Finite Group Databasebeta

← #10702942

Subgroups — #10702942

Order 768.

922 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5131C211yesmaximal, characteristic
2C2×#5131C211yesmaximal, characteristic
3C22×#823C211yesmaximal, characteristic
4C3×#1738C211yesmaximal, characteristic, complemented
5C2×#4559C211yesmaximal, characteristic, complemented
6C2×#4556C211yesmaximal, characteristic, complemented
7C2×#4557C211yesmaximal, characteristic, complemented
8#11181—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C22×#208C2211yescharacteristic
16#5131—22no—
17#5131—22no—
18#5131—22no—
19#5131C414yescomplemented
20C2×#823—22no—
21C2×#823—22no—
22C2×#823—22no—
23C2×#821—22no—
24C2×#821—22no—
25C3×#341—22no—
26#4559—22no—
27#4557—22no—
28#4556—22no—
29C2×#418—31no—
30C2×#418—31no—
31C22×#100—31no—
32C2×#344—31no—
33C2×#344—31no—
34#1738D311yescharacteristic, complemented
35C2×H(3,ℤ/4)—31no—
36C24×D3—21no—
37C23×C3×C4C2311yescharacteristic
38C22×D12—21no—
39C22×D12—21no—
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—21no—
43C22×D12—21no—
44C22×D12—21no—
45C22×D12—21no—
46C22×D12—21no—
47C22×D12—21no—
48C22×D12—21no—
49C22×D12C2×C412yes—
50C22×D12D412yes—
51C22×D12D412yes—
52C2×C3×M16—21no—
53C2×C3×M16—21no—
54C2×C3×M16—21no—
55C2×C3×M16—21no—
56C2×C3×M16—21no—
57C2×C3×M16—21no—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
63C2×C3×M16—21no—
64C2×C3×M16—21no—
65C2×C3×M16—21no—
66C2×C3×M16—21no—
67C2×C3×M16—21no—
68C2×C3×M16C2×C412yes—
69C2×C3×M16D412yes—
70C22×C3×C8—21no—
71C22×C3×C8—21no—
72C22×C3×C8D412yes—
73C2×#217—21no—
74C2×#217—21no—
75#926—21no—
76#926—21no—
77#926—21no—
78#926—21no—
79#926—21no—
80#926—21no—
81#926C2×C412yes—
82#926D412yes—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4×Dic3—21no—
86C2×C4×Dic3—21no—
87C2×#208—21no—
88C2×#208—21no—
89C2×#208—21no—
90C2×#208—21no—
91C2×#208—21no—
92C2×#208D412yes—
93C22×C3⋊C8—21no—
94#823—28no—
95#821—28no—
96#821—28no—
97C23×D4—31no—
98C22×M16—31no—
99C22×M16C2×D311yescharacteristic
100C22×M16C2×D311yescharacteristic
101C22×M16C2×D311yescharacteristic
102C2×#117—31no—
103C2×#117—31no—
104#418—34no—
105#418—62no—
106#418—62no—
107#418—62no—
108C2×#102—62no—
109C2×#102—62no—
110C2×#100—62no—
111C2×#100—62no—
112C2×#100—62no—
113#344—62no—
114#344—62no—
115#341—22no—
116H(3,ℤ/4)—62no—
117C24×C3#4111yescharacteristic
118C23×D3—22no—
119C23×D3—22no—
120C23×D3—22no—
121C23×D3—41no—
122C23×D3—41no—
123C23×D3—41no—
124C23×D3—41no—
125C23×D3—84no—
126C22×C3×C4C22×C411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×Dic3—41no—
141C22×Dic3—41no—
142C2×D12—22no—
143C2×D12—22no—
144C2×D12—22no—
145C2×D12—22no—
146C2×D12—22no—
147C2×D12—22no—
148C2×D12—22no—
149C2×D12—22no—
150C2×D12—22no—
151C2×D12—22no—
152C2×D12—22no—
153C2×D12—22no—
154C2×D12—22no—
155C2×D12—22no—
156C2×D12—22no—
157C2×D12—41no—
158C2×D12—41no—
159C2×D12—41no—
160C2×D12—41no—
161C2×D12—41no—
162C2×D12—41no—
