Finite Group Databasebeta

← #10804127

Subgroups — #10804127

Order 768.

1570 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#929C211yesmaximal, characteristic
2C3×#2280C211yesmaximal, characteristic, complemented
3#84027C211yesmaximal, characteristic, complemented
4#71595C214yesmaximal, complemented
5#69535C214yesmaximal, complemented
6#69498C212yesmaximal
7#69495C212yesmaximal, complemented
8#21747—31nomaximal, Sylow
9C24×C3×C4C2211yescharacteristic
10C23×#67C2211yescharacteristic
11C2×C3×#117C2212yescomplemented
12C22×#217C2211yescharacteristic
13C2×#929—22no—
14C2×#929—22no—
15C2×#929—22no—
16C2×#929—22no—
17C2×#929C414yescomplemented
18C2×#929C2212yes—
19C2×#929C2212yes—
20C22×(C2×D3)⋊C4C412yescomplemented
21C2×#926C2212yescomplemented
22C22×Dic3⋊C4C412yescomplemented
23C3×#381C2212yescomplemented
24C3×#380—22no—
25C3×#379C2212yes—
26C3×C4×C22⋊C4—22no—
27#5247C2212yes—
28#5212—22no—
29#5211C2212yescomplemented
30C4×#217—22no—
31C2×#806C2218yes—
32#4527C2218yescomplemented
33C22×#123—31no—
34#2280—31no—
35#2280D311yescharacteristic, complemented
36#1899—34no—
37#1899—34no—
38#1819—32no—
39#1781—32no—
40C25×C3C2311yescharacteristic
41C24×D3C2×C411yescharacteristic
42C23×C3×C4—44no—
43C23×C3×C4C2×C412yes—
44C23×C3×C4C2×C412yes—
45C23×C3×C4D412yes—
46C23×C3×C4D412yes—
47C23×C3×C4D412yes—
48C23×C3×C4D412yes—
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C22×#67—22no—
52C22×#67—22no—
53C22×#67—44no—
54C22×#67C2×C412yes—
55C22×#67C2312yes—
56C23×Dic3C2×C411yescharacteristic
57C23×Dic3C2×C412yes—
58C3×#117—44no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C22⋊C4—24no—
61C2×C3×C22⋊C4C2314yes—
62C2×C3×C42—22no—
63C2×#217—22no—
64C2×#217—22no—
65C2×#217—24no—
66C2×#217C2×C412yes—
67C2×#217C2314yes—
68C2×#217C2312yes—
69#929—24no—
70#929—24no—
71#929—24no—
72#929—24no—
73#929—24no—
74#929—22no—
75#929—22no—
76#929—24no—
77#929—22no—
78#929—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4C2×C414yes—
84C2×(C2×D3)⋊C4C2×C414yes—
85#926—44no—
86C2×C4⋊Dic3—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4C2×C414yes—
92C2×Dic3⋊C4C2×C414yes—
93C2×C4×Dic3—22no—
94C3×#95—24no—
95C3⋊#95—24no—
96#806—48no—
97C23×D4—31no—
98C24×C4C2×D311yescharacteristic
99C2×#123—34no—
100C2×#123—32no—
101C2×#123—32no—
102C2×#123—62no—
103C2×#123—62no—
104C2×#123—62no—
105C2×#123—62no—
106C2×#117—32no—
107C2×#117C2×D312yescomplemented
108C22×C4⋊C4—32no—
109C22×C22⋊C4—31no—
110C22×C22⋊C4—32no—
111C2×#99—38no—
112#381—32no—
113#381C2×D312yescomplemented
114#380—22no—
115#380—62no—
116#379—32no—
117#379C2×D312yes—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—62no—
120#342—38no—
121C24×C3—24no—
122C24×C3—44no—
123C24×C3C22×C411yescharacteristic
124C24×C3C22×C411yescharacteristic
125C24×C3C2×D412yes—
126C24×C3C2×D412yes—
127C24×C3C2411yescharacteristic
128C23×D3—22no—
129C23×D3—42no—
130C23×D3—44no—
131C23×D3C22×C412yes—
132C22×C3×C4—24no—
133C22×C3×C4—24no—
134C22×C3×C4—24no—
135C22×C3×C4—44no—
136C22×C3×C4—44no—
137C22×C3×C4—44no—
138C22×C3×C4—44no—
139C22×C3×C4—44no—
140C22×C3×C4—44no—
141C22×C3×C4C22⋊C412yes—
142C22×C3×C4C22⋊C412yes—
143C22×C3×C4C22⋊C412yes—
144C22×C3×C4C22⋊C412yes—
145C22×C3×C4C22⋊C412yes—
146C22×C3×C4C22⋊C412yes—
147C22×C3×C4C22⋊C412yes—
148C22×C3×C4C22⋊C412yes—
149C22×C3×C4C22×C412yes—
