Finite Group Databasebeta

← #10765320

Subgroups — #10765320

Order 768.

626 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79309C211yesmaximal, characteristic
2#79290C211yesmaximal, characteristic
3#79263C211yesmaximal, characteristic, complemented
4C3×#1453C211yesmaximal, characteristic, complemented
5#70751C211yesmaximal, characteristic, complemented
6#69857C211yesmaximal, characteristic, complemented
7#69853C211yesmaximal, characteristic
8#68088C212yesmaximal, complemented
9#68085C212yesmaximal, complemented
10#68076C212yesmaximal, complemented
11C8×#204C212yesmaximal, complemented
12#17923—31nomaximal, Sylow
13#5477C2211yescharacteristic
14C2×C4×Dic6C2211yescharacteristic
15C3×#431C2211yescharacteristic
16C3×#431C2211yescharacteristic
17C3×C4×M16C2211yescharacteristic
18#4926C2211yescharacteristic
19#4925C2211yescharacteristic
20#4851C412yes—
21#4848C412yes—
22#4836—22no—
23#4830—22no—
24#4738C412yes—
25#4735C412yes—
26#4727—22no—
27#4718—22no—
28C4×#204C2212yes—
29C4×#204C2212yes—
30#4688C2212yes—
31C8×D12C2212yes—
32C8×Dic6C2212yes—
33C8×Dic6C2212yes—
34C3×#322C2211yescharacteristic
35C3×#322C2211yescharacteristic
36C3×#322C2211yescharacteristic
37C3×C82C2211yescharacteristic
38#4459C2212yes—
39#4459C2212yes—
40#4458C2212yes—
41#4457C2212yes—
42#4456C2212yes—
43#4456C2212yes—
44#3103—31no—
45#2982—31no—
46#2981—31no—
47#1588—32no—
48#1583—32no—
49SD16⋊C8—32no—
50C8×SD16—32no—
51#1572—31no—
52#1491—31no—
53#1484—31no—
54#1453D311yescharacteristic, complemented
55C2×#235—21no—
56C22×Dic6C2×C411yescharacteristic
57C2×C3×M16—21no—
58C2×C3×C42C2311yescharacteristic
59#930—21no—
60#930C2×C412yes—
61C2×C4⋊Dic3C2×C411yescharacteristic
62C2×Dic3⋊C4—21no—
63C2×C4×Dic3—21no—
64C2×#204C2×C412yes—
65C2×#204C2×C412yes—
66C2×ΓL(1,𝔽25)—22no—
67C2×D3×C8—22no—
68#875—21no—
69#874C2×C411yescharacteristic
70C4×D12—21no—
71C4×D12C2312yes—
72#869C2×C411yescharacteristic
73C4×Dic6—21no—
74C4×Dic6—21no—
75C4×Dic6C2×C412yes—
76C4×Dic6C2312yes—
77C3×C4⋊C8C2×C412yes—
78C3×C4⋊C8D412yes—
79C3×C22⋊C8C2×C412yes—
80C3×C22⋊C8D412yes—
81C3×#97—21no—
82C3×C4×C8C2311yescharacteristic
83C3×C4×C8C2311yescharacteristic
84C3×C4×C8C2311yescharacteristic
85C3×C4×C8C2311yescharacteristic
86C3×C4×C8C2311yescharacteristic
87C3×C4×C8C2311yescharacteristic
88#830—21no—
89#830—21no—
90#821C2×C412yes—
91#821C2×C412yes—
92(C2×D3)⋊C8—22no—
93(C2×D3)⋊C8—22no—
94#817C2×C412yes—
95#817C2×C412yes—
96#816C2×C412yes—
97#816C2×C412yes—
98Dic3⋊C8—22no—
99Dic3⋊C8—22no—
100C8×Dic3—22no—
101C8×Dic3—22no—
102C4⋊(C3⋊C8)C2312yes—
103C4⋊(C3⋊C8)C2312yes—
104#803—21no—
105C4×C3⋊C8—21no—
106#532—31no—
107C2×C4×Q8—31no—
108#481—32no—
109#475—32no—
110#460—32no—
111#448—32no—
112C8×Q8—32no—
113C8×Q8—32no—
114C8×Q8—62no—
115C4×SD16—32no—
116C4×SD16—32no—
117#436—32no—
118#435—62no—
119C8×D4—32no—
120#432—31no—
121#431—31no—
