Finite Group Databasebeta

← #10822735

Subgroups — #10822735

Order 768.

1610 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5805C211yesmaximal, characteristic
2C2×C3×#418C211yesmaximal, characteristic, complemented
3C22×#832C211yesmaximal, characteristic, complemented
4C2×#5129C211yesmaximal, characteristic, complemented
5#82270C211yesmaximal, characteristic, complemented
6C2×#4998C211yesmaximal, characteristic, complemented
7#71985C211yesmaximal, characteristic, complemented
8#70207C214yesmaximal, complemented
9#70206C214yesmaximal, complemented
10#23486—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×#953—22no—
16#5805—24no—
17#5805—24no—
18C2×#940—21no—
19C2×#938C2214yes—
20C2×#936—21no—
21C2×#926C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24C3×#418—24no—
25C2×C3×#102C2214yes—
26C2×#832—24no—
27C2×#832C2212yes—
28C2×#832C2212yes—
29#5129—24no—
30C2×(C2×D3)⋊C8C2212yescomplemented
31C2×(C2×D3)⋊C8C2212yescomplemented
32#4998—24no—
33#4987C2214yescomplemented
34C2×#808C2212yescomplemented
35C2×#807C2212yescomplemented
36#4958—22no—
37#4939—21no—
38#4929—21no—
39#4551C2214yescomplemented
40C2×#535—31no—
41C2×#427—31no—
42C2×#418D311yescharacteristic, complemented
43C22×#102—31no—
44C2×#408—31no—
45#2311—31no—
46#2034—34no—
47#2033—34no—
48#1736—31no—
49C25×C3—21no—
50C22×C3×D4—22no—
51C22×C3×D4—21no—
52C22×C3×D4—22no—
53C22×C3×D4—22no—
54C22×C3×D4—22no—
55C22×C3×D4—22no—
56C22×C3×D4D414yes—
57C22×C3×D4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C22×#67—24no—
60C22×#67D412yes—
61C23×Dic3—21no—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C22×D12D411yescharacteristic
69C22×C4×D3C2312yes—
70C22×Dic6D411yescharacteristic
71C2×C3×M16—22no—
72C2×C3×M16—22no—
73C2×C3×M16—22no—
74C22×C3×C8C2312yes—
75C3×#117—22no—
76C3×#117—22no—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—21no—
79C2×C3×C42—21no—
80#953—216no—
81C2×#217—21no—
82C2×#217—24no—
83#940—28no—
84#938—28no—
85#938—28no—
86#936—28no—
87C2×(C2×D3)⋊C4—24no—
88#926—22no—
89#926D414yes—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2312yes—
95C2×C4×Dic3—21no—
96C22×C3⋊C8—22no—
97C22×C3⋊C8—22no—
98C22×C3⋊C8D414yescomplemented
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101C3×#102—28no—
102#832—24no—
103#832—24no—
104C3⋊#95—24no—
105(C2×D3)⋊C8—24no—
106(C2×D3)⋊C8—24no—
107#808—24no—
108#807—24no—
109C22×#41—31no—
110C23×D4C2×D311yescharacteristic
111C22×M16C2×D311yescharacteristic
112C23×C8—31no—
113#535—64no—
114#535—64no—
115C2×#127—61no—
116C2×#124—61no—
117C2×#121—34no—
118C2×#120—62no—
119C2×#117—31no—
120C2×#117C2×D311yescharacteristic
121C22×C4⋊C4—31no—
122#427—64no—
123C2×#107—32no—
124C2×#106—32no—
125#418—24no—
126C2×#102—32no—
127C2×#102—32no—
128C2×#102—64no—
129C2×#102C2×D314yes—
130#408—64no—
131C2×C22⋊C8—32no—
132C2×C22⋊C8—32no—
133#391—61no—
134#390—61no—
135#386—62no—
136#380—34no—
137#340—34no—
138C24×C3—22no—
139C24×C3—24no—
140C24×C3—42no—
141C24×C3—42no—
142C24×C3—82no—
143C24×C3—82no—
144C24×C3C2×D411yescharacteristic
145C23×D3C2×D412yes—
146C2×C3×D4—24no—
147C2×C3×D4—28no—
148C2×C3×D4—28no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—44no—
155C2×C3×D4—42no—
156C2×C3×D4—42no—
157C2×C3×D4—44no—
158C2×C3×D4C2×D414yes—
159C22×C3×C4—22no—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4#4112yes—
165C22×C3×C4#4112yes—
166C22×C3×C4#4112yes—
167C22×C3×C4C2411yescharacteristic
168C2×#67—24no—
169C2×#67—24no—
170C2×#67—24no—
171C2×#67—216no—
172C2×#67—84no—
173C2×#67—84no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—42no—
