Finite Group Databasebeta

← #10796355

Subgroups — #10796355

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×C3×#532C211yesmaximal, characteristic
2C22×#948C211yesmaximal, characteristic, complemented
3C2×#5234C211yesmaximal, characteristic, complemented
4C2×#5234C211yesmaximal, characteristic, complemented
5C2×#5234C211yesmaximal, characteristic, complemented
6#71611C214yesmaximal, complemented
7#71610C214yesmaximal, complemented
8#71527C211yesmaximal, characteristic, complemented
9C2×#4524C211yesmaximal, characteristic, complemented
10#20900—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C22×C3×#41C2211yescharacteristic
13C3×#532—24no—
14C3×#532—24no—
15C2×C3×C4⋊Q8C411yescharacteristic
16C2×C3×#127C411yescharacteristic
17C2×C3×#124C411yescharacteristic
18C2×C3×#124C411yescharacteristic
19C2×C3×#121C414yes—
20C2×C3×C4×D4C2214yes—
21C22×C3×C42C2211yescharacteristic
22C2×#948—24no—
23C2×#948—24no—
24C2×#948—24no—
25C22×#208C2211yescharacteristic
26C22×#208C2212yes—
27C22×#208C2211yescharacteristic
28C23×C3⋊C8C2211yescharacteristic
29C23×C3⋊C8C2212yes—
30C23×C3⋊C8C2211yescharacteristic
31#5234—24no—
32#5234—24no—
33#5234—24no—
34C2×#830C2212yes—
35C2×#830C2214yes—
36C2×#830C2214yes—
37C2×#830C2212yes—
38#4524C2212yes—
39#4524C2212yes—
40#4524C2214yes—
41C22×#131—31no—
42C2×#532D311yescharacteristic
43C2×#408—31no—
44C2×#408—31no—
45C2×#408—31no—
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×#41—22no—
55C2×C3×#41—22no—
56C2×C3×#41—22no—
57C2×C3×#41—22no—
58C2×C3×#41—22no—
59C2×C3×#41—22no—
60C2×C3×#41D414yes—
61C2×C3×#41D414yes—
62C22×C3×Q8C2×C411yescharacteristic
63C22×C3×Q8C2×C411yescharacteristic
64C22×C3×D4C2×C412yes—
65C22×C3×D4C2×C411yescharacteristic
66C22×C3×D4C2×C414yes—
67C22×C3×D4C2×C412yes—
68C22×C3×D4C2×C411yescharacteristic
69C23×C3×C4C2×C412yes—
70C23×C3×C4C2312yes—
71C23×C3×C4C2312yes—
72C23×C3×C4C2311yescharacteristic
73C3×C4⋊Q8—24no—
74C3×#127—24no—
75C3×#124—24no—
76C3×#124—24no—
77C3×#121—216no—
78C3×C4×D4—28no—
79C3×C4×D4—28no—
80C2×C3×C4⋊C4C2×C414yes—
81C2×C3×C22⋊C4C2×C414yes—
82C2×C3×C22⋊C4C2×C414yes—
83C2×C3×C42—22no—
84C2×C3×C42—22no—
85C2×C3×C42C2×C411yescharacteristic
86C2×C3×C42C2×C411yescharacteristic
87C2×C3×C42C2312yes—
88#948—232no—
89C2×#208—22no—
90C2×#208—24no—
91C2×#208—22no—
92C2×#208—22no—
93C2×#208—22no—
94C2×#208—22no—
95C2×#208—24no—
96C2×#208—24no—
97C2×#208D414yescomplemented
98C22×C3⋊C8—22no—
99C22×C3⋊C8—24no—
100C22×C3⋊C8—22no—
101C22×C3⋊C8—24no—
102C22×C3⋊C8—24no—
103C22×C3⋊C8—22no—
104C22×C3⋊C8—22no—
105C22×C3⋊C8—22no—
106C22×C3⋊C8D414yescomplemented
107C22×C3⋊C8C2312yes—
108C22×C3⋊C8C2312yes—
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2312yes—
111C22×C3⋊C8C2311yescharacteristic
112#830—28no—
113#830—28no—
114#830—24no—
115#830—24no—
116C22×#41C2×D311yescharacteristic
117C22×#41C2×D311yescharacteristic
118C2×#131—64no—
119C2×#131—64no—
120C2×#131—64no—
121C22×M16—31no—
122C22×M16—31no—
123C22×M16—32no—
124C23×C8—31no—
125C23×C8—31no—
126C23×C8—32no—
127#532—24no—
128#532—24no—
129C2×C4⋊Q8Dic311yescharacteristic
130C2×#127Dic311yescharacteristic
131C2×#124Dic311yescharacteristic
132C2×#124Dic311yescharacteristic
133C2×#121Dic314yes—
134C2×C4×D4C2×D314yes—
135C22×C42C2×D311yescharacteristic
136#408—64no—
137#408—64no—
138#408—64no—
139C2×C22⋊C8—32no—
140C2×C22⋊C8—34no—
141C2×C22⋊C8—34no—
142C2×C22⋊C8—32no—
143#336—32no—
144#336—34no—
145#336—32no—
146C24×C3C22×C412yes—
147C24×C3C22×C412yes—
148C24×C3C22×C411yescharacteristic
149C3×#41—28no—
150C3×#41—28no—
151C3×#41—28no—
152C3×#41—28no—
153C3×#41—44no—
154C3×#41—44no—
155C3×#41—44no—
