Finite Group Databasebeta

← #10767969

Subgroups — #10767969

Order 768.

1250 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1250 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
2#82348C211yesmaximal, characteristic, complemented
3#79355C212yesmaximal, complemented
4C3×#1466C211yesmaximal, characteristic, complemented
5#70832C212yesmaximal, complemented
6#68110C214yesmaximal, complemented
7#68108C214yesmaximal, complemented
8#18179—31nomaximal, Sylow
9C2×#1004—21no—
10C2×C3×C4×D4C2211yescharacteristic
11C2×#930—22no—
12#5555—22no—
13#5554—24no—
14D4×D12—22no—
15#5510—21no—
16#5493C2214yes—
17#5477C2212yes—
18C2×C4×Dic6C2211yescharacteristic, complemented
19C3×#432C2212yescomplemented
20#5222—21no—
21C4×#217—22no—
22#4985—21no—
23#4953C2212yescomplemented
24#4947C2212yescomplemented
25#4932C2211yescharacteristic, complemented
26#4850—22no—
27#4848—22no—
28#4738C2214yescomplemented
29#4736C2214yescomplemented
30#4710—24no—
31C3×#323C2214yescomplemented
32#4470C2218yescomplemented
33#3496—31no—
34#3112—32no—
35#2398—31no—
36#1607—34no—
37#1601—34no—
38#1505—32no—
39#1466D311yescharacteristic, complemented
40C22×C3×D4D411yescharacteristic
41C23×C3×C4D412yes—
42C22×#67—24no—
43#1004—48no—
44C2×#237—24no—
45C2×D3×D4—24no—
46C2×#235—22no—
47C22×D12—22no—
48C22×Dic6C2311yescharacteristic
49C3×C4×D4—22no—
50C3×C4×D4—22no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4C2312yes—
53C2×C3×C42C2311yescharacteristic
54C2×#217—22no—
55C2×#217—22no—
56#930—24no—
57#930—24no—
58#930—22no—
59#930—24no—
60#930C2314yes—
61C2×(C2×D3)⋊C4—24no—
62C2×C4⋊Dic3—22no—
63C2×C4⋊Dic3C2311yescharacteristic
64C2×Dic3⋊C4—21no—
65C2×C4×Dic3—21no—
66C2×C4×Dic3—21no—
67C2×#204—24no—
68#897—24no—
69#895—24no—
70#886—28no—
71#882—28no—
72#875—22no—
73#874D412yes—
74C4×D12—22no—
75C4×D12D414yes—
76#869D412yes—
77C4×Dic6—22no—
78C4×Dic6—22no—
79C3×C4⋊C8—22no—
80C3×C22⋊C8C2314yes—
81C3×#97—22no—
82C3⋊#95—21no—
83C3⋊#95—22no—
84C3⋊#95—22no—
85C3⋊#95C2312yes—
86#821—24no—
87#818—24no—
88#816—24no—
89C2×H(5,𝔽2)—61no—
90#547—62no—
91#546—64no—
92D42—62no—
93#537—61no—
94#534—34no—
95#532—32no—
96C2×#121—62no—
97C2×C4×Q8—31no—
98C2×C4×D4C2×D311yescharacteristic
99#480—34no—
100#477—62no—
101#460—62no—
102#448—34no—
103#440—64no—
104#432C2×D312yescomplemented
105#400—61no—
106#397—32no—
107#394—61no—
108#393—32no—
109C4×C22⋊C4—62no—
110#376—31no—
111#327—38no—
112#323C2×D314yescomplemented
113C24×C3C2×D412yes—
114C23×D3—24no—
115C2×C3×D4—24no—
116C2×C3×D4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—44no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D412yes—
123C22×C3×C4C2411yescharacteristic
124C2×#67—28no—
125C2×#67—24no—
126C2×#67—48no—
127C2×#67—48no—
128C22×Dic3—24no—
129C22×Dic3—21no—
130C22×Dic3—22no—
131C22×Dic3—21no—
132#237—416no—
133D3×D4—416no—
134#235—48no—
135C2×D12—22no—
136C2×D12—22no—
137C2×D12—24no—
138C2×D12—84no—
139C2×D12C2×D414yes—
140C2×C4×D3—24no—
141C2×Dic6—22no—
142C2×Dic6—22no—
143C2×Dic6—42no—
144C2×C3×C8—24no—
145C3×C4⋊C4—22no—
146C3×C22⋊C4—24no—
147C3×C42C2×D412yes—
148C3×C42C2×D412yes—
149#217—44no—
150#217—44no—
151(C2×D3)⋊C4—28no—
152(C2×D3)⋊C4—24no—
153(C2×D3)⋊C4—24no—
154C4⋊Dic3—24no—
155C4⋊Dic3—22no—
156C4⋊Dic3—22no—
157C4⋊Dic3C2×D414yes—
158Dic3⋊C4—42no—
159C4×Dic3—42no—
160C4×Dic3—41no—
161C4×Dic3—41no—
162#204—48no—
163H(5,𝔽2)—128no—
164C2×#41—64no—
165C2×#41—62no—
166C22×Q8—31no—
167C22×D4—62no—
168C22×D4—64no—
169C22×D4—64no—
170C22×D4D1211yescharacteristic
171C23×C4D1212yes—
172C2×SD16—64no—
173C4⋊Q8—32no—
174#127—32no—
175#124—62no—
176#123—68no—
177C22⋊Q8—64no—
178#121—34no—
179#121—64no—
180#121—64no—
181#121—64no—
182#121—62no—
183#121—64no—
184#120—68no—
185C4×Q8—62no—
186C4×Q8—62no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—34no—
190C4×D4—62no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4D1211yescharacteristic
195C2×C22⋊C4—64no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—62no—
198C2×C22⋊C4C22×D312yes—
199C2×C42—61no—
200C2×C42—61no—
201C2×C42C22×D311yescharacteristic
202#107—64no—
203C4⋊C8—22no—
204#103—64no—
205#102—64no—
206C22⋊C8C22×D314yes—
207#97—22no—
208#95—32no—
209#95—61no—
