Finite Group Databasebeta

← #10822713

Subgroups — #10822713

Order 768.

1418 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic
2#84087C211yesmaximal, characteristic
3C2×#5234C211yesmaximal, characteristic
4C2×#5225C211yesmaximal, characteristic, complemented
5C22×#835C211yesmaximal, characteristic, complemented
6C2×#4995C211yesmaximal, characteristic, complemented
7#71695C214yesmaximal, complemented
8#71694C214yesmaximal, complemented
9#71580C211yesmaximal, characteristic, complemented
10#23484—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×Q8C2211yescharacteristic
13C3×#536—24no—
14C3×#536—24no—
15C2×C3×C4⋊Q8—21no—
16C2×C3×#124—21no—
17C2×C3×#123—22no—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×#117C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C22×#208C2211yescharacteristic
24C23×C3⋊C8C2211yescharacteristic
25#5234—24no—
26#5229—21no—
27#5225—24no—
28C2×#835—24no—
29C2×#835C2214yes—
30C2×#835C2214yes—
31#5221—21no—
32#5220—22no—
33#5212C2214yes—
34C2×#830C2214yes—
35#4995—24no—
36C2×#808C2212yes—
37C2×#807C2212yes—
38#4529C2212yes—
39#4529C2212yes—
40C2×#536D311yescharacteristic
41C2×#428—31no—
42C2×#419—31no—
43C22×#103—31no—
44C2×#408—31no—
45#2311—31no—
46#2036—34no—
47#2035—34no—
48#1735—31no—
49C2×C3×#41—22no—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C22×C3×Q8—21no—
56C22×C3×Q8—22no—
57C22×C3×Q8—22no—
58C22×C3×Q8—22no—
59C22×C3×Q8—22no—
60C22×C3×Q8—22no—
61C22×C3×Q8D414yes—
62C22×C3×Q8D411yescharacteristic
63C22×C3×Q8C2312yes—
64C22×C3×D4D412yes—
65C22×C3×D4D411yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×Dic3—21no—
70C3×C4⋊Q8—28no—
71C3×#124—28no—
72C3×#123—216no—
73C3×C22⋊Q8—24no—
74C3×C22⋊Q8—24no—
75C3×C22⋊Q8—24no—
76C3×C22⋊Q8—24no—
77C3×#117—22no—
78C3×#117D414yes—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C42—21no—
87C2×#217—21no—
88C2×#217—24no—
89#926—22no—
90#926—22no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3C2312yes—
95C2×C4⋊Dic3C2312yes—
96C2×C4×Dic3—21no—
97C2×#208—22no—
98C2×#208—22no—
99C2×#208D414yes—
100C22×C3⋊C8—22no—
101C22×C3⋊C8—22no—
102C22×C3⋊C8—22no—
103C22×C3⋊C8C2312yes—
104C22×C3⋊C8C2312yes—
105#835—28no—
106#835—28no—
107C3⋊#95—24no—
108#830—28no—
109#808—24no—
110#807—24no—
111C22×#41C2×D311yescharacteristic
112C23×Q8C2×D311yescharacteristic
113C22×M16—31no—
114C23×C8—31no—
115#536—24no—
116#536—24no—
117C2×C4⋊Q8—21no—
118C2×#124—21no—
119C2×#123—22no—
120C2×C22⋊Q8C2×D312yes—
121C2×C22⋊Q8C2×D312yes—
122C2×#117—31no—
123C2×#117C2×D311yescharacteristic
124C22×C4⋊C4—31no—
125#428—64no—
126C2×#107—32no—
127C2×#106—32no—
128#419—64no—
129C2×#103—34no—
130C2×#103—34no—
131C2×#103—64no—
132#408—64no—
133C2×C22⋊C8—34no—
134#391—61no—
135#390—61no—
136#386—62no—
137#380—34no—
138#340—32no—
139#340—32no—
140C24×C3C2×D412yes—
141C24×C3C2×D411yescharacteristic
142C3×#41—44no—
143C3×#41—44no—
144C3×#41—44no—
145C3×#41—44no—
146C2×C3×Q8—24no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—28no—
152C2×C3×Q8—28no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—44no—
158C2×C3×Q8—42no—
159C2×C3×Q8—42no—
160C2×C3×Q8—44no—
161C2×C3×Q8C2×D414yes—
162C2×C3×D4—24no—
163C2×C3×D4—24no—
