Finite Group Databasebeta

← #10767965

Subgroups — #10767965

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
2C3×#2398C211yesmaximal, characteristic, complemented
3#82444C212yesmaximal, complemented
4#71603C211yesmaximal, characteristic, complemented
5#69708C214yesmaximal, complemented
6#69699C214yesmaximal, complemented
7#69432C212yesmaximal, complemented
8#18179—31nomaximal, Sylow
9C2×#1004—21no—
10C2×C3×C4×Q8C2211yescharacteristic, complemented
11C2×C3×C4×D4C2211yescharacteristic
12C2×#930—22no—
13#5555—22no—
14#5554—24no—
15D4×D12—22no—
16#5510—21no—
17#5493C2214yes—
18#5477C2212yes—
19C3×#400—21no—
20C3×#397C2212yescomplemented
21C3×#394—21no—
22C3×#393C2212yescomplemented
23C3×C4×C22⋊C4—22no—
24C3×#376C2211yescharacteristic, complemented
25#5227C2214yescomplemented
26#5027—22no—
27#5026—24no—
28#5021—22no—
29#4967C2214yescomplemented
30#4926C2212yescomplemented
31#4525C2214yescomplemented
32#4451C2218yescomplemented
33#3496—31no—
34#3112—32no—
35#2398D311yescharacteristic, complemented
36#1607—34no—
37#1601—34no—
38#1505—32no—
39#1466—31no—
40C22×C3×Q8C2311yescharacteristic
41C22×C3×D4D411yescharacteristic
42C23×C3×C4D412yes—
43C22×#67—24no—
44#1004—48no—
45C2×#237—24no—
46C2×D3×D4—24no—
47C2×#235—22no—
48C22×D12—22no—
49C3×C4×Q8—22no—
50C3×C4×Q8—22no—
51C3×C4×D4—22no—
52C3×C4×D4—22no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C4⋊C4C2311yescharacteristic
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—22no—
58C2×C3×C22⋊C4C2312yes—
59C2×C3×C42—21no—
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62C2×#215—24no—
63#930—24no—
64#930—24no—
65#930—22no—
66#930—24no—
67#930C2314yes—
68C2×(C2×D3)⋊C4—24no—
69C2×C4⋊Dic3—22no—
70#897—24no—
71#895—24no—
72#886—28no—
73#882—28no—
74#875—22no—
75#874D412yes—
76C4×D12—22no—
77C4×D12D414yes—
78#869D412yes—
79C3×#95—21no—
80C3×#95—22no—
81C3×#95—22no—
82C3×#95C2312yes—
83#835—24no—
84#830C2314yes—
85#809—24no—
86#807—24no—
87C4⋊(C3⋊C8)—22no—
88#803—22no—
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×Q8C2×D311yescharacteristic, complemented
98C2×C4×D4C2×D311yescharacteristic
99#480—34no—
100#477—62no—
101#460—62no—
102#448—34no—
103#440—64no—
104#432—32no—
105#400—21no—
106#397C2×D312yescomplemented
107#394—21no—
108#393C2×D312yescomplemented
109C4×C22⋊C4—22no—
110#376C2×D311yescharacteristic, complemented
111#327—38no—
112#323—34no—
113C24×C3C2×D412yes—
114C23×D3—24no—
115C2×C3×Q8—22no—
116C2×C3×Q8—42no—
117C2×C3×D4—24no—
118C2×C3×D4—22no—
119C22×C3×C4—21no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—21no—
124C22×C3×C4—44no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D412yes—
128C22×C3×C4C2411yescharacteristic
129C2×#67—28no—
130C2×#67—24no—
131C2×#67—48no—
132C2×#67—48no—
133C22×Dic3—24no—
134#237—416no—
135D3×D4—416no—
136#235—48no—
137C2×D12—22no—
138C2×D12—22no—
139C2×D12—24no—
140C2×D12—84no—
141C2×D12C2×D414yes—
142C2×C4×D3—24no—
143C2×Dic6—22no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—42no—
147C3×C22⋊C4—24no—
148C3×C22⋊C4—44no—
149C3×C22⋊C4—44no—
150C3×C42—42no—
151C3×C42—41no—
152C3×C42—41no—
153C3×C42C2×D412yes—
154C3×C42C2×D412yes—
155#215—48no—
156(C2×D3)⋊C4—28no—
157(C2×D3)⋊C4—24no—
158(C2×D3)⋊C4—24no—
159C4⋊Dic3—24no—
160C4⋊Dic3—22no—
161C4⋊Dic3C2×D414yes—
162C2×C3⋊C8—24no—
163H(5,𝔽2)—128no—
164C2×#41—64no—
165C2×#41—62no—
166C22×Q8C22×D311yescharacteristic
167C22×D4—62no—
168C22×D4—64no—
169C22×D4—64no—
170C22×D4#6711yescharacteristic
171C23×C4#6712yes—
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—22no—
