Finite Group Databasebeta

← #10690210

Subgroups — #10690210

Order 768.

1224 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74004C211yesmaximal, characteristic, complemented
2C2×#4671C211yesmaximal, characteristic
3#69959C211yesmaximal, characteristic, complemented
4C2×#4468C211yesmaximal, characteristic, complemented
5#69521C211yesmaximal, characteristic, complemented
6C3×#1294C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9020—31nomaximal, Sylow
9C23×D12—21no—
10C23×D12C2211yescharacteristic
11C22×(C2×D3)⋊C4—21no—
12C22×(C2×D3)⋊C4C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×#380C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16#5212C2211yescharacteristic
17C2×#830C2211yescharacteristic
18C2×#821—22no—
19#4990—22no—
20C2×#809—22no—
21#4955—21no—
22Dic3×C22⋊C4—21no—
23#4671—22no—
24#4671C414yescomplemented
25#4670—22no—
26#4468—22no—
27C2×#392—31no—
28#2312—31no—
29#1786—31no—
30#1786—31no—
31C2×#323—31no—
32#1294—31no—
33#1294D311yescharacteristic, complemented
34C24×D3—21no—
35C24×D3D411yescharacteristic
36C23×C3×C4—21no—
37C23×C3×C4C2311yescharacteristic
38C23×Dic3D411yescharacteristic
39C22×D12—21no—
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—21no—
43C22×D12—24no—
44C22×D12—21no—
45C22×D12—44no—
46C22×D12—44no—
47C22×D12—44no—
48C22×D12—42no—
49C22×D12—42no—
50C22×D12C2×C412yes—
51C22×C3×C8—21no—
52C2×C3×C4⋊C4D412yescomplemented
53C2×C3×C22⋊C4—21no—
54C2×#217—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—24no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4—44no—
63C2×(C2×D3)⋊C4C2×C412yes—
64C2×(C2×D3)⋊C4D412yes—
65C2×C4⋊Dic3D412yescomplemented
66C2×C4×Dic3—22no—
67C22×C3⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#95—21no—
71C3×#95—21no—
72C3×#95C2×C412yes—
73C3⋊#95—22no—
74C3⋊#95—21no—
75#830—21no—
76#830—21no—
77#821—44no—
78#809—44no—
79C23×D4—31no—
80C23×D4—61no—
81C22×C22⋊C4—31no—
82C22×C22⋊C4—61no—
83C2×#102—62no—
84C2×#102—62no—
85C2×C22⋊C8—31no—
86C2×C22⋊C8C2×D311yescharacteristic
87#392—34no—
88#392—62no—
89#388—62no—
90#388—62no—
91#380—31no—
92#380C2×D311yescharacteristic
93#379—61no—
94C4×C22⋊C4—61no—
95C2×#95C2×D311yescharacteristic
96#323—62no—
97C24×C3C2×D411yescharacteristic
98C23×D3—24no—
99C23×D3—22no—
100C23×D3—21no—
101C23×D3—21no—
102C23×D3—21no—
103C23×D3—21no—
104C23×D3—21no—
105C23×D3—21no—
106C23×D3—42no—
107C23×D3—42no—
108C23×D3—42no—
109C23×D3—42no—
110C23×D3—42no—
111C23×D3—41no—
112C23×D3—41no—
113C23×D3—41no—
114C23×D3—41no—
115C23×D3—41no—
116C23×D3—41no—
117C23×D3C22⋊C412yes—
118C22×C3×C4—24no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—42no—
122C22×C3×C4—42no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4C22×C411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—22no—
134C22×Dic3—42no—
135C22×Dic3C22⋊C412yes—
136C2×D12—24no—
137C2×D12—24no—
138C2×D12—22no—
139C2×D12—22no—
140C2×D12—44no—
141C2×D12—44no—
142C2×D12—44no—
143C2×D12—42no—
144C2×D12—42no—
145C2×D12—48no—
146C2×D12—44no—
147C2×D12—48no—
148C2×D12—48no—
149C2×D12—416no—
150C2×D12—416no—
151C2×D12—41no—
152C2×D12—41no—
153C2×C3×C8—42no—
154C2×C3×C8—41no—
155C3×C4⋊C4—42no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158#217—41no—
159#217—41no—
160(C2×D3)⋊C4—22no—
161(C2×D3)⋊C4—22no—
162(C2×D3)⋊C4—22no—
163(C2×D3)⋊C4—22no—
164(C2×D3)⋊C4—48no—
165(C2×D3)⋊C4—82no—
166(C2×D3)⋊C4—82no—
167C4⋊Dic3—42no—
168C4×Dic3—82no—
169C2×C3⋊C8—42no—
170C2×C3⋊C8—41no—
171C25—31no—
172C25—61no—
173C22×D4—32no—
174C22×D4—64no—
175C22×D4—61no—
176C22×D4—61no—
177C22×D4—61no—