164C2×C3×D4—24no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C22×C3×C4—24no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—42no—
179C22×C3×C4—42no—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4#4112yes—
186C22×C3×C4#4112yes—
187C22×C3×C4#4112yes—
188C22×C3×C4C2411yescharacteristic
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—42no—
192C22×Dic3—82no—
193C3×C4⋊C4—28no—
194C3×C4⋊C4—28no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4C2×D412yes—
198C3×C4⋊C4C2×D412yes—
199C3×C22⋊C4—28no—
200C3×C22⋊C4—28no—
201C3×C22⋊C4—24no—
202C3×C42—24no—
203#217—42no—
204#217—84no—
205C4⋊Dic3—22no—
206C4⋊Dic3—22no—
207C4⋊Dic3—22no—
208C4⋊Dic3—22no—
209C4⋊Dic3—42no—
210C4×Dic3—42no—
211#208—28no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—44no—
218C2×C3⋊C8C2×D414yes—
219C2×#41—22no—
220C2×#41—22no—
221C2×#41—22no—
222C2×#41—22no—
223C2×#41—22no—
224C2×#41—22no—
225C22×Q8—21no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8#6714yes—
232C22×Q8#6711yescharacteristic
233C22×Q8C22×D312yes—
234C22×D4#6712yes—
235C22×D4#6711yescharacteristic
236C23×C4—21no—
237C23×C4—61no—
238C23×C4C22×D312yes—
239C23×C4C22×D311yescharacteristic
240C2×M16—34no—
241C2×M16—62no—
242C2×M16—62no—
243C22×C8—32no—
244C22×C8—32no—
245C22×C8—62no—
246C22×C8—62no—
247C22×C8—62no—
248C4⋊Q8—28no—
249#124—28no—
250#123—216no—
251C22⋊Q8—24no—
252C22⋊Q8—24no—
253C22⋊Q8—24no—
254C22⋊Q8—24no—
255#117—22no—
256#117—62no—
257#117—62no—
258#117#6714yes—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—32no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—64no—
273C2×C42—21no—
274C2×C42—61no—
275#107—64no—
276#106—64no—
277#103—68no—
278#103—68no—
279C22⋊C8—68no—
280#95—64no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—24no—
287C23×C3—24no—
288C23×C3#12012yes—
289C23×C3#12112yes—
290C23×C3#12312yes—
291C23×C3C22×D411yescharacteristic
292C3×Q8—24no—
293C3×Q8—24no—
294C3×Q8—42no—
295C3×Q8—42no—
296C3×Q8—48no—
297C3×Q8—44no—
298C3×D4—48no—
299C3×D4—44no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—48no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4#12012yes—
316C2×C3×C4#12114yes—
317C2×C3×C4#12114yes—
318C2×C3×C4#12112yes—
319C2×C3×C4#12314yes—
320C2×C3×C4#12312yes—
321C2×C3×C4#12414yes—
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327C2×C3×C4C2×#4111yescharacteristic
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—44no—
332C2×Dic3—42no—
333C2×Dic3—84no—
334C3⋊C8—24no—
335C3⋊C8—42no—
336C24C2×#6712yes—
337C24C2×#6711yescharacteristic
338#41—44no—
339#41—44no—
340#41—44no—
341#41—44no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—24no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—28no—
348C2×Q8—28no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—42no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—42no—
357C2×Q8C2×#6714yes—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C22×C4—24no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—122no—
379C22×C4—242no—
380C22×C4D3×D412yes—
381C22×C4#23712yes—
382C22×C4#23712yes—
383C22×C4#23712yes—
384C22×C4C2×#6712yes—
385C22×C4C2×#6711yescharacteristic
386C22×C4C2×#6712yes—
387C22×C4C2×#6712yes—
388C22×C4C23×D311yescharacteristic