177C22×Dic3—82no—
178C22×Dic3C2×D412yes—
179#235—44no—
180#235—44no—
181#235—44no—
182#235—44no—
183C2×D12—22no—
184C2×D12—22no—
185C2×D12—22no—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×Dic6—22no—
193C2×Dic6—22no—
194C2×Dic6—22no—
195C3×M16—44no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C22⋊C4—42no—
202C3×C42—42no—
203#217—216no—
204#217—24no—
205(C2×D3)⋊C4—84no—
206C4⋊Dic3—22no—
207C4⋊Dic3—22no—
208C4⋊Dic3—22no—
209C4⋊Dic3—42no—
210C4⋊Dic3C2×D414yes—
211C4×Dic3—24no—
212C2×C3⋊C8—28no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8C2×D412yes—
218C2×C3⋊C8C2×D412yes—
219C25—21no—
220C2×#41—62no—
221C2×#41—62no—
222C2×#41—62no—
223C2×#41—62no—
224C2×#41—62no—
225C2×#41—62no—
226C22×Q8—31no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—32no—
234C22×D4—31no—
235C22×D4—64no—
236C22×D4#6714yes—
237C22×D4C22×D312yes—
238C23×C4—32no—
239C23×C4—61no—
240C23×C4C22×D311yescharacteristic
241C2×M16—22no—
242C2×M16—22no—
243C2×M16—22no—
244C22×C8—34no—
245C22×C8—31no—
246C22×C8—31no—
247C22×C8—62no—
248C22×C8—62no—
249C22×C8C22×D312yes—
250#127—68no—
251#124—68no—
252#121—68no—
253#121—68no—
254#120—616no—
255#117—22no—
256#117—22no—
257#117—34no—
258#117—62no—
259C2×C4⋊C4—32no—
260C2×C4⋊C4—32no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4C22×D312yes—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—64no—
267C2×C22⋊C4—61no—
268C2×C22⋊C4—64no—
269C2×C42—21no—
270C2×C42—61no—
271#107—64no—
272#106—64no—
273#102—28no—
274#102—64no—
275#102—64no—
276C22⋊C8—64no—
277C22⋊C8—64no—
278#95—64no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—24no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—44no—
285C23×C3—44no—
286C23×C3—48no—
287C23×C3—88no—
288C23×C3—88no—
289C23×C3—84no—
290C23×C3—84no—
291C23×C3#12012yes—
292C23×C3#12112yes—
293C23×C3#12312yes—
294C23×C3C22×D411yescharacteristic
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—22no—
298C22×D3—24no—
299C22×D3—24no—
300C3×D4—24no—
301C3×D4—24no—
302C3×D4—42no—
303C3×D4—42no—
304C3×D4—48no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4#12012yes—
309C2×C3×C4#12114yes—
310C2×C3×C4#12112yes—
311C2×C3×C4#12114yes—
312C2×C3×C4#12312yes—
313C2×C3×C4#12314yes—
314C2×C3×C4#12414yes—
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321#67—48no—
322#67—816no—
323C2×Dic3—24no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—44no—
331C2×Dic3—84no—
332D12—44no—
333C4×D3—42no—
334C4×D3—42no—
335C4×D3—42no—
336C4×D3—42no—
337Dic6—44no—
338C3×C8—42no—
339C3⋊C8—22no—
340C3⋊C8—22no—
341C24—22no—
342C24—24no—
343C24—32no—
344C24—42no—
345C24—42no—
346C24—82no—
347C24—82no—
348C24D3×D411yescharacteristic
349#41—124no—
350#41—124no—
351#41—124no—
352#41—124no—
353C2×Q8—62no—
354C2×Q8—62no—
355C2×Q8—62no—
356C2×D4—28no—
357C2×D4—24no—
358C2×D4—28no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—64no—
369C2×D4—64no—
370C2×D4—64no—
371C2×D4—616no—
372C2×D4—62no—
373C2×D4—62no—
374C2×D4—62no—
375C2×D4—244no—
376C2×D4—244no—
377C2×D4C2×#6714yes—
378C22×C4—22no—
379C22×C4—32no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—122no—
389C22×C4—242no—
390C22×C4#23512yes—
391C22×C4#23512yes—
392C22×C4D3×D411yescharacteristic
393C22×C4D3×D412yes—
394C22×C4D3×D412yes—
395C22×C4#23712yes—
396C22×C4C2×#6712yes—
397C22×C4C23×D311yescharacteristic
398M16—44no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—32no—
403C2×C8—32no—
404C2×C8—68no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—61no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—34no—
412C4⋊C4—62no—
413C4⋊C4—62no—
