Finite Group Databasebeta

← #10849345

Subgroups — #10849345

Order 768.

1048 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×#556C211yesmaximal, characteristic
2C2×#5055C211yesmaximal, characteristic
3#82475C211yesmaximal, characteristic, complemented
4C2×#5021C211yesmaximal, characteristic, complemented
5#71781C212yesmaximal, complemented
6#71722C212yesmaximal
7#71721C211yesmaximal, characteristic
8#71635C211yesmaximal, characteristic, complemented
9#71631C211yesmaximal, characteristic, complemented
10#71628C212yesmaximal, complemented
11#71606C212yesmaximal
12#25661—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×#944—21no—
27C2×#869C2211yescharacteristic
28C2×#835C2212yes—
29C2×#835C2212yes—
30C2×#835C2212yes—
31C2×#835C2212yes—
32C2×#832C2212yes—
33C2×#832C2212yes—
34#5055—24no—
35#5021—24no—
36#4981—21no—
37#4981—21no—
38#4973—21no—
39C2×#807C2212yes—
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×#477—31no—
51#2546—31no—
52#2043—32no—
53#2042—32no—
54#1985—31no—
55#1983—31no—
56#1863—32no—
57#1862—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
101#944—44no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2312yes—
104C2×C4⋊Dic3C2312yes—
105C2×Dic3⋊C4—22no—
106C2×Dic3⋊C4—22no—
107C2×C4×Dic3—21no—
108C22×C3⋊C8C2312yes—
109C22×C3⋊C8C2312yes—
110#869—22no—
111#869—22no—
112#835—24no—
113#835—24no—
114#835—24no—
115#835—24no—
116#832—24no—
117#832—24no—
118C3⋊#95—22no—
119C3⋊#95—22no—
120C3⋊#95—21no—
121#807—24no—
122C4⋊(C3⋊C8)—22no—
123#803—22no—
124#556—24no—
125#556—24no—
126C2×C4⋊Q8—31no—
127C2×C4⋊Q8—61no—
128C2×C4⋊Q8C2×D311yescharacteristic
129C2×#126—21no—
130C2×#126—21no—
131C2×#124C2×D311yescharacteristic
132C2×#124C2×D311yescharacteristic
133C2×C22⋊Q8—21no—
134C2×C4×Q8C2×D311yescharacteristic
135C2×C4×Q8C2×D311yescharacteristic
136C2×C4×Q8C2×D311yescharacteristic
137C2×C4×D4C2×D311yescharacteristic
138C2×#117—21no—
139#489—64no—
140#477—64no—
141C2×#107—32no—
142C2×C4⋊C8—31no—
143C2×#103—32no—
144C2×#103—32no—
145C2×#103—32no—
146C2×#103—32no—
147C2×#102—32no—
148C2×#102—32no—
149C2×#97—31no—
150#398—61no—
151#398—61no—
152#391—61no—
153#389—32no—
154#389—32no—
155C4×C4⋊C4—31no—
156#340—32no—
157#336—32no—
158#336—32no—
159C24×C3—21no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—41no—
169C2×C3×Q8C2×D412yes—
170C2×C3×Q8C2×D412yes—
171C2×C3×D4—21no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4C2×D412yes—
175C22×C3×C4—24no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—42no—
180C22×C3×C4—42no—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C22×Dic3—22no—
189C22×Dic3—22no—
190C2×Dic6—42no—
191C2×Dic6—42no—
192C2×Dic6—42no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4C2×D412yes—
202C3×C22⋊C4—24no—
203C3×C22⋊C4—24no—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—24no—
207C3×C42—24no—
208C3×C42—24no—
209C3×C42—21no—
210C3×C42—21no—
211C3×C42—21no—
212C3×C42—21no—
213C3×C42C2×D412yes—
214C3×C42C2×D412yes—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3—42no—
219C4⋊Dic3#4114yes—
220Dic3⋊C4—44no—
221Dic3⋊C4—44no—
222C4×Dic3—42no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C22×Q8—61no—
230C22×Q8#6711yescharacteristic
231C22×Q8C22×D311yescharacteristic
232C22×Q8C22×D311yescharacteristic
233C22×D4C22×D311yescharacteristic
234C23×C4—21no—
235C22×C8—32no—
236C22×C8—32no—
237C4⋊Q8—22no—
238C4⋊Q8—22no—
239C4⋊Q8—62no—
240C4⋊Q8—62no—
241C4⋊Q8—124no—
242#126—28no—
243#126—28no—
244#124—22no—
245#124—22no—
246#124—22no—
247#124#6714yes—
248C22⋊Q8—28no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8#6714yes—
255C4×D4—22no—
256C4×D4—22no—
257#117—28no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—62no—
267C2×C4⋊C4#6711yescharacteristic
268C2×C4⋊C4#6711yescharacteristic
269C2×C4⋊C4#6711yescharacteristic
270C2×C4⋊C4C22×D311yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—61no—
279C2×C42C22×D311yescharacteristic
280C2×C42C22×D311yescharacteristic
281C2×C42C22×D311yescharacteristic
282#107—64no—
283C4⋊C8—62no—
284#103—64no—
285#103—64no—
286#103—64no—
287#103—64no—
288#102—64no—