163C2×D12—41no—
164C2×D12—41no—
165C2×D12—41no—
166C2×D12—41no—
167C2×D12—41no—
168C2×D12—41no—
169C2×D12C22⋊C414yes—
170C3×M16—22no—
171C3×M16—22no—
172C3×M16—22no—
173C3×M16—22no—
174C3×M16—41no—
175C3×M16—41no—
176C3×M16—41no—
177C3×M16—41no—
178C3×M16—41no—
179C3×M16—41no—
180C3×M16—41no—
181C3×M16—41no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—41no—
188C2×C3×C8—41no—
189C2×C3×C8—41no—
190C2×C3×C8—41no—
191#217—22no—
192#217—22no—
193#217—41no—
194#217—41no—
195C4⋊Dic3—22no—
196C4⋊Dic3—22no—
197C4⋊Dic3—41no—
198C4⋊Dic3—41no—
199C4×Dic3—22no—
200C4×Dic3—22no—
201C4×Dic3—41no—
202C4×Dic3—41no—
203#208—22no—
204#208—22no—
205#208—41no—
206#208—41no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—41no—
209C2×C3⋊C8—41no—
210C25—61no—
211C22×D4—32no—
212C22×D4—32no—
213C22×D4—32no—
214C22×D4—61no—
215C22×D4—61no—
216C22×D4—61no—
217C22×D4—61no—
218C22×D4—61no—
219C22×D4—61no—
220C22×D4—61no—
221C22×D4—61no—
222C22×D4—61no—
223C22×D4—61no—
224C22×D4—61no—
225C23×C4C22×D311yescharacteristic
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16—21no—
239C2×M16—21no—
240C2×M16—21no—
241C2×M16—21no—
242C2×M16—32no—
243C2×M16—61no—
244C2×M16—61no—
245C2×M16—61no—
246C2×M16—61no—
247C2×M16—61no—
248C2×M16C4×D312yes—
249C2×M16D1212yes—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—61no—
253C22×C8#6712yes—
254#117—32no—
255#117—32no—
256#117—61no—
257#117—61no—
258#117—61no—
259#117—61no—
260#117—61no—
261#117—61no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—61no—
264C2×C22⋊C4—61no—
265C2×C22⋊C4—61no—
266C2×C42—61no—
267C2×C42—61no—
268#102—68no—
269#102—68no—
270#100—68no—
271C23×C3—42no—
272C23×C3C4×D411yescharacteristic
273C23×C3#12111yescharacteristic
274C23×C3#12311yescharacteristic
275C23×C3#12311yescharacteristic
276C23×C3#12411yescharacteristic
277C23×C3#12611yescharacteristic
278C23×C3#12611yescharacteristic
279C22×D3—24no—
280C22×D3—42no—
281C22×D3—42no—
282C22×D3—42no—
283C22×D3—42no—
284C22×D3—42no—
285C22×D3—42no—
286C22×D3—84no—
287C22×D3—84no—
288C22×D3—84no—
289C22×D3—84no—
290C22×D3—84no—
291C22×D3—84no—
292C22×D3—84no—
293C2×C3×C4C2×C22⋊C411yescharacteristic
294C2×C3×C4#11711yescharacteristic
295C2×C3×C4#11711yescharacteristic
296C2×C3×C4C4×D411yescharacteristic
297C2×C3×C4C4×D411yescharacteristic
298C2×C3×C4C4×D411yescharacteristic
299C2×C3×C4#12011yescharacteristic
300C2×C3×C4#12011yescharacteristic
301C2×C3×C4#12111yescharacteristic
302C2×C3×C4#12111yescharacteristic
303C2×C3×C4#12111yescharacteristic
304C2×C3×C4#12111yescharacteristic
305C2×C3×C4#12111yescharacteristic
306C2×C3×C4#12111yescharacteristic
307C2×C3×C4#12111yescharacteristic
308C2×C3×C4#12311yescharacteristic
309C2×C3×C4#12311yescharacteristic
310C2×C3×C4#12311yescharacteristic
311C2×C3×C4#12311yescharacteristic
312C2×C3×C4#12311yescharacteristic
313C2×C3×C4#12311yescharacteristic
314C2×C3×C4#12311yescharacteristic
315C2×C3×C4#12311yescharacteristic
