Finite Group Databasebeta

← #10822716

Subgroups — #10822716

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×#5816C211yesmaximal, characteristic
2C2×C3×#419C211yesmaximal, characteristic, complemented
3C22×#835C211yesmaximal, characteristic, complemented
4C2×#5129C211yesmaximal, characteristic
5C2×#5121C211yesmaximal, characteristic, complemented
6#82270C211yesmaximal, characteristic
7#71580C211yesmaximal, characteristic, complemented
8#70244C214yesmaximal, complemented
9#70243C214yesmaximal, complemented
10#23484—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15#5816—24no—
16#5816—24no—
17C2×#947—21no—
18C2×#946C2212yes—
19C2×#946C2212yes—
20C2×#944—21no—
21C2×#929—22no—
22C2×#926C2211yescharacteristic
23C22×C4⋊Dic3C2211yescharacteristic
24C23×C3⋊C8C2211yescharacteristic
25C3×#419—24no—
26C2×C3×#103C2214yes—
27C2×#835—24no—
28C2×#835C2214yes—
29#5129—24no—
30#5121—24no—
31C2×(C2×D3)⋊C8C2214yes—
32C2×#818C2212yes—
33C2×#817C2212yes—
34#4987C2214yes—
35#4958—22no—
36#4939—21no—
37#4929—21no—
38#4529C2212yes—
39#4529C2212yes—
40C2×#536—31no—
41C2×#428—31no—
42C2×#419D311yescharacteristic, complemented
43C22×#103—31no—
44C2×#408—31no—
45#2311—31no—
46#2036—34no—
47#2035—34no—
48#1735—31no—
49C22×C3×Q8—21no—
50C22×C3×Q8—22no—
51C22×C3×Q8—22no—
52C22×C3×Q8—22no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8D414yes—
56C22×C3×Q8C2312yes—
57C23×C3×C4—21no—
58C23×C3×C4C2311yescharacteristic
59C22×#67D412yes—
60C23×Dic3—21no—
61C2×#235—22no—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C22×D12D411yescharacteristic
68C22×C4×D3C2312yes—
69C22×Dic6D411yescharacteristic
70C2×C3×M16—22no—
71C2×C3×M16—22no—
72C2×C3×M16D414yes—
73C22×C3×C8C2312yes—
74C3×#117—22no—
75C3×#117—22no—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—21no—
78C2×C3×C42—21no—
79C2×#217—21no—
80#947—28no—
81#946—24no—
82#946—24no—
83#946—24no—
84#946—24no—
85#944—28no—
86#929—216no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—24no—
90#926—22no—
91#926D414yes—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2312yes—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4—22no—
100C2×C4×Dic3—21no—
101C22×C3⋊C8—22no—
102C22×C3⋊C8—22no—
103C22×C3⋊C8—22no—
104C22×C3⋊C8C2312yes—
105C3×#103—28no—
106#835—28no—
107C3⋊#95—24no—
108(C2×D3)⋊C8—28no—
109#818—24no—
110#817—24no—
111C22×#41—31no—
112C23×Q8C2×D311yescharacteristic
113C22×M16C2×D311yescharacteristic
114C23×C8—31no—
115#536—64no—
116#536—64no—
117C2×C4⋊Q8—61no—
118C2×#124—61no—
119C2×#123—62no—
120C2×C22⋊Q8—32no—
121C2×C22⋊Q8—32no—
122C2×#117—31no—
123C2×#117C2×D311yescharacteristic
124C22×C4⋊C4—31no—
125#428—64no—
126C2×#107—32no—
127C2×#106—32no—
128#419—24no—
129C2×#103—34no—
130C2×#103—64no—
131C2×#103C2×D314yes—
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×D411yescharacteristic
141C23×D3C2×D412yes—
142C2×C3×Q8—24no—
143C2×C3×Q8—22no—
144C2×C3×Q8—28no—
145C2×C3×Q8—28no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—42no—
151C2×C3×Q8—44no—
152C2×C3×Q8—42no—
153C2×C3×Q8—44no—
154C2×C3×Q8C2×D414yes—
155C22×C3×C4—24no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D411yescharacteristic
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—
171C22×Dic3—21no—
172C22×Dic3—21no—
173C22×Dic3—22no—
174C22×Dic3—42no—
175C22×Dic3—82no—
176C22×Dic3C2×D412yes—
177#235—44no—