178C22×D4—61no—
179C22×D4—61no—
180C22×D4—124no—
181C22×D4—124no—
182C22×D4—124no—
183C22×D4—122no—
184C22×D4—122no—
185C23×C4—21no—
186C23×C4—31no—
187C23×C4C22×D311yescharacteristic
188C22×C8—21no—
189C22×C8—61no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4D1212yescomplemented
192C2×C22⋊C4—21no—
193C2×C22⋊C4—32no—
194C2×C22⋊C4—32no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—64no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—62no—
202C2×C22⋊C4—62no—
203C2×C22⋊C4—124no—
204C2×C42—62no—
205#102—124no—
206#102—124no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211#95—21no—
212#95—21no—
213#95—62no—
214#95—61no—
215#95C4×D312yes—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3C2×C22⋊C411yescharacteristic
219C23×C3#12011yescharacteristic
220C23×C3#12311yescharacteristic
221C23×C3#13511yescharacteristic
222C23×C3#13511yescharacteristic
223C23×C3AGL(1,ℤ/8)11yescharacteristic
224C23×C3AGL(1,ℤ/8)11yescharacteristic
225C22×D3—24no—
226C22×D3—22no—
227C22×D3—22no—
228C22×D3—22no—
229C22×D3—42no—
230C22×D3—42no—
231C22×D3—42no—
232C22×D3—44no—
233C22×D3—44no—
234C22×D3—44no—
235C22×D3—44no—
236C22×D3—42no—
237C22×D3—42no—
238C22×D3—44no—
239C22×D3—44no—
240C22×D3—44no—
241C22×D3—42no—
242C22×D3—42no—
243C22×D3—42no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—42no—
247C22×D3—42no—
248C22×D3—44no—
249C22×D3—41no—
250C22×D3—41no—
251C22×D3—81no—
252C22×D3—81no—
253C22×D3—82no—
254C22×D3—82no—
255C22×D3—82no—
256C22×D3—82no—
257C22×D3—82no—
258C22×D3—81no—
259C22×D3—81no—
260C22×D3—82no—
261C22×D3—81no—
262C22×D3—81no—
263C22×D3—82no—
264C22×D3—82no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—24no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—44no—
273C2×C3×C4—44no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4C4×D411yescharacteristic
280C2×C3×C4C4×D411yescharacteristic
281C2×C3×C4#12111yescharacteristic
282C2×C3×C4#12111yescharacteristic
283C2×Dic3—22no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—42no—
289C2×Dic3—81no—
290D12—24no—
291D12—24no—
292D12—44no—
293D12—44no—
294D12—42no—
295D12—42no—
296D12—432no—
297C3×C8—81no—
298C3⋊C8—81no—
299C24—32no—
300C24—64no—
301C24—62no—
302C24—61no—
303C24—61no—
304C24—61no—
305C24—61no—
306C24—61no—
307C24—61no—
308C24—122no—
309C24—122no—
310C24—122no—
311C24—122no—
312C24—122no—
313C24—121no—
314C24—121no—
315C24—121no—
316C24—121no—
317C24—121no—
318C24—121no—
319C24D3×D411yescharacteristic
320C2×D4—64no—
321C2×D4—64no—
322C2×D4—62no—
323C2×D4—62no—
324C2×D4—122no—
325C2×D4—122no—
326C2×D4—128no—
327C2×D4—124no—
328C2×D4—124no—
329C2×D4—124no—
330C2×D4—124no—
331C2×D4—128no—
332C2×D4—128no—
333C2×D4—1216no—
334C2×D4—1216no—
335C2×D4—121no—
336C2×D4—121no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—32no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4—62no—
348C22×C4—122no—
349C22×C4C2×C4×D311yescharacteristic
350C22×C4C2×D1211yescharacteristic
351C22×C4#23511yescharacteristic
352C22×C4D3×D411yescharacteristic
353C22×C4D3×D411yescharacteristic
354C22×C4#23911yescharacteristic
355C2×C8—42no—
356C2×C8—41no—
357C2×C8—122no—
358C2×C8—121no—
359C4⋊C4—42no—
360C4⋊C4—122no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—62no—
364C22⋊C4—62no—
365C22⋊C4—62no—
366C22⋊C4—62no—
367C22⋊C4—128no—
368C22⋊C4—121no—
369C22⋊C4—121no—
370C22⋊C4—242no—
371C22⋊C4—242no—
372C42—242no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3#38611yescharacteristic
382C22×C3#41611yescharacteristic
383C22×C3#41811yescharacteristic
