Finite Group Databasebeta

← #10823380

Subgroups — #10823380

Order 768.

1156 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×C4×#216C212yesmaximal, complemented
3C4×#924C211yesmaximal, characteristic
4#71794C211yesmaximal, characteristic, complemented
5#71793C211yesmaximal, characteristic, complemented
6#71782C211yesmaximal, characteristic, complemented
7#71781C211yesmaximal, characteristic, complemented
8#71700C212yesmaximal
9#71695C212yesmaximal
10C4×#835C212yesmaximal, complemented
11C4×#830C211yesmaximal, characteristic
12#23571—31nomaximal, Sylow
13C3×#533—28no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125—21no—
17C2×C3×C22⋊Q8—21no—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×#117—21no—
22C22×C3×C42C2211yescharacteristic
23C22×#216C2212yes—
24C2×#924C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×#835C2212yes—
27C2×#835C2212yes—
28C2×#835C2212yes—
29#5212C2211yescharacteristic
30#5210—21no—
31C4×#217—21no—
32C2×#830C2212yes—
33C4×#216—28no—
34C2×#810C2212yes—
35C2×#807C2212yes—
36C4×C4⋊Dic3C2211yescharacteristic
37#4933—21no—
38C4×Dic3⋊C4—21no—
39#4929C2211yescharacteristic
40#4928C2211yescharacteristic
41C2×C4×C3⋊C8C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2211yescharacteristic
45#4524C2211yescharacteristic
46C2×#533D311yescharacteristic
47C2×C4×Q16—32no—
48C4×C22⋊Q8—31no—
49#2057—31no—
50#2043—31no—
51#2036—32no—
52#2024—31no—
53#2016—31no—
54#2001—32no—
55C4×#103—32no—
56C4×C22⋊C8—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8C2312yes—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×Dic6C2311yescharacteristic
63C3×C4⋊Q8—24no—
64C3×C4⋊Q8—24no—
65C3×#125—28no—
66C3×C22⋊Q8—28no—
67C3×C22⋊Q8—24no—
68C3×C22⋊Q8—24no—
69C3×C4×Q8—28no—
70C3×C4×Q8—24no—
71C3×C4×Q8D418yes—
72C3×#117—28no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—22no—
80C2×C3×C42—22no—
81C2×C3×C42—24no—
82C2×C3×C42D411yescharacteristic
83C2×C3×C42D411yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×#217—21no—
87C2×#217—21no—
88C2×#216—24no—
89C2×#216—24no—
90C2×#216—24no—
91C2×C4⋊Dic3C2311yescharacteristic
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2311yescharacteristic
94#924—22no—
95#924—22no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×C4×Dic3—21no—
100C2×C4×Dic3—21no—
101C22×C3⋊C8C2312yes—
102C22×C3⋊C8C2312yes—
103C4×Dic6—24no—
104#835—24no—
105#835—24no—
106#835—24no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112#830—24no—
113#810—24no—
114#807—24no—
115C4×C3⋊C8—24no—
116C22×Q16—32no—
117#533—28no—
118C2×C4⋊Q8C2×D311yescharacteristic
119C2×C4⋊Q8C2×D311yescharacteristic
120C2×#125—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—31no—
123C2×C22⋊Q8C2×D312yes—
124C2×C4×Q8—21no—
125C2×C4×Q8—31no—
126C2×C4×Q8C2×D312yes—
127C2×#117—21no—
128C22×C42C2×D311yescharacteristic
129C4×Q16—68no—
130C2×#107—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103—32no—
134C2×#103—32no—
135C2×C22⋊C8—32no—
136C2×C4×C8—32no—
137#391—31no—
138#389—31no—
139#387—61no—
140#385—61no—
141#380—31no—
142C4×C4⋊C4—31no—
143C4×C4⋊C4—61no—
144C4×C22⋊C4—61no—
145#340—32no—
146#340—32no—
147#336—31no—
148#336—31no—
149C24×C3C2×D411yescharacteristic
150C2×C3×Q8—24no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—42no—
157C2×C3×Q8—42no—
158C2×C3×Q8C2×D414yes—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—41no—
170C22×C3×C4Q1614yescomplemented
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2411yescharacteristic
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—21no—
180C22×Dic3—21no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—21no—
184C2×Dic6#4112yes—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—28no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—28no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4C2×D414yes—
192C3×C22⋊C4—28no—
193C3×C42—22no—
194C3×C42—22no—
195C3×C42—28no—
196C3×C42—22no—
197C3×C42C2×D414yes—
198#217—24no—
199#217—42no—
200#216—216no—
201C4⋊Dic3—21no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204C4⋊Dic3#4112yes—
205Dic3⋊C4—24no—
206Dic3⋊C4—42no—
207Dic3⋊C4—42no—
208C4×Dic3—42no—
209C4×Dic3—42no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8#4114yes—
216C22×Q8—21no—
217C22×Q8—31no—
218C22×Q8C22×D312yes—
219C23×C4#6711yescharacteristic
220C23×C4#6711yescharacteristic
221C23×C4C22×D311yescharacteristic
222C2×Q16—64no—
223C2×Q16—64no—
224C2×Q16—64no—
225C22×C8—32no—
226C22×C8—32no—
227C4⋊Q8—24no—
228C4⋊Q8—24no—
229#125—28no—
230C22⋊Q8—28no—
231C22⋊Q8—24no—
232C22⋊Q8—24no—
233C22⋊Q8—62no—
234C22⋊Q8—62no—
235C4×Q8—28no—
236C4×Q8—24no—
237C4×Q8—64no—
238C4×Q8#6718yes—
239#117—28no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—31no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4C22×D312yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—61no—
