Finite Group Databasebeta

← #10804450

Subgroups — #10804450

Order 768.

1538 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1538 conjugacy classes of subgroups — too many to draw. Only its 235 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
1C2×#5804C211yesmaximal, characteristic
2#73820C212yesmaximal, complemented
3C3×#1781C211yesmaximal, characteristic, complemented
4#72130C211yesmaximal, characteristic, complemented
5#70030C218yesmaximal, complemented
6#69498C212yesmaximal, complemented
7#21795—31nomaximal, Sylow
8C23×C3×D4C2211yescharacteristic
9C23×#67C2211yescharacteristic
10C2×#953C412yescomplemented
11C22×#217C2211yescharacteristic
12#5804—24no—
13#5804—24no—
14C2×#939C2214yes—
15C2×#937C412yescomplemented
16C2×#935C412yescomplemented
17C2×#929C412yescomplemented
18C2×C3×#99C2214yescomplemented
19C3×#379C2212yes—
20#5247C2212yes—
21C2×#834C2214yescomplemented
22C2×#806C2218yescomplemented
23#4945—24no—
24#4944—24no—
25#4942C2214yescomplemented
26Dic3×C22⋊C4C2214yescomplemented
27C2×#534—31no—
28#2312—32no—
29#1964—38no—
30#1781—32no—
31#1781—31no—
32#1781D311yescharacteristic, complemented
33C25×C3C2×C411yescharacteristic
34C25×C3C2311yescharacteristic
35C24×D3C2×C411yescharacteristic
36C22×C3×D4—22no—
37C22×C3×D4—22no—
38C22×C3×D4—44no—
39C22×C3×D4C2×C412yes—
40C22×C3×D4C2312yes—
41C23×C3×C4C2×C411yescharacteristic
42C22×#67—22no—
43C22×#67—22no—
44C22×#67—44no—
45C22×#67C2×C412yes—
46C22×#67C2×C414yes—
47C22×#67C2312yes—
48C23×Dic3C2×C411yescharacteristic
49C23×Dic3C2312yes—
50C2×C3×C22⋊C4—22no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4C2314yes—
53#953—28no—
54C2×#217—22no—
55C2×#217—22no—
56C2×#217C2×C414yes—
57C2×#217C2×C412yes—
58C2×#217C2×C412yes—
59C2×#217D414yes—
60C2×#217D414yes—
61C2×#217C2314yes—
62#939—28no—
63#939—28no—
64#937—24no—
65#937—24no—
66#935—24no—
67#935—24no—
68#929—28no—
69C2×(C2×D3)⋊C4C2×C414yes—
70C2×Dic3⋊C4—24no—
71C2×Dic3⋊C4C2×C414yes—
72C2×C4×Dic3—24no—
73C3×#99—44no—
74#834—44no—
75C3⋊#95—24no—
76C3⋊#95—24no—
77#806—48no—
78C23×D4—31no—
79C23×D4C2×D311yescharacteristic
80#534—64no—
81#534—64no—
82C2×#123—32no—
83C2×#123—32no—
84C2×#121—34no—
85C2×#120—32no—
86C2×#120—32no—
87C22×C22⋊C4—31no—
88C2×#99—34no—
89C2×#99—38no—
90C2×#99C2×D314yescomplemented
91#388—64no—
92#388—64no—
93#380—34no—
94#379—32no—
95#379C2×D312yes—
96C4×C22⋊C4—34no—
97C24×C3—24no—
98C24×C3—22no—
99C24×C3—42no—
100C24×C3—82no—
101C24×C3—82no—
102C24×C3C22×C412yes—
103C24×C3C22×C411yescharacteristic
104C24×C3C22×C411yescharacteristic
105C24×C3C2×D412yes—
106C24×C3C2×D412yes—
107C24×C3C2411yescharacteristic
108C23×D3—21no—
109C23×D3—21no—
110C23×D3—42no—
111C23×D3—82no—
112C23×D3C22×C412yes—
113C2×C3×D4—22no—
114C2×C3×D4—22no—
115C2×C3×D4—24no—
116C2×C3×D4—24no—
117C2×C3×D4—24no—
118C2×C3×D4—42no—
119C2×C3×D4—42no—
120C2×C3×D4—48no—
121C2×C3×D4—48no—
122C22×C3×C4—22no—
123C22×C3×C4—24no—
124C22×C3×C4—42no—
125C22×C3×C4C22×C412yes—
126C2×#67—24no—
127C2×#67—24no—
128C2×#67—28no—
129C2×#67—28no—
130C2×#67—44no—
131C2×#67—48no—
132C2×#67—48no—
