Finite Group Databasebeta

← #10796350

Subgroups — #10796350

Order 768.

1854 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5832C211yesmaximal, characteristic
2C22×#948C211yesmaximal, characteristic, complemented
3C2×C3×#408C211yesmaximal, characteristic, complemented
4C2×#5129C211yesmaximal, characteristic, complemented
5C2×#5110C211yesmaximal, characteristic, complemented
6#71979C211yesmaximal, characteristic, complemented
7C2×#4550C211yesmaximal, characteristic, complemented
8#69953C214yesmaximal, complemented
9#69952C214yesmaximal, complemented
10#20900—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15#5832—24no—
16#5832—24no—
17C2×#948—24no—
18C2×#948—24no—
19C2×#948—24no—
20C2×#947C411yescharacteristic
21C2×#944C411yescharacteristic
22C2×#940C411yescharacteristic
23C2×#939C414yes—
24C2×#936C411yescharacteristic
25C2×C4×#67C2214yes—
26C22×C4×Dic3C2211yescharacteristic
27C22×#208C2212yes—
28C22×#208C2211yescharacteristic
29C23×C3⋊C8C2212yes—
30C23×C3⋊C8C2211yescharacteristic
31C3×#408—24no—
32C2×C3×C22⋊C8C2214yes—
33#5129—24no—
34#5110—24no—
35C2×(C2×D3)⋊C8C2214yes—
36C2×(C2×D3)⋊C8C2212yes—
37C2×(C2×D3)⋊C8C2212yes—
38#4550C2212yes—
39#4550C2212yes—
40#4550C2214yes—
41C22×#131—31no—
42C2×#532—31no—
43C2×#408—31no—
44C2×#408—31no—
45C2×#408D311yescharacteristic, complemented
46#1870—34no—
47#1869—34no—
48#1728—31no—
49C2×#336—31no—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41D414yes—
55C2×C3×#41D414yes—
56C22×C3×Q8C2×C411yescharacteristic
57C22×C3×D4C2×C411yescharacteristic
58C22×C3×D4C2×C412yes—
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C22×#67C2×C412yes—
62C22×#67C2×C414yes—
63C23×Dic3C2×C412yes—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235—22no—
69C2×#235—22no—
70C22×D12C2×C411yescharacteristic
71C22×C4×D3C2312yes—
72C22×Dic6C2×C411yescharacteristic
73C2×C3×M16—22no—
74C2×C3×M16—22no—
75C2×C3×M16—22no—
76C22×C3×C8—22no—
77C22×C3×C8—22no—
78C22×C3×C8—22no—
79C22×C3×C8C2312yes—
80C22×C3×C8C2312yes—
81C2×#217C2×C414yes—
82#948—232no—
83#947—24no—
84#944—24no—
85#940—24no—
86#939—216no—
87#936—24no—
88C4×#67—28no—
89C4×#67—28no—
90C2×(C2×D3)⋊C4C2×C414yes—
91C2×Dic3⋊C4C2×C414yes—
92C2×C4×Dic3—22no—
93C2×C4×Dic3—22no—
94C2×C4×Dic3C2×C411yescharacteristic
95C2×C4×Dic3C2×C411yescharacteristic
96C2×C4×Dic3C2312yes—
97C2×#208—24no—
98C2×#208—22no—
99C2×#208—22no—
100C2×#208—24no—
101C2×#208—24no—
102C2×#208D414yescomplemented
103C22×C3⋊C8—24no—
104C22×C3⋊C8—22no—
105C22×C3⋊C8—22no—
106C22×C3⋊C8—24no—
107C22×C3⋊C8—24no—
108C22×C3⋊C8D414yescomplemented
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2312yes—
111C22×C3⋊C8C2311yescharacteristic
112C3×C22⋊C8—28no—
113(C2×D3)⋊C8—24no—
114(C2×D3)⋊C8—24no—
115(C2×D3)⋊C8—28no—
116C22×#41—31no—
117C22×#41C2×D311yescharacteristic
118C2×#131—64no—
119C2×#131—64no—
120C2×#131—64no—
121C22×M16—31no—
122C22×M16—32no—
