Finite Group Databasebeta

← #10765312

Subgroups — #10765312

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#82442C211yesmaximal, characteristic
2#82435C211yesmaximal, characteristic, complemented
3#79338C211yesmaximal, characteristic
4C3×#1491C211yesmaximal, characteristic, complemented
5#70796C211yesmaximal, characteristic, complemented
6#69853C211yesmaximal, characteristic
7#69848C211yesmaximal, characteristic, complemented
8#68223C212yesmaximal, complemented
9#68217C212yesmaximal, complemented
10#68214C212yesmaximal, complemented
11#68212C212yesmaximal, complemented
12#17923—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14#5477C2211yescharacteristic
15C3×#432C2211yescharacteristic
16C3×C4×M16C2211yescharacteristic
17#5227C412yes—
18#5027C412yes—
19C4×#215C2212yes—
20C4×#215C2212yes—
21Q8×C3⋊C8—22no—
22Q8×C3⋊C8C2212yes—
23Q8×C3⋊C8C2212yes—
24#5020C412yes—
25#4968C412yes—
26#4925—22no—
27#4925C2211yescharacteristic
28#4925C2211yescharacteristic
29#4925C2211yescharacteristic
30#4838C2212yes—
31#4834C2212yes—
32#4727—22no—
33#4706—22no—
34C3×Q8⋊C8C2212yes—
35C3×Q8⋊C8C2212yes—
36#4480C2212yes—
37#4479C2212yes—
38#4459C2212yes—
39#4459C2212yes—
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#1572—31no—
52#1491D311yescharacteristic, complemented
53#1484—31no—
54#1453—31no—
55C22×C3×Q8C2×C411yescharacteristic
56C2×#235—21no—
57C2×C3×M16—21no—
58C3×C4×Q8—21no—
59C3×C4×Q8—21no—
60C3×C4×Q8C2×C412yes—
61C3×C4×Q8C2312yes—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4C2×C411yescharacteristic
64C2×C3×C42—21no—
65C2×C3×C42C2311yescharacteristic
66C2×#215C2×C412yes—
67C2×#215C2×C412yes—
68#930—21no—
69#930C2×C412yes—
70C2×ΓL(1,𝔽25)—22no—
71C2×D3×C8—22no—
72#875—21no—
73#874C2×C411yescharacteristic
74C4×D12—21no—
75C4×D12C2312yes—
76#869C2×C411yescharacteristic
77C3×C4⋊C8C2312yes—
78C3×C22⋊C8—21no—
79C3×#97—21no—
80C3×#97—21no—
81C3×C4×C8C2311yescharacteristic
82C3×C4×C8C2311yescharacteristic
83#835C2×C412yes—
84#835C2×C412yes—
85#830—21no—
86#830C2×C412yes—
87#830D412yes—
88(C2×D3)⋊C8—22no—
89(C2×D3)⋊C8—22no—
90#809C2×C412yes—
91#809C2×C412yes—
92#808C2×C412yes—
93#808C2×C412yes—
94C4⋊(C3⋊C8)—22no—
95C4⋊(C3⋊C8)—22no—
96C4⋊(C3⋊C8)C2×C412yes—
97C4⋊(C3⋊C8)D412yes—
98C4⋊(C3⋊C8)C2312yes—
99C4×C3⋊C8—21no—
100C4×C3⋊C8—22no—
101C4×C3⋊C8—22no—
102C4×C3⋊C8C2311yescharacteristic
103C4×C3⋊C8C2311yescharacteristic
104C4×C3⋊C8C2311yescharacteristic
105C4×C3⋊C8C2311yescharacteristic
106#532—31no—
107C2×C4×Q8C2×D311yescharacteristic
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—31no—
123#431—31no—
124#431—62no—
125#408—62no—
126C4×M16C2×D311yescharacteristic
127#335—32no—
128#334—32no—
129Q8⋊C8C2×D312yes—
130Q8⋊C8C2×D312yes—
131D4⋊C8—32no—
132D4⋊C8—32no—
133#322—31no—
134#322—31no—
135#322—31no—
136C82—31no—
137C2×C3×Q8—24no—
138C2×C3×Q8—21no—
139C2×C3×Q8C22×C412yes—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4C22×C411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C2×#67—22no—
146#235—82no—
147C2×D12—21no—
148C2×D12—21no—
149C2×D12C22×C412yes—
150C2×C4×D3—22no—
151C2×Dic6—21no—
152C3×M16—42no—
153C2×C3×C8—21no—
154C2×C3×C8—22no—
155C2×C3×C8—21no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4C22×C412yes—
160C3×C42—22no—
161C3×C42—22no—
162C3×C42C22×C411yescharacteristic
163C3×C42C2×D411yescharacteristic
164C3×C42#4111yescharacteristic
165C3×C42C2411yescharacteristic
166#215—24no—
167#215—24no—
168(C2×D3)⋊C4—22no—
169(C2×D3)⋊C4—22no—
170C4⋊Dic3—21no—
171C4⋊Dic3C22×C412yes—
172C2×C3⋊C8—22no—
173C2×C3⋊C8C22×C412yes—
174C2×C3⋊C8C22×C412yes—
175C2×C3⋊C8C2×D412yes—
176C2×C3⋊C8C2×D412yes—
177ΓL(1,𝔽25)—82no—
178D3×C8—82no—
179C2×#41—61no—
180C22×Q8C4×D311yescharacteristic
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—21no—
