Finite Group Databasebeta

← #10804435

Subgroups — #10804435

Order 768.

1442 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5524C211yesmaximal, characteristic
2C2×C3×#433C211yesmaximal, characteristic, complemented
3C2×#5000C211yesmaximal, characteristic, complemented
4#70471C212yesmaximal, complemented
5#70470C212yesmaximal, complemented
6#69956C212yesmaximal, complemented
7#69955C212yesmaximal, complemented
8#69837C212yesmaximal, complemented
9#69787C212yesmaximal, complemented
10#21792—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C23×C4×D3C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14#5524—24no—
15#5524—24no—
16C2×#897C412yes—
17C2×#895C412yes—
18C2×#886C412yes—
19C2×#882C412yes—
20C2×C4×D12C2212yes—
21C2×C4×D12C2212yes—
22C3×#433—24no—
23C2×C3×C4⋊C8C2212yescomplemented
24C2×C3×C22⋊C8C2212yes—
25C2×C3×#97C2211yescharacteristic, complemented
26C2×C3×C4×C8C2211yescharacteristic, complemented
27C2×#830C2212yes—
28C2×(C2×D3)⋊C8C2212yes—
29C2×(C2×D3)⋊C8C2212yes—
30C2×(C2×D3)⋊C8C2212yes—
31C2×(C2×D3)⋊C8C2212yes—
32C2×(C2×D3)⋊C8C2212yescomplemented
33C2×(C2×D3)⋊C8C2212yescomplemented
34C2×(C2×D3)⋊C8C2212yescomplemented
35C2×(C2×D3)⋊C8C2212yescomplemented
36#5000—24no—
37C2×C4⋊(C3⋊C8)C2212yescomplemented
38C2×#803C2211yescharacteristic, complemented
39C2×C4×C3⋊C8C2211yescharacteristic, complemented
40C2×#525—31no—
41C2×#433—31no—
42C2×#433D311yescharacteristic, complemented
43#1955—32no—
44#1955—32no—
45#1954—32no—
46#1954—32no—
47#1780—32no—
48#1779—32no—
49C24×D3C2×C411yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C22×#67C2×C412yes—
52C23×Dic3C2×C411yescharacteristic
53C2×#235—22no—
54C2×#235—22no—
55C2×#235—42no—
56C2×#235—42no—
57C22×D12C2×C412yes—
58C22×D12C2×C412yes—
59C22×D12C2×C411yescharacteristic
60C22×C4×D3—22no—
61C22×C4×D3—22no—
62C22×C4×D3—42no—
63C22×C4×D3—42no—
64C22×C4×D3C2×C412yes—
65C22×C4×D3C2312yes—
66C22×C4×D3C2312yes—
67C22×Dic6C2×C411yescharacteristic
68C22×C3×C8C2312yes—
69C22×C3×C8C2312yes—
70C3×#117D414yes—
71C3×#117D414yes—
72C2×C3×C4⋊C4C2×C412yes—
73C2×C3×C22⋊C4C2×C412yes—
74C2×C3×C42C2311yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×(C2×D3)⋊C4C2×C412yes—
77C2×(C2×D3)⋊C4C2×C412yes—
78C2×(C2×D3)⋊C4C2×C412yes—
79C2×(C2×D3)⋊C4C2×C412yes—
80C2×C4⋊Dic3C2×C412yes—
81C2×C4⋊Dic3C2×C412yes—
82C22×C3⋊C8C2312yes—
83C22×C3⋊C8C2312yes—
84#897—28no—
85#895—28no—
86#886—28no—
87#882—28no—
88C4×D12—24no—
89C4×D12—24no—
90C4×D12—24no—
91C4×D12—24no—
92C3×C4⋊C8—24no—
93C3×C22⋊C8—24no—
94C3×#97—22no—
95C3×C4×C8—22no—
96#830—24no—
97(C2×D3)⋊C8—42no—
98(C2×D3)⋊C8—42no—
99(C2×D3)⋊C8—42no—
100(C2×D3)⋊C8—42no—
101(C2×D3)⋊C8—42no—
102(C2×D3)⋊C8—42no—
103(C2×D3)⋊C8—42no—
104(C2×D3)⋊C8—42no—
105C4⋊(C3⋊C8)—24no—
106#803—22no—
107C4×C3⋊C8—22no—
108C22×#41—31no—
109C24×C4—31no—
