Finite Group Databasebeta

← #10876216

Subgroups — #10876216

Order 768.

954 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82269C211yesmaximal, characteristic
2#82231C211yesmaximal, characteristic, complemented
3C3×#1818C211yesmaximal, characteristic, complemented
4#69959C212yesmaximal
5#69929C212yesmaximal, complemented
6#68680C212yesmaximal, complemented
7#68635C212yesmaximal, complemented
8#68626C212yesmaximal, complemented
9#68625C212yesmaximal, complemented
10#27591—31nomaximal, Sylow
11C23×D12C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C2×C3×C22⋊C8C2212yes—
15#5212C2212yes—
16C2×#821C2212yes—
17C2×#821C2212yes—
18C2×#821C2212yes—
19C2×#821C2212yes—
20C2×#818C2212yescomplemented
21C2×#817C2212yescomplemented
22#4988—22no—
23#4980—22no—
24#4955—22no—
25#4951—22no—
26#4939C2212yes—
27#4939C2212yes—
28#4936C2211yescharacteristic, complemented
29#4935C2211yescharacteristic, complemented
30C4×C4⋊Dic3C2212yescomplemented
31C3×#336C2211yescharacteristic, complemented
32C3×#336C2211yescharacteristic, complemented
33C3×#336C2212yescomplemented
34#4551C2212yescomplemented
35#4551C2212yescomplemented
36#2314—31no—
37#2309—31no—
38#2092—32no—
39#2071—32no—
40#1926—32no—
41#1925—32no—
42#1854—32no—
43#1818D311yescharacteristic, complemented
44#1786—32no—
45C24×D3—21no—
46C23×C3×C4C2311yescharacteristic
47C22×D12—21no—
48C22×D12—24no—
49C22×D12—24no—
50C22×D12—41no—
51C22×D12—41no—
52C22×D12C2312yes—
53C22×C3×C8C2312yes—
54C22×C3×C8C2312yes—
55C3×#117—24no—
56C2×C3×C4⋊C4D412yes—
57C2×C3×C22⋊C4D412yes—
58C2×C3×C42C2311yescharacteristic
59C2×C3×C42C2311yescharacteristic
60C2×#217—21no—
61C2×#217—22no—
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—22no—
65C2×(C2×D3)⋊C4—22no—
66#926—41no—
67#926—41no—
68C2×C4⋊Dic3—22no—
69C2×C4⋊Dic3C2312yes—
70C2×C4⋊Dic3C2312yes—
71C2×C4⋊Dic3C2312yes—
72C2×C4×Dic3—21no—
73C2×C4×Dic3—22no—
74C3×C22⋊C8—24no—
75C3⋊#95—22no—
76C3⋊#95—22no—
77C3⋊#95—22no—
78#821—24no—
79#821—24no—
80#821—42no—
81#821—42no—
82#818—42no—
83#817—42no—
84C23×D4—31no—
85C2×#117—31no—
86C2×#117C2×D311yescharacteristic
87C2×#107—32no—
88C2×#106—32no—
89C2×#102—32no—
90C2×#102—32no—
91C2×#102—32no—
92C2×#102—32no—
93C2×C22⋊C8C2×D312yes—
94#390—32no—
95#390—32no—
96#390—62no—
97#388—62no—
98#387—62no—
99#384—31no—
100#382—31no—
101#380—32no—
102#379—62no—
103C4×C4⋊C4—32no—
104#340—32no—
105#340—32no—
106#336C2×D312yescomplemented
107#336C2×D311yescharacteristic, complemented
108#336C2×D311yescharacteristic, complemented
109C24×C3C2×D411yescharacteristic
110C23×D3—22no—
111C23×D3—24no—
112C23×D3—42no—
113C23×D3—42no—
114C23×D3—82no—
115C23×D3—82no—
116C22×C3×C4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4C2×D412yes—
120C22×C3×C4C2×D412yes—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4#4114yes—
123C22×C3×C4C2411yescharacteristic
124C22×Dic3—22no—
125C22×Dic3—22no—
