Finite Group Databasebeta

← #10780366

Subgroups — #10780366

Order 768.

792 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79303C211yesmaximal, characteristic, complemented
2#79286C211yesmaximal, characteristic, complemented
3#79055C211yesmaximal, characteristic, complemented
4C2×#4843C211yesmaximal, characteristic, complemented
5C3×#1571C211yesmaximal, characteristic, complemented
6#70502C211yesmaximal, characteristic, complemented
7#70431C211yesmaximal, characteristic, complemented
8#68580C214yesmaximal, complemented
9#68576C214yesmaximal, complemented
10#19386—31nomaximal, Sylow
11C2×#908—21no—
12C22×D24—21no—
13#5477C2211yescharacteristic
14C2×#874C2211yescharacteristic
15C2×C4×D12C2211yescharacteristic
16C3×#431C2211yescharacteristic
17C3×#423C2211yescharacteristic
18C2×C3×C4⋊C8C2211yescharacteristic
19#5133—21no—
20C2×#821—21no—
21#4925C2211yescharacteristic
22#4844—22no—
23#4843—22no—
24#4843—22no—
25#4843C2214yes—
26#4843C2212yescomplemented
27#4841C2212yescomplemented
28#4834—22no—
29#4732—22no—
30D3×C22⋊C8—22no—
31#4695C2212yescomplemented
32#4694C2212yescomplemented
33C8×D12C2214yescomplemented
34C3×#335C2214yescomplemented
35#4485C2212yescomplemented
36#4485C2214yescomplemented
37#4482C2212yescomplemented
38#3205—31no—
39#3036—31no—
40C2×#459—31no—
41#2972—31no—
42#1681—34no—
43#1667—34no—
44#1571D311yescharacteristic, complemented
45#1545—31no—
46#1540—31no—
47C2×#235—21no—
48C22×D12—21no—
49C22×D12—21no—
50C22×D12—21no—
51C22×D12D411yescharacteristic
52C22×C4×D3—21no—
53C2×C3×M16—21no—
54C22×C3×C8—21no—
55C2×C3×C42C2311yescharacteristic
56#930—21no—
57#930—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×C4⋊Dic3D411yescharacteristic
60#908—44no—
61C2×D24—24no—
62C2×D24—22no—
63C2×D24—24no—
64C2×D24—44no—
65C2×#204—22no—
66C2×D3×C8—24no—
67#875—21no—
68#874—21no—
69#874—21no—
70#874—21no—
71#874C2312yes—
72#874C2311yescharacteristic
73C4×D12—21no—
74C4×D12—21no—
75C4×D12—21no—
76C4×D12—21no—
77C4×D12D412yes—
78C4×D12C2312yes—
79#869C2311yescharacteristic
80C3×C4⋊C8—21no—
81C3×C4⋊C8—21no—
82C3×C4⋊C8D412yes—
83C3×C4⋊C8C2312yes—
84C3×C4⋊C8C2312yes—
85C3×C22⋊C8D412yes—
86C3×C4×C8C2312yes—
87#830—21no—
88#821—24no—
89#821—22no—
90#821—24no—
91(C2×D3)⋊C8—24no—
92#816—22no—
93C4⋊(C3⋊C8)C2312yes—
94C4×C3⋊C8—21no—
95C2×AGL(1,ℤ/8)—61no—
96C22×D8—61no—
97#532—31no—
98C2×#127—31no—
99C2×C4×D4—31no—
100#493—32no—
101#486—32no—
102#463—62no—
103#461—32no—
104#459—32no—
105#459—34no—
106#459—62no—
107#459—62no—
108#447—62no—
109C8×D4—34no—
110C8×D4—62no—
111#431—31no—
112#431C2×D311yescharacteristic
113#423C2×D311yescharacteristic
114C2×C4⋊C8C2×D311yescharacteristic
115#417—61no—
116C2×#102—61no—
117C2×C22⋊C8—62no—
118#335C2×D314yescomplemented
119#332—32no—
120#331—32no—
121#331—34no—
122C23×D3—21no—
123C23×D3—41no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C2×#67—41no—
128C22×Dic3—21no—
129#235—42no—
130#235—42no—
131C2×D12—21no—
132C2×D12—22no—
133C2×D12—22no—
134C2×D12—22no—
135C2×D12—21no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—22no—
139C2×D12—42no—
140C2×D12—42no—
