Finite Group Databasebeta

← #10900917

Subgroups — #10900917

Order 768.

1292 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#4774C212yesmaximal
3#76475C211yesmaximal, characteristic
4C3×#2023C211yesmaximal, characteristic, complemented
5#71782C211yesmaximal, characteristic, complemented
6#70163C212yesmaximal, complemented
7#69638C211yesmaximal, characteristic, complemented
8#68871C212yesmaximal, complemented
9#68826C212yesmaximal, complemented
10#68664C211yesmaximal, characteristic, complemented
11#68632C211yesmaximal, characteristic, complemented
12#29638—31nomaximal, Sylow
13C22×D3×Q8—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×#944—21no—
16C2×#938C2211yescharacteristic
17D3×C4⋊Q8—28no—
18C2×D3×C4⋊C4—21no—
19C2×D3×C4⋊C4C2212yes—
20C2×#888—21no—
21C2×#888C2212yes—
22C2×#883—21no—
23C2×C42×D3C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C3×#102C2212yes—
26C3×#390C2211yescharacteristic
27#5221C2211yescharacteristic
28C2×#832C2212yes—
29C2×(C2×D3)⋊C8C2212yes—
30C2×#818C2212yes—
31#4987C2211yescharacteristic
32#4984C2211yescharacteristic
33C2×#807C2212yes—
34#4956—21no—
35#4951—21no—
36#4945—21no—
37#4928C2211yescharacteristic
38#4774—28no—
39C3×#340C2212yes—
40C3×#336C2211yescharacteristic
41#4551C2212yes—
42#4550C2212yes—
43#4529C2212yes—
44#4529C2212yes—
45#4524C2211yescharacteristic
46C22×C4⋊Q8—31no—
47C2×#480—32no—
48#2854—31no—
49#2057—32no—
50#2057—31no—
51#2057—31no—
52#2023—32no—
53#2023—31no—
54#2023D311yescharacteristic, complemented
55#1823—32no—
56#1817—31no—
57C22×C3×Q8—21no—
58C22×C3×D4C2311yescharacteristic
59C22×#67—21no—
60C2×D3×Q8—28no—
61C2×D3×Q8—44no—
62C2×D3×Q8—44no—
63C22×C4×D3—22no—
64C22×C4×D3D411yescharacteristic
65C22×C4×D3D411yescharacteristic
66C22×C4×D3C2311yescharacteristic
67C22×Dic6—21no—
68C22×Dic6—22no—
69C22×C3×C8C2312yes—
70C3×C4⋊Q8—24no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C42C2311yescharacteristic
77C2×#217—21no—
78C2×#217—21no—
79C2×#217—21no—
80#944—28no—
81#938—24no—
82C2×(C2×D3)⋊C4—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2312yes—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—21no—
91C2×C4×Dic3D411yescharacteristic
92C2×C4×Dic3D411yescharacteristic
93C2×C4×Dic3C2311yescharacteristic
94C22×C3⋊C8C2312yes—
95D3×C4⋊C4—28no—
96D3×C4⋊C4—24no—
97D3×C4⋊C4D418yes—
98#888—28no—
99#888—24no—
100#888—24no—
101#883—44no—
102C42×D3—24no—
103#869—24no—
104C3×#102—24no—
105#832—24no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109(C2×D3)⋊C8—24no—
110#818—24no—
111#807—24no—
112C23×Q8—61no—
113C2×C4⋊Q8—32no—
114C2×C4⋊Q8—31no—
115C2×C4⋊Q8—61no—
116C2×C4⋊Q8—61no—
117C2×C4⋊Q8—68no—
118C2×C4⋊Q8C2×D311yescharacteristic
119C2×#123—61no—
120C2×#121—31no—
121C22×C4⋊C4—32no—
122C22×C4⋊C4—61no—
123C22×C42—31no—
124#480—68no—
125C2×#107—32no—
126C2×#107—32no—
127C2×#102—32no—
128C2×#102C2×D312yes—
129C2×C22⋊C8—32no—
130#390—31no—
131#390C2×D311yescharacteristic
132#389—31no—
133#389—31no—
134#388—61no—
135#388—61no—
136#386—61no—
137#380—31no—
138#340—32no—
139#340—32no—
140#340—32no—
141#340C2×D312yes—
142#336—32no—
143#336—31no—
144#336C2×D311yescharacteristic
145C24×C3—21no—
146C23×D3C2×D411yescharacteristic
147C2×C3×Q8—24no—
148C2×C3×Q8—42no—
149C2×C3×Q8—42no—
150C2×C3×D4—22no—
151C2×C3×D4—21no—
152C2×C3×D4—22no—
153C2×C3×D4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C2×#67—42no—
160C2×#67—42no—
161C2×#67—42no—
162C22×Dic3—22no—
163C22×Dic3—21no—
164C22×Dic3—21no—
165C22×Dic3C2×D411yescharacteristic
166C22×Dic3C2×D411yescharacteristic
167C22×Dic3C2×D411yescharacteristic
168D3×Q8—432no—
169C2×C4×D3—22no—
170C2×C4×D3—22no—
171C2×C4×D3—22no—
172C2×C4×D3—22no—
173C2×C4×D3—21no—
174C2×C4×D3—22no—
175C2×C4×D3—21no—
176C2×C4×D3—22no—
177C2×C4×D3—28no—
178C2×C4×D3—44no—
179C2×C4×D3—44no—
180C2×C4×D3D814yes—
181C2×C4×D3C2×D414yes—
182C2×Dic6—24no—
183C2×Dic6—28no—
184C2×Dic6—44no—
185C2×Dic6—42no—
186C2×Dic6—42no—
187C2×Dic6—44no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4C2×D414yes—