186C4×Q8—22no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—34no—
190C4×D4—62no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4#6711yescharacteristic
194C2×C4⋊C4C22×D311yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—64no—
198C2×C22⋊C4C22×D312yes—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42C22×D311yescharacteristic
202#107—64no—
203C4⋊C8—62no—
204#103—64no—
205#102—64no—
206C22⋊C8—34no—
207#97—62no—
208#95—21no—
209#95—22no—
210#95—22no—
211#95C22×D312yes—
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×Q8—24no—
223C3×Q8—42no—
224C3×D4—28no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—44no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—82no—
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12012yes—
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4C22×D412yes—
244#67—416no—
245#67—48no—
246C2×Dic3—24no—
247C2×Dic3—48no—
248D12—44no—
249D12—44no—
250D12—44no—
251D12—84no—
252C4×D3—48no—
253Dic6—44no—
254C3⋊C8—44no—
255C24—64no—
256C24C2×#6712yes—
257#41—128no—
258#41—1216no—
259C2×Q8—22no—
260C2×Q8—42no—
261C2×Q8—62no—
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—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—44no—
280C22×C4—64no—
281C22×C4—64no—
282C22×C4C2×#6711yescharacteristic
283C22×C4C2×#6711yescharacteristic
284C22×C4C2×#6712yes—
285C22×C4C23×D311yescharacteristic
286SD16—128no—
287C2×C8—64no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—34no—
291C4⋊C4—42no—
292C4⋊C4—64no—
293C4⋊C4—62no—
294C22⋊C4—24no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—68no—
298C22⋊C4—64no—
299C22⋊C4—64no—
300C42—42no—
301C42—41no—
302C42—41no—
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—42no—
323C3×C4—82no—
324C3×C4—81no—
325C3×C4#45512yes—
326Dic3—44no—
327C23—22no—
328C23—22no—
329C23—42no—
330C23—42no—
331C23—64no—
332C23—128no—
333C23—244no—
334C23—244no—
335C23#94914yes—
336C23C22×#6711yescharacteristic
337Q8—24no—
338Q8—42no—
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—22no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—64no—
362C2×C4—82no—
363C2×C4—128no—
364C2×C4—128no—
365C2×C4#93711yescharacteristic
366C2×C4#93712yes—
367C2×C4#93711yescharacteristic
368C2×C4C2×D3×D412yes—
369C8—124no—
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×#93711yescharacteristic
388C22C2×#94912yes—
389C4—21no—
390C4—22no—
391C4—22no—
392C4—42no—
393C4—42no—
394C4—82no—
395C4—81no—
396C4—124no—
397C4#506112yes—
398C3#1817911yescharacteristic, complemented, Sylow
399C2—22no—
400C2—44no—
401C2—244no—
402C2#8292711yescharacteristic
403C2#8293611yescharacteristic
404C2C2×#506111yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2398C2
socleC22×C3C2×#120
prime radical#85786C2
hypercenter#2398D3
hypocenterC3#18179
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2398

Complements

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

NH ≅ G/Ntype
2. C3×#2398C2semidirect
3. #82444C2semidirect
4. #71603C2semidirect
5. #69708C2semidirect
6. #69699C2semidirect
7. #69432C2semidirect
10. C2×C3×C4×Q8C22semidirect
20. C3×#397C22semidirect
22. C3×#393C22semidirect
24. C3×#376C22semidirect
25. #5227C22semidirect
29. #4967C22semidirect
30. #4926C22semidirect
31. #4525C22semidirect
32. #4451C22semidirect
35. #2398D3semidirect
97. C2×C4×Q8C2×D3semidirect
106. #397C2×D3semidirect
108. #393C2×D3semidirect
110. #376C2×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
C3×#23982
#824442
#716032
#697082
#696992
#694322
#181793