122#431—62no—
123#431C2×D311yescharacteristic
124#431C2×D311yescharacteristic
125#408—62no—
126C4×M16C2×D311yescharacteristic
127#335—32no—
128#334—32no—
129Q8⋊C8—32no—
130Q8⋊C8—32no—
131D4⋊C8—32no—
132D4⋊C8—32no—
133#322C2×D311yescharacteristic
134#322C2×D311yescharacteristic
135#322C2×D311yescharacteristic
136C82C2×D311yescharacteristic
137C22×C3×C4C22×C411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C2×#67—22no—
141C22×Dic3—21no—
142C22×Dic3—21no—
143#235—82no—
144C2×D12—21no—
145C2×D12—21no—
146C2×D12C22×C412yes—
147C2×C4×D3—22no—
148C2×Dic6—21no—
149C2×Dic6—24no—
150C2×Dic6—21no—
151C2×Dic6C22×C412yes—
152C3×M16—42no—
153C2×C3×C8—21no—
154C2×C3×C8—21no—
155C2×C3×C8C22×C412yes—
156C2×C3×C8C22×C412yes—
157C2×C3×C8C2×D412yes—
158C2×C3×C8C2×D412yes—
159C3×C42C22×C411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42#4111yescharacteristic
162C3×C42C2411yescharacteristic
163(C2×D3)⋊C4—22no—
164(C2×D3)⋊C4—22no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3C22×C412yes—
168C4⋊Dic3C22×C412yes—
169Dic3⋊C4—22no—
170Dic3⋊C4—22no—
171C4×Dic3—22no—
172C4×Dic3—22no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—22no—
175#204—24no—
176#204—24no—
177ΓL(1,𝔽25)—82no—
178D3×C8—82no—
179C2×#41—61no—
180C22×Q8—31no—
181C2×SD16—32no—
182C2×SD16—32no—
183C2×M16—21no—
184C2×M16—62no—
185C22×C8—62no—
186C4⋊Q8—31no—
187#127—31no—
188#124—61no—
189#121—32no—
190#121—61no—
191C4×Q8—32no—
192C4×Q8—32no—
193C4×Q8—61no—
194C4×Q8—61no—
195C4×D4—32no—
196C4×D4—61no—
197C2×C4⋊C4—31no—
198C2×C4⋊C4—61no—
199C2×C42—61no—
200C2×C42C22×D311yescharacteristic
201#106—32no—
202#106—32no—
203C4⋊C8—32no—
204C4⋊C8—32no—
205C4⋊C8—62no—
206C4⋊C8—62no—
207C4⋊C8C4×D312yes—
208C4⋊C8D1212yes—
209#103—32no—
210#103—32no—
211#102—32no—
212#102—32no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8—62no—
216C22⋊C8—62no—
217C22⋊C8C4×D312yes—
218C22⋊C8D1212yes—
219#97—21no—
220#97—61no—
221C4×C8—62no—
222C4×C8—62no—
223C4×C8—61no—
224C4×C8C22×D311yescharacteristic
225C4×C8C22×D311yescharacteristic
226C4×C8C22×D311yescharacteristic
227C4×C8C22×D311yescharacteristic
228C4×C8C22×D311yescharacteristic
229C4×C8C22×D311yescharacteristic
230C23×C3C4×D411yescharacteristic
231C22×D3—22no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—41no—
235C2×C3×C4C4×D411yescharacteristic
236C2×C3×C4C4×D411yescharacteristic
237C2×C3×C4C4×D411yescharacteristic
238C2×C3×C4C23×C411yescharacteristic
239C2×C3×C4C22×D411yescharacteristic
240C2×C3×C4C2×#4111yescharacteristic
241#67—82no—
242C2×Dic3—24no—
243C2×Dic3—22no—
244C2×Dic3—22no—
245C2×Dic3—22no—
246C2×Dic3—81no—
247D12—22no—
248D12—22no—
249D12—81no—
250D12—81no—
251C4×D3—42no—
252C4×D3—42no—
253Dic6—24no—
254Dic6—22no—
255Dic6—81no—
256Dic6#13114yes—
257C3×C8—22no—
258C3×C8—22no—
259C3×C8—22no—
260C3×C8—22no—
261C3×C8—22no—
262C3×C8—41no—
263C3⋊C8—42no—