110#525—64no—
111#525—64no—
112C2×#123—32no—
113C2×C22⋊Q8—32no—
114C2×#121—32no—
115C2×#120—32no—
116C2×C4×D4—32no—
117C2×C4×D4—32no—
118C2×#117C2×D311yescharacteristic
119#433—24no—
120#433—64no—
121C2×C4⋊C8—32no—
122C2×C4⋊C8C2×D312yescomplemented
123C2×C22⋊C8—32no—
124C2×C22⋊C8—32no—
125C2×C22⋊C8—32no—
126C2×C22⋊C8—32no—
127C2×C22⋊C8—32no—
128C2×C22⋊C8—32no—
129C2×C22⋊C8—32no—
130C2×C22⋊C8—32no—
131C2×C22⋊C8—32no—
132C2×C22⋊C8C2×D312yes—
133C2×#97—31no—
134C2×#97C2×D311yescharacteristic, complemented
135C2×C4×C8—31no—
136C2×C4×C8C2×D311yescharacteristic, complemented
137C24×C3C22×C411yescharacteristic
138C23×D3—22no—
139C23×D3—24no—
140C23×D3—41no—
141C23×D3—41no—
142C23×D3—42no—
143C23×D3C22×C412yes—
144C23×D3C22×C412yes—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4C22×C412yes—
148C22×C3×C4C22×C412yes—
149C22×C3×C4C22×C411yescharacteristic
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2411yescharacteristic
153C2×#67—24no—
154C2×#67—42no—
155C2×#67—42no—
156C22×Dic3—22no—
157C22×Dic3—41no—
158C22×Dic3—41no—
159C22×Dic3C22×C412yes—
160C22×Dic3C22×C412yes—
161#235—48no—
162#235—48no—
163C2×D12—24no—
164C2×D12—24no—
165C2×D12—24no—
166C2×D12—24no—
167C2×D12—22no—
168C2×D12—42no—
169C2×D12—42no—
170C2×D12—41no—
171C2×D12—41no—
172C2×C4×D3—24no—
173C2×C4×D3—22no—
174C2×C4×D3—22no—
175C2×C4×D3—22no—
176C2×C4×D3—22no—
177C2×C4×D3—22no—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—42no—
186C2×C4×D3—42no—
187C2×C4×D3—48no—
188C2×C4×D3—48no—
189C2×Dic6—22no—
190C2×Dic6—41no—
191C2×Dic6—41no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C3×C4⋊C4C22⋊C418yes—
199C3×C22⋊C4C22⋊C418yes—
200C3×C42C2×D412yes—
201C3×C42C2×D412yes—
202C3×C42C2×D412yes—
203C3×C42C2×D412yes—
204(C2×D3)⋊C4—24no—
205(C2×D3)⋊C4—24no—
206(C2×D3)⋊C4—24no—
207(C2×D3)⋊C4—24no—
208C4⋊Dic3—24no—
209C4⋊Dic3—24no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C25—31no—
217C2×#41—62no—
218C2×#41—62no—
219C2×#41—122no—
220C2×#41—122no—
221C22×Q8—31no—
222C22×D4—32no—
223C22×D4—32no—
224C22×D4—32no—
225C22×D4—31no—
226C23×C4—31no—
227C23×C4—32no—
228C23×C4—32no—
229C23×C4—32no—
230C23×C4—62no—
231C23×C4—62no—
232C23×C4—122no—
233C23×C4—122no—
234C23×C4C22×D311yescharacteristic
235C22×C8—32no—
236C22×C8—32no—
237C22×C8C22×D312yes—
238C22×C8C22×D312yes—
239#123—68no—
240C22⋊Q8—68no—
241#121—68no—
242#120—68no—
243C4×D4—64no—
244C4×D4—64no—
245C4×D4—64no—
246C4×D4—64no—
247#117D1214yes—
248#117#6714yes—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4C4×D312yes—
252C2×C22⋊C4—32no—
253C2×C22⋊C4—32no—
254C2×C22⋊C4—32no—
255C2×C22⋊C4—32no—
256C2×C22⋊C4C4×D312yes—
257C2×C42C22×D311yescharacteristic
258C2×C42C22×D311yescharacteristic
259C4⋊C8—24no—
260C4⋊C8—64no—
261C22⋊C8—24no—
262C22⋊C8—64no—
263C22⋊C8—122no—
264C22⋊C8—122no—