178#235—44no—
179#235—44no—
180#235—44no—
181C2×D12—22no—
182C2×D12—22no—
183C2×D12—22no—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×Dic6—22no—
191C2×Dic6—22no—
192C2×Dic6—22no—
193C3×M16—28no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8C2×D414yes—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C22⋊C4—42no—
200C3×C42—42no—
201#217—24no—
202(C2×D3)⋊C4—28no—
203(C2×D3)⋊C4—28no—
204(C2×D3)⋊C4—84no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—22no—
208C4⋊Dic3—22no—
209C4⋊Dic3—42no—
210C4⋊Dic3C2×D412yes—
211C4⋊Dic3C2×D412yes—
212Dic3⋊C4—28no—
213Dic3⋊C4—28no—
214C4×Dic3—24no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—44no—
219C2×#41—62no—
220C2×#41—62no—
221C2×#41—62no—
222C2×#41—62no—
223C2×#41—62no—
224C2×#41—62no—
225C22×Q8—21no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—31no—
232C22×Q8D1214yes—
233C22×Q8C22×D312yes—
234C22×D4—32no—
235C22×D4—31no—
236C23×C4—21no—
237C23×C4—32no—
238C23×C4—61no—
239C23×C4C22×D311yescharacteristic
240C2×M16—22no—
241C2×M16—22no—
242C2×M16#6714yes—
243C22×C8—32no—
244C22×C8—62no—
245C22×C8—62no—
246C22×C8—62no—
247C22×C8C22×D312yes—
248C4⋊Q8—68no—
249#124—68no—
250#123—616no—
251C22⋊Q8—64no—
252C22⋊Q8—64no—
253C22⋊Q8—64no—
254C22⋊Q8—64no—
255#117—22no—
256#117—22no—
257#117—34no—
258#117—62no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—32no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—62no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4C22×D312yes—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—62no—
270C2×C22⋊C4—62no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—64no—
273C2×C42—21no—
274C2×C42—61no—
275#107—64no—
276#106—64no—
277#103—28no—
278#103—68no—
279C22⋊C8—68no—
280#95—64no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3#12012yes—
284C23×C3#12112yes—
285C23×C3#12312yes—
286C23×C3C22×D411yescharacteristic
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—22no—
290C22×D3—24no—
291C22×D3—24no—
292C3×Q8—24no—
293C3×Q8—24no—
294C3×Q8—42no—
295C3×Q8—42no—
296C3×Q8—48no—
297C2×C3×C4—24no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—48no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4#12012yes—
305C2×C3×C4#12114yes—
306C2×C3×C4#12112yes—
307C2×C3×C4#12114yes—
308C2×C3×C4#12312yes—
309C2×C3×C4#12314yes—
310C2×C3×C4#12414yes—
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317#67—48no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—44no—
327C2×Dic3—84no—
328D12—44no—
329C4×D3—42no—
330C4×D3—42no—
331C4×D3—42no—
332C4×D3—42no—
333Dic6—44no—
334C3×C8—24no—
335C3⋊C8—42no—
336C24—32no—
337C24D3×D411yescharacteristic
338#41—124no—
339#41—124no—
340#41—124no—
341#41—124no—
342C2×Q8—24no—
343C2×Q8—22no—
344C2×Q8—28no—
345C2×Q8—28no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—42no—
351C2×Q8—44no—
352C2×Q8—42no—
353C2×Q8—44no—
354C2×Q8—62no—
355C2×Q8—62no—
356C2×Q8—62no—
357C2×Q8C2×D1214yes—
358C2×D4—64no—
359C2×D4—64no—
360C2×D4—64no—
361C2×D4—62no—
362C2×D4—62no—
363C2×D4—62no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—32no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—61no—
371C22×C4—61no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—122no—
380C22×C4—242no—
381C22×C4C2×D1212yes—
382C22×C4#23512yes—
383C22×C4D3×D411yescharacteristic
384C22×C4D3×D412yes—