289#102—64no—
290#97—62no—
291#95—61no—
292#95—62no—
293#95—62no—
294C23×C3—22no—
295C23×C3—42no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—42no—
299C23×C3C22×D411yescharacteristic
300C3×Q8—22no—
301C3×Q8—22no—
302C3×Q8—24no—
303C3×Q8—44no—
304C3×Q8—41no—
305C3×Q8—41no—
306C3×Q8—42no—
307C3×D4—24no—
308C3×D4—41no—
309C3×D4—41no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—22no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—48no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4#12012yes—
331C2×C3×C4#12012yes—
332C2×C3×C4AGL(1,ℤ/8)12yes—
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C2×#4112yes—
336C2×C3×C42−1+412yes—
337C2×Dic3—24no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343Dic6—84no—
344C3⋊C8—42no—
345C3⋊C8—42no—
346C24—21no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—41no—
356C2×Q8—122no—
357C2×Q8—122no—
358C2×Q8—122no—
359C2×Q8C2×#6712yes—
360C2×Q8C2×#6712yes—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—21no—
364C2×D4C2×#6712yes—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4C2×#6711yescharacteristic
374C22×C4C2×#6711yescharacteristic
375C22×C4C2×#6711yescharacteristic
376C22×C4C2×#6711yescharacteristic
377C22×C4C2×#6711yescharacteristic
378C22×C4C2×#6711yescharacteristic
379C22×C4C23×D311yescharacteristic
380C2×C8—62no—
381C2×C8—62no—
382C2×C8—62no—
383C2×C8—62no—
384C2×C8—62no—
385C2×C8—62no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—34no—
395C4⋊C4—62no—
396C4⋊C4—62no—
397C4⋊C4—62no—
398C4⋊C4—124no—
399C4⋊C4—124no—
400C4⋊C4—122no—
401C4⋊C4C2×#6712yes—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C42—24no—
408C42—24no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—122no—
414C42C2×#6712yes—
415C42C2×#6712yes—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—82no—
420C22×C3—82no—
421C22×C3C2×#12011yescharacteristic
422C22×C3#54812yes—
423C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
424C22×C3#57511yescharacteristic
425C22×C3#57812yes—
426C3×C4—22no—
427C3×C4—22no—
428C3×C4—22no—
429C3×C4—42no—
430C3×C4—42no—
431C3×C4—41no—
432C3×C4—42no—
433C3×C4—41no—
434C3×C4—41no—
435C3×C4#53211yescharacteristic
436C3×C4#53611yescharacteristic
437C3×C4#54712yes—
438Dic3—44no—
439Dic3—82no—
440C23—22no—
441C23—41no—
442C23—41no—
443C23—42no—
444C23—42no—
445C23C22×#6711yescharacteristic
446Q8—24no—
447Q8—22no—
448Q8—22no—
449Q8—44no—
450Q8—42no—
451Q8—41no—
452Q8—41no—
453Q8—244no—
454D4—24no—
455D4—41no—
456D4—41no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—48no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—64no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4#94912yes—
484C2×C4#95312yes—
485C2×C4#95312yes—
486C2×C4D3×#4112yes—
487C2×C4#101012yes—
488C2×C4C22×#6711yescharacteristic
489C2×C4C22×#6711yescharacteristic
490C8—122no—
491C8—122no—
492C2×C3—82no—
493C2×C3#243211yescharacteristic
494C2×C3#300611yescharacteristic
495C2×C3#301111yescharacteristic
496C2×C3#336712yes—
497C2×C3#337712yes—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—82no—
502C22—82no—
503C22C2×#94911yescharacteristic
504C22#582211yescharacteristic
505C22#583112yes—
506C22#583812yes—
507C22C2×#95311yescharacteristic
508C4—22no—
509C4—22no—
510C4—22no—
511C4—41no—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—41no—
516C4—41no—
517C4—124no—
518C4—242no—
519C4#582412yes—
520C4#582511yescharacteristic
521C4#583211yescharacteristic
522C3#2566111yescharacteristic, complemented, Sylow
523C2—82no—
524C2#8366812yes—
525C2#8367812yes—
526C2#8408211yescharacteristic
527C2#8409311yescharacteristic
528C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#556C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#556D3
hypocenterC3#25661
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. #82475C2semidirect
4. C2×#5021C2semidirect
5. #71781C2semidirect
8. #71635C2semidirect
9. #71631C2semidirect
10. #71628C2semidirect
522. C3#25661semidirect

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
#824752
C2×#50212
#717812
#717222
#717212
#716352
#716312
#716282
#716062
#256613