264C3⋊C8—42no—
265#41—242no—
266C2×Q8—32no—
267C2×Q8—64no—
268C2×Q8—61no—
269C2×Q8—61no—
270C2×D4—32no—
271C2×D4—62no—
272C2×D4—61no—
273C2×D4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—62no—
277C22×C4C2×C4×D311yescharacteristic
278C22×C4C2×D1211yescharacteristic
279C22×C4#23511yescharacteristic
280SD16—64no—
281SD16—64no—
282M16—42no—
283M16—242no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—62no—
287C2×C8—62no—
288C2×C8—242no—
289C2×C8C2×C4×D312yes—
290C2×C8C2×C4×D312yes—
291C2×C8C2×D1212yes—
292C2×C8C2×D1212yes—
293C4⋊C4—32no—
294C4⋊C4—32no—
295C4⋊C4—61no—
296C4⋊C4—62no—
297C4⋊C4—61no—
298C4⋊C4—62no—
299C22⋊C4—62no—
300C22⋊C4—62no—
301C42—62no—
302C42—62no—
303C42C2×C4×D311yescharacteristic
304C42C2×D1211yescharacteristic
305C42#23511yescharacteristic
306C42C23×D311yescharacteristic
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3C2×C4×D411yescharacteristic
311C2×D3—42no—
312C2×D3—42no—
313C2×D3—42no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4C2×#13112yes—
319C3×C4#57712yes—
320Dic3—24no—
321Dic3—42no—
322Dic3—42no—
323C23—62no—
324C23C4×D1211yescharacteristic
325Q8—34no—
326Q8—64no—
327Q8—62no—
328Q8—241no—
329D4—62no—
330D4—62no—
331D4—241no—
332D4—241no—
333D4—242no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—64no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—241no—
344C2×C4C4×D1211yescharacteristic
345C2×C4C4×D1211yescharacteristic
346C2×C4C4×D1211yescharacteristic
347C2×C4C22×C4×D311yescharacteristic
348C2×C4C22×D1211yescharacteristic
349C2×C4C2×#23511yescharacteristic
350C8—22no—
351C8—22no—
352C8—22no—
353C8—22no—
354C8—22no—
355C8—41no—
356C8—122no—
357C8—122no—
358C2×C3—41no—
359C2×C3#292911yescharacteristic
360C2×C3#294911yescharacteristic
361C2×C3#295611yescharacteristic
362D3—82no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—122no—
367C22—122no—
368C22—122no—
369C22C2×C4×D1211yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—64no—
375C4—122no—
376C4—122no—
377C4D3×#13112yes—
378C4#575112yes—
379C3#1792311yescharacteristic, complemented, Sylow
380C2—41no—
381C2—242no—
382C2#7854111yescharacteristic
383C2#7855311yescharacteristic
384C2#7857211yescharacteristic
subgroupG/H
centerC2×C4C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1453C2
socleC22×C3C2×C4×D4
prime radical#5477C22
cosocleC3×C42C24
hypercenter#1453D3
hypocenterC3#17923
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
3. #79263C2semidirect
4. C3×#1453C2semidirect
5. #70751C2semidirect
6. #69857C2semidirect
8. #68088C2semidirect
9. #68085C2semidirect
10. #68076C2semidirect
11. C8×#204C2semidirect
54. #1453D3semidirect
379. C3#17923semidirect

Maximal subgroups

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

subgroupindex
#793092
#792902
#792632
C3×#14532
#707512
#698572
#698532
#680882
#680852
#680762
C8×#2042
#179233