Finite Group Databasebeta

← #10690899

Subgroups — #10690899

Order 768.

1270 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82834C211yesmaximal, characteristic
2C2×D3×#95C211yesmaximal, characteristic
3C2×C3×#328C211yesmaximal, characteristic, complemented
4C2×#4538C211yesmaximal, characteristic, complemented
5#69955C211yesmaximal, characteristic
6#67675C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9143—31nomaximal, Sylow
9C23×C4×D3—21no—
10C23×C4×D3C2211yescharacteristic
11C2×C3×C22⋊Q8C2211yescharacteristic
12C2×#946C412yescomplemented
13C2×C3×C22⋊C8C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15#5212C2211yescharacteristic
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2211yescharacteristic
19C2×(C2×D3)⋊C8—22no—
20#4989—22no—
21#4987C412yescomplemented
22#4955—21no—
23#4946—21no—
24D3×#95—22no—
25D3×#95—22no—
26C3×#328—22no—
27#4538—22no—
28#2848—31no—
29C22×#95—31no—
30#1779—31no—
31C2×#328—31no—
32C2×#328D311yescharacteristic, complemented
33#1294—31no—
34#1294—31no—
35C24×D3D411yescharacteristic
36C22×C3×Q8C2×C411yescharacteristic, complemented
37C23×C3×C4—21no—
38C23×C3×C4C2311yescharacteristic
39C23×Dic3—21no—
40C23×Dic3D411yescharacteristic
41C22×C4×D3—24no—
42C22×C4×D3—28no—
43C22×C4×D3—21no—
44C22×C4×D3—21no—
45C22×C4×D3—21no—
46C22×C4×D3—21no—
47C22×C4×D3—44no—
48C22×C4×D3—44no—
49C22×C4×D3—42no—
50C22×C4×D3—42no—
51C22×C4×D3C2×C412yes—
52C22×C4×D3D412yes—
53C22×C3×C8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
60C2×C3×C22⋊C4—21no—
61C2×#217—21no—
62#946—44no—
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4⋊Dic3C2×C412yescomplemented
66C2×Dic3⋊C4—22no—
67C22×C3⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#95—21no—
71C3×#95D412yes—
72C3⋊#95—24no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—22no—
76C3⋊#95D412yes—
77#830—21no—
78#830—21no—
79(C2×D3)⋊C8—44no—
80C24×C4—31no—
81C24×C4—61no—
82C2×C22⋊Q8—32no—
83C2×C22⋊Q8C2×D311yescharacteristic
84C2×C22⋊C8—31no—
85C2×C22⋊C8—62no—
86C2×C22⋊C8C2×D311yescharacteristic
87#385—61no—
88#385—62no—
89#380—31no—
90#380—32no—
91#379—61no—
92C2×#95—31no—
93C2×#95—62no—
94C2×#95—61no—
95C2×#95—62no—
96C2×#95C2×D311yescharacteristic
97#328—22no—
98#328—62no—
99C24×C3C2×D411yescharacteristic
100C23×D3—21no—
101C23×D3—21no—
102C23×D3—21no—
103C23×D3—21no—
104C23×D3—24no—
105C23×D3—24no—
106C23×D3C22⋊C412yes—
107C23×D3SD1612yes—
108C23×D3Q1612yes—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—42no—
112C22×C3×C4—24no—
113C22×C3×C4—22no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—42no—
117C22×C3×C4—42no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×Dic3—24no—
127C22×Dic3—21no—
128C22×Dic3—22no—
129C22×Dic3—21no—
130C22×Dic3—22no—
131C22×Dic3—42no—
132C22×Dic3—42no—
133C22×Dic3—41no—
134C22×Dic3—41no—
135C22×Dic3C22⋊C412yes—
136C2×C4×D3—216no—
137C2×C4×D3—24no—
138C2×C4×D3—24no—
139C2×C4×D3—22no—
140C2×C4×D3—22no—
141C2×C4×D3—22no—
142C2×C4×D3—22no—
143C2×C4×D3—42no—
144C2×C4×D3—42no—
145C2×C4×D3—44no—
146C2×C4×D3—44no—
147C2×C4×D3—44no—
148C2×C4×D3—44no—
149C2×C4×D3—48no—
150C2×C4×D3—416no—
151C2×C4×D3—416no—
152C2×C4×D3—48no—
153C2×C3×C8—42no—
154C2×C3×C8—41no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—41no—
158C3×C4⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161#217—41no—
162#217—41no—
163(C2×D3)⋊C4—44no—
164(C2×D3)⋊C4—44no—
165C4⋊Dic3—42no—
166Dic3⋊C4—44no—