414C4⋊C4—62no—
415C4⋊C4—122no—
416C22⋊C4—42no—
417C22⋊C4—616no—
418C22⋊C4—64no—
419C22⋊C4—244no—
420C42—42no—
421C42—64no—
422C22×C3—24no—
423C22×C3—24no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—44no—
427C22×C3—48no—
428C22×C3—816no—
429C22×C3—84no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3#39414yes—
433C22×C3C2×#12011yescharacteristic
434C22×C3C2×#12111yescharacteristic
435C22×C3C2×#12311yescharacteristic
436C22×C3#57612yes—
437C22×C3#57712yes—
438C2×D3—44no—
439C2×D3—42no—
440C2×D3—44no—
441C2×D3—42no—
442C2×D3—42no—
443C3×C4—82no—
444C3×C4#39414yes—
445C3×C4C2×#12111yescharacteristic
446C3×C4C2×#12111yescharacteristic
447C3×C4C2×#12311yescharacteristic
448C3×C4C2×#12411yescharacteristic
449Dic3—42no—
450Dic3—44no—
451Dic3—82no—
452C23—22no—
453C23—22no—
454C23—24no—
455C23—48no—
456C23—42no—
457C23—42no—
458C23—44no—
459C23—44no—
460C23—62no—
461C23—62no—
462C23—64no—
463C23—64no—
464C23—62no—
465C23—88no—
466C23—88no—
467C23—84no—
468C23—84no—
469C23#88312yes—
470C23#88412yes—
471C23#93712yes—
472C23C2×D3×D411yescharacteristic
473Q8—124no—
474D4—24no—
475D4—24no—
476D4—48no—
477D4—42no—
478D4—42no—
479D4—128no—
480D4—124no—
481D4—2416no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—64no—
488C2×C4—62no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—124no—
497C2×C4—244no—
498C2×C4#88312yes—
499C2×C4#88412yes—
500C2×C4#88414yes—
501C2×C4#88514yes—
502C2×C4#92914yes—
503C2×C4#93712yes—
504C2×C4#93914yes—
505C2×C4C2×#23511yescharacteristic
506C2×C4C2×#23511yescharacteristic
507C2×C4C2×D3×D411yescharacteristic
508C2×C4C2×D3×D411yescharacteristic
509C2×C4C2×#23711yescharacteristic
510C2×C4C22×#6711yescharacteristic
511C8—42no—
512C8—62no—
513C8—62no—
514C2×C3—24no—
515C2×C3—84no—
516C2×C3C2×#39411yescharacteristic
517C2×C3#301212yes—
518C2×C3#305711yescharacteristic
519C2×C3#305811yescharacteristic
520C2×C3#309612yes—
521D3—82no—
522C22—24no—
523C22—22no—
524C22—22no—
525C22—24no—
526C22—44no—
527C22—48no—
528C22—816no—
529C22—84no—
530C22—82no—
531C22—82no—
532C22—124no—
533C22—122no—
534C22—122no—
535C22—122no—
536C22—124no—
537C22#495914yes—
538C22C2×#88311yescharacteristic
539C22C2×#88411yescharacteristic
540C22#577412yes—
541C22#577512yes—
542C22C2×#93711yescharacteristic
543C4—82no—
544C4—122no—
545C4—124no—
546C4—242no—
547C4#495914yes—
548C4C2×#88411yescharacteristic
549C4C2×#88511yescharacteristic
550C4C2×#92911yescharacteristic
551C4C2×#93911yescharacteristic
552C3#2348611yescharacteristic, complemented, Sylow
553C2—24no—
554C2—84no—
555C2—242no—
556C2#7891912yes—
557C2#7893111yescharacteristic
558C2#7893211yescharacteristic
559C2C2×#495911yescharacteristic
560C2#8394912yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#418C2
socleC23×C3C22×D4
prime radicalC2×#5805C2
hypercenterC2×#418D3
hypocenterC3#23486
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#418

Complements

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

NH ≅ G/Ntype
2. C2×C3×#418C2semidirect
3. C22×#832C2semidirect
4. C2×#5129C2semidirect
5. #82270C2semidirect
6. C2×#4998C2semidirect
7. #71985C2semidirect
8. #70207C2semidirect
9. #70206C2semidirect
30. C2×(C2×D3)⋊C8C22semidirect
31. C2×(C2×D3)⋊C8C22semidirect
33. #4987C22semidirect
34. C2×#808C22semidirect
35. C2×#807C22semidirect
39. #4551C22semidirect
42. C2×#418D3semidirect
98. C22×C3⋊C8D4semidirect
552. C3#23486semidirect

Maximal subgroups

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

subgroupindex
C2×#58052
C2×C3×#4182
C22×#8322
C2×#51292
#822702
C2×#49982
#719852
#702072
#702062
#234863