Finite Group Databasebeta

← #10849356

Subgroups — #10849356

Order 768.

1028 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic
2C2×#5055C211yesmaximal, characteristic
3#82478C211yesmaximal, characteristic, complemented
4C2×#5020C211yesmaximal, characteristic, complemented
5#71781C212yesmaximal, complemented
6#71722C212yesmaximal
7#71721C211yesmaximal, characteristic
8#71635C211yesmaximal, characteristic, complemented
9#71631C211yesmaximal, characteristic, complemented
10#71629C212yesmaximal, complemented
11#71607C212yesmaximal
12#25662—31nomaximal, Sylow
13C3×#556—24no—
14C3×#556—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—21no—
17C2×C3×#126—21no—
18C2×C3×#124C2211yescharacteristic
19C2×C3×#124C2211yescharacteristic
20C2×C3×C22⋊Q8—21no—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×#117—21no—
26C2×#869C2211yescharacteristic
27#5229—21no—
28C2×#835C2212yes—
29C2×#835C2212yes—
30C2×#835C2212yes—
31C2×#835C2212yes—
32C2×#832C2212yes—
33C2×#832C2212yes—
34#5055—24no—
35#5020—24no—
36#4979—21no—
37#4979—21no—
38#4975—21no—
39C2×#808C2212yes—
40C4×C4⋊Dic3C2211yescharacteristic
41#4928C2212yes—
42#4928C2212yes—
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4529C2212yes—
46#4524C2212yes—
47#4524C2212yes—
48C2×#556D311yescharacteristic
49C2×#489—31no—
50C2×#475—31no—
51#2567—31no—
52#2043—32no—
53#2042—32no—
54#1985—31no—
55#1983—31no—
56#1865—32no—
57#1864—32no—
58#1763—31no—
59C22×C3×Q8D411yescharacteristic
60C22×C3×Q8C2311yescharacteristic
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C22×Dic6—21no—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8—22no—
67C3×#126—28no—
68C3×#126—28no—
69C3×#124—22no—
70C3×#124—22no—
71C3×#124—22no—
72C3×#124D414yes—
73C3×C22⋊Q8—28no—
74C3×C4×Q8—22no—
75C3×C4×Q8—22no—
76C3×C4×Q8—22no—
77C3×C4×Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×Q8D414yes—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×#117—28no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42—21no—
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2312yes—
103C2×C4⋊Dic3C2312yes—
104C2×Dic3⋊C4—22no—
105C2×Dic3⋊C4—22no—
106C2×C4×Dic3—21no—
107C22×C3⋊C8C2312yes—
108C22×C3⋊C8C2312yes—
109#869—22no—
110#869—22no—
111#835—24no—
112#835—24no—
113#835—24no—
114#835—24no—
115#832—24no—
116#832—24no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119C3⋊#95—21no—
120#808—24no—
121C4⋊(C3⋊C8)—22no—
122#803—22no—
123#556—24no—
124#556—24no—
125C2×C4⋊Q8—31no—
126C2×C4⋊Q8C2×D311yescharacteristic
127C2×#126—21no—
128C2×#126—21no—
129C2×#124C2×D311yescharacteristic
130C2×#124C2×D311yescharacteristic
131C2×C22⋊Q8—21no—
132C2×C4×Q8C2×D311yescharacteristic
133C2×C4×Q8C2×D311yescharacteristic
134C2×C4×Q8C2×D311yescharacteristic
135C2×C4×D4C2×D311yescharacteristic
136C2×#117—21no—
137#489—64no—
138#475—64no—
139C2×#106—32no—
140C2×C4⋊C8—31no—
141C2×#103—32no—
142C2×#103—32no—
143C2×#103—32no—
144C2×#103—32no—
145C2×#102—32no—
146C2×#102—32no—
147C2×#97—31no—
148#395—61no—
149#395—61no—
150#391—61no—
151#389—32no—
152#389—32no—
153#384—61no—
154C4×C4⋊C4—31no—
155#340—32no—
156#336—32no—
157#336—32no—
158C24×C3—21no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—41no—
168C2×C3×Q8C2×D412yes—
169C2×C3×Q8C2×D412yes—
170C2×C3×D4—21no—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4C2×D412yes—
174C22×C3×C4—24no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—42no—
179C22×C3×C4—42no—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C22×Dic3—22no—
188C22×Dic3—22no—
189C2×Dic6—42no—
190C2×Dic6—82no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4C2×D412yes—
200C3×C22⋊C4—24no—
201C3×C22⋊C4—24no—
202C3×C22⋊C4—24no—
203C3×C22⋊C4—24no—
204C3×C22⋊C4—24no—
205C3×C42—24no—
206C3×C42—24no—
207C3×C42—21no—
208C3×C42—21no—
209C3×C42—21no—
210C3×C42—21no—
211C3×C42C2×D412yes—
212C3×C42C2×D412yes—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3—42no—
217C4⋊Dic3#4114yes—
218Dic3⋊C4—82no—
219Dic3⋊C4—82no—
220C4×Dic3—42no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C22×Q8—61no—
228C22×Q8#6711yescharacteristic
229C22×Q8C22×D311yescharacteristic
230C22×Q8C22×D311yescharacteristic
231C22×D4C22×D311yescharacteristic
232C23×C4—21no—
233C22×C8—32no—
234C22×C8—32no—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237C4⋊Q8—62no—
238C4⋊Q8—62no—
239#126—28no—
240#126—28no—
241#124—22no—
242#124—22no—
243#124—22no—
244#124#6714yes—
245C22⋊Q8—28no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8#6714yes—
252C4×D4—22no—
253C4×D4—22no—
254#117—28no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—32no—