265C22⋊C8—122no—
266C22⋊C8—122no—
267C22⋊C8—122no—
268C22⋊C8—122no—
269C22⋊C8—122no—
270C22⋊C8—122no—
271#97—22no—
272#97—62no—
273C4×C8—22no—
274C4×C8—62no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—22no—
278C23×C3—22no—
279C23×C3C2×C22⋊C412yes—
280C23×C3C23×C411yescharacteristic
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—22no—
284C22×D3—22no—
285C22×D3—22no—
286C22×D3—22no—
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—22no—
293C22×D3—22no—
294C22×D3—44no—
295C22×D3—48no—
296C22×D3—48no—
297C22×D3—44no—
298C22×D3—44no—
299C22×D3#13114yes—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4C2×C22⋊C414yes—
309C2×C3×C4C2×C22⋊C414yes—
310C2×C3×C4C2×C22⋊C412yes—
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4H(5,𝔽2)12yes—
314C2×C3×C4H(5,𝔽2)12yes—
315#67—48no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—44no—
323D12—48no—
324D12—48no—
325D12—44no—
326C4×D3—48no—
327C4×D3—42no—
328C4×D3—42no—
329C4×D3—42no—
330C4×D3—42no—
331C4×D3—42no—
332C4×D3—42no—
333C4×D3—42no—
334C4×D3—42no—
335Dic6—44no—
336C3×C8—42no—
337C3×C8—42no—
338C3⋊C8—42no—
339C3⋊C8—42no—
340C24—32no—
341C24—32no—
342C24—64no—
343C24—62no—
344C24—121no—
345C24—121no—
346C24—122no—
347C24C2×C4×D311yescharacteristic
348#41—128no—
349#41—128no—
350C2×Q8—62no—
351C2×Q8—121no—
352C2×Q8—121no—
353C2×D4—64no—
354C2×D4—64no—
355C2×D4—64no—
356C2×D4—64no—
357C2×D4—64no—
358C2×D4—62no—
359C2×D4—121no—
360C2×D4—121no—
361C2×D4—122no—
362C2×D4—122no—
363C2×D4—122no—
364C2×D4—122no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—32no—
368C22×C4—32no—
369C22×C4—62no—
370C22×C4—64no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—122no—
384C22×C4—122no—
385C22×C4—128no—
386C22×C4—128no—
387C22×C4—121no—
388C22×C4—121no—
389C22×C4C2×C4×D312yes—
390C22×C4C2×C4×D312yes—
391C22×C4C2×C4×D311yescharacteristic
392C22×C4C2×D1212yes—
393C22×C4C2×#6712yes—
394C22×C4C23×D311yescharacteristic
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—62no—
402C2×C8—62no—
403C2×C8—62no—
404C2×C8—62no—
405C2×C8—62no—
406C2×C8—62no—
407C4⋊C4—64no—
408C4⋊C4—64no—
409C4⋊C4(C2×D3)⋊C418yes—
410C22⋊C4—64no—
411C22⋊C4—64no—
412C22⋊C4—64no—
413C22⋊C4—64no—
414C22⋊C4(C2×D3)⋊C418yes—
415C42C2×D1212yes—
416C42C2×D1212yes—
417C42C2×#6712yes—
418C42C2×#6712yes—
419C22×C3—22no—
420C22×C3—24no—
421C22×C3—42no—
422C22×C3—41no—
423C22×C3—41no—
424C22×C3C22×C22⋊C411yescharacteristic
425C22×C3#51811yescharacteristic
426C22×C3#51811yescharacteristic
427C22×C3C2×#13112yes—
428C22×C3#56812yes—
429C2×D3—24no—
430C2×D3—24no—
431C2×D3—24no—
432C2×D3—416no—
433C2×D3—42no—
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—42no—
438C2×D3—42no—
439C2×D3—42no—
440C2×D3—42no—
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—42no—
444C3×C4—22no—
445C3×C4—24no—