192C4×Q8—21no—
193C4×Q8C4×D312yes—
194C4×Q8C22×D312yes—
195C4×D4—32no—
196C4×D4—61no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C4×D311yescharacteristic
199C2×C42—21no—
200C2×C42C22×D311yescharacteristic
201#106—32no—
202#106—32no—
203C4⋊C8—32no—
204C4⋊C8—32no—
205C4⋊C8—32no—
206C4⋊C8—62no—
207C4⋊C8—62no—
208C4⋊C8C22×D312yes—
209#103—32no—
210#103—32no—
211#102—32no—
212#102—32no—
213C22⋊C8—21no—
214C22⋊C8—32no—
215C22⋊C8—32no—
216C22⋊C8—62no—
217C22⋊C8—62no—
218C22⋊C8—61no—
219#97—21no—
220#97—21no—
221C4×C8—31no—
222C4×C8—31no—
223C4×C8—31no—
224C4×C8—31no—
225C4×C8—62no—
226C4×C8—62no—
227C4×C8—61no—
228C4×C8C22×D311yescharacteristic
229C4×C8C22×D311yescharacteristic
230C23×C3C4×D411yescharacteristic
231C22×D3—22no—
232C3×Q8—24no—
233C3×Q8—22no—
234C3×Q8#13114yes—
235C2×C3×C4—24no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—41no—
241C2×C3×C4—81no—
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—
250D12—22no—
251D12—22no—
252D12—81no—
253D12—81no—
254C4×D3—42no—
255C4×D3—42no—
256Dic6—81no—
257C3×C8—22no—
258C3×C8—41no—
259C3×C8—42no—
260C3⋊C8—22no—
261C3⋊C8—22no—
262C3⋊C8—22no—
263C3⋊C8—22no—
264C3⋊C8—42no—
265#41—242no—
266C2×Q8—24no—
267C2×Q8—21no—
268C2×Q8—61no—
269C2×Q8C2×C4×D312yes—
270C2×D4—32no—
271C2×D4—62no—
272C2×D4—61no—
273C2×D4—61no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—62no—
277C22×C4C2×C4×D311yescharacteristic
278C22×C4D3×D411yescharacteristic
279C22×C4#23711yescharacteristic
280SD16—64no—
281SD16—64no—
282M16—42no—
283M16—242no—
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—21no—
287C2×C8—32no—
288C2×C8—32no—
289C2×C8—32no—
290C2×C8—32no—
291C2×C8—62no—
292C2×C8—242no—
293C4⋊C4—22no—
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—32no—
297C4⋊C4—61no—
298C4⋊C4C2×C4×D312yes—
299C22⋊C4—62no—
300C22⋊C4—62no—
301C42—22no—
302C42—22no—
303C42C2×C4×D311yescharacteristic
304C42D3×D411yescharacteristic
305C42#23711yescharacteristic
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—24no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—42no—
320C3×C4C2×#13112yes—
321C3×C4#57712yes—
322Dic3—42no—
323C23—62no—
324C23#67⋊C411yescharacteristic
325Q8—24no—
326Q8—22no—
327Q8—241no—
328Q8#90714yes—
329D4—62no—
330D4—62no—
331D4—241no—
332D4—241no—
333D4—242no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—62no—
341C2×C4—81no—
342C2×C4—122no—
343C2×C4—122no—
344C2×C4#67⋊C411yescharacteristic
345C2×C4#67⋊C411yescharacteristic
346C2×C4#67⋊C411yescharacteristic
347C2×C4C22×C4×D311yescharacteristic
348C2×C4C2×D3×D411yescharacteristic
349C2×C4C2×#23711yescharacteristic
350C8—22no—
351C8—41no—
352C8—42no—
353C8—62no—
354C8—62no—
355C8—62no—
356C8—62no—
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×#67⋊C411yescharacteristic
370C4—24no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—122no—
377C4C2×#90712yes—
378C4#577812yes—
379C3#1792311yescharacteristic, complemented, Sylow
380C2—41no—
381C2—242no—
382C2#7862611yescharacteristic
383C2#7894711yescharacteristic
384C2#7898011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1491C2
socleC22×C3C2×C4×D4
prime radical#5477C22
cosocleC3×C42C24
hypercenter#1491D3
hypocenterC3#17923
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1491

Complements

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

NH ≅ G/Ntype
2. #82435C2semidirect
4. C3×#1491C2semidirect
5. #70796C2semidirect
7. #69848C2semidirect
8. #68223C2semidirect
9. #68217C2semidirect
10. #68214C2semidirect
11. #68212C2semidirect
52. #1491D3semidirect
379. C3#17923semidirect

Maximal subgroups

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

subgroupindex
#824422
#824352
#793382
C3×#14912
#707962
#698532
#698482
#682232
#682172
#682142
#682122
#179233