Finite Group Databasebeta

← #10767958

Subgroups — #10767958

Order 768.

1386 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1386 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85786C211yesmaximal, characteristic, complemented
2C3×#2397C211yesmaximal, characteristic, complemented
3#82310C212yesmaximal, complemented
4#71603C211yesmaximal, characteristic, complemented
5#69714C214yesmaximal, complemented
6#69696C214yesmaximal, complemented
7#69432C212yesmaximal, complemented
8#18178—31nomaximal, Sylow
9C2×#1004—21no—
10C2×C3×C4×D4—22no—
11C2×C3×C4×D4C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic, complemented
13C2×#930—22no—
14#5555—22no—
15#5554—24no—
16D4×D12—22no—
17#5510—21no—
18#5493C2214yes—
19#5477C2212yes—
20C3×#399—21no—
21C3×#399C2212yescomplemented
22C3×#392—21no—
23C3×#392C2212yescomplemented
24C3×#376C2211yescharacteristic, complemented
25#5217C2214yescomplemented
26#5015—22no—
27#5014—24no—
28#5008—22no—
29#4968C2214yescomplemented
30#4926C2212yescomplemented
31#4525C2214yescomplemented
32#4451C2218yescomplemented
33#3496—31no—
34#3109—32no—
35#2397D311yescharacteristic, complemented
36#1613—34no—
37#1595—34no—
38#1505—32no—
39#1466—31no—
40C22×C3×D4—22no—
41C22×C3×D4D411yescharacteristic
42C22×C3×D4C2311yescharacteristic
43C23×C3×C4—21no—
44C23×C3×C4D412yes—
45C22×#67—24no—
46#1004—48no—
47C2×#237—24no—
48C2×D3×D4—24no—
49C2×#235—22no—
50C22×D12—22no—
51C3×C4×D4—22no—
52C3×C4×D4—22no—
53C3×C4×D4—22no—
54C3×C4×D4—22no—
55C3×C4×D4—48no—
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C4⋊C4C2311yescharacteristic
59C2×C3×C22⋊C4—22no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4C2312yes—
63C2×C3×C42—21no—
64C2×C3×C42C2311yescharacteristic
65C2×#213—24no—
66#930—24no—
67#930—24no—
68#930—22no—
69#930—24no—
70#930C2314yes—
71C2×(C2×D3)⋊C4—24no—
72C2×C4⋊Dic3—22no—
73#897—24no—
74#895—24no—
75#886—28no—
76#882—28no—
77#875—22no—
78#874D412yescomplemented
79C4×D12—22no—
80C4×D12D414yescomplemented
81#869D412yescomplemented
82C3×#95—21no—
83C3×#95C2312yes—
84#832—24no—
85#830C2314yes—
86#809—24no—
87#808—24no—
88C4⋊(C3⋊C8)—22no—
89#803—22no—
90C2×H(5,𝔽2)—61no—
91#547—62no—
92#546—64no—
93D42—62no—
94#537—61no—
95#534—34no—
96#532—32no—
97C2×#121—62no—
98C2×C4×D4—22no—
99C2×C4×D4C2×D311yescharacteristic
100C2×C4×D4C2×D311yescharacteristic, complemented
101#481—34no—
102#476—62no—
103#459—62no—
104#447—34no—
105#442—64no—
106#432—32no—
107#399—21no—
108#399C2×D312yescomplemented
109#392—21no—
110#392C2×D312yescomplemented
111#376C2×D311yescharacteristic, complemented
112#327—38no—
113#323—34no—
114C24×C3—21no—
115C24×C3C2×D412yes—
116C23×D3—24no—
117C2×C3×D4—24no—
118C2×C3×D4—22no—
119C2×C3×D4—22no—
120C2×C3×D4—44no—
121C2×C3×D4—42no—
122C2×C3×D4—44no—
123C2×C3×D4—44no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—44no—
129C22×C3×C4—44no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2411yescharacteristic
136C2×#67—28no—
137C2×#67—24no—
138C2×#67—48no—
139C2×#67—48no—
140C22×Dic3—24no—
141#237—416no—
142D3×D4—416no—
143#235—48no—
144C2×D12—22no—
145C2×D12—22no—
146C2×D12—24no—
147C2×D12—84no—
148C2×D12C2×D414yes—
149C2×C4×D3—24no—
150C2×Dic6—22no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—44no—
154C3×C22⋊C4—24no—
155C3×C22⋊C4—44no—
156C3×C22⋊C4—44no—
157C3×C22⋊C4—42no—
158C3×C42—24no—
159C3×C42C2×D412yes—
160C3×C42C2×D412yes—
161#213—48no—
162(C2×D3)⋊C4—28no—
163(C2×D3)⋊C4—24no—
164(C2×D3)⋊C4—24no—
165C4⋊Dic3—24no—
166C4⋊Dic3—22no—
167C4⋊Dic3C2×D414yes—
168C2×C3⋊C8—24no—
169H(5,𝔽2)—128no—
170C2×#41—64no—
171C2×#41—62no—
172C22×D4—22no—
173C22×D4—62no—
174C22×D4—64no—
175C22×D4—64no—
176C22×D4#6711yescharacteristic
177C22×D4C22×D311yescharacteristic
178C23×C4—21no—
179C23×C4#6712yes—
180C2×D8—64no—
181C4⋊Q8—32no—
182#127—32no—
183#124—62no—
184#123—68no—
185C22⋊Q8—64no—
186#121—34no—
187#121—64no—
188#121—64no—
189#121—64no—
190#121—62no—
191#121—64no—
192#120—68no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—34no—
198C4×D4—48no—
199C4×D4—62no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—62no—
202C2×C4⋊C4#6711yescharacteristic
203C2×C4⋊C4C22×D311yescharacteristic
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—64no—
208C2×C22⋊C4C22×D312yes—
209C2×C42—21no—
210C2×C42C22×D311yescharacteristic
211#106—64no—
212C4⋊C8—62no—
213#102—64no—
214#102—64no—
215C22⋊C8—34no—
216#97—62no—
217#95—21no—
218#95C22×D312yes—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—22no—