389M16—68no—
390C2×C8—34no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—62no—
396C2×C8—124no—
397C4⋊C4—28no—
398C4⋊C4—28no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—62no—
402C4⋊C4—62no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—122no—
406C4⋊C4C2×#6712yes—
407C4⋊C4C2×#6712yes—
408C22⋊C4—28no—
409C22⋊C4—28no—
410C22⋊C4—24no—
411C22⋊C4—122no—
412C22⋊C4—244no—
413C42—24no—
414C42—122no—
415C22×C3—24no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—24no—
419C22×C3—44no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—44no—
424C22×C3#39414yes—
425C22×C3C2×#12011yescharacteristic
426C22×C3C2×#12111yescharacteristic
427C22×C3C2×#12311yescharacteristic
428C22×C3#57712yes—
429C22×C3#57812yes—
430C3×C4—42no—
431C3×C4—44no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4#39414yes—
435C3×C4C2×#12111yescharacteristic
436C3×C4C2×#12111yescharacteristic
437C3×C4C2×#12311yescharacteristic
438C3×C4C2×#12411yescharacteristic
439Dic3—82no—
440Dic3—82no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—24no—
446C23—24no—
447C23—22no—
448C23#93512yes—
449C23#93912yes—
450C23#95312yes—
451C23C22×#6711yescharacteristic
452Q8—24no—
453Q8—24no—
454Q8—48no—
455Q8—44no—
456Q8—42no—
457Q8—42no—
458D4—48no—
459D4—44no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—48no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—124no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—244no—
481C2×C4#93512yes—
482C2×C4#93514yes—
483C2×C4#93614yes—
484C2×C4#93814yes—
485C2×C4#93912yes—
486C2×C4#93914yes—
487C2×C4#95312yes—
488C2×C4C2×D3×D411yescharacteristic
489C2×C4C2×#23711yescharacteristic
490C2×C4C2×#23711yescharacteristic
491C2×C4C2×#23711yescharacteristic
492C2×C4C22×#6711yescharacteristic
493C2×C4C22×#6711yescharacteristic
494C8—64no—
495C8—122no—
496C2×C3—24no—
497C2×C3—82no—
498C2×C3C2×#39411yescharacteristic
499C2×C3#301011yescharacteristic
500C2×C3#301111yescharacteristic
501C2×C3#306312yes—
502C2×C3#309712yes—
503C22—22no—
504C22—22no—
505C22—24no—
506C22—24no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—44no—
511C22—44no—
512C22#522214yes—
513C22C2×#93511yescharacteristic
514C22C2×#93911yescharacteristic
515C22#583712yes—
516C22#583812yes—
517C22C2×#95311yescharacteristic
518C4—42no—
519C4—44no—
520C4—42no—
521C4—42no—
522C4—242no—
523C4—242no—
524C4#522214yes—
525C4C2×#93511yescharacteristic
526C4C2×#93611yescharacteristic
527C4C2×#93811yescharacteristic
528C4C2×#93911yescharacteristic
529C3#2348411yescharacteristic, complemented, Sylow
530C2—24no—
531C2—82no—
532C2#8339712yes—
533C2#8340012yes—
534C2C2×#522211yescharacteristic
535C2#8411311yescharacteristic
536C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#536C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenterC2×#536D3
hypocenterC3#23484
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
4. C2×#5225C2semidirect
5. C22×#835C2semidirect
6. C2×#4995C2semidirect
7. #71695C2semidirect
8. #71694C2semidirect
9. #71580C2semidirect
529. C3#23484semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5362
#840872
C2×#52342
C2×#52252
C22×#8352
C2×#49952
#716952
#716942
#715802
#234843