Finite Group Databasebeta

← #10798799

Subgroups — #10798799

Order 768.

1718 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5828C211yesmaximal, characteristic
2C2×C3×#408C211yesmaximal, characteristic, complemented
3C2×#5234C211yesmaximal, characteristic, complemented
4C2×#5126C211yesmaximal, characteristic, complemented
5C2×#5122C211yesmaximal, characteristic, complemented
6C22×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7C2×#5102C211yesmaximal, characteristic, complemented
8#69917C214yesmaximal, complemented
9#69916C214yesmaximal, complemented
10#21191—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C4×D3C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15#5828—24no—
16#5828—24no—
17C2×#946C412yes—
18C2×#938C412yes—
19C2×#937C412yes—
20C2×#935C412yes—
21C2×C4×#67C2214yes—
22C2×#926C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C23×C3⋊C8C2211yescharacteristic
25C3×#408—24no—
26C2×C3×C22⋊C8C2214yescomplemented
27#5234—24no—
28C2×#830C2212yescomplemented
29C2×#830C2212yescomplemented
30#5126—24no—
31#5122—24no—
32C2×(C2×D3)⋊C8—24no—
33C2×(C2×D3)⋊C8C2212yes—
34C2×(C2×D3)⋊C8C2212yes—
35C2×(C2×D3)⋊C8C2214yes—
36#5102—24no—
37C2×#815C2212yescomplemented
38C2×Dic3⋊C8C2214yescomplemented
39C2×C8×Dic3C2212yescomplemented
40C2×#525—31no—
41C2×#424—31no—
42C2×#408—31no—
43C2×#408D311yescharacteristic, complemented
44C2×#407—31no—
45C22×C22⋊C8—31no—
46C2×#405—31no—
47#1955—34no—
48#1954—34no—
49C24×D3C2×C411yescharacteristic
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C22×C3×Q8C2×C411yescharacteristic
57C22×C3×D4C2×C412yes—
58C22×C3×D4C2×C411yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C22×#67C2×C414yes—
62C23×Dic3C2×C411yescharacteristic
63C22×C4×D3—22no—
64C22×C4×D3—22no—
65C22×C4×D3—22no—
66C22×C4×D3—22no—
67C22×C4×D3—22no—
68C22×C4×D3—22no—
69C22×C4×D3C2×C412yes—
70C22×C4×D3C2312yes—
71C2×C3×M16—22no—
72C2×C3×M16—22no—
73C2×C3×M16D414yescomplemented
74C22×C3×C8—22no—
75C22×C3×C8—22no—
76C22×C3×C8—22no—
77C22×C3×C8C2312yes—
78C22×C3×C8C2312yes—
79C2×#217C2×C414yes—
80C2×#217C2×C412yes—
81#946—28no—
82#938—28no—
83#937—28no—
84#935—28no—
85C4×#67—28no—
86C4×#67—28no—
87C2×(C2×D3)⋊C4C2×C414yes—
88#926D414yes—
89#926D414yes—
90C2×C4⋊Dic3C2×C412yes—
91C2×Dic3⋊C4C2×C414yes—
92C2×C4×Dic3C2312yes—
93C2×#208—22no—
94C2×#208—22no—
95C2×#208—22no—
96C22×C3⋊C8—22no—
97C22×C3⋊C8—22no—
98C22×C3⋊C8D414yescomplemented
99C22×C3⋊C8C2312yes—
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2311yescharacteristic
102C3×C22⋊C8—28no—
103#830—24no—
104#830—24no—
105(C2×D3)⋊C8—28no—
106(C2×D3)⋊C8—24no—
107(C2×D3)⋊C8—24no—
108#815—24no—
109Dic3⋊C8—28no—
110C8×Dic3—24no—
111C22×#41C2×D311yescharacteristic
112C24×C4—31no—
113C22×M16—31no—
114C22×M16C2×D311yescharacteristic
115C23×C8—31no—
116C23×C8C2×D311yescharacteristic
117#525—64no—
118#525—64no—
119C2×#123—32no—
120C2×C22⋊Q8—32no—
121C2×#121—32no—
122C2×#120—32no—
123C2×C4×D4—34no—
124C2×#117—31no—
125#424—64no—
126C2×C4⋊C8—34no—
127#408—24no—
128#408—64no—
129#407—64no—
130C2×C22⋊C8—32no—
131C2×C22⋊C8—32no—
132C2×C22⋊C8—32no—
133C2×C22⋊C8—32no—
134C2×C22⋊C8—34no—
135C2×C22⋊C8—64no—
136C2×C22⋊C8C2×D314yescomplemented
137#405—64no—
138C2×#97—32no—
139C2×C4×C8—32no—
140C24×C3C22×C412yes—
141C24×C3C22×C411yescharacteristic
142C23×D3—22no—
143C23×D3—24no—
144C23×D3—22no—
145C23×D3—22no—
146C23×D3—42no—
147C23×D3C22×C412yes—
148C3×#41—44no—
149C3×#41—44no—
150C3×#41—44no—
151C3×#41—44no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×D4—24no—
156C2×C3×D4—24no—
157C2×C3×D4—24no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4C22×C412yes—
168C22×C3×C4C22×C411yescharacteristic
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4#4112yes—
174C22×C3×C4#4112yes—
175C22×C3×C4C2411yescharacteristic
