Finite Group Databasebeta

← #10710684

Subgroups — #10710684

Order 768.

1214 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1214 conjugacy classes of subgroups — too many to draw. Only its 139 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×C3×#535C211yesmaximal, characteristic
2#72130C212yesmaximal, complemented
3C2×#4545C212yesmaximal, complemented
4C2×#4541C212yesmaximal, complemented
5#12696—31nomaximal, Sylow
6C22×C3×#41C411yescharacteristic, complemented
7C23×C3×D4C2211yescharacteristic
8C3×#535—24no—
9C3×#535—24no—
10C2×C3×#127C411yescharacteristic, complemented
11C2×C3×#127C2211yescharacteristic
12C2×C3×#124C411yescharacteristic, complemented
13C2×C3×#124C2211yescharacteristic
14C2×C3×#121—22no—
15C2×C3×#120—22no—
16C2×C3×#117C411yescharacteristic, complemented
17#5247—22no—
18C2×#834—22no—
19C2×#834C2214yes—
20#4545—24no—
21#4541—24no—
22C2×#535D311yescharacteristic
23C2×#354—32no—
24C2×#353—32no—
25#1781—32no—
26C25×C3D412yes—
27C2×C3×#41—22no—
28C2×C3×#41—22no—
29C2×C3×#41—42no—
30C2×C3×#41—42no—
31C22×C3×Q8C2×C411yescharacteristic
32C22×C3×D4—21no—
33C22×C3×D4—22no—
34C22×C3×D4—22no—
35C22×C3×D4—22no—
36C22×C3×D4—22no—
37C22×C3×D4—42no—
38C22×C3×D4—42no—
39C22×C3×D4C2×C411yescharacteristic
40C22×C3×D4C2×C411yescharacteristic
41C22×C3×D4D412yes—
42C22×C3×D4C2311yescharacteristic
43C23×C3×C4—21no—
44C23×C3×C4C2×C411yescharacteristic
45C3×#127—22no—
46C3×#127—22no—
47C3×#127—22no—
48C3×#127D414yes—
49C3×#124—22no—
50C3×#124—22no—
51C3×#124—22no—
52C3×#124D414yes—
53C3×#121—216no—
54C3×#120—216no—
55C3×#117—22no—
56C3×#117—22no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—22no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C42C2×C411yescharacteristic
62C2×C3×C42C2×C411yescharacteristic
63C2×#217—24no—
64C2×#217—44no—
65#834—24no—
66#834—24no—
67#834—44no—
68C22×#41Dic311yescharacteristic, complemented
69C23×D4C2×D311yescharacteristic
70#535—24no—
71#535—24no—
72C2×#127Dic311yescharacteristic, complemented
73C2×#127C2×D311yescharacteristic
74C2×#124Dic311yescharacteristic, complemented
75C2×#124C2×D311yescharacteristic
76C2×#121—22no—
77C2×#120—22no—
78C2×#117Dic311yescharacteristic, complemented
79C2×#99—34no—
80C2×#99—62no—
81#379—62no—
82#354—64no—
83#353—64no—
84C24×C3—21no—
85C24×C3—24no—
86C24×C3—24no—
87C24×C3—42no—
88C24×C3—84no—
89C24×C3C22⋊C412yes—
90C24×C3C22⋊C411yescharacteristic
91C24×C3C2×D412yes—
92C3×#41—48no—
93C3×#41—48no—
94C2×C3×Q8—21no—
95C2×C3×Q8—41no—
96C2×C3×Q8—41no—
97C2×C3×Q8C22⋊C412yescomplemented
98C2×C3×D4—24no—
99C2×C3×D4—24no—
100C2×C3×D4—28no—
101C2×C3×D4—28no—
102C2×C3×D4—24no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4—41no—
107C2×C3×D4—41no—
108C2×C3×D4—42no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—48no—
114C2×C3×D4—48no—
115C2×C3×D4—42no—
116C2×C3×D4—42no—
117C2×C3×D4—84no—
118C2×C3×D4—84no—
119C2×C3×D4C22⋊C412yescomplemented
120C2×C3×D4C2×D412yes—
121C2×C3×D4C2×D412yes—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—24no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—42no—
131C22×C3×C4—42no—
132C22×C3×C4C22⋊C411yescharacteristic
133C22×C3×C4C22×C411yescharacteristic
134C22×Dic3—44no—
135C3×C4⋊C4—24no—
136C3×C22⋊C4—24no—
137C3×C22⋊C4—28no—
138C3×C22⋊C4—28no—
139C3×C42—21no—
140C3×C42—21no—
141C3×C42C22⋊C412yescomplemented
142C3×C42C22⋊C412yescomplemented
143#217—44no—
144#217—44no—
145#217—88no—
146C25#6712yes—
147C2×#41—22no—
148C2×#41—22no—
149C2×#41—42no—
150C2×#41—42no—
151C22×Q8C2×Dic311yescharacteristic
152C22×D4—21no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4C2×Dic311yescharacteristic
160C22×D4C2×Dic311yescharacteristic
161C22×D4#6712yes—
162C22×D4C22×D311yescharacteristic
163C23×C4—21no—
164C23×C4C2×Dic311yescharacteristic
165#127—22no—
166#127—22no—
167#127—22no—
168#127#6714yes—
169#124—22no—
170#124—22no—
171#124—22no—
172#124#6714yes—
173#121—216no—
174#120—216no—
175#117—22no—
176#117—22no—
177C2×C4⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—64no—
182C2×C22⋊C4—124no—
183C2×C42C2×Dic311yescharacteristic
184C2×C42C2×Dic311yescharacteristic
185#99—64no—
186#99—64no—
187#99—124no—
188C23×C3—21no—
189C23×C3—22no—
190C23×C3—21no—
191C23×C3—22no—
192C23×C3—22no—
193C23×C3—22no—
194C23×C3—22no—
195C23×C3—22no—
196C23×C3—42no—
197C23×C3—48no—
198C23×C3—44no—
199C23×C3—44no—
200C23×C3—48no—
