Finite Group Databasebeta

← #10945072

Subgroups — #10945072

Order 768.

1442 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1442 conjugacy classes of subgroups — too many to draw. Only its 203 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#87260C211yesmaximal, characteristic, complemented
2#84096C211yesmaximal, characteristic
3#74129C212yesmaximal, complemented
4C3×#1782C211yesmaximal, characteristic, complemented
5#70022C214yesmaximal, complemented
6#70021C214yesmaximal, complemented
7#69534C212yesmaximal, complemented
8#33021—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10#5827—22no—
11#5827—22no—
12Dic3×#41—24no—
13#5812—21no—
14C2×#937C2212yes—
15#5798—21no—
16#5798C2214yes—
17C22×(C2×D3)⋊C4C2211yescharacteristic
18C2×#926—22no—
19C2×#926C2212yes—
20C2×C3×#99C2214yes—
21C3×#381C2212yes—
22#5231—21no—
23#5223—21no—
24#5220C2212yes—
25#5213—22no—
26#5211C2211yescharacteristic
27#4955C2214yes—
28#4950C2214yes—
29#4677—22no—
30#4674—22no—
31#4670—24no—
32#4527C2218yes—
33#3428—31no—
34#2482—32no—
35#2404—31no—
36#1889—34no—
37#1867—34no—
38#1820—32no—
39#1782D311yescharacteristic, complemented
40C24×D3C2311yescharacteristic
41C2×C3×#41—28no—
42C2×C3×#41—22no—
43C2×C3×#41—22no—
44C22×C3×Q8D411yescharacteristic
45C22×C3×D4D411yescharacteristic
46C22×C3×D4C2312yes—
47C23×C3×C4D412yes—
48C23×C3×C4C2311yescharacteristic
49C22×#67—22no—
50C23×Dic3C2311yescharacteristic
51C2×C3×C22⋊C4C2314yes—
52C2×#217—24no—
53C2×#217—22no—
54C2×#217C2312yes—
55C2×#217C2314yes—
56Q8×Dic3—24no—
57Q8×Dic3—24no—
58#937—28no—
59D4×Dic3—24no—
60D4×Dic3—24no—
61D4×Dic3—28no—
62D4×Dic3—28no—
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—22no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—24no—
69C2×(C2×D3)⋊C4—24no—
70#926—24no—
71#926—24no—
72#926—22no—
73#926—24no—
74#926—22no—
75#926—22no—
76#926—22no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3—24no—
80C2×C4⋊Dic3D412yescomplemented
81C2×C4⋊Dic3D412yescomplemented
82C2×C4×Dic3—24no—
83C2×C4×Dic3D414yescomplemented
84C3×#99—28no—
85C3×#95—22no—
86C3×#95—22no—
87C3⋊#95—21no—
88C3⋊#95—21no—
89C3⋊#95—22no—
90C3⋊#95—24no—
91C22×#41C2×D311yescharacteristic
92C2×#120—32no—
93#520—62no—
94#520—62no—
95#519—61no—
96#518—34no—
97#518—61no—
98C4×#41—64no—
99C2×#117—32no—
100C2×#117—62no—
101C22×C22⋊C4—31no—
102C2×#99C2×D314yes—
103#399—62no—
104#397—62no—
105#393—61no—
106#392—61no—
107#388—64no—
108#386—32no—
109#386—62no—
110#385—34no—
111#381—31no—
112#381C2×D312yes—
113#379—34no—
114#342—38no—
115C24×C3C2×D412yes—
116C24×C3C2411yescharacteristic
117C23×D3—22no—
118C23×D3—21no—
119C23×D3—24no—
120C23×D3—42no—
121C23×D3—81no—
122C3×#41—216no—
123C3×#41—216no—
124C2×C3×Q8—22no—
125C2×C3×Q8—24no—
126C2×C3×D4—24no—
127C2×C3×D4—24no—
128C2×C3×D4—24no—
129C2×C3×D4—22no—
130C2×C3×D4#4118yes—
131C22×C3×C4—24no—
132C22×C3×C4—24no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—24no—