176C2×#67—28no—
177C2×#67—28no—
178C2×#67—44no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3C22×C412yes—
183C22×Dic3C22×C414yes—
184C2×C4×D3—24no—
185C2×C4×D3—24no—
186C2×C4×D3—24no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—44no—
194C2×C4×D3—44no—
195C2×C4×D3—44no—
196C2×C4×D3—44no—
197C3×M16—28no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—44no—
204C2×C3×C8C2×D414yes—
205#217—28no—
206#217C22⋊C418yes—
207(C2×D3)⋊C4—28no—
208C4⋊Dic3C22⋊C418yes—
209Dic3⋊C4—28no—
210C4×Dic3C2×D414yes—
211C4×Dic3C2×D414yes—
212#208—44no—
213C2×C3⋊C8—28no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8C2×D412yes—
222C2×C3⋊C8C2×D412yes—
223C25—31no—
224C2×#41—22no—
225C2×#41—22no—
226C2×#41—22no—
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—22no—
230C22×Q8C4×D311yescharacteristic
231C22×D4—34no—
232C22×D4C4×D312yes—
233C22×D4C4×D311yescharacteristic
234C23×C4—31no—
235C23×C4—32no—
236C23×C4—32no—
237C23×C4—62no—
238C23×C4—62no—
239C23×C4—62no—
240C23×C4—62no—
241C23×C4—62no—
242C23×C4—62no—
243C23×C4C22×D312yes—
244C23×C4C22×D311yescharacteristic
245C2×M16—22no—
246C2×M16—22no—
247C2×M16—62no—
248C2×M16—62no—
249C2×M16—62no—
250C2×M16#6714yescomplemented
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—34no—
255C22×C8—31no—
256C22×C8—31no—
257C22×C8—32no—
258C22×C8—62no—
259C22×C8—62no—
260C22×C8C22×D312yes—
261C22×C8C22×D312yes—
262#123—68no—
263C22⋊Q8—68no—
264#121—68no—
265#120—68no—
266C4×D4—68no—
267C4×D4—68no—
268#117—34no—
269#117—34no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—34no—
272C2×C22⋊C4—32no—
273C2×C22⋊C4—34no—
274C2×C22⋊C4—34no—
275C2×C42—32no—
276C4⋊C8—68no—
277C22⋊C8—28no—
278C22⋊C8—64no—
279C22⋊C8—64no—
280C22⋊C8—64no—
281C22⋊C8—64no—
282C22⋊C8—68no—
283#97—64no—
284C4×C8—64no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—24no—
291C23×C3—24no—
292C23×C3C2×C22⋊C412yes—
293C23×C3C4×D412yes—
294C23×C3C4×D412yes—
295C23×C3C23×C411yescharacteristic
296C22×D3—28no—
297C22×D3—22no—
298C22×D3—22no—
299C22×D3—22no—
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—22no—
303C22×D3—48no—
304C22×D3—48no—
305C22×D3—44no—
306C22×D3—44no—
307C22×D3#13114yes—
308C3×Q8—44no—
309C3×D4—48no—
310C3×D4—44no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4C2×C22⋊C412yes—
319C2×C3×C4C4×D412yes—
320C2×C3×C4C4×D412yes—
321C2×C3×C4#12114yes—
322C2×C3×C4#12114yes—
323C2×C3×C4#12414yes—
324C2×C3×C4#12714yes—
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C2×#4111yescharacteristic
330C2×C3×C4C2×#4111yescharacteristic
331#67—416no—
332C2×Dic3—24no—
333C2×Dic3—24no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—44no—
338C2×Dic3C2×C22⋊C418yes—
339C4×D3—42no—
340C4×D3—42no—
341C4×D3—42no—
342C4×D3—42no—
343C4×D3—48no—
344C3×C8—24no—
345C3×C8—42no—
346C3⋊C8—22no—
347C3⋊C8—22no—
348C3⋊C8—42no—
349C24—32no—
350C24—64no—
351C24—62no—
352C24—62no—
353C24—62no—
354C24—122no—
355C24C2×C4×D312yes—
356C24C2×C4×D311yescharacteristic
357#41—44no—
358#41—44no—
359#41—44no—
360#41—44no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—68no—
371C2×D4—68no—
372C2×D4—124no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—34no—
380C22×C4—32no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—64no—
385C22×C4—64no—
386C22×C4—64no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—62no—
393C22×C4—124no—
394C22×C4—124no—
395C22×C4—124no—
396C22×C4—124no—
397C22×C4C2×C4×D312yes—
398C22×C4C2×C4×D311yescharacteristic
399C22×C4#23512yes—
400C22×C4D3×D412yes—
401C22×C4D3×D412yes—
402C22×C4D3×D412yes—
403C22×C4#23712yes—
404C22×C4C2×#6712yes—
405C22×C4C23×D311yescharacteristic
406M16—28no—
407M16—124no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—32no—
414C2×C8—32no—
415C2×C8—44no—