254C2×C22⋊C4—61no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—61no—
258C2×C42—61no—
259C2×C42#6711yescharacteristic
260C2×C42#6711yescharacteristic
261C2×C42C22×D311yescharacteristic
262C2×C42C22×D311yescharacteristic
263#107—64no—
264#103—64no—
265#103—64no—
266#103—64no—
267#103—64no—
268C22⋊C8—64no—
269C4×C8—64no—
270#95—61no—
271#95—61no—
272#95—61no—
273#95—61no—
274#95—61no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3C2×Q1612yes—
282C23×C3C22×D411yescharacteristic
283C3×Q8—24no—
284C3×Q8—24no—
285C3×Q8—48no—
286C3×Q8—42no—
287C3×Q8—42no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—28no—
295C2×C3×C4—21no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—44no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4#12012yes—
308C2×C3×C4#12012yes—
309C2×C3×C4C2×Q1612yes—
310C2×C3×C4C2×Q1612yes—
311C2×C3×C4#13512yes—
312C2×C3×C4#13712yes—
313C2×C3×C4C22×D412yes—
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4C2×#4112yes—
316C2×C3×C4C2×#4111yescharacteristic
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329Dic6—24no—
330Dic6—42no—
331C3⋊C8—24no—
332C3⋊C8—42no—
333C24C2×#6711yescharacteristic
334C2×Q8—24no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—32no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—61no—
344C2×Q8—62no—
345C2×Q8—62no—
346C2×Q8C2×#6714yes—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—41no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—61no—
361C22×C4—61no—
362C22×C4#21614yescomplemented
363C22×C4C2×#6711yescharacteristic
364C22×C4C2×#6711yescharacteristic
365C22×C4C2×#6711yescharacteristic
366C22×C4C2×#6711yescharacteristic
367C22×C4C2×#6711yescharacteristic
368C22×C4C23×D311yescharacteristic
369Q16—616no—
370C2×C8—34no—
371C2×C8—62no—
372C2×C8—62no—
373C2×C8—62no—
374C2×C8—62no—
375C2×C8—62no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—28no—
379C4⋊C4—24no—
380C4⋊C4—28no—
381C4⋊C4—22no—
382C4⋊C4—32no—
383C4⋊C4—64no—
384C4⋊C4—61no—
385C4⋊C4—62no—
386C4⋊C4—62no—
387C4⋊C4—122no—
388C4⋊C4—122no—
389C4⋊C4C2×#6714yes—
390C22⋊C4—28no—
391C22⋊C4—64no—
392C22⋊C4—122no—
393C42—22no—
394C42—22no—
395C42—28no—
396C42—22no—
397C42—122no—
398C42—122no—
399C42C2×#6714yes—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3C2×#12011yescharacteristic
408C22×C3#52511yescharacteristic
409C22×C3#53211yescharacteristic
410C22×C3C22×Q1611yescharacteristic
411C22×C3C2×#13511yescharacteristic
412C22×C3C2×#13711yescharacteristic
413C22×C3#57511yescharacteristic
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—41no—
418C3×C4—44no—
419C3×C4—44no—
420C3×C4#45114yes—
421C3×C4#52512yes—
422C3×C4#53212yes—
423Dic3—42no—
424Dic3—42no—
425Dic3—81no—
426Dic3—81no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23C2×#21612yes—
434C23C22×#6711yescharacteristic
435Q8—24no—
436Q8—24no—
437Q8—48no—
438Q8—42no—
439Q8—42no—
440Q8—64no—
441Q8—122no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—28no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—121no—
470C2×C4—121no—
471C2×C4—121no—
472C2×C4—121no—
473C2×C4#94212yes—
474C2×C4C2×#21612yes—
475C2×C4C2×#21612yes—
476C2×C4#95012yes—
477C2×C4#95312yes—
478C2×C4#95312yes—
479C2×C4D3×#4111yescharacteristic
480C2×C4D3×#4112yes—
481C2×C4D3×#4111yescharacteristic
482C2×C4C22×#6712yes—
483C8—64no—
484C8—122no—
485C2×C3—42no—
486C2×C3#238811yescharacteristic
487C2×C3C2×#45111yescharacteristic
488C2×C3#300211yescharacteristic
489C2×C3#310411yescharacteristic
490C2×C3#310711yescharacteristic
491C2×C3#317011yescharacteristic
492C2×C3#317211yescharacteristic
493C22—22no—
494C22—22no—
495C22—22no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22C2×#94211yescharacteristic
501C22C22×#21611yescharacteristic
502C22C2×#95011yescharacteristic
503C22#582211yescharacteristic
504C22#582811yescharacteristic
505C22#583211yescharacteristic
506C22C2×#95311yescharacteristic
507C4—22no—
508C4—22no—
509C4—22no—
510C4—44no—
511C4—41no—
512C4—44no—
513C4—122no—
514C4—122no—
515C4—241no—
516C4—241no—
517C4#522814yes—
518C4#582812yes—
519C4#583212yes—
520C3#2357111yescharacteristic, complemented, Sylow
521C2—42no—
522C2#8334411yescharacteristic
523C2#8334511yescharacteristic
524C2#8335211yescharacteristic
525C2#8336811yescharacteristic
526C2C2×#522811yescharacteristic
527C2#8408011yescharacteristic
528C2#8409511yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#533C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#533D3
hypocenterC3#23571
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
2. C2×C4×#216C2semidirect
4. #71794C2semidirect
5. #71793C2semidirect
6. #71782C2semidirect
7. #71781C2semidirect
10. C4×#835C2semidirect
170. C22×C3×C4Q16semidirect
362. C22×C4#216semidirect
520. C3#23571semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×C4×#2162
C4×#9242
#717942
#717932
#717822
#717812
#717002
#716952
C4×#8352
C4×#8302
#235713