156C3×#41—44no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8C22⋊C414yes—
163C2×C3×D4—24no—
164C2×C3×D4—24no—
165C2×C3×D4—28no—
166C2×C3×D4—24no—
167C2×C3×D4—24no—
168C2×C3×D4—24no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—28no—
175C2×C3×D4—44no—
176C2×C3×D4C22⋊C418yes—
177C2×C3×D4C22⋊C414yes—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—24no—
183C22×C3×C4—24no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—42no—
191C22×C3×C4C22×C412yes—
192C22×C3×C4C22×C412yes—
193C22×C3×C4C22×C412yes—
194C22×C3×C4C22×C412yes—
195C22×C3×C4C22×C411yescharacteristic
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×D414yes—
199C22×C3×C4C2×D414yes—
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2×D412yes—
202C22×C3×C4#4112yes—
203C22×C3×C4#4112yes—
204C22×C3×C4C2411yescharacteristic
205C3×C4⋊C4—28no—
206C3×C22⋊C4—28no—
207C3×C22⋊C4—28no—
208C3×C42—22no—
209C3×C42—22no—
210C3×C42—22no—
211C3×C42—22no—
212#208—216no—
213#208—28no—
214#208—44no—
215C2×C3⋊C8—216no—
216C2×C3⋊C8—28no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—44no—
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—22no—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41#6714yes—
244C2×#41#6714yes—
245C22×Q8C2×Dic311yescharacteristic
246C22×Q8C2×Dic311yescharacteristic
247C22×D4C2×Dic312yes—
248C22×D4C2×Dic311yescharacteristic
249C22×D4C2×Dic314yes—
250C22×D4C2×Dic312yes—
251C22×D4C2×Dic311yescharacteristic
252C23×C4C2×Dic312yes—
253C23×C4C22×D312yes—
254C23×C4C22×D312yes—
255C23×C4C22×D311yescharacteristic
256#131—632no—
257C2×M16—34no—
258C2×M16—64no—
259C2×M16—64no—
260C2×M16—62no—
261C2×M16—64no—
262C2×M16—62no—
263C2×M16—62no—
264C2×M16—62no—
265C2×M16—62no—
266C22×C8—34no—
267C22×C8—31no—
268C22×C8—31no—
269C22×C8—32no—
270C22×C8—32no—
271C22×C8—32no—
272C22×C8—64no—
273C22×C8—64no—
274C22×C8—62no—
275C22×C8—64no—
276C22×C8—62no—
277C22×C8—62no—
278C22×C8—62no—
279C22×C8—62no—
280C4⋊Q8—24no—
281#127—24no—
282#124—24no—
283#124—24no—
284#121—216no—
285C4×D4—28no—
286C4×D4—28no—
287C2×C4⋊C4C2×Dic314yes—
288C2×C22⋊C4C2×Dic314yes—
289C2×C22⋊C4C2×Dic314yes—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42C2×Dic311yescharacteristic
293C2×C42C2×Dic311yescharacteristic
294C2×C42C22×D312yes—
295C22⋊C8—64no—
296C22⋊C8—64no—
297C22⋊C8—68no—
298C22⋊C8—68no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3C2×C22⋊C414yes—
311C23×C3C2×C22⋊C412yes—
312C23×C3C4×D412yes—
313C23×C3C4×D412yes—
314C23×C3C23×C411yescharacteristic
315C3×Q8—28no—
316C3×Q8—44no—
317C3×D4—216no—
318C3×D4—28no—
319C3×D4—416no—
320C3×D4—48no—
321C3×D4—44no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—48no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4C2×C22⋊C414yes—
342C2×C3×C4C2×C22⋊C412yes—
343C2×C3×C4C4×D412yes—
344C2×C3×C4C4×D412yes—
345C2×C3×C4#12014yes—
346C2×C3×C4#12114yes—
347C2×C3×C4#12114yes—
348C2×C3×C4#12314yes—
349C2×C3×C4#13114yescomplemented
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C3⋊C8—22no—
357C3⋊C8—24no—
358C3⋊C8—22no—
359C3⋊C8—42no—
360C3⋊C8—42no—
361C24C22×Dic312yes—
362C24C22×Dic312yes—
363C24C22×Dic311yescharacteristic
364#41—28no—
365#41—28no—
366#41—28no—
367#41—28no—
368#41—44no—
369#41—44no—
370#41—44no—
371#41—44no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8#21714yes—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—28no—
383C2×D4—24no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—28no—
389C2×D4—22no—
390C2×D4—44no—
391C2×D4#21718yes—
392C2×D4#21714yes—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—42no—
406C22×C4D3×D412yes—
407C22×C4D3×D412yes—
408C22×C4#23712yes—
409C22×C4#23712yes—