136C22×C3×C4—24no—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2×D412yes—
142C2×#67—44no—
143C2×#67—44no—
144C2×#67—82no—
145C22×Dic3—24no—
146C22×Dic3—24no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—21no—
150C22×Dic3—81no—
151C22×Dic3C2×D414yes—
152C22×Dic3C2×D414yes—
153C3×C22⋊C4—28no—
154#217—28no—
155#217—24no—
156#217—24no—
157#217—24no—
158#217—82no—
159(C2×D3)⋊C4—48no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—42no—
162(C2×D3)⋊C4—41no—
163(C2×D3)⋊C4—44no—
164(C2×D3)⋊C4—84no—
165C4⋊Dic3—24no—
166C4⋊Dic3—24no—
167C4⋊Dic3—22no—
168C4⋊Dic3—22no—
169C4⋊Dic3—22no—
170C4⋊Dic3—22no—
171C4⋊Dic3—84no—
172C4×Dic3—28no—
173C4×Dic3—24no—
174C4×Dic3—24no—
175C25—31no—
176C2×#41—28no—
177C2×#41—22no—
178C2×#41—22no—
179C22×Q8D1211yescharacteristic
180C22×D4—62no—
181C22×D4D1211yescharacteristic
182C22×D4C22×D312yes—
183C23×C4—31no—
184C23×C4D1212yes—
185C23×C4C22×D311yescharacteristic
186#120—68no—
187C4×Q8—64no—
188C4×Q8—64no—
189C4×D4—68no—
190C4×D4—68no—
191C4×D4—64no—
192C4×D4—64no—
193#117—64no—
194#117—64no—
195#117—62no—
196#117—62no—
197#117—64no—
198#117—62no—
199#117—62no—
200C2×C4⋊C4—32no—
201C2×C4⋊C4—32no—
202C2×C4⋊C4—62no—
203C2×C4⋊C4—62no—
204C2×C4⋊C4—64no—
205C2×C22⋊C4—34no—
206C2×C22⋊C4—32no—
207C2×C22⋊C4—64no—
208C2×C22⋊C4—64no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—62no—
214C2×C22⋊C4—64no—
215C2×C22⋊C4—62no—
216C2×C22⋊C4C22×D314yes—
217C2×C42—34no—
218C2×C42—64no—
219#99—28no—
220#95—22no—
221#95—22no—
222#95—64no—
223#95—61no—
224#95—61no—
225#95—62no—
226C23×C3—22no—
227C23×C3—22no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3#12314yes—
233C23×C3C22×D411yescharacteristic
234C23×C3C22×D412yes—
235C23×C3C2×#4114yes—
236C22×D3—24no—
237C22×D3—22no—
238C22×D3—44no—
239C22×D3—42no—
240C22×D3—42no—
241C22×D3—41no—
242C22×D3—41no—
243C22×D3—42no—
244C22×D3—44no—
245C22×D3—88no—
246C3×Q8—28no—
247C3×D4—28no—
248C3×D4—216no—
249C2×C3×C4—28no—
250C2×C3×C4—28no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—44no—
258C2×C3×C4—44no—
259C2×C3×C4—44no—
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12012yes—
262C2×C3×C4#12011yescharacteristic
263C2×C3×C4#12314yes—
264#67—88no—
265C2×Dic3—24no—
266C2×Dic3—24no—
267C2×Dic3—44no—
268C2×Dic3—44no—
269C2×Dic3—44no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—41no—
274C2×Dic3—42no—
275C2×Dic3—84no—
276C2×Dic3—84no—
277C24—61no—
278C24—62no—
279C24—64no—
280C24—122no—
281C24—241no—
282C24C2×D1212yes—
283C24C23×D311yescharacteristic
284#41—216no—
285#41—216no—
286C2×Q8—22no—
287C2×Q8—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—22no—
292C2×D4—124no—
293C2×D4—124no—
294C2×D4—242no—
295C2×D4#23718yes—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—34no—
303C22×C4—34no—
304C22×C4—64no—
305C22×C4—64no—
306C22×C4—62no—
307C22×C4—61no—
308C22×C4—62no—
309C22×C4—241no—
310C22×C4C2×D1211yescharacteristic
311C22×C4C2×D1211yescharacteristic
312C22×C4C2×D1212yes—
313C22×C4D3×D412yes—