260C2×C4⋊C4—32no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4#6711yescharacteristic
265C2×C4⋊C4#6711yescharacteristic
266C2×C4⋊C4#6711yescharacteristic
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—61no—
276C2×C42C22×D311yescharacteristic
277C2×C42C22×D311yescharacteristic
278C2×C42C22×D311yescharacteristic
279#106—64no—
280C4⋊C8—62no—
281#103—64no—
282#103—64no—
283#103—64no—
284#103—64no—
285#102—64no—
286#102—64no—
287#97—62no—
288#95—61no—
289#95—62no—
290#95—62no—
291C23×C3—22no—
292C23×C3—42no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—42no—
296C23×C3C22×D411yescharacteristic
297C3×Q8—22no—
298C3×Q8—22no—
299C3×Q8—24no—
300C3×Q8—44no—
301C3×Q8—41no—
302C3×Q8—41no—
303C3×Q8—42no—
304C3×D4—24no—
305C3×D4—41no—
306C3×D4—41no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—21no—
313C2×C3×C4—22no—
314C2×C3×C4—21no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—48no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4#12012yes—
328C2×C3×C4#12012yes—
329C2×C3×C4#13712yes—
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C2×#4112yes—
333C2×C3×C42−1+412yes—
334C2×Dic3—24no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340Dic6—84no—
341C3⋊C8—42no—
342C3⋊C8—42no—
343C24—21no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—41no—
353C2×Q8—122no—
354C2×Q8—242no—
355C2×Q8C2×#6712yes—
356C2×Q8C2×#6712yes—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—21no—
360C2×D4C2×#6712yes—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4C2×#6711yescharacteristic
370C22×C4C2×#6711yescharacteristic
371C22×C4C2×#6711yescharacteristic
372C22×C4C2×#6711yescharacteristic
373C22×C4C2×#6711yescharacteristic
374C22×C4C2×#6711yescharacteristic
375C22×C4C23×D311yescharacteristic
376C2×C8—62no—
377C2×C8—62no—
378C2×C8—62no—
379C2×C8—62no—
380C2×C8—62no—
381C2×C8—62no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—34no—
391C4⋊C4—62no—
392C4⋊C4—62no—
393C4⋊C4—62no—
394C4⋊C4—122no—
395C4⋊C4—242no—
396C4⋊C4—242no—
397C4⋊C4C2×#6712yes—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C42—24no—
404C42—24no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—122no—
410C42C2×#6712yes—
411C42C2×#6712yes—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—82no—
416C22×C3—82no—
417C22×C3C2×#12011yescharacteristic
418C22×C3#54812yes—
419C22×C3C2×#13711yescharacteristic
420C22×C3#57511yescharacteristic
421C22×C3#57712yes—
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—22no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4—41no—
428C3×C4—42no—
429C3×C4—41no—
430C3×C4—41no—
431C3×C4#53211yescharacteristic
432C3×C4#53611yescharacteristic
433C3×C4#54712yes—
434Dic3—44no—
435Dic3—82no—
436C23—22no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23—42no—
441C23C22×#6711yescharacteristic
442Q8—24no—
443Q8—22no—
444Q8—22no—
445Q8—44no—
446Q8—42no—
447Q8—41no—
448Q8—41no—
449Q8—244no—
450D4—24no—
451D4—41no—
452D4—41no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—48no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—64no—
474C2×C4—122no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4#95112yes—
480C2×C4#95312yes—
481C2×C4#95312yes—
482C2×C4D3×#4112yes—
483C2×C4#101012yes—
484C2×C4C22×#6711yescharacteristic
485C2×C4C22×#6711yescharacteristic
486C8—122no—
487C8—122no—
488C2×C3—82no—
489C2×C3#243211yescharacteristic
490C2×C3#300711yescharacteristic
491C2×C3#301011yescharacteristic
492C2×C3#336912yes—
493C2×C3#337612yes—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—82no—
498C22—82no—
499C22#582211yescharacteristic
500C22C2×#95111yescharacteristic
501C22#583112yes—
502C22#583712yes—
503C22C2×#95311yescharacteristic
504C4—22no—
505C4—22no—
506C4—22no—
507C4—41no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—41no—
512C4—41no—
513C4—124no—
514C4—242no—
515C4#582412yes—
516C4#582511yescharacteristic
517C4#583211yescharacteristic
518C3#2566211yescharacteristic, complemented, Sylow
519C2—82no—
520C2#8366912yes—
521C2#8367712yes—
522C2#8408411yescharacteristic
523C2#8409311yescharacteristic
524C2#8411311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#556C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#556D3
hypocenterC3#25662
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
3. #82478C2semidirect
4. C2×#5020C2semidirect
5. #71781C2semidirect
8. #71635C2semidirect
9. #71631C2semidirect
10. #71629C2semidirect
518. C3#25662semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×#50552
#824782
C2×#50202
#717812
#717222
#717212
#716352
#716312
#716292
#716072
#256623