384C22×C3#44911yescharacteristic
385C22×C3#44911yescharacteristic
386C22×C3#45312yes—
387C22×C3B(4,𝔽2)12yes—
388C22×C3#48211yescharacteristic
389C22×C3#48211yescharacteristic
390C2×D3—44no—
391C2×D3—44no—
392C2×D3—44no—
393C2×D3—42no—
394C2×D3—42no—
395C2×D3—44no—
396C2×D3—42no—
397C2×D3—82no—
398C2×D3—82no—
399C2×D3—82no—
400C2×D3—82no—
401C2×D3—84no—
402C2×D3—84no—
403C2×D3—82no—
404C2×D3—82no—
405C2×D3—84no—
406C2×D3—84no—
407C2×D3—82no—
408C2×D3—82no—
409C2×D3—82no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—44no—
415C3×C4—82no—
416Dic3—42no—
417Dic3—82no—
418C23—22no—
419C23—22no—
420C23—64no—
421C23—62no—
422C23—62no—
423C23—62no—
424C23—122no—
425C23—122no—
426C23—122no—
427C23—122no—
428C23—124no—
429C23—122no—
430C23—124no—
431C23—124no—
432C23—124no—
433C23—124no—
434C23—124no—
435C23—124no—
436C23—121no—
437C23—121no—
438C23—122no—
439C23—122no—
440C23—124no—
441C23—122no—
442C23—121no—
443C23—122no—
444C23—121no—
445C23—122no—
446C23—241no—
447C23—241no—
448C23—242no—
449C23—242no—
450C23—242no—
451C23—242no—
452C23—242no—
453C23—242no—
454C23—242no—
455C23—241no—
456C23—241no—
457C23—241no—
458C23—241no—
459C23—242no—
460C23D3×C22⋊C411yescharacteristic
461C23#88211yescharacteristic
462C23#89411yescharacteristic
463C23#91111yescharacteristic
464C23#91411yescharacteristic
465C23#91611yescharacteristic
466C23#91911yescharacteristic
467D4—64no—
468D4—64no—
469D4—124no—
470D4—124no—
471D4—122no—
472D4—122no—
473D4—1232no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—62no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—241no—
495C2×C4C4×D1211yescharacteristic
496C2×C4#88411yescharacteristic
497C2×C4#89311yescharacteristic
498C2×C4#89511yescharacteristic
499C8—81no—
500C8—241no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3#160012yes—
506C2×C3#160012yes—
507C2×C3#187511yescharacteristic
508C2×C3#188711yescharacteristic
509C2×C3#188911yescharacteristic
510D3—82no—
511D3—84no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—124no—
521C22—124no—
522C22—124no—
523C22—122no—
524C22—124no—
525C22—122no—
526C22—122no—
527C22—242no—
528C22—242no—
529C22—242no—
530C22—242no—
531C22—244no—
532C22—242no—
533C22—244no—
534C22—242no—
535C22—242no—
536C22—242no—
537C22—244no—
538C22—242no—
539C22—244no—
540C22#466511yescharacteristic
541C22#474112yes—
542C22#477011yescharacteristic
543C22#478111yescharacteristic
544C22#480411yescharacteristic
545C22#481011yescharacteristic
546C22#482212yes—
547C22#486611yescharacteristic
548C22#488511yescharacteristic
549C4—21no—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—44no—
554C4—82no—
555C4—122no—
556C4—242no—
557C3#902011yescharacteristic, complemented, Sylow
558C2—21no—
559C2—21no—
560C2—21no—
561C2—21no—
562C2—242no—
563C2—244no—
564C2#6828512yes—
565C2#6831712yes—
566C2#6876011yescharacteristic
567C2#6888111yescharacteristic
568C2#6896211yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3#120
prime radicalC2×#4671C2
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9020
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
1. #74004C2semidirect
3. #69959C2semidirect
4. C2×#4468C2semidirect
5. #69521C2semidirect
6. C3×#1294C2semidirect
7. #67612C2semidirect
24. #4671C4semidirect
33. #1294D3semidirect
52. C2×C3×C4⋊C4D4semidirect
65. C2×C4⋊Dic3D4semidirect
191. C2×C4⋊C4D12semidirect
557. C3#9020semidirect

Maximal subgroups

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

subgroupindex
#740042
C2×#46712
#699592
C2×#44682
#695212
C3×#12942
#676122
#90203