150C22×C3×C4C22×C412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C2×#67—24no—
158C2×#67—22no—
159C2×#67—22no—
160C2×#67—28no—
161C2×#67—44no—
162C2×#67—48no—
163C2×#67—48no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—24no—
168C22×Dic3—42no—
169C22×Dic3—44no—
170C22×Dic3C22×C412yes—
171C22×Dic3C22×C414yes—
172C3×C4⋊C4—44no—
173C3×C22⋊C4—28no—
174C3×C22⋊C4—48no—
175C3×C42—44no—
176#217—24no—
177#217—22no—
178#217—22no—
179#217—28no—
180#217—48no—
181(C2×D3)⋊C4—24no—
182(C2×D3)⋊C4—24no—
183(C2×D3)⋊C4—24no—
184(C2×D3)⋊C4—24no—
185C4⋊Dic3—44no—
186Dic3⋊C4—24no—
187Dic3⋊C4—24no—
188Dic3⋊C4—24no—
189Dic3⋊C4—24no—
190C4×Dic3—44no—
191C25—31no—
192C25C22×D311yescharacteristic
193C22×D4—32no—
194C22×D4—32no—
195C22×D4—62no—
196C22×D4—62no—
197C22×D4—124no—
198C23×C4—32no—
199C23×C4—31no—
200C23×C4—44no—
201C23×C4C4×D312yes—
202C23×C4C4×D312yes—
203C23×C4D1212yes—
204C23×C4D1212yes—
205C23×C4#6712yes—
206C23×C4#6712yes—
207C23×C4C22×D311yescharacteristic
208C23×C4C22×D311yescharacteristic
209#123—64no—
210#123—64no—
211#123—64no—
212#123—64no—
213#123—64no—
214#123—62no—
215#123—62no—
216#123—64no—
217#123—62no—
218#123—62no—
219#117—44no—
220#117—124no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—34no—
223C2×C4⋊C4—34no—
224C2×C4⋊C4—62no—
225C2×C4⋊C4—62no—
226C2×C4⋊C4—62no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4—62no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—32no—
231C2×C22⋊C4—32no—
232C2×C22⋊C4—34no—
233C2×C22⋊C4—34no—
234C2×C22⋊C4—34no—
235C2×C22⋊C4—62no—
236C2×C22⋊C4—62no—
237C2×C22⋊C4—62no—
238C2×C22⋊C4—62no—
239C2×C22⋊C4—62no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4—64no—
242C2×C22⋊C4C22×D314yes—
243C2×C42—22no—
244C2×C42—62no—
245#99—128no—
246#95—24no—
247#95—64no—
248C23×C3—22no—
249C23×C3—24no—
250C23×C3—22no—
251C23×C3—24no—
252C23×C3—24no—
253C23×C3—24no—
254C23×C3—24no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—416no—
260C23×C3C2×C22⋊C412yes—
261C23×C3C2×C22⋊C412yes—
262C23×C3C23×C411yescharacteristic
263C23×C3C22×D411yescharacteristic
264C23×C3C22×D411yescharacteristic
265C23×C3H(5,𝔽2)14yes—
266C22×D3—22no—
267C22×D3—24no—
268C22×D3—44no—
269C22×D3—44no—
270C22×D3—44no—
271C22×D3—44no—
272C22×D3—88no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—48no—
290C2×C3×C4—44no—
291C2×C3×C4C2×C22⋊C412yes—
292C2×C3×C4C2×C22⋊C412yes—
293C2×C3×C4C2×C22⋊C412yes—
294C2×C3×C4C2×C22⋊C412yes—
295#67—48no—
296#67—44no—
297#67—44no—
298C2×Dic3—22no—
299C2×Dic3—24no—
300C2×Dic3—24no—
301C2×Dic3—44no—
302C2×Dic3—44no—
303C2×Dic3—44no—
304C2×Dic3—44no—
305C2×Dic3—48no—
306C2×Dic3—44no—
307C24—24no—
308C24—32no—
309C24—44no—
310C24—62no—
311C24—122no—
312C24—124no—
313C24C2×C4×D311yescharacteristic
314C24C2×C4×D311yescharacteristic
315C24C2×D1212yes—
316C24C2×#6712yes—
317C24C23×D311yescharacteristic
318C2×D4—64no—
319C2×D4—62no—
320C2×D4—68no—
321C2×D4—62no—
322C2×D4—124no—
323C2×D4—128no—
324C2×D4—128no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—32no—
329C22×C4—34no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—62no—
337C22×C4—62no—
338C22×C4—62no—
339C22×C4—64no—
340C22×C4—122no—
341C22×C4—124no—
342C22×C4(C2×D3)⋊C412yes—
343C22×C4(C2×D3)⋊C412yes—
344C22×C4(C2×D3)⋊C412yes—
345C22×C4(C2×D3)⋊C412yes—
346C22×C4(C2×D3)⋊C412yes—
347C22×C4(C2×D3)⋊C412yes—