133C2×#67—84no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—24no—
137C22×Dic3—42no—
138C22×Dic3—82no—
139C22×Dic3C22×C412yes—
140C22×Dic3C22×C414yes—
141C22×Dic3C2×D414yes—
142C22×Dic3C2×D414yes—
143C3×C22⋊C4—28no—
144C3×C22⋊C4—82no—
145C3×C22⋊C4—82no—
146#217—28no—
147#217—28no—
148#217—82no—
149#217—82no—
150#217C22⋊C418yes—
151#217C22⋊C414yes—
152#217C22⋊C414yes—
153(C2×D3)⋊C4—28no—
154Dic3⋊C4—28no—
155Dic3⋊C4—84no—
156C4×Dic3—84no—
157C25—31no—
158C25C4×D311yescharacteristic
159C25C22×D311yescharacteristic
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—32no—
163C22×D4—32no—
164C22×D4—34no—
165C22×D4—44no—
166C22×D4—62no—
167C22×D4—62no—
168C22×D4—124no—
169C22×D4C4×D312yes—
170C22×D4C22×D312yes—
171C23×C4—32no—
172C23×C4—31no—
173C23×C4C4×D311yescharacteristic
174#123—68no—
175#123—64no—
176#123—64no—
177#121—68no—
178#121—68no—
179#120—64no—
180#120—64no—
181#120—68no—
182C2×C4⋊C4—34no—
183C2×C4⋊C4—64no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—32no—
187C2×C22⋊C4—32no—
188C2×C22⋊C4—34no—
189C2×C22⋊C4—34no—
190C2×C22⋊C4—34no—
191C2×C22⋊C4—34no—
192C2×C22⋊C4—34no—
193C2×C22⋊C4—62no—
194C2×C22⋊C4—62no—
195C2×C22⋊C4C22×D314yes—
196C2×C42—64no—
197#99—44no—
198#99—128no—
199#99—124no—
200#95—64no—
201#95—64no—
202C23×C3—22no—
203C23×C3—22no—
204C23×C3—24no—
205C23×C3—24no—
206C23×C3—22no—
207C23×C3—22no—
208C23×C3—22no—
209C23×C3—42no—
210C23×C3—42no—
211C23×C3—42no—
212C23×C3—42no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3—44no—
217C23×C3—44no—
218C23×C3—82no—
219C23×C3—82no—
220C23×C3—88no—
221C23×C3—88no—
222C23×C3C2×C22⋊C411yescharacteristic
223C23×C3C2×C22⋊C411yescharacteristic
224C23×C3C2×C22⋊C412yes—
225C23×C3C23×C411yescharacteristic
226C23×C3C22×D412yes—
227C23×C3H(5,𝔽2)14yes—
228C22×D3—22no—
229C22×D3—24no—
230C22×D3—42no—
231C22×D3—42no—
232C22×D3—44no—
233C22×D3—82no—
234C22×D3—82no—
235C22×D3—88no—
236C3×D4—44no—
237C3×D4—44no—
238C3×D4—48no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—44no—
243C2×C3×C4—44no—
244C2×C3×C4—44no—
245C2×C3×C4—44no—
246#67—48no—
247#67—48no—
248#67—88no—
249C2×Dic3—22no—
250C2×Dic3—24no—
251C2×Dic3—44no—
252C2×Dic3—44no—
253C2×Dic3—44no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—44no—
257C2×Dic3—44no—
258C2×Dic3—44no—
259C2×Dic3C2×C22⋊C418yes—
260C24—24no—
261C24—22no—
262C24—32no—
263C24—42no—
264C24—61no—
265C24—61no—
266C24—82no—
267C24—82no—
268C24—122no—
269C24—242no—
270C24C2×C4×D312yes—
271C24C2×C4×D311yescharacteristic
272C24C2×C4×D311yescharacteristic
273C24D3×D412yes—
274C24D3×D412yes—
275C24C23×D311yescharacteristic
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—48no—
284C2×D4—48no—
285C2×D4—68no—
286C2×D4—64no—
287C2×D4—64no—
288C2×D4—68no—
289C2×D4—124no—
290C2×D4—128no—
291C2×D4—128no—
292C2×D4—244no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—34no—
296C22×C4—34no—
297C22×C4—34no—
298C22×C4—32no—
299C22×C4—42no—
300C22×C4—61no—
301C22×C4—61no—
302C22×C4—64no—
303C22×C4—122no—
304C22×C4—242no—
305C22×C4C2×C4×D312yes—
306C4⋊C4—68no—
307C4⋊C4—244no—