123C22×M16C2×D311yescharacteristic
124C23×C8—31no—
125C23×C8—32no—
126C23×C8C2×D311yescharacteristic
127#532—64no—
128#532—64no—
129C2×C4⋊Q8—31no—
130C2×#127—31no—
131C2×#124—31no—
132C2×#124—31no—
133C2×#121—34no—
134C2×C4×D4—34no—
135C22×C42—31no—
136#408—24no—
137#408—64no—
138#408—64no—
139C2×C22⋊C8—32no—
140C2×C22⋊C8—34no—
141C2×C22⋊C8—32no—
142C2×C22⋊C8C2×D314yes—
143#336—32no—
144#336—34no—
145#336—32no—
146C24×C3C22×C412yes—
147C24×C3C22×C411yescharacteristic
148C23×D3C22×C412yes—
149C3×#41—28no—
150C3×#41—28no—
151C3×#41—28no—
152C3×#41—28no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8C22⋊C414yes—
156C2×C3×D4—24no—
157C2×C3×D4—24no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4C22⋊C414yes—
161C2×C3×D4C22⋊C418yes—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4C22×C412yes—
167C22×C3×C4C22×C411yescharacteristic
168C22×C3×C4C2×D414yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D414yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4#4112yes—
175C22×C3×C4#4112yes—
176C22×C3×C4C2411yescharacteristic
177C2×#67—28no—
178C2×#67—24no—
179C2×#67—24no—
180C2×#67—24no—
181C2×#67—28no—
182C2×#67—44no—
183C22×Dic3—24no—
184C22×Dic3—24no—
185C22×Dic3—42no—
186C22×Dic3C22×C412yes—
187C22×Dic3C22×C412yes—
188C22×Dic3C22×C412yes—
189#235—44no—
190#235—44no—
191#235—44no—
192#235—44no—
193C2×D12—22no—
194C2×D12—22no—
195C2×D12—22no—
196C2×C4×D3—22no—
197C2×C4×D3—22no—
198C2×C4×D3—22no—
199C2×C4×D3—22no—
200C2×C4×D3—22no—
201C2×C4×D3—22no—
202C2×Dic6—22no—
203C2×Dic6—22no—
204C2×Dic6—22no—
205C3×M16—44no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—44no—
213#217—28no—
214(C2×D3)⋊C4—28no—
215Dic3⋊C4—28no—
216C4×Dic3—22no—
217C4×Dic3—22no—
218C4×Dic3—22no—
219C4×Dic3—22no—
220#208—216no—
221#208—28no—
222C2×C3⋊C8—216no—
223C2×C3⋊C8—28no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8—21no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8C2×D412yes—
231C2×C3⋊C8C2×D414yes—
232C2×C3⋊C8C2×D412yes—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—62no—
238C2×#41—62no—
239C2×#41—62no—
240C2×#41—62no—
241C2×#41—62no—
242C2×#41—62no—
243C2×#41D1214yes—
244C2×#41#6714yes—
245C22×Q8—31no—
246C22×Q8C4×D311yescharacteristic
247C22×D4—32no—
248C22×D4—34no—
249C22×D4—31no—
250C22×D4C4×D312yes—
251C22×D4C4×D311yescharacteristic
252C23×C4—32no—
253C23×C4—32no—
254C23×C4C22×D312yes—
255C23×C4C22×D311yescharacteristic
256#131—632no—
257C2×M16—22no—
258C2×M16—22no—
259C2×M16—22no—
260C2×M16—34no—
261C2×M16—64no—
262C2×M16—64no—
263C2×M16—64no—
264C2×M16—62no—
265C2×M16—62no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—34no—
270C22×C8—31no—
271C22×C8—31no—
272C22×C8—32no—
273C22×C8—64no—
274C22×C8—64no—
275C22×C8—64no—
276C22×C8—62no—
277C22×C8—62no—
278C22×C8C22×D312yes—
279C22×C8C22×D312yes—
280C4⋊Q8—64no—
281#127—64no—
282#124—64no—
283#124—64no—
284#121—616no—
285C4×D4—68no—