210#95—62no—
211#95—62no—
212C23×C3—22no—
213C23×C3—22no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3AGL(1,ℤ/8)14yes—
217C23×C3C22×D411yescharacteristic
218C22×D3—24no—
219C22×D3—48no—
220C22×D3—84no—
221C22×D3—84no—
222C3×D4—28no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—41no—
227C2×C3×C4—44no—
228C2×C3×C4—42no—
229C2×C3×C4—82no—
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12012yes—
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4C22×D412yes—
234#67—416no—
235#67—48no—
236C2×Dic3—22no—
237C2×Dic3—24no—
238C2×Dic3—41no—
239C2×Dic3—48no—
240C2×Dic3—41no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—41no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246D12—44no—
247D12—44no—
248D12—44no—
249D12—84no—
250C4×D3—48no—
251Dic6—24no—
252Dic6—42no—
253Dic6—44no—
254C3×C8—44no—
255C24—64no—
256C24C2×D1212yes—
257#41—128no—
258#41—1216no—
259C2×Q8—62no—
260C2×Q8—62no—
261C2×Q8—122no—
262C2×D4—24no—
263C2×D4—22no—
264C2×D4—34no—
265C2×D4—64no—
266C2×D4—68no—
267C2×D4—64no—
268C2×D4—62no—
269C2×D4—62no—
270C2×D4—1216no—
271C2×D4—128no—
272C2×D4—128no—
273C2×D4—244no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—44no—
277C22×C4—64no—
278C22×C4—61no—
279C22×C4—64no—
280C22×C4—62no—
281C22×C4—61no—
282C22×C4C2×D1211yescharacteristic
283C22×C4C2×D1211yescharacteristic
284C22×C4C2×D1212yes—
285C22×C4C23×D311yescharacteristic
286SD16—128no—
287C2×C8—24no—
288C4⋊C4—22no—
289C4⋊C4—34no—
290C4⋊C4—64no—
291C4⋊C4—62no—
292C4⋊C4—62no—
293C4⋊C4—122no—
294C22⋊C4—24no—
295C22⋊C4—68no—
296C22⋊C4—64no—
297C22⋊C4—64no—
298C22⋊C4—124no—
299C22⋊C4—124no—
300C42—122no—
301C42—121no—
302C42—121no—
303C42D3×D412yes—
304C42D3×D412yes—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—24no—
308C22×C3—44no—
309C22×C3—42no—
310C22×C3—42no—
311C22×C3—82no—
312C22×C3C2×#12011yescharacteristic
313C22×C3C2×AGL(1,ℤ/8)12yes—
314C2×D3—44no—
315C2×D3—44no—
316C2×D3—44no—
317C2×D3—168no—
318C3×C4—22no—
319C3×C4—21no—
320C3×C4—22no—
321C3×C4—42no—
322C3×C4#45512yes—
323Dic3—44no—
324Dic3—42no—
325Dic3—82no—
326Dic3—81no—
327C23—22no—
328C23—22no—
329C23—42no—
330C23—42no—
331C23—64no—
332C23—128no—
333C23—244no—
334C23—244no—
335C23#90814yes—
336C23C22×D1211yescharacteristic
337Q8—64no—
338Q8—122no—
339Q8—124no—
340D4—28no—
341D4—124no—
342D4—124no—
343D4—1216no—
344D4—128no—
345D4—124no—
346D4—244no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—62no—
354C2×C4—64no—
355C2×C4—82no—
356C2×C4—128no—
357C2×C4—128no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4—121no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4#88211yescharacteristic
366C2×C4#88212yes—
367C2×C4#88211yescharacteristic
368C2×C4C2×D3×D412yes—
369C8—44no—
370C2×C3—22no—
371C2×C3—44no—
372C2×C3#300311yescharacteristic
373C2×C3C2×#45511yescharacteristic
374C2×C3#302511yescharacteristic
375D3—84no—
376C22—21no—
377C22—21no—
378C22—24no—
379C22—44no—
380C22—42no—
381C22—42no—
382C22—82no—
383C22—124no—
384C22—124no—
385C22—124no—
386C22—488no—
387C22C2×#88211yescharacteristic
388C22C2×#90812yes—
389C4—22no—
390C4—22no—
391C4—21no—
392C4—42no—
393C4—124no—
394C4—122no—
395C4—242no—
396C4—241no—
397C4#474612yes—
398C3#1817911yescharacteristic, complemented, Sylow
399C2—22no—
400C2—44no—
401C2—244no—
402C2#7863711yescharacteristic
403C2#7869011yescharacteristic
404C2C2×#474611yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1466C2
socleC22×C3C2×#120
prime radical#85786C2
hypercenter#1466D3
hypocenterC3#18179
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
2. #82348C2semidirect
3. #79355C2semidirect
4. C3×#1466C2semidirect
5. #70832C2semidirect
6. #68110C2semidirect
7. #68108C2semidirect
18. C2×C4×Dic6C22semidirect
19. C3×#432C22semidirect
23. #4953C22semidirect
24. #4947C22semidirect
25. #4932C22semidirect
28. #4738C22semidirect
29. #4736C22semidirect
31. C3×#323C22semidirect
32. #4470C22semidirect
39. #1466D3semidirect
104. #432C2×D3semidirect
112. #323C2×D3semidirect
398. C3#18179semidirect

Maximal subgroups

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

subgroupindex
#857862
#823482
#793552
C3×#14662
#708322
#681102
#681082
#181793