126C22×Dic3—22no—
127C2×D12—22no—
128C2×D12—22no—
129C2×D12—22no—
130C2×D12—22no—
131C2×D12—24no—
132C2×D12—22no—
133C2×D12—48no—
134C2×D12—48no—
135C2×D12—42no—
136C2×D12—42no—
137C2×D12#4114yes—
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8—22no—
143C2×C3×C8—22no—
144C3×C4⋊C4—24no—
145C3×C22⋊C4—24no—
146C3×C42—22no—
147C3×C42—22no—
148#217—42no—
149#217—82no—
150(C2×D3)⋊C4—82no—
151(C2×D3)⋊C4—82no—
152(C2×D3)⋊C4—82no—
153(C2×D3)⋊C4—82no—
154C4⋊Dic3—22no—
155C4⋊Dic3—22no—
156C4⋊Dic3—22no—
157C4⋊Dic3—22no—
158C4⋊Dic3—22no—
159C4⋊Dic3—22no—
160C4⋊Dic3—82no—
161C4×Dic3—42no—
162C4×Dic3—82no—
163C25—61no—
164C22×D4—32no—
165C22×D4—64no—
166C22×D4—61no—
167C22×D4—64no—
168C22×D4—121no—
169C22×D4—121no—
170C23×C4C22×D311yescharacteristic
171C22×C8C22×D312yes—
172C22×C8C22×D312yes—
173#117—24no—
174#117—121no—
175#117—121no—
176C2×C4⋊C4—32no—
177C2×C4⋊C4—32no—
178C2×C4⋊C4—32no—
179C2×C4⋊C4—62no—
180C2×C4⋊C4D1212yes—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—62no—
183C2×C22⋊C4—62no—
184C2×C22⋊C4—62no—
185C2×C22⋊C4—62no—
186C2×C22⋊C4—62no—
187C2×C22⋊C4D1212yes—
188C2×C42—61no—
189C2×C42—62no—
190C2×C42C22×D311yescharacteristic
191C2×C42C22×D311yescharacteristic
192#107—122no—
193#106—122no—
194#102—64no—
195#102—64no—
196#102—122no—
197#102—122no—
198C22⋊C8—24no—
199#95—62no—
200#95—62no—
201#95—62no—
202C23×C3—21no—
203C23×C3—21no—
204C23×C3—22no—
205C23×C3—22no—
206C23×C3#12312yes—
207C23×C3C22×D411yescharacteristic
208C22×D3—24no—
209C22×D3—48no—
210C22×D3—42no—
211C22×D3—42no—
212C22×D3—42no—
213C22×D3—42no—
214C22×D3—42no—
215C22×D3—42no—
216C22×D3—84no—
217C22×D3—84no—
218C22×D3—88no—
219C22×D3—88no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4#12312yes—
231C2×C3×C4C2×#4112yes—
232C2×C3×C4C2×#4112yes—
233C2×C3×C4H(5,𝔽2)11yescharacteristic
234C2×C3×C4H(5,𝔽2)11yescharacteristic
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244D12—24no—
245D12—24no—
246D12—48no—
247D12—42no—
248D12—42no—
249C3×C8—42no—
250C3×C8—42no—
251C24—64no—
252C24—62no—
253C24—122no—
254C24—122no—
255C24—242no—
256C24—242no—
257C24C2×D1211yescharacteristic
258C2×D4—34no—
259C2×D4—64no—
260C2×D4—62no—
261C2×D4—62no—
262C2×D4—62no—
263C2×D4—62no—
264C2×D4—62no—
265C2×D4—128no—
266C2×D4—128no—
267C2×D4—122no—
268C2×D4—122no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—62no—
273C22×C4—62no—
274C22×C4—62no—
275C22×C4C2×D1212yes—
276C22×C4C2×D1212yes—
277C22×C4C2×D1211yescharacteristic
278C22×C4#23714yes—
279C22×C4C23×D311yescharacteristic
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C4⋊C4—24no—
287C4⋊C4—62no—
288C4⋊C4—62no—
289C4⋊C4—62no—
290C4⋊C4—62no—
291C4⋊C4—62no—
292C4⋊C4—62no—
293C4⋊C4—242no—
294C22⋊C4—24no—
295C22⋊C4—122no—
296C22⋊C4—242no—
297C22⋊C4—242no—
298C22⋊C4—242no—
299C22⋊C4—242no—
300C22⋊C4—242no—
301C42—22no—
302C42—22no—
303C42—122no—
304C42—242no—
305C22×C3—24no—