201C23×C3—44no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—84no—
205C23×C3—88no—
206C23×C3—88no—
207C23×C3#9912yes—
208C23×C3C2×C22⋊C411yescharacteristic
209C23×C3#12014yes—
210C3×Q8—42no—
211C3×Q8—42no—
212C3×D4—42no—
213C3×D4—42no—
214C3×D4—48no—
215C3×D4—42no—
216C3×D4—42no—
217C3×D4—42no—
218C3×D4—42no—
219C3×D4—48no—
220C3×D4—88no—
221C3×D4—88no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—48no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4#9912yes—
237C2×C3×C4C2×C22⋊C411yescharacteristic
238C2×C3×C4C2×C22⋊C411yescharacteristic
239C2×Dic3—84no—
240C2×Dic3—88no—
241C24—21no—
242C24—24no—
243C24—24no—
244C24—42no—
245C24—84no—
246C24#21712yes—
247C24#21711yescharacteristic
248C24C2×#6712yes—
249#41—48no—
250#41—48no—
251C2×Q8—21no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8#21712yescomplemented
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—28no—
258C2×D4—28no—
259C2×D4—24no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—48no—
271C2×D4—48no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—84no—
275C2×D4—84no—
276C2×D4#21712yescomplemented
277C2×D4C2×#6712yes—
278C2×D4C2×#6712yes—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—24no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—124no—
290C22×C4#21711yescharacteristic
291C22×C4C22×Dic311yescharacteristic
292C4⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—28no—
295C22⋊C4—28no—
296C22⋊C4—124no—
297C22⋊C4—124no—
298C22⋊C4—248no—
299C42—21no—
300C42—21no—
301C42#21712yescomplemented
302C42#21712yescomplemented
303C22×C3—22no—
304C22×C3—24no—
305C22×C3—24no—
306C22×C3—22no—
307C22×C3—44no—
308C22×C3—42no—
309C22×C3—42no—
310C22×C3—44no—
311C22×C3—44no—
312C22×C3—44no—
313C22×C3—44no—
314C22×C3—42no—
315C22×C3—42no—
316C22×C3—82no—
317C22×C3—82no—
318C22×C3—82no—
319C22×C3—816no—
320C22×C3—82no—
321C22×C3#37911yescharacteristic
322C22×C3C2×#9911yescharacteristic
323C22×C3C2×#9911yescharacteristic
324C3×C4—22no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—42no—
328C3×C4—42no—
329C3×C4—42no—
330Dic3—164no—
331C23—21no—
332C23—22no—
333C23—21no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—42no—
340C23—48no—
341C23—44no—
342C23—44no—
343C23—48no—
344C23—44no—
345C23—41no—
346C23—41no—
347C23—84no—
348C23—88no—
349C23—88no—
350C23#83412yes—
351C23C2×#21711yescharacteristic
352C23#95314yes—
353Q8—42no—
354Q8—42no—
355D4—42no—
356D4—42no—
357D4—48no—
358D4—42no—
359D4—42no—
360D4—42no—
361D4—42no—
362D4—48no—
363D4—88no—
364D4—88no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—48no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—244no—
380C2×C4—248no—
381C2×C4#83412yes—
382C2×C4C2×#21711yescharacteristic
383C2×C4C2×#21711yescharacteristic
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—44no—
387C2×C3—42no—
388C2×C3—82no—
389C2×C3—82no—
390C2×C3#178111yescharacteristic
391C2×C3#212712yes—
392C22—22no—
393C22—24no—
394C22—22no—
395C22—24no—
396C22—44no—
397C22—42no—
398C22—42no—
399C22—44no—
400C22—44no—
401C22—44no—
402C22—44no—
403C22—42no—
404C22—42no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22—816no—
409C22—82no—
410C22C2×#83411yescharacteristic
411C22C2×#83411yescharacteristic
412C22#524711yescharacteristic
413C4—22no—
414C4—21no—
415C4—21no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—484no—
420C3#1269611yescharacteristic, complemented, Sylow
421C2—21no—
422C2—21no—
423C2—44no—
424C2—42no—
425C2—82no—
426C2—82no—
427C2#7186012yes—
428C2#7213011yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC22×D4C22×D3
Fitting, prime radicalC2×C3×#535C2
socleC22×C3C2×#99
cosocleC22×C3×D4C23
hypercenterC2×#535D3
hypocenterC3#12696
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
2. #72130C2semidirect
3. C2×#4545C2semidirect
4. C2×#4541C2semidirect
6. C22×C3×#41C4semidirect
10. C2×C3×#127C4semidirect
12. C2×C3×#124C4semidirect
16. C2×C3×#117C4semidirect
68. C22×#41Dic3semidirect
72. C2×#127Dic3semidirect
74. C2×#124Dic3semidirect
78. C2×#117Dic3semidirect
97. C2×C3×Q8C22⋊C4semidirect
119. C2×C3×D4C22⋊C4semidirect
141. C3×C42C22⋊C4semidirect
142. C3×C42C22⋊C4semidirect
254. C2×Q8#217semidirect
276. C2×D4#217semidirect
301. C42#217semidirect
302. C42#217semidirect
420. C3#12696semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5352
#721302
C2×#45452
C2×#45412
#126963