348C22×C4(C2×D3)⋊C412yes—
349C22×C4(C2×D3)⋊C412yes—
350C22×C4C2×C4×D312yes—
351C22×C4C2×C4×D312yes—
352C22×C4C2×D1212yes—
353C22×C4C2×D1211yescharacteristic
354C22×C4C2×D1211yescharacteristic
355C22×C4C2×#6712yes—
356C22×C4C2×#6711yescharacteristic
357C22×C4C2×#6711yescharacteristic
358C4⋊C4—44no—
359C4⋊C4—64no—
360C4⋊C4—64no—
361C4⋊C4—64no—
362C4⋊C4—64no—
363C4⋊C4—124no—
364C22⋊C4—28no—
365C22⋊C4—48no—
366C22⋊C4—64no—
367C22⋊C4—62no—
368C22⋊C4—62no—
369C22⋊C4—64no—
370C22⋊C4—64no—
371C22⋊C4—64no—
372C22⋊C4—64no—
373C22⋊C4—68no—
374C22⋊C4—128no—
375C42—44no—
376C42—124no—
377C22×C3—28no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—44no—
387C22×C3—48no—
388C22×C3—48no—
389C22×C3—44no—
390C22×C3#41014yes—
391C22×C3C22×C22⋊C411yescharacteristic
392C22×C3#51812yes—
393C22×C3#53412yes—
394C22×C3#53412yes—
395C2×D3—42no—
396C2×D3—44no—
397C2×D3—88no—
398C2×D3—84no—
399C3×C4—22no—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—22no—
403C3×C4—84no—
404Dic3—42no—
405Dic3—44no—
406Dic3—84no—
407C23—22no—
408C23—24no—
409C23—22no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—416no—
419C23—62no—
420C23—64no—
421C23—124no—
422C23—124no—
423C23—124no—
424C23—124no—
425C23—248no—
426C23C2×(C2×D3)⋊C412yes—
427C23C2×(C2×D3)⋊C412yes—
428C23C22×C4×D311yescharacteristic
429C23C22×D1211yescharacteristic
430C23#100414yes—
431C23C22×#6711yescharacteristic
432D4—128no—
433D4—124no—
434D4—124no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—48no—
452C2×C4—44no—
453C2×C4—64no—
454C2×C4—64no—
455C2×C4—62no—
456C2×C4—124no—
457C2×C4—124no—
458C2×C4—124no—
459C2×C4—124no—
460C2×C4—128no—
461C2×C4—124no—
462C2×C4C2×(C2×D3)⋊C412yes—
463C2×C4C2×(C2×D3)⋊C412yes—
464C2×C4C2×(C2×D3)⋊C412yes—
465C2×C4C2×(C2×D3)⋊C412yes—
466C2×C3—22no—
467C2×C3—22no—
468C2×C3—22no—
469C2×C3—24no—
470C2×C3—42no—
471C2×C3#230811yescharacteristic
472C2×C3C2×#41012yes—
473D3—82no—
474C22—28no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—44no—
484C22—48no—
485C22—48no—
486C22—44no—
487C22—124no—
488C22—122no—
489C22—244no—
490C22—248no—
491C22#500614yes—
492C22#548612yes—
493C22#549312yes—
494C22C22×(C2×D3)⋊C411yescharacteristic
495C22#580412yes—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—22no—
500C4—84no—
501C4—124no—
502C4—122no—
503C4—244no—
504C3#2174711yescharacteristic, complemented, Sylow
505C2—22no—
506C2—22no—
507C2—22no—
508C2—24no—
509C2—42no—
510C2—242no—
511C2#8212111yescharacteristic
512C2C2×#500612yes—
subgroupG/H
centerC22C22×(C2×D3)⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#2280C2
socleC22×C3C22×C22⋊C4
prime radicalC23×#67C22
cosocleC24×C3C24
hypercenter#2280D3
hypocenterC3#21747
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
2. C3×#2280C2semidirect
3. #84027C2semidirect
4. #71595C2semidirect
5. #69535C2semidirect
7. #69495C2semidirect
11. C2×C3×#117C22semidirect
17. C2×#929C4semidirect
20. C22×(C2×D3)⋊C4C4semidirect
21. C2×#926C22semidirect
22. C22×Dic3⋊C4C4semidirect
23. C3×#381C22semidirect
29. #5211C22semidirect
32. #4527C22semidirect
35. #2280D3semidirect
107. C2×#117C2×D3semidirect
113. #381C2×D3semidirect
504. C3#21747semidirect

Maximal subgroups

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

subgroupindex
C22×#9292
C3×#22802
#840272
#715952
#695352
#694982
#694952
#217473