316C2×C3×C4#12411yescharacteristic
317C2×C3×C4#12411yescharacteristic
318C2×C3×C4#12411yescharacteristic
319C2×C3×C4#12611yescharacteristic
320C2×C3×C4#12611yescharacteristic
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—82no—
324C2×Dic3—82no—
325D12—24no—
326D12—24no—
327D12—42no—
328D12—42no—
329D12—42no—
330D12—42no—
331D12—42no—
332D12—42no—
333C3×C8—42no—
334C3×C8—42no—
335C3×C8—42no—
336C3⋊C8—42no—
337C24—62no—
338C24—62no—
339C24—62no—
340C24—121no—
341C24—121no—
342C24—121no—
343C24—121no—
344C24—244no—
345C24#23911yescharacteristic
346C2×D4—34no—
347C2×D4—62no—
348C2×D4—62no—
349C2×D4—62no—
350C2×D4—62no—
351C2×D4—62no—
352C2×D4—62no—
353C2×D4—62no—
354C2×D4—62no—
355C2×D4—62no—
356C2×D4—62no—
357C2×D4—62no—
358C2×D4—62no—
359C2×D4—62no—
360C2×D4—62no—
361C2×D4—62no—
362C2×D4—121no—
363C2×D4—121no—
364C2×D4—121no—
365C2×D4—121no—
366C2×D4—121no—
367C2×D4—121no—
368C2×D4—121no—
369C2×D4—121no—
370C2×D4—121no—
371C2×D4—121no—
372C2×D4—121no—
373C2×D4—121no—
374C22×C4—121no—
375C22×C4—121no—
376C22×C4C2×C4×D311yescharacteristic
377C22×C4C2×D1211yescharacteristic
378C22×C4#23511yescharacteristic
379C22×C4#23511yescharacteristic
380C22×C4#23511yescharacteristic
381C22×C4#23511yescharacteristic
382C22×C4D3×D411yescharacteristic
383C22×C4D3×D411yescharacteristic
384C22×C4D3×D411yescharacteristic
385C22×C4D3×D411yescharacteristic
386C22×C4#23711yescharacteristic
387C22×C4#23711yescharacteristic
388C22×C4#23911yescharacteristic
389C22×C4C2×#6711yescharacteristic
390M16—22no—
391M16—22no—
392M16—22no—
393M16—22no—
394M16—41no—
395M16—41no—
396M16—41no—
397M16—41no—
398M16—41no—
399M16—41no—
400M16—41no—
401M16—41no—
402M16—62no—
403M16—62no—
404M16—121no—
405M16—121no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—62no—
416C2×C8—121no—
417C2×C8—121no—
418C4⋊C4—62no—
419C4⋊C4—62no—
420C4⋊C4—121no—
421C4⋊C4—121no—
422C22⋊C4—62no—
423C22⋊C4—62no—
424C22⋊C4—121no—
425C22⋊C4—121no—
426C42—62no—
427C42—62no—
428C42—121no—
429C42—121no—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3#38611yescharacteristic
438C22×C3#38811yescharacteristic
439C22×C3#38811yescharacteristic
440C22×C3#39411yescharacteristic
441C22×C3#39711yescharacteristic
442C22×C3#39711yescharacteristic
443C22×C3#39711yescharacteristic
444C2×D3—44no—
445C2×D3—84no—
446C2×D3—84no—
447C2×D3—84no—
448C2×D3—84no—
449C2×D3—84no—
450C2×D3—84no—
451C2×D3—84no—
452C3×C4#38111yescharacteristic
453C3×C4#38611yescharacteristic
454C3×C4#38811yescharacteristic
455C3×C4#38811yescharacteristic
456C3×C4#39411yescharacteristic
457C3×C4#39411yescharacteristic
458C3×C4#39411yescharacteristic
459C3×C4#39711yescharacteristic
460Dic3—82no—
461Dic3—82no—
462C23—42no—
463C23—64no—
464C23—122no—
465C23—122no—
466C23—122no—
467C23—122no—
468C23—122no—
469C23—122no—
470C23—244no—
471C23—244no—
472C23—244no—
473C23—244no—
474C23—244no—
475C23—244no—
476C23—244no—
477C23#89311yescharacteristic