446C3×C4—24no—
447C3×C4#53512yes—
448C3×C4#53512yes—
449Dic3—42no—
450Dic3—42no—
451C23—22no—
452C23—22no—
453C23—21no—
454C23—21no—
455C23—34no—
456C23—62no—
457C23—62no—
458C23—62no—
459C23—62no—
460C23—62no—
461C23—62no—
462C23—62no—
463C23—62no—
464C23—62no—
465C23—62no—
466C23—62no—
467C23—62no—
468C23—62no—
469C23—128no—
470C23—124no—
471C23—124no—
472C23—128no—
473C23—124no—
474C23C2×(C2×D3)⋊C412yes—
475C23C22×C4×D311yescharacteristic
476Q8—124no—
477D4—128no—
478D4—128no—
479D4—124no—
480D4—128no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—128no—
496C2×C4—124no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4C2×(C2×D3)⋊C414yes—
506C2×C4C2×(C2×D3)⋊C414yes—
507C2×C4C2×(C2×D3)⋊C412yes—
508C2×C4C22×D1211yescharacteristic
509C2×C4#100912yes—
510C2×C4#100912yes—
511C2×C4C22×#6711yescharacteristic
512C8—42no—
513C8—42no—
514C8—122no—
515C8—122no—
516C2×C3—42no—
517C2×C3#230911yescharacteristic
518C2×C3#288012yes—
519C2×C3#297612yes—
520C2×C3#297711yescharacteristic
521C2×C3#297811yescharacteristic
522D3—44no—
523D3—82no—
524C22—24no—
525C22—22no—
526C22—42no—
527C22—41no—
528C22—41no—
529C22—64no—
530C22—64no—
531C22—64no—
532C22—1216no—
533C22—122no—
534C22—122no—
535C22—122no—
536C22—122no—
537C22—122no—
538C22—122no—
539C22—122no—
540C22—122no—
541C22—122no—
542C22—122no—
543C22—122no—
544C22#552111yescharacteristic
545C22#552111yescharacteristic
546C22D3×#13112yes—
547C22#575012yes—
548C22C22×(C2×D3)⋊C411yescharacteristic
549C4—24no—
550C4—24no—
551C4—22no—
552C4—122no—
553C4—122no—
554C4#552512yes—
555C4#583012yes—
556C3#2179211yescharacteristic, complemented, Sylow
557C2—42no—
558C2—124no—
559C2—242no—
560C2#7937612yes—
561C2#7937711yescharacteristic
562C2#7937811yescharacteristic
563C2#8226911yescharacteristic
564C2#8379312yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC2×C3×#433C2
prime radicalC2×#5524C2
cosocleC22×C3×C4C24
hypercenterC2×#433D3
hypocenterC3#21792
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
2. C2×C3×#433C2semidirect
3. C2×#5000C2semidirect
4. #70471C2semidirect
5. #70470C2semidirect
6. #69956C2semidirect
7. #69955C2semidirect
8. #69837C2semidirect
9. #69787C2semidirect
23. C2×C3×C4⋊C8C22semidirect
25. C2×C3×#97C22semidirect
26. C2×C3×C4×C8C22semidirect
32. C2×(C2×D3)⋊C8C22semidirect
33. C2×(C2×D3)⋊C8C22semidirect
34. C2×(C2×D3)⋊C8C22semidirect
35. C2×(C2×D3)⋊C8C22semidirect
37. C2×C4⋊(C3⋊C8)C22semidirect
38. C2×#803C22semidirect
39. C2×C4×C3⋊C8C22semidirect
42. C2×#433D3semidirect
122. C2×C4⋊C8C2×D3semidirect
134. C2×#97C2×D3semidirect
136. C2×C4×C8C2×D3semidirect
556. C3#21792semidirect

Maximal subgroups

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

subgroupindex
C2×#55242
C2×C3×#4332
C2×#50002
#704712
#704702
#699562
#699552
#698372
#697872
#217923