286C4×D4—68no—
287C2×C4⋊C4—34no—
288C2×C22⋊C4—34no—
289C2×C22⋊C4—34no—
290C2×C42—31no—
291C2×C42—31no—
292C2×C42—32no—
293C2×C42—62no—
294C2×C42—62no—
295C22⋊C8—28no—
296C22⋊C8—68no—
297C22⋊C8—64no—
298C22⋊C8—64no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3C2×C22⋊C414yes—
306C23×C3C2×C22⋊C412yes—
307C23×C3C4×D412yes—
308C23×C3C4×D412yes—
309C23×C3C23×C411yescharacteristic
310C22×D3—22no—
311C22×D3—22no—
312C22×D3—22no—
313C22×D3—24no—
314C22×D3—24no—
315C3×Q8—28no—
316C3×D4—216no—
317C3×D4—28no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4C2×C22⋊C414yes—
325C2×C3×C4C2×C22⋊C412yes—
326C2×C3×C4C4×D412yes—
327C2×C3×C4C4×D412yes—
328C2×C3×C4#12014yes—
329C2×C3×C4#12114yes—
330C2×C3×C4#12114yes—
331C2×C3×C4#12314yes—
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C2×#4111yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338#67—416no—
339#67—48no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—48no—
349C2×Dic3#13114yescomplemented
350D12—44no—
351C4×D3—42no—
352C4×D3—42no—
353C4×D3—42no—
354C4×D3—42no—
355Dic6—44no—
356C3×C8—42no—
357C3×C8—42no—
358C3⋊C8—22no—
359C3⋊C8—24no—
360C3⋊C8—22no—
361C24—32no—
362C24C2×C4×D312yes—
363C24C2×C4×D311yescharacteristic
364#41—28no—
365#41—28no—
366#41—28no—
367#41—28no—
368#41—124no—
369#41—124no—
370#41—124no—
371#41—124no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—62no—
375C2×Q8—62no—
376C2×Q8—62no—
377C2×Q8(C2×D3)⋊C414yes—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—68no—
383C2×D4—64no—
384C2×D4—64no—
385C2×D4—68no—
386C2×D4—64no—
387C2×D4—62no—
388C2×D4—62no—
389C2×D4—62no—
390C2×D4—124no—
391C2×D4(C2×D3)⋊C418yes—
392C2×D4(C2×D3)⋊C414yes—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—32no—
398C22×C4—32no—
399C22×C4—32no—
400C22×C4—64no—
401C22×C4—64no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—122no—
409C22×C4C2×C4×D312yes—
410C22×C4C2×C4×D311yescharacteristic
411C22×C4C2×D1214yes—
412C22×C4C2×D1212yes—
413C22×C4D3×D412yes—
414C22×C4D3×D412yes—
415C22×C4#23712yes—
416C22×C4#23712yes—
417C22×C4C2×#6714yes—
418C22×C4C2×#6712yes—
419C22×C4C23×D311yescharacteristic
420M16—44no—
421M16—616no—
422M16—68no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—32no—
430C2×C8—32no—
431C2×C8—34no—
432C2×C8—44no—
433C2×C8—616no—
434C2×C8—68no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—62no—
438C2×C8—61no—
439C2×C8—62no—
440C2×C8—61no—
441C4⋊C4—68no—
442C22⋊C4—68no—
443C22⋊C4—68no—
444C42—62no—
445C42—62no—
446C42—62no—
447C42—62no—
448C22×C3—24no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—24no—
452C22×C3—28no—
453C22×C3—24no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—44no—
457C22×C3#38614yes—
458C22×C3C22×C22⋊C411yescharacteristic
459C22×C3C2×C4×D411yescharacteristic
460C22×C3C2×C4×D411yescharacteristic
461C22×C3C2×#13112yes—
462C22×C3#56812yes—
463C2×D3—44no—
464C2×D3—42no—