385C22×C4#23712yes—
386C22×C4#23712yes—
387C22×C4C2×#6712yes—
388C22×C4C23×D311yescharacteristic
389M16—28no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—124no—
396C2×C8C2×#6714yes—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—32no—
400C4⋊C4—32no—
401C4⋊C4—68no—
402C4⋊C4—68no—
403C4⋊C4—61no—
404C4⋊C4—61no—
405C4⋊C4—62no—
406C4⋊C4—62no—
407C4⋊C4—122no—
408C22⋊C4—42no—
409C22⋊C4—68no—
410C22⋊C4—68no—
411C22⋊C4—64no—
412C22⋊C4—244no—
413C42—42no—
414C42—64no—
415C22×C3—24no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—24no—
419C22×C3#39414yes—
420C22×C3C2×#12011yescharacteristic
421C22×C3C2×#12111yescharacteristic
422C22×C3C2×#12311yescharacteristic
423C22×C3#57712yes—
424C22×C3#57812yes—
425C2×D3—44no—
426C2×D3—42no—
427C2×D3—42no—
428C2×D3—42no—
429C2×D3—44no—
430C3×C4—44no—
431C3×C4—82no—
432C3×C4#39414yes—
433C3×C4C2×#12111yescharacteristic
434C3×C4C2×#12111yescharacteristic
435C3×C4C2×#12311yescharacteristic
436C3×C4C2×#12411yescharacteristic
437Dic3—42no—
438Dic3—42no—
439Dic3—42no—
440Dic3—82no—
441C23—22no—
442C23—22no—
443C23—62no—
444C23—62no—
445C23—64no—
446C23—64no—
447C23—62no—
448C23#88212yes—
449C23#88312yes—
450C23#93812yes—
451C23C2×D3×D411yescharacteristic
452Q8—24no—
453Q8—24no—
454Q8—48no—
455Q8—42no—
456Q8—42no—
457Q8—124no—
458D4—128no—
459D4—124no—
460C2×C4—24no—
461C2×C4—48no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4—122no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—124no—
480C2×C4—244no—
481C2×C4#88212yes—
482C2×C4#88312yes—
483C2×C4#88514yes—
484C2×C4#88614yes—
485C2×C4#93014yes—
486C2×C4#93812yes—
487C2×C4#93914yes—
488C2×C4C22×D1211yescharacteristic
489C2×C4C2×#23511yescharacteristic
490C2×C4C2×D3×D411yescharacteristic
491C2×C4C2×#23711yescharacteristic
492C2×C4C2×#23711yescharacteristic
493C2×C4C22×#6711yescharacteristic
494C8—24no—
495C8—122no—
496C2×C3—24no—
497C2×C3C2×#39411yescharacteristic
498C2×C3#301011yescharacteristic
499C2×C3#301111yescharacteristic
500C2×C3#306312yes—
501C2×C3#309712yes—
502D3—82no—
503C22—24no—
504C22—24no—
505C22—22no—
506C22—22no—
507C22—124no—
508C22—124no—
509C22—122no—
510C22—122no—
511C22—122no—
512C22#496114yes—
513C22C2×#88211yescharacteristic
514C22C2×#88311yescharacteristic
515C22#577812yes—
516C22#577912yes—
517C22C2×#93811yescharacteristic
518C4—44no—
519C4—82no—
520C4—122no—
521C4—122no—
522C4—122no—
523C4—242no—
524C4#496114yes—
525C4C2×#88511yescharacteristic
526C4C2×#88611yescharacteristic
527C4C2×#93011yescharacteristic
528C4C2×#93911yescharacteristic
529C3#2348411yescharacteristic, complemented, Sylow
530C2—24no—
531C2—242no—
532C2#7895112yes—
533C2#7896011yescharacteristic
534C2#7896111yescharacteristic
535C2C2×#496111yescharacteristic
536C2#8398612yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#419C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenterC2×#419D3
hypocenterC3#23484
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. C2×C3×#419C2semidirect
3. C22×#835C2semidirect
5. C2×#5121C2semidirect
7. #71580C2semidirect
8. #70244C2semidirect
9. #70243C2semidirect
42. C2×#419D3semidirect
529. C3#23484semidirect

Maximal subgroups

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

subgroupindex
C2×#58162
C2×C3×#4192
C22×#8352
C2×#51292
C2×#51212
#822702
#715802
#702442
#702432
#234843