141C2×D12—41no—
142C2×D12—41no—
143C2×D12—41no—
144C2×D12—82no—
145C2×D12—82no—
146C2×D12D814yescomplemented
147C2×D12C2×D412yes—
148C2×C4×D3—22no—
149C2×C4×D3—22no—
150C2×C4×D3—41no—
151C2×C4×D3—82no—
152C2×Dic6—21no—
153C3×M16—42no—
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C2×C3×C8—42no—
157C2×C3×C8C2×D414yes—
158C3×C42C2×D411yescharacteristic
159C3×C42C2×D411yescharacteristic
160C3×C42#4111yescharacteristic
161C3×C42C2411yescharacteristic
162(C2×D3)⋊C4—22no—
163(C2×D3)⋊C4—22no—
164(C2×D3)⋊C4—41no—
165(C2×D3)⋊C4—41no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C4⋊Dic3C2×D412yes—
170C2×C3⋊C8—22no—
171D24—48no—
172D24—44no—
173D24—84no—
174#204—44no—
175D3×C8—84no—
176C2×#41—61no—
177C22×D4—31no—
178C22×D4—61no—
179C22×D4—61no—
180C22×D4—61no—
181C23×C4—61no—
182AGL(1,ℤ/8)—124no—
183C2×SD16—62no—
184C2×D8—64no—
185C2×D8—62no—
186C2×D8—64no—
187C2×D8—124no—
188C2×M16—21no—
189C22×C8—21no—
190C22×C8—64no—
191C4⋊Q8—31no—
192#127—32no—
193#127—31no—
194#127—61no—
195#127—61no—
196#127—61no—
197#124—61no—
198#121—61no—
199#121—61no—
200C4×D4—32no—
201C4×D4—32no—
202C4×D4—61no—
203C4×D4—61no—
204C4×D4—61no—
205C4×D4—61no—
206C2×C4⋊C4—31no—
207C2×C22⋊C4—61no—
208C2×C42C22×D311yescharacteristic
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—32no—
212C4⋊C8D1212yes—
213C4⋊C8C22×D312yes—
214C4⋊C8C22×D312yes—
215#103—62no—
216#102—64no—
217#102—62no—
218#102—64no—
219C22⋊C8—61no—
220C22⋊C8—64no—
221C22⋊C8D1212yes—
222C4×C8—61no—
223C4×C8C22×D312yes—
224C23×C3#12111yescharacteristic
225C22×D3—22no—
226C22×D3—24no—
227C22×D3—41no—
228C22×D3—42no—
229C22×D3—81no—
230C22×D3—81no—
231C22×D3—82no—
232C22×D3—82no—
233C22×D3—82no—
234C2×C3×C4—22no—
235C2×C3×C4—41no—
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4C2×D812yes—
240C2×C3×C4#13512yes—
241C2×C3×C4C22×D411yescharacteristic
242C2×C3×C4C22×D411yescharacteristic
243C2×C3×C4C2×#4111yescharacteristic
244#67—82no—
245C2×Dic3—22no—
246C2×Dic3—41no—
247C2×Dic3—81no—
248D12—22no—
249D12—24no—
250D12—42no—
251D12—41no—
252D12—42no—
253D12—41no—
254D12—42no—
255D12—44no—
256D12—81no—
257D12—82no—
258D12—82no—
259D12—82no—
260D12—82no—
261D12C2×D814yes—
262C4×D3—44no—
263C4×D3—82no—
264C4×D3—82no—
265Dic6—41no—
266Dic6—41no—
267C3×C8—24no—
268C3×C8—42no—
269C3×C8—42no—
270C3⋊C8—42no—
271C24—61no—
272C24—121no—
273#41—122no—
274#41—122no—
275C2×Q8—61no—
276C2×D4—34no—
277C2×D4—32no—
278C2×D4—61no—
279C2×D4—62no—
280C2×D4—62no—
281C2×D4—61no—
282C2×D4—61no—
283C2×D4—62no—
284C2×D4—61no—
285C2×D4—62no—
286C2×D4—121no—
287C2×D4—121no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4—122no—
291C2×D4—122no—
292C2×D4—242no—
293C2×D4—242no—
294C22×C4—61no—
295C22×C4—62no—
296C22×C4—62no—
297C22×C4—121no—
298C22×C4—242no—
299C22×C4C2×D1211yescharacteristic
300C22×C4D3×D411yescharacteristic
301C22×C4#23911yescharacteristic
302SD16—124no—
303D8—128no—
304D8—124no—
305D8—244no—
306M16—42no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—42no—
310C2×C8—62no—
311C2×C8—244no—
312C2×C8C2×D1214yes—
313C4⋊C4—32no—
314C4⋊C4—61no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C22⋊C4—62no—