314C22×C4D3×D412yes—
315C4⋊C4—64no—
316C4⋊C4—64no—
317C4⋊C4—62no—
318C4⋊C4—62no—
319C4⋊C4—62no—
320C4⋊C4—62no—
321C4⋊C4—244no—
322C22⋊C4—28no—
323C22⋊C4—68no—
324C22⋊C4—64no—
325C22⋊C4—64no—
326C22⋊C4—64no—
327C22⋊C4—128no—
328C22⋊C4—121no—
329C22⋊C4—122no—
330C22⋊C4—121no—
331C22⋊C4—124no—
332C22⋊C4—244no—
333C22⋊C4—242no—
334C42—68no—
335C42—64no—
336C42—64no—
337C22×C3—22no—
338C22×C3—21no—
339C22×C3—22no—
340C22×C3—21no—
341C22×C3—24no—
342C22×C3—44no—
343C22×C3—44no—
344C22×C3—42no—
345C22×C3—42no—
346C22×C3—82no—
347C22×C3C2×#12011yescharacteristic
348C22×C3C2×#12312yes—
349C22×C3#52514yes—
350C2×D3—44no—
351C2×D3—48no—
352C2×D3—42no—
353C2×D3—82no—
354C2×D3—84no—
355C2×D3—81no—
356C2×D3—81no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—22no—
360C3×C4—24no—
361C3×C4—84no—
362Dic3—44no—
363Dic3—44no—
364Dic3—82no—
365Dic3—81no—
366C23—22no—
367C23—22no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—42no—
372C23—64no—
373C23—62no—
374C23—124no—
375C23—122no—
376C23—122no—
377C23—121no—
378C23—121no—
379C23—122no—
380C23—124no—
381C23—248no—
382C23#88614yes—
383C23C22×D1211yescharacteristic
384C23C2×D3×D412yes—
385C23C2×#23714yes—
386Q8—28no—
387D4—28no—
388D4—216no—
389D4—248no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—64no—
402C2×C4—64no—
403C2×C4—124no—
404C2×C4—124no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—124no—
408C2×C4—122no—
409C2×C4—121no—
410C2×C4—122no—
411C2×C4—244no—
412C2×C4—244no—
413C2×C4#88211yescharacteristic
414C2×C4#88212yes—
415C2×C4#88211yescharacteristic
416C2×C4#88614yes—
417C2×C3—22no—
418C2×C3—24no—
419C2×C3—44no—
420C2×C3#284911yescharacteristic
421C2×C3#302511yescharacteristic
422C2×C3#302711yescharacteristic
423D3—82no—
424C22—22no—
425C22—21no—
426C22—22no—
427C22—21no—
428C22—24no—
429C22—44no—
430C22—44no—
431C22—42no—
432C22—42no—
433C22—82no—
434C22—124no—
435C22—128no—
436C22—122no—
437C22—244no—
438C22—242no—
439C22—241no—
440C22—241no—
441C22C2×#88211yescharacteristic
442C22C2×#88612yes—
443C22#558614yes—
444C4—21no—
445C4—21no—
446C4—22no—
447C4—24no—
448C4—84no—
449C4—124no—
450C4—124no—
451C4—242no—
452C4—241no—
453C3#3302111yescharacteristic, complemented, Sylow
454C2—24no—
455C2—22no—
456C2—44no—
457C2—242no—
458C2#7384711yescharacteristic
459C2#7869011yescharacteristic
460C2#7869111yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#1782C2
socleC22×C3C2×#120
prime radical#84096C2
hypercenter#1782D3
hypocenterC3#33021
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
1. #87260C2semidirect
3. #74129C2semidirect
4. C3×#1782C2semidirect
5. #70022C2semidirect
6. #70021C2semidirect
7. #69534C2semidirect
39. #1782D3semidirect
80. C2×C4⋊Dic3D4semidirect
81. C2×C4⋊Dic3D4semidirect
83. C2×C4×Dic3D4semidirect
453. C3#33021semidirect

Maximal subgroups

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

subgroupindex
#872602
#840962
#741292
C3×#17822
#700222
#700212
#695342
#330213