167C2×C3⋊C8—42no—
168C2×C3⋊C8—41no—
169C25—31no—
170C22×Q8C4×D311yescharacteristic, complemented
171C23×C4—21no—
172C23×C4—31no—
173C23×C4—32no—
174C23×C4—32no—
175C23×C4—64no—
176C23×C4—68no—
177C23×C4—61no—
178C23×C4—61no—
179C23×C4—61no—
180C23×C4—61no—
181C23×C4—61no—
182C23×C4—124no—
183C23×C4—122no—
184C23×C4—124no—
185C23×C4—122no—
186C23×C4C22×D311yescharacteristic
187C22×C8—21no—
188C22×C8—61no—
189C22⋊Q8—21no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193C22⋊Q8—124no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—32no—
196C2×C4⋊C4—62no—
197C2×C4⋊C4C4×D311yescharacteristic, complemented
198C2×C22⋊C4—21no—
199C2×C22⋊C4—62no—
200C2×C22⋊C4—62no—
201C2×C22⋊C4—61no—
202C22⋊C8—21no—
203C22⋊C8—21no—
204C22⋊C8—61no—
205C22⋊C8—61no—
206C22⋊C8—124no—
207#95—21no—
208#95—32no—
209#95—62no—
210#95—64no—
211#95—61no—
212#95—61no—
213#95#6712yes—
214C23×C3—22no—
215C23×C3—22no—
216C23×C3C2×C22⋊C411yescharacteristic
217C23×C3#12011yescharacteristic
218C23×C3#12311yescharacteristic
219C23×C3C2×SD1611yescharacteristic
220C23×C3C2×Q1611yescharacteristic
221C23×C3AGL(1,ℤ/8)11yescharacteristic
222C23×C3#13711yescharacteristic
223C22×D3—24no—
224C22×D3—24no—
225C22×D3—28no—
226C22×D3—24no—
227C22×D3—24no—
228C22×D3—28no—
229C22×D3—22no—
230C22×D3—22no—
231C22×D3—22no—
232C22×D3—22no—
233C22×D3—22no—
234C22×D3—22no—
235C22×D3—22no—
236C22×D3—22no—
237C22×D3—28no—
238C22×D3—28no—
239C22×D3#9914yes—
240C22×D3#10314yes—
241C22×D3#10414yes—
242C3×Q8—81no—
243C3×Q8—81no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—24no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—44no—
253C2×C3×C4—44no—
254C2×C3×C4—42no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4C4×D411yescharacteristic
260C2×C3×C4C4×D411yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×Dic3—24no—
264C2×Dic3—22no—
265C2×Dic3—22no—
266C2×Dic3—22no—
267C2×Dic3—42no—
268C2×Dic3—44no—
269C2×Dic3—44no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—42no—
276C4×D3—44no—
277C4×D3—44no—
278C4×D3—44no—
279C4×D3—44no—
280C4×D3—432no—
281C3×C8—81no—
282C3⋊C8—81no—
283C24—32no—
284C24—32no—
285C24—32no—
286C24—64no—
287C24—64no—
288C24—61no—
289C24—61no—
290C24—61no—
291C24—61no—
292C24D3×D411yescharacteristic
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—42no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—32no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—64no—
306C22×C4—64no—
307C22×C4—616no—
308C22×C4—64no—
309C22×C4—61no—
310C22×C4—62no—
311C22×C4—61no—
312C22×C4—62no—
313C22×C4—62no—
314C22×C4—62no—
315C22×C4—62no—
316C22×C4—62no—
317C22×C4—122no—
318C22×C4—122no—
319C22×C4—124no—
320C22×C4—124no—
321C22×C4—124no—
322C22×C4—124no—
323C22×C4—1216no—
324C22×C4—128no—
325C22×C4—1216no—
326C22×C4—128no—
327C22×C4—122no—
328C22×C4—122no—
329C22×C4—121no—
330C22×C4—121no—
331C22×C4C2×C4×D311yescharacteristic
332C22×C4#23511yescharacteristic
333C22×C4D3×D411yescharacteristic
334C22×C4D3×D411yescharacteristic
335C22×C4#23711yescharacteristic
336C22×C4C2×#6711yescharacteristic
337C2×C8—42no—
338C2×C8—41no—
339C2×C8—122no—
340C2×C8—121no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—122no—
346C4⋊C4—124no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—124no—
350C22⋊C4—124no—
351C22⋊C4—121no—
352C22⋊C4—121no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3#38611yescharacteristic
