2×D4—61no—
282C2×D4—61no—
283C22×C4—61no—
284C22×C4—62no—
285C22×C4—62no—
286C22×C4—62no—
287C22×C4—242no—
288C22×C4C2×C4×D311yescharacteristic
289C22×C4C2×D1211yescharacteristic
290C22×C4#23511yescharacteristic
291D8—64no—
292D8—64no—
293M16—42no—
294M16—242no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—62no—
298C2×C8—62no—
299C2×C8—242no—
300C2×C8—242no—
301C2×C8—242no—
302C2×C8C2×C4×D312yes—
303C2×C8C2×C4×D312yes—
304C2×C8C2×D1212yes—
305C2×C8C2×D1212yes—
306C4⋊C4—32no—
307C4⋊C4—32no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C22⋊C4—62no—
311C22⋊C4—62no—
312C22⋊C4—62no—
313C22⋊C4—62no—
314C42C2×C4×D311yescharacteristic
315C42C2×D1211yescharacteristic
316C42#23511yescharacteristic
317C42C23×D311yescharacteristic
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3C2×C4×D411yescharacteristic
322C2×D3—24no—
323C2×D3—42no—
324C2×D3—42no—
325C2×D3—42no—
326C2×D3—42no—
327C2×D3—42no—
328C2×D3—84no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4C2×#13112yes—
334C3×C4#57612yes—
335Dic3—42no—
336Dic3—42no—
337C23—64no—
338C23—62no—
339C23—62no—
340C23—241no—
341C23—241no—
342C23C4×D1211yescharacteristic
343Q8—241no—
344D4—34no—
345D4—64no—
346D4—62no—
347D4—62no—
348D4—62no—
349D4—241no—
350D4—241no—
351D4—242no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—62no—
356C2×C4—62no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—242no—
362C2×C4—241no—
363C2×C4C4×D1211yescharacteristic
364C2×C4C4×D1211yescharacteristic
365C2×C4C4×D1211yescharacteristic
366C2×C4C22×C4×D311yescharacteristic
367C2×C4C22×D1211yescharacteristic
368C2×C4C2×#23511yescharacteristic
369C8—22no—
370C8—22no—
371C8—22no—
372C8—22no—
373C8—22no—
374C8—41no—
375C8—122no—
376C8—122no—
377C2×C3—41no—
378C2×C3#292911yescharacteristic
379C2×C3#294811yescharacteristic
380C2×C3#295611yescharacteristic
381D3—44no—
382D3—82no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—64no—
387C22—122no—
388C22—122no—
389C22—122no—
390C22—122no—
391C22—122no—
392C22—244no—
393C22C2×C4×D1211yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—122no—
399C4—122no—
400C4D3×#13112yes—
401C4#575212yes—
402C3#1792211yescharacteristic, complemented, Sylow
403C2—41no—
404C2—124no—
405C2—242no—
406C2#7854111yescharacteristic
407C2#7855311yescharacteristic
408C2#7856111yescharacteristic
subgroupG/H
centerC2×C4C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1453C2
socleC22×C3C2×C4×D4
prime radical#79297C2
cosocleC3×C42C24
hypercenter#1453D3
hypocenterC3#17922
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1453

Complements

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

NH ≅ G/Ntype
2. #79290C2semidirect
3. #79286C2semidirect
4. C3×#1453C2semidirect
5. #70751C2semidirect
6. #69855C2semidirect
7. #69853C2semidirect
8. #68081C2semidirect
9. #68078C2semidirect
10. #68077C2semidirect
11. C8×D24C2semidirect
30. #4688C22semidirect
33. C8×D12C22semidirect
34. C3×#322C22semidirect
35. C3×#322C22semidirect
36. C3×#322C22semidirect
37. C3×C82C22semidirect
40. #4459C22semidirect
41. #4459C22semidirect
42. #4458C22semidirect
43. #4457C22semidirect
54. #1453D3semidirect
133. #322C2×D3semidirect
134. #322C2×D3semidirect
135. #322C2×D3semidirect
136. C82C2×D3semidirect
402. C3#17922semidirect

Maximal subgroups

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

subgroupindex
#792972
#792902
#792862
C3×#14532
#707512
#698552
#698532
#680812
#680782
#680772
C8×D242
#179223