416C2×C8—68no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—62no—
424C2×C8C2×#6714yes—
425C4⋊C4—38no—
426C4⋊C4—68no—
427C22⋊C4—38no—
428C22⋊C4—68no—
429C22⋊C4—68no—
430C42—34no—
431C42—34no—
432C22×C3—24no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—24no—
436C22×C3—44no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—44no—
440C22×C3—42no—
441C22×C3#39014yes—
442C22×C3C22×C22⋊C411yescharacteristic
443C22×C3C2×C4×D411yescharacteristic
444C22×C3C2×C4×D411yescharacteristic
445C22×C3C2×#13112yes—
446C22×C3#56812yes—
447C2×D3—24no—
448C2×D3—24no—
449C2×D3—24no—
450C2×D3—416no—
451C2×D3—42no—
452C2×D3—42no—
453C2×D3—42no—
454C2×D3—42no—
455C3×C4—42no—
456C3×C4#39014yes—
457C3×C4C2×#12111yescharacteristic
458C3×C4C2×#12111yescharacteristic
459C3×C4C2×#12411yescharacteristic
460C3×C4C2×#12711yescharacteristic
461Dic3—28no—
462Dic3—42no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—34no—
471C23—68no—
472C23—62no—
473C23—62no—
474C23—62no—
475C23—62no—
476C23—62no—
477C23—62no—
478C23—128no—
479C23—128no—
480C23—124no—
481C23—124no—
482C23D3×C22⋊C412yes—
483C23#67⋊C412yes—
484C23C4×#6712yes—
485C23C22×C4×D311yescharacteristic
486Q8—44no—
487D4—48no—
488D4—44no—
489D4—1216no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—38no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—64no—
499C2×C4—64no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—124no—
504C2×C4—128no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4D3×C22⋊C412yes—
510C2×C4#67⋊C412yes—
511C2×C4#88414yes—
512C2×C4#88514yes—
513C2×C4C4×#6712yes—
514C2×C4#93914yes—
515C2×C4#94014yes—
516C2×C4C2×#23511yescharacteristic
517C2×C4C2×D3×D411yescharacteristic
518C2×C4C2×D3×D411yescharacteristic
519C2×C4C2×D3×D411yescharacteristic
520C2×C4C2×#23711yescharacteristic
521C2×C4C22×#6711yescharacteristic
522C8—24no—
523C8—42no—
524C8—62no—
525C8—62no—
526C8—122no—
527C2×C3—24no—
528C2×C3—82no—
529C2×C3C2×#39011yescharacteristic
530C2×C3#288012yes—
531C2×C3#292911yescharacteristic
532C2×C3#293011yescharacteristic
533C2×C3#293312yes—
534D3—44no—
535C22—22no—
536C22—22no—
537C22—24no—
538C22—24no—
539C22—42no—
540C22—42no—
541C22—44no—
542C22—44no—
543C22—42no—
544C22—64no—
545C22—64no—
546C22—64no—
547C22—1216no—
548C22—122no—
549C22—122no—
550C22—122no—
551C22—122no—
552C22#495714yes—
553C22C2×D3×C22⋊C411yescharacteristic
554C22C2×#67⋊C411yescharacteristic
555C22D3×#13112yes—
556C22#575012yes—
557C22C2×C4×#6711yescharacteristic
558C4—42no—
559C4—68no—
560C4—122no—
561C4#495714yes—
562C4C2×#88411yescharacteristic
563C4C2×#88511yescharacteristic
564C4C2×#93911yescharacteristic
565C4C2×#94011yescharacteristic
566C3#2119111yescharacteristic, complemented, Sylow
567C2—24no—
568C2—82no—
569C2—124no—
570C2#7865511yescharacteristic
571C2#7865611yescharacteristic
572C2#7866312yes—
573C2C2×#495711yescharacteristic
574C2#8379612yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC2×C3×#408C2
prime radicalC2×#5828C2
cosocleC22×C3×C4C24
hypercenterC2×#408D3
hypocenterC3#21191
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
2. C2×C3×#408C2semidirect
3. C2×#5234C2semidirect
4. C2×#5126C2semidirect
5. C2×#5122C2semidirect
6. C22×(C2×D3)⋊C8C2semidirect
7. C2×#5102C2semidirect
8. #69917C2semidirect
9. #69916C2semidirect
26. C2×C3×C22⋊C8C22semidirect
28. C2×#830C22semidirect
29. C2×#830C22semidirect
37. C2×#815C22semidirect
38. C2×Dic3⋊C8C22semidirect
39. C2×C8×Dic3C22semidirect
43. C2×#408D3semidirect
73. C2×C3×M16D4semidirect
98. C22×C3⋊C8D4semidirect
136. C2×C22⋊C8C2×D3semidirect
250. C2×M16#67semidirect
566. C3#21191semidirect

Maximal subgroups

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

subgroupindex
C2×#58282
C2×C3×#4082
C2×#52342
C2×#51262
C2×#51222
C22×(C2×D3)⋊C82
C2×#51022
#699172
#699162
#211913