318C22⋊C4—62no—
319C22⋊C4—121no—
320C22⋊C4—121no—
321C42C2×D1211yescharacteristic
322C42D3×D411yescharacteristic
323C42#23911yescharacteristic
324C42C23×D311yescharacteristic
325C22×C3—21no—
326C22×C3—21no—
327C22×C3#46612yes—
328C22×C3C2×#12111yescharacteristic
329C2×D3—24no—
330C2×D3—42no—
331C2×D3—42no—
332C2×D3—84no—
333C2×D3—82no—
334C2×D3—81no—
335C2×D3—81no—
336C2×D3—82no—
337C2×D3—82no—
338C2×D3—81no—
339C2×D3—164no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4#46612yes—
344C3×C4C22×D811yescharacteristic
345C3×C4C2×#13511yescharacteristic
346C3×C4#57612yes—
347Dic3—42no—
348Dic3—81no—
349C23—62no—
350C23—64no—
351C23—121no—
352C23—122no—
353C23—241no—
354C23—241no—
355C23—242no—
356C23—242no—
357C23—242no—
358C23#89511yescharacteristic
359Q8—121no—
360Q8—121no—
361D4—34no—
362D4—64no—
363D4—62no—
364D4—122no—
365D4—124no—
366D4—122no—
367D4—121no—
368D4—122no—
369D4—121no—
370D4—242no—
371D4—241no—
372D4—242no—
373D4—242no—
374D4—242no—
375D4—242no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—62no—
379C2×C4—124no—
380C2×C4—121no—
381C2×C4—242no—
382C2×C4—241no—
383C2×C4—242no—
384C2×C4#89511yescharacteristic
385C2×C4#89511yescharacteristic
386C2×C4#89511yescharacteristic
387C2×C4D3×D812yes—
388C2×C4#91912yes—
389C2×C4C22×D1211yescharacteristic
390C2×C4C2×D3×D411yescharacteristic
391C2×C4C2×#23911yescharacteristic
392C8—24no—
393C8—42no—
394C8—42no—
395C8—122no—
396C2×C3—22no—
397C2×C3#304111yescharacteristic
398C2×C3C2×#46611yescharacteristic
399C2×C3#306811yescharacteristic
400D3—44no—
401D3—161no—
402D3—162no—
403C22—21no—
404C22—21no—
405C22—64no—
406C22—122no—
407C22—122no—
408C22—244no—
409C22—241no—
410C22—242no—
411C22—242no—
412C22—241no—
413C22—242no—
414C22—241no—
415C22—484no—
416C22#488312yes—
417C22C2×#89511yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—122no—
422C4—241no—
423C4#488312yes—
424C4#575212yes—
425C4C2×D3×D811yescharacteristic
426C4C2×#91911yescharacteristic
427C3#1938611yescharacteristic, complemented, Sylow
428C2—22no—
429C2—124no—
430C2—482no—
431C2—481no—
432C2#7908611yescharacteristic
433C2C2×#488311yescharacteristic
434C2#7926111yescharacteristic
subgroupG/H
centerC22C2×#895
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1571C2
socleC22×C3C2×#121
hypercenter#1571D3
hypocenterC3#19386
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
1. #79303C2semidirect
2. #79286C2semidirect
3. #79055C2semidirect
4. C2×#4843C2semidirect
5. C3×#1571C2semidirect
6. #70502C2semidirect
7. #70431C2semidirect
8. #68580C2semidirect
9. #68576C2semidirect
26. #4843C22semidirect
27. #4841C22semidirect
31. #4695C22semidirect
32. #4694C22semidirect
33. C8×D12C22semidirect
34. C3×#335C22semidirect
35. #4485C22semidirect
36. #4485C22semidirect
37. #4482C22semidirect
44. #1571D3semidirect
118. #335C2×D3semidirect
146. C2×D12D8semidirect
427. C3#19386semidirect

Maximal subgroups

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

subgroupindex
#793032
#792862
#790552
C2×#48432
C3×#15712
#705022
#704312
#685802
#685762
#193863