306C22×C3—22no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—42no—
310C22×C3C2×#12311yescharacteristic
311C22×C3#52511yescharacteristic
312C22×C3#53511yescharacteristic
313C22×C3#57612yes—
314C22×C3#57712yes—
315C2×D3—44no—
316C2×D3—44no—
317C2×D3—44no—
318C2×D3—816no—
319C2×D3—82no—
320C2×D3—82no—
321C2×D3—82no—
322C2×D3—82no—
323C3×C4—22no—
324C3×C4—42no—
325C3×C4—42no—
326C3×C4#55112yes—
327C3×C4#55112yes—
328Dic3—82no—
329Dic3—82no—
330Dic3—82no—
331C23—22no—
332C23—22no—
333C23—21no—
334C23—21no—
335C23—64no—
336C23—128no—
337C23—122no—
338C23—122no—
339C23—122no—
340C23—122no—
341C23—122no—
342C23—122no—
343C23—248no—
344C23—244no—
345C23—244no—
346C23—248no—
347C23#88612yes—
348C23C22×D1211yescharacteristic
349D4—64no—
350D4—64no—
351D4—128no—
352D4—122no—
353D4—122no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4—122no—
367C2×C4—122no—
368C2×C4—122no—
369C2×C4—122no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4#88612yes—
374C2×C4C2×#23712yes—
375C2×C4D3×#4112yes—
376C2×C4#100911yescharacteristic
377C2×C4#100911yescharacteristic
378C8—42no—
379C8—42no—
380C2×C3—42no—
381C2×C3#248211yescharacteristic
382C2×C3#309612yes—
383C2×C3#312211yescharacteristic
384C2×C3#312311yescharacteristic
385C2×C3#322412yes—
386D3—84no—
387C22—24no—
388C22—22no—
389C22—41no—
390C22—41no—
391C22—42no—
392C22—124no—
393C22—124no—
394C22—124no—
395C22—2416no—
396C22—242no—
397C22—242no—
398C22—242no—
399C22—242no—
400C22C2×#88611yescharacteristic
401C22#552411yescharacteristic
402C22#552511yescharacteristic
403C22#575112yes—
404C22#575212yes—
405C4—22no—
406C4—42no—
407C4—42no—
408C4—242no—
409C4—242no—
410C4—242no—
411C4#565812yes—
412C4#565812yes—
413C3#2759111yescharacteristic, complemented, Sylow
414C2—42no—
415C2—244no—
416C2#7412911yescharacteristic
417C2#7865112yes—
418C2#7938611yescharacteristic
419C2#7938711yescharacteristic
420C2#7939212yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1818C2
socleC23×C3C22×D4
prime radicalC23×D12C22
hypercenter#1818D3
hypocenterC3#27591
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1818

Complements

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

NH ≅ G/Ntype
2. #82231C2semidirect
3. C3×#1818C2semidirect
5. #69929C2semidirect
6. #68680C2semidirect
7. #68635C2semidirect
8. #68626C2semidirect
9. #68625C2semidirect
20. C2×#818C22semidirect
21. C2×#817C22semidirect
28. #4936C22semidirect
29. #4935C22semidirect
30. C4×C4⋊Dic3C22semidirect
31. C3×#336C22semidirect
32. C3×#336C22semidirect
33. C3×#336C22semidirect
34. #4551C22semidirect
35. #4551C22semidirect
43. #1818D3semidirect
106. #336C2×D3semidirect
107. #336C2×D3semidirect
108. #336C2×D3semidirect
413. C3#27591semidirect

Maximal subgroups

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

subgroupindex
#822692
#822312
C3×#18182
#699592
#699292
#686802
#686352
#686262
#686252
#275913