308C22⋊C4—28no—
309C22⋊C4—38no—
310C22⋊C4—34no—
311C22⋊C4—34no—
312C22⋊C4—68no—
313C22⋊C4—68no—
314C22⋊C4—68no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—242no—
318C22⋊C4—242no—
319C42—244no—
320C22×C3—22no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—44no—
331C22×C3—44no—
332C22×C3—44no—
333C22×C3—42no—
334C22×C3—44no—
335C22×C3—84no—
336C22×C3—84no—
337C22×C3—82no—
338C22×C3—82no—
339C22×C3—88no—
340C22×C3C22×C22⋊C411yescharacteristic
341C22×C3#51812yes—
342C22×C3#53514yes—
343C2×D3—42no—
344C2×D3—44no—
345C2×D3—84no—
346C2×D3—84no—
347C2×D3—84no—
348C3×C4—42no—
349C3×C4—84no—
350Dic3—28no—
351Dic3—42no—
352Dic3—84no—
353C23—22no—
354C23—22no—
355C23—24no—
356C23—24no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—44no—
368C23—44no—
369C23—62no—
370C23—64no—
371C23—82no—
372C23—82no—
373C23—88no—
374C23—88no—
375C23—122no—
376C23—122no—
377C23—124no—
378C23—242no—
379C23—242no—
380C23—248no—
381C23D3×C22⋊C412yes—
382C23D3×C22⋊C411yescharacteristic
383C23D3×C22⋊C411yescharacteristic
384C23C22×C4×D311yescharacteristic
385C23C2×D3×D412yes—
386C23#100414yes—
387D4—44no—
388D4—44no—
389D4—48no—
390D4—128no—
391D4—128no—
392D4—248no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—38no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—62no—
402C2×C4—64no—
403C2×C4—124no—
404C2×C4—124no—
405C2×C4—124no—
406C2×C4—122no—
407C2×C4—122no—
408C2×C4—124no—
409C2×C4—124no—
410C2×C4—124no—
411C2×C3—24no—
412C2×C3—22no—
413C2×C3—41no—
414C2×C3—41no—
415C2×C3—42no—
416C2×C3—82no—
417C2×C3#230911yescharacteristic
418C2×C3#288312yes—
419D3—82no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—44no—
431C22—44no—
432C22—44no—
433C22—42no—
434C22—44no—
435C22—84no—
436C22—84no—
437C22—82no—
438C22—82no—
439C22—88no—
440C22—122no—
441C22—124no—
442C22—244no—
443C22—244no—
444C22—244no—
445C22C2×D3×C22⋊C411yescharacteristic
446C22#548612yes—
447C22#559214yes—
448C4—42no—
449C4—68no—
450C4—84no—
451C4—122no—
452C4—244no—
453C3#2179511yescharacteristic, complemented, Sylow
454C2—24no—
455C2—22no—
456C2—42no—
457C2—41no—
458C2—41no—
459C2—82no—
460C2—242no—
461C2#7381911yescharacteristic
462C2#7868812yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1781C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5804C2
cosocleC24×C3C24
hypercenter#1781D3
hypocenterC3#21795
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #1781

Complements

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

NH ≅ G/Ntype
2. #73820C2semidirect
3. C3×#1781C2semidirect
4. #72130C2semidirect
5. #70030C2semidirect
6. #69498C2semidirect
10. C2×#953C4semidirect
15. C2×#937C4semidirect
16. C2×#935C4semidirect
17. C2×#929C4semidirect
18. C2×C3×#99C22semidirect
21. C2×#834C22semidirect
22. C2×#806C22semidirect
25. #4942C22semidirect
26. Dic3×C22⋊C4C22semidirect
32. #1781D3semidirect
90. C2×#99C2×D3semidirect
453. C3#21795semidirect

Maximal subgroups

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

subgroupindex
C2×#58042
#738202
C3×#17812
#721302
#700302
#694982
#217953