410C22×C4C22×Dic312yes—
411C22×C4C22×Dic312yes—
412C22×C4C22×Dic312yes—
413C22×C4C22×Dic312yes—
414C22×C4C22×Dic311yescharacteristic
415C22×C4C2×#6714yes—
416C22×C4C2×#6714yes—
417C22×C4C2×#6712yes—
418C22×C4C2×#6712yes—
419C22×C4C23×D311yescharacteristic
420M16—616no—
421M16—68no—
422M16—124no—
423C2×C8—32no—
424C2×C8—32no—
425C2×C8—34no—
426C2×C8—616no—
427C2×C8—68no—
428C2×C8—61no—
429C2×C8—62no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—62no—
434C2×C8—62no—
435C2×C8—62no—
436C2×C8—62no—
437C2×C8—62no—
438C2×C8—62no—
439C2×C8—62no—
440C2×C8—124no—
441C4⋊C4—28no—
442C22⋊C4—28no—
443C22⋊C4—28no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
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—44no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—44no—
461C22×C3—42no—
462C22×C3#38614yes—
463C22×C3C22×C22⋊C411yescharacteristic
464C22×C3C2×C4×D411yescharacteristic
465C22×C3C2×C4×D411yescharacteristic
466C22×C3C2×#13112yes—
467C22×C3#56812yes—
468C3×C4—24no—
469C3×C4—24no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4#38614yes—
473C3×C4C2×#12011yescharacteristic
474C3×C4C2×#12111yescharacteristic
475C3×C4C2×#12111yescharacteristic
476C3×C4C2×#12311yescharacteristic
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23D4×Dic312yes—
489C23D4×Dic312yes—
490C23C2×#21714yes—
491C23C2×#21712yes—
492C23C23×Dic311yescharacteristic
493Q8—28no—
494Q8—44no—
495D4—216no—
496D4—28no—
497D4—416no—
498D4—48no—
499D4—44no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—48no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4D4×Dic312yes—
520C2×C4D4×Dic312yes—
521C2×C4#93514yes—
522C2×C4#93714yes—
523C2×C4#93914yes—
524C2×C4#93914yes—
525C2×C4#94814yescomplemented
526C2×C4C2×#21714yes—
527C2×C4C2×#21712yes—
528C2×C4C2×D3×D411yescharacteristic
529C2×C4C2×D3×D411yescharacteristic
530C2×C4C2×#23711yescharacteristic
531C2×C4C2×#23711yescharacteristic
532C2×C4C22×#6711yescharacteristic
533C2×C4C22×#6711yescharacteristic
534C8—62no—
535C8—62no—
536C8—64no—
537C8—122no—
538C8—122no—
539C2×C3—24no—
540C2×C3—44no—
541C2×C3—82no—
542C2×C3C2×#38611yescharacteristic
543C2×C3#288012yes—
544C2×C3#293111yescharacteristic
545C2×C3#293211yescharacteristic
546C2×C3#293512yes—
547C22—22no—
548C22—22no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—28no—
553C22—42no—
554C22—42no—
555C22—44no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—44no—
560C22—44no—
561C22#522014yes—
562C22C2×D4×Dic311yescharacteristic
563C22C2×D4×Dic311yescharacteristic
564C22C2×#94812yes—
565C22#581912yes—
566C22C22×#21711yescharacteristic
567C4—24no—
568C4—24no—
569C4—42no—
570C4—42no—
571C4#522014yes—
572C4C2×#93511yescharacteristic
573C4C2×#93711yescharacteristic
574C4C2×#93911yescharacteristic
575C4C2×#93911yescharacteristic
576C3#2090011yescharacteristic, complemented, Sylow
577C2—24no—
578C2—44no—
579C2—82no—
580C2#8327511yescharacteristic
581C2#8327611yescharacteristic
582C2#8327712yes—
583C2C2×#522011yescharacteristic
584C2#8407512yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
Fitting, prime radicalC2×C3×#532C2
cosocleC22×C3×C4C24
hypercenterC2×#532D3
hypocenterC3#20900
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
2. C22×#948C2semidirect
3. C2×#5234C2semidirect
4. C2×#5234C2semidirect
5. C2×#5234C2semidirect
6. #71611C2semidirect
7. #71610C2semidirect
8. #71527C2semidirect
9. C2×#4524C2semidirect
97. C2×#208D4semidirect
106. C22×C3⋊C8D4semidirect
349. C2×C3×C4#131semidirect
525. C2×C4#948semidirect
576. C3#20900semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C22×#9482
C2×#52342
#716112
#716102
#715272
C2×#45242
#209003