362C22×C3C2×#9912yes—
363C22×C3C2×#10311yescharacteristic
364C22×C3#41711yescharacteristic
365C22×C3C2×#10412yes—
366C22×C3#45011yescharacteristic
367C22×C3#45111yescharacteristic
368C22×C3#48311yescharacteristic
369C22×C3#48411yescharacteristic
370C2×D3—28no—
371C2×D3—28no—
372C2×D3—28no—
373C2×D3—28no—
374C2×D3—24no—
375C2×D3—24no—
376C2×D3—24no—
377C2×D3—24no—
378C2×D3—24no—
379C2×D3—24no—
380C2×D3#32818yescomplemented
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—44no—
386C3×C4—81no—
387C3×C4—81no—
388Dic3—44no—
389Dic3—42no—
390Dic3—82no—
391C23—22no—
392C23—22no—
393C23—34no—
394C23—34no—
395C23—34no—
396C23—68no—
397C23—68no—
398C23—64no—
399C23—64no—
400C23—64no—
401C23—68no—
402C23—68no—
403C23—64no—
404C23—62no—
405C23—62no—
406C23—62no—
407C23—62no—
408C23—62no—
409C23—62no—
410C23—62no—
411C23—62no—
412C23D3×C22⋊C411yescharacteristic
413C23#88311yescharacteristic
414C23D3×SD1611yescharacteristic
415C23#91411yescharacteristic
416C23D3×Q1611yescharacteristic
417C23#91811yescharacteristic
418C23#93711yescharacteristic
419Q8—81no—
420Q8—81no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—64no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4—62no—
440C2×C4—124no—
441C2×C4—124no—
442C2×C4—124no—
443C2×C4—124no—
444C2×C4—1232no—
445C2×C4—122no—
446C2×C4—124no—
447C2×C4—124no—
448C2×C4—122no—
449C2×C4—122no—
450C2×C4—122no—
451C2×C4—122no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4#67⋊C411yescharacteristic
455C2×C4#88411yescharacteristic
456C2×C4C4×#6711yescharacteristic
457C2×C4#93811yescharacteristic
458C8—81no—
459C8—241no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3C2×#32812yes—
465C2×C3#150412yes—
466C2×C3#187211yescharacteristic
467C2×C3#189611yescharacteristic
468C2×C3#189811yescharacteristic
469D3—28no—
470C22—21no—
471C22—21no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—21no—
477C22—21no—
478C22—38no—
479C22—68no—
480C22—68no—
481C22—68no—
482C22—68no—
483C22—64no—
484C22—64no—
485C22—64no—
486C22—64no—
487C22—64no—
488C22—64no—
489C22D3×#9912yes—
490C22D3×#10311yescharacteristic
491C22#480311yescharacteristic
492C22#480511yescharacteristic
493C22#481111yescharacteristic
494C22D3×#10412yes—
495C22#495611yescharacteristic
496C22#516911yescharacteristic
497C22#518611yescharacteristic
498C4—21no—
499C4—21no—
500C4—21no—
501C4—21no—
502C4—44no—
503C4—81no—
504C4—81no—
505C4—124no—
506C4—122no—
507C4—242no—
508C3#914311yescharacteristic, complemented, Sylow
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2—68no—
514C2D3×#32812yes—
515C2#6833412yes—
516C2#7003611yescharacteristic
517C2#7026811yescharacteristic
518C2#7038211yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#328C2
socleC23×C3C2×C22⋊C4
prime radicalC24×D3D4
cosocleC23×C3×C4C23
hypercenterC2×#328D3
hypocenterC3#9143
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
3. C2×C3×#328C2semidirect
4. C2×#4538C2semidirect
6. #67675C2semidirect
7. #67612C2semidirect
12. C2×#946C4semidirect
21. #4987C4semidirect
32. C2×#328D3semidirect
36. C22×C3×Q8C2×C4semidirect
59. C2×C3×C4⋊C4C2×C4semidirect
65. C2×C4⋊Dic3C2×C4semidirect
170. C22×Q8C4×D3semidirect
197. C2×C4⋊C4C4×D3semidirect
380. C2×D3#328semidirect
508. C3#9143semidirect

Maximal subgroups

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

subgroupindex
#828342
C2×D3×#952
C2×C3×#3282
C2×#45382
#699552
#676752
#676122
#91433