222C23×C3—42no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—42no—
228C23×C3AGL(1,ℤ/8)14yes—
229C23×C3C22×D411yescharacteristic
230C22×D3—24no—
231C22×D3—48no—
232C22×D3—84no—
233C22×D3—84no—
234C3×D4—28no—
235C3×D4—24no—
236C3×D4—42no—
237C3×D4—416no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—22no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—44no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—41no—
249C2×C3×C4—82no—
250C2×C3×C4—82no—
251C2×C3×C4—82no—
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12012yes—
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4C22×D412yes—
256#67—416no—
257#67—48no—
258C2×Dic3—24no—
259C2×Dic3—48no—
260D12—44no—
261D12—44no—
262D12—44no—
263D12—84no—
264C4×D3—48no—
265Dic6—44no—
266C3⋊C8—44no—
267C24—21no—
268C24—64no—
269C24C2×#6712yes—
270#41—128no—
271#41—1216no—
272C2×Q8—62no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—34no—
277C2×D4—44no—
278C2×D4—42no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—64no—
282C2×D4—68no—
283C2×D4—64no—
284C2×D4—62no—
285C2×D4—62no—
286C2×D4—1216no—
287C2×D4—128no—
288C2×D4—128no—
289C2×D4—244no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—64no—
299C22×C4—64no—
300C22×C4C2×#6711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302C22×C4C2×#6712yes—
303C22×C4C23×D311yescharacteristic
304D8—128no—
305C2×C8—64no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—34no—
309C4⋊C4—44no—
310C4⋊C4—64no—
311C4⋊C4—62no—
312C22⋊C4—24no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—42no—
316C22⋊C4—68no—
317C22⋊C4—64no—
318C22⋊C4—64no—
319C42—24no—
320C42D3×D412yes—
321C42D3×D412yes—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—24no—
325C22×C3—44no—
326C22×C3—44no—
327C22×C3—42no—
328C22×C3—42no—
329C22×C3—42no—
330C22×C3—82no—
331C22×C3—82no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3C2×#12011yescharacteristic
335C22×C3C2×AGL(1,ℤ/8)12yes—
336C2×D3—44no—
337C2×D3—44no—
338C2×D3—44no—
339C2×D3—168no—
340C3×C4—22no—
341C3×C4—21no—
342C3×C4—22no—
343C3×C4—44no—
344C3×C4—42no—
345C3×C4—81no—
346C3×C4#45312yes—
347Dic3—44no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—64no—
358C23—128no—
359C23—244no—
360C23—244no—
361C23#94914yes—
362C23C22×#6711yescharacteristic
363Q8—124no—
364D4—28no—
365D4—24no—
366D4—42no—
367D4—416no—
368D4—124no—
369D4—124no—
370D4—1216no—
371D4—128no—
372D4—124no—
373D4—244no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—64no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—128no—
390C2×C4—128no—
391C2×C4#93712yes—
392C2×C4#93711yescharacteristic
393C2×C4#93711yescharacteristic
394C2×C4C2×D3×D412yes—
395C8—124no—
396C2×C3—22no—
397C2×C3—44no—
398C2×C3—82no—
399C2×C3#300311yescharacteristic
400C2×C3C2×#45311yescharacteristic
401C2×C3#302511yescharacteristic
402D3—84no—
403C22—21no—
404C22—21no—
405C22—24no—
406C22—44no—
407C22—44no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—82no—
412C22—82no—
413C22—81no—
414C22—81no—
415C22—124no—
416C22—124no—
417C22—124no—
418C22—488no—
419C22C2×#93711yescharacteristic
420C22C2×#94912yes—
421C4—21no—
422C4—22no—
423C4—22no—
424C4—44no—
425C4—42no—
426C4—81no—
427C4—124no—
428C4#507712yes—
429C3#1817811yescharacteristic, complemented, Sylow
430C2—22no—
431C2—44no—
432C2—82no—
433C2—244no—
434C2#8292711yescharacteristic
435C2#8293611yescharacteristic
436C2C2×#507711yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2397C2
socleC22×C3C2×#120
hypercenter#2397D3
hypocenterC3#18178
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2397

Complements

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

NH ≅ G/Ntype
1. #85786C2semidirect
2. C3×#2397C2semidirect
3. #82310C2semidirect
4. #71603C2semidirect
5. #69714C2semidirect
6. #69696C2semidirect
7. #69432C2semidirect
12. C2×C3×C4×D4C22semidirect
21. C3×#399C22semidirect
23. C3×#392C22semidirect
24. C3×#376C22semidirect
25. #5217C22semidirect
29. #4968C22semidirect
30. #4926C22semidirect
31. #4525C22semidirect
32. #4451C22semidirect
35. #2397D3semidirect
78. #874D4semidirect
80. C4×D12D4semidirect
81. #869D4semidirect
100. C2×C4×D4C2×D3semidirect
108. #399C2×D3semidirect
110. #392C2×D3semidirect
111. #376C2×D3semidirect
429. C3#18178semidirect

Maximal subgroups

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

subgroupindex
#857862
C3×#23972
#823102
#716032
#697142
#696962
#694322
#181783