465C2×D3—42no—
466C2×D3—44no—
467C2×D3—42no—
468C3×C4—24no—
469C3×C4#38614yes—
470C3×C4C2×#12011yescharacteristic
471C3×C4C2×#12111yescharacteristic
472C3×C4C2×#12111yescharacteristic
473C3×C4C2×#12311yescharacteristic
474Dic3—24no—
475Dic3—42no—
476Dic3—42no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—24no—
483C23—62no—
484C23—62no—
485C23—64no—
486C23—64no—
487C23—62no—
488C23#67⋊C412yes—
489C23#67⋊C412yes—
490C23C2×(C2×D3)⋊C414yes—
491C23C2×(C2×D3)⋊C412yes—
492C23C22×C4×D311yescharacteristic
493Q8—28no—
494Q8—124no—
495D4—216no—
496D4—28no—
497D4—1216no—
498D4—128no—
499D4—124no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—34no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—128no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4#67⋊C412yes—
521C2×C4#67⋊C412yes—
522C2×C4#88214yes—
523C2×C4#88614yes—
524C2×C4C2×(C2×D3)⋊C414yes—
525C2×C4C2×(C2×D3)⋊C412yes—
526C2×C4#93914yes—
527C2×C4#93914yes—
528C2×C4C22×D1211yescharacteristic
529C2×C4C2×D3×D411yescharacteristic
530C2×C4C2×D3×D411yescharacteristic
531C2×C4C2×#23711yescharacteristic
532C2×C4C2×#23711yescharacteristic
533C2×C4C22×#6711yescharacteristic
534C8—42no—
535C8—42no—
536C8—62no—
537C8—62no—
538C8—64no—
539C2×C3—24no—
540C2×C3—44no—
541C2×C3C2×#38611yescharacteristic
542C2×C3#288012yes—
543C2×C3#293111yescharacteristic
544C2×C3#293211yescharacteristic
545C2×C3#293512yes—
546D3—82no—
547C22—22no—
548C22—22no—
549C22—24no—
550C22—24no—
551C22—28no—
552C22—24no—
553C22—42no—
554C22—42no—
555C22—44no—
556C22—122no—
557C22—122no—
558C22—124no—
559C22—124no—
560C22—122no—
561C22#495814yes—
562C22C2×#67⋊C411yescharacteristic
563C22C2×#67⋊C411yescharacteristic
564C22D3×#13112yes—
565C22#575012yes—
566C22C22×(C2×D3)⋊C411yescharacteristic
567C4—24no—
568C4—64no—
569C4—122no—
570C4—122no—
571C4#495814yes—
572C4C2×#88211yescharacteristic
573C4C2×#88611yescharacteristic
574C4C2×#93911yescharacteristic
575C4C2×#93911yescharacteristic
576C3#2090011yescharacteristic, complemented, Sylow
577C2—24no—
578C2—44no—
579C2—242no—
580C2#7865711yescharacteristic
581C2#7865811yescharacteristic
582C2#7866012yes—
583C2C2×#495811yescharacteristic
584C2#8379312yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC2×C3×#408C2
prime radicalC2×#5832C2
cosocleC22×C3×C4C24
hypercenterC2×#408D3
hypocenterC3#20900
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
2. C22×#948C2semidirect
3. C2×C3×#408C2semidirect
4. C2×#5129C2semidirect
5. C2×#5110C2semidirect
6. #71979C2semidirect
7. C2×#4550C2semidirect
8. #69953C2semidirect
9. #69952C2semidirect
45. C2×#408D3semidirect
102. C2×#208D4semidirect
108. C22×C3⋊C8D4semidirect
349. C2×Dic3#131semidirect
576. C3#20900semidirect

Maximal subgroups

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

subgroupindex
C2×#58322
C22×#9482
C2×C3×#4082
C2×#51292
C2×#51102
#719792
C2×#45502
#699532
#699522
#209003