478C23#89411yescharacteristic
479C23#89411yescharacteristic
480C23#89511yescharacteristic
481C23#89811yescharacteristic
482C23#89811yescharacteristic
483C23#94711yescharacteristic
484D4—64no—
485D4—64no—
486D4—122no—
487D4—122no—
488D4—122no—
489D4—122no—
490D4—122no—
491D4—122no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—242no—
495C2×C4—242no—
496C2×C4#87211yescharacteristic
497C2×C4C4×D1211yescharacteristic
498C2×C4#87511yescharacteristic
499C2×C4#87611yescharacteristic
500C2×C4D3×C22⋊C411yescharacteristic
501C2×C4#67⋊C411yescharacteristic
502C2×C4#88211yescharacteristic
503C2×C4#88311yescharacteristic
504C2×C4#88311yescharacteristic
505C2×C4#88311yescharacteristic
506C2×C4#88411yescharacteristic
507C2×C4#88411yescharacteristic
508C2×C4#88411yescharacteristic
509C2×C4#88511yescharacteristic
510C2×C4#88511yescharacteristic
511C2×C4#88611yescharacteristic
512C2×C4#89211yescharacteristic
513C2×C4#89411yescharacteristic
514C2×C4#89411yescharacteristic
515C2×C4#89411yescharacteristic
516C2×C4#89511yescharacteristic
517C2×C4#89811yescharacteristic
518C2×C4C4×#6711yescharacteristic
519C2×C4#92911yescharacteristic
520C2×C4#93011yescharacteristic
521C2×C4#93711yescharacteristic
522C2×C4#93811yescharacteristic
523C2×C4#93911yescharacteristic
524C8—42no—
525C8—42no—
526C8—42no—
527C8—122no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3#144011yescharacteristic
535C2×C3#208112yes—
536D3—84no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—124no—
545C22—244no—
546C22—244no—
547C22—244no—
548C22—244no—
549C22—244no—
550C22—244no—
551C22—244no—
552C22#466511yescharacteristic
553C22#466911yescharacteristic
554C22#467411yescharacteristic
555C22#467511yescharacteristic
556C22#499011yescharacteristic
557C22#499111yescharacteristic
558C22#499211yescharacteristic
559C4—242no—
560C4—242no—
561C4#466411yescharacteristic
562C4#467011yescharacteristic
563C4#467211yescharacteristic
564C4#467511yescharacteristic
565C4#494011yescharacteristic
566C4#495611yescharacteristic
567C4#495911yescharacteristic
568C4#496111yescharacteristic
569C3#1118111yescharacteristic, complemented, Sylow
570C2—21no—
571C2—21no—
572C2—21no—
573C2—21no—
574C2—21no—
575C2—21no—
576C2—244no—
577C2#6942911yescharacteristic
578C2#7073112yes—
subgroupG/H
centerC22#4992
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1738C2
socleC22×C3#397
prime radicalC23×D12C22
cosocleC23×C3×C4C23
hypercenter#1738D3
hypocenterC3#11181
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1738

Complements

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

NH ≅ G/Ntype
4. C3×#1738C2semidirect
5. C2×#4559C2semidirect
6. C2×#4556C2semidirect
7. C2×#4557C2semidirect
19. #5131C4semidirect
34. #1738D3semidirect
569. C3#11181semidirect

Maximal subgroups

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

subgroupindex
C2×#51312
C22×#8232
C3×#17382
C2×#45592
C2×#45562
C2×#45572
#111813