Finite Group Databasebeta

← #10765316

Subgroups — #10765316

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#79338C211yesmaximal, characteristic
2#79309C211yesmaximal, characteristic
3#79263C211yesmaximal, characteristic, complemented
4C3×#1572C211yesmaximal, characteristic, complemented
5#70802C211yesmaximal, characteristic, complemented
6#70761C211yesmaximal, characteristic
7#70743C211yesmaximal, characteristic, complemented
8#68562C212yesmaximal, complemented
9#68558C212yesmaximal, complemented
10#68520C212yesmaximal, complemented
11#68516C212yesmaximal, complemented
12#17923—31nomaximal, Sylow
13#5477C2211yescharacteristic
14C2×C4×Dic6C2211yescharacteristic
15C3×#432C2211yescharacteristic
16C3×#431C2211yescharacteristic
17C3×#431C2211yescharacteristic
18#4925C2211yescharacteristic
19C4×#208C2211yescharacteristic
20#4851C412yes—
21#4848C412yes—
22#4838C2212yes—
23#4834C2212yes—
24#4830C2212yes—
25#4830C2212yes—
26#4738C412yes—
27#4735C412yes—
28#4727—22no—
29#4718—22no—
30#4706—22no—
31C4×#204C2212yes—
32C4×#204C2212yes—
33C8×Dic6—22no—
34C3×#335C2212yes—
35C3×#334C2212yes—
36#4484C2212yes—
37#4484C2212yes—
38#4483C2212yes—
39#4483C2212yes—
40#4455C2211yescharacteristic
41#4454C2211yescharacteristic
42#4454C2211yescharacteristic
43C8×C3⋊C8C2211yescharacteristic
44#3103—31no—
45#2982—31no—
46#2981—31no—
47#1588—32no—
48#1583—32no—
49SD16⋊C8—32no—
50C8×SD16—32no—
51#1572D311yescharacteristic, complemented
52#1491—31no—
53#1484—31no—
54#1453—31no—
55C2×#235—21no—
56C22×Dic6C2×C411yescharacteristic
57C2×C3×C42C2311yescharacteristic
58#930—21no—
59#930C2×C412yes—
60C2×C4⋊Dic3C2×C411yescharacteristic
61C2×Dic3⋊C4—21no—
62C2×C4×Dic3—21no—
63C2×#208—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⋊C8C2312yes—
79C3×C4⋊C8C2312yes—
80C3×C22⋊C8—21no—
81C3×C22⋊C8—21no—
82C3×C22⋊C8C2×C412yes—
83C3×#97—21no—
84C3×C4×C8—21no—
85C3×C4×C8C2311yescharacteristic
86C3×C4×C8C2311yescharacteristic
87#830D412yes—
88#821C2×C412yes—
89#821C2×C412yes—
90(C2×D3)⋊C8—22no—
91(C2×D3)⋊C8—22no—
92#817C2×C412yes—
93#817C2×C412yes—
94#816C2×C412yes—
95#816C2×C412yes—
96Dic3⋊C8—22no—
97Dic3⋊C8—22no—
98C8×Dic3—22no—
99C8×Dic3—22no—
100C4⋊(C3⋊C8)D412yes—
101#803—21no—
102C4×C3⋊C8C2311yescharacteristic
103C4×C3⋊C8C2311yescharacteristic
104C4×C3⋊C8C2311yescharacteristic
105C4×C3⋊C8C2311yescharacteristic
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#432C2×D311yescharacteristic
121#431—31no—
122#431—62no—
123#431C2×D311yescharacteristic
124#431C2×D311yescharacteristic
125#408—62no—
126C4×M16—31no—
127#335C2×D312yes—
128#334C2×D312yes—
129Q8⋊C8—32no—
130Q8⋊C8—32no—
131D4⋊C8—32no—
132D4⋊C8—32no—
133#322—31no—
134#322—31no—
135#322—31no—
136C82—31no—
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—
152C2×C3×C8—22no—
153C2×C3×C8—22no—
154C2×C3×C8C22×C412yes—
155C2×C3×C8C22×C412yes—
156C3×C42C22×C411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42#4111yescharacteristic
159C3×C42C2411yescharacteristic
160(C2×D3)⋊C4—22no—
161(C2×D3)⋊C4—22no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3C22×C412yes—
165C4⋊Dic3C22×C412yes—
166Dic3⋊C4—22no—
167Dic3⋊C4—22no—
168C4×Dic3—22no—
169C4×Dic3—22no—
170#208—42no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8—21no—
173C2×C3⋊C8C2×D412yes—
174C2×C3⋊C8C2×D412yes—
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—62no—
184C2×M16—61no—
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—62no—
205C4⋊C8—62no—
206C4⋊C8C4×D312yes—
207C4⋊C8C22×D312yes—
208C4⋊C8C22×D312yes—
209#103—32no—
210#103—32no—
211#102—32no—
212#102—32no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—32no—
216C22⋊C8—62no—
217C22⋊C8—62no—
218C22⋊C8C4×D312yes—
219#97—21no—
220#97—61no—
221C4×C8—21no—
222C4×C8—31no—
223C4×C8—31no—
224C4×C8—31no—
225C4×C8—31no—
226C4×C8—62no—
227C4×C8—62no—
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—42no—
260C3×C8—42no—
261C3⋊C8—22no—
262C3⋊C8—22no—
263C3⋊C8—22no—
264C3⋊C8—41no—
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×C4D3×D411yescharacteristic
279C22×C4#23911yescharacteristic
280SD16—64no—
281SD16—64no—
282M16—122no—
283M16—242no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—32no—
287C2×C8—32no—
288C2×C8—61no—
289C2×C8—61no—
290C2×C8—242no—
291C2×C8C2×C4×D312yes—
292C2×C8C2×C4×D312yes—
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
304C42D3×D411yescharacteristic
305C42#23911yescharacteristic
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—
324C23#89311yescharacteristic
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×C4#89311yescharacteristic
345C2×C4#89311yescharacteristic
346C2×C4#89311yescharacteristic
347C2×C4C22×C4×D311yescharacteristic
348C2×C4C2×D3×D411yescharacteristic
349C2×C4C2×#23911yescharacteristic
350C8—22no—
351C8—22no—
352C8—42no—
353C8—42no—
354C8—62no—
355C8—62no—
356C8—62no—
357C8—121no—
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×#89311yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—64no—
375C4—122no—
376C4—122no—
377C4D3×#13112yes—
378C4#577012yes—
379C3#1792311yescharacteristic, complemented, Sylow
380C2—41no—
381C2—242no—
382C2#7907311yescharacteristic
383C2#7916811yescharacteristic
384C2#7925811yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1572C2
socleC22×C3C2×C4×D4
prime radical#5477C22
cosocleC3×C42C24
hypercenter#1572D3
hypocenterC3#17923
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1572

Complements

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

NH ≅ G/Ntype
3. #79263C2semidirect
4. C3×#1572C2semidirect
5. #70802C2semidirect
7. #70743C2semidirect
8. #68562C2semidirect
9. #68558C2semidirect
10. #68520C2semidirect
11. #68516C2semidirect
51. #1572D3semidirect
379. C3#17923semidirect

Maximal subgroups

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

subgroupindex
#793382
#793092
#792632
C3×#15722
#708022
#707612
#707432
#685622
#685582
#685202
#685162
#179233