195C3×C22⋊C4—42no—
196C3×C22⋊C4—42no—
197C3×C42—22no—
198#217—42no—
199#217—42no—
200#217—42no—
201(C2×D3)⋊C4—42no—
202C4⋊Dic3—24no—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—42no—
208C4⋊Dic3C2×D414yes—
209Dic3⋊C4—28no—
210Dic3⋊C4—28no—
211Dic3⋊C4—42no—
212C4×Dic3—22no—
213C4×Dic3—22no—
214C4×Dic3—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C22×Q8—21no—
219C22×Q8—62no—
220C22×Q8—68no—
221C22×Q8—61no—
222C22×Q8—124no—
223C22×Q8—124no—
224C22×D4—61no—
225C22×D4C22×D311yescharacteristic
226C23×C4—31no—
227C23×C4—31no—
228C23×C4—31no—
229C23×C4—62no—
230C22×C8—32no—
231C22×C8C22×D312yes—
232C4⋊Q8—24no—
233C4⋊Q8—64no—
234C4⋊Q8—64no—
235C4⋊Q8—68no—
236C4⋊Q8—68no—
237C4⋊Q8—64no—
238#123—124no—
239#121—64no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—32no—
242C2×C4⋊C4—38no—
243C2×C4⋊C4—31no—
244C2×C4⋊C4—31no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—68no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4C22×D311yescharacteristic
253C2×C4⋊C4C22×D312yes—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—61no—
257C2×C22⋊C4—61no—
258C2×C22⋊C4—61no—
259C2×C22⋊C4—61no—
260C2×C42—31no—
261C2×C42—31no—
262C2×C42—31no—
263C2×C42—64no—
264C2×C42C22×D311yescharacteristic
265#107—64no—
266#107—64no—
267#102—24no—
268#102—64no—
269C22⋊C8—64no—
270#95—61no—
271#95—61no—
272#95—61no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3—41no—
280C23×C3C22×D411yescharacteristic
281C22×D3—21no—
282C22×D3—21no—
283C22×D3—21no—
284C22×D3—21no—
285C22×D3#12012yes—
286C22×D3C2×D812yes—
287C22×D3AGL(1,ℤ/8)12yes—
288C3×Q8—48no—
289C3×D4—42no—
290C3×D4—42no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—24no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4C2×D812yes—
302C2×C3×C4#13712yes—
303C2×C3×C4C22×D412yes—
304C2×C3×C42−1+411yescharacteristic
305C2×C3×C42−1+412yes—
306C2×C3×C42−1+411yescharacteristic
307#67—84no—
308C2×Dic3—21no—
309C2×Dic3—21no—
310C2×Dic3—21no—
311C2×Dic3—21no—
312C2×Dic3—21no—
313C2×Dic3—21no—
314C2×Dic3—24no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3#12012yes—
324C2×Dic3C2×D812yes—
325C2×Dic3#13712yes—
326C4×D3—28no—
327C4×D3—28no—
328C4×D3—24no—
329C4×D3—24no—
330C4×D3—416no—
331Dic6—416no—
332Dic6—48no—
333C3×C8—42no—
334C3⋊C8—42no—
335C24—21no—
336C24—31no—
337C2×Q8—24no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—64no—
341C2×Q8—68no—
342C2×Q8—122no—
343C2×Q8—124no—
344C2×Q8—124no—
345C2×Q8—1232no—
346C2×Q8—122no—
347C2×D4—22no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—22no—
351C2×D4—122no—
352C2×D4—122no—
353C2×D4—122no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—31no—
357C22×C4—31no—
358C22×C4—31no—
359C22×C4—34no—
360C22×C4—34no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—68no—
366C22×C4—62no—
367C22×C4—61no—
368C22×C4—62no—
369C22×C4—61no—
370C22×C4—62no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4—124no—
374C22×C4—124no—
375C22×C4D3×D411yescharacteristic
376C22×C4D3×D411yescharacteristic
377C22×C4C23×D311yescharacteristic
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—62no—
382C2×C8—62no—
383C2×C8—62no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—34no—
388C4⋊C4—68no—
389C4⋊C4—68no—
390C4⋊C4—64no—
391C4⋊C4—62no—
392C4⋊C4—62no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4—122no—
396C4⋊C4—122no—
397C4⋊C4D3×D414yes—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—122no—
401C22⋊C4—122no—
402C22⋊C4—122no—
403C22⋊C4—122no—
404C42—22no—
405C42—62no—
406C42—62no—
407C42—62no—
408C22×C3—81no—
409C22×C3—81no—
410C22×C3—81no—
411C22×C3—81no—
412C22×C3—81no—
413C22×C3—81no—
414C22×C3—81no—
415C22×C3C2×#12011yescharacteristic
416C22×C3#53611yescharacteristic
417C22×C3#53611yescharacteristic
418C22×C3C22×D811yescharacteristic
419C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
420C22×C3C2×#13711yescharacteristic
421C22×C3C2×#13711yescharacteristic
422C2×D3—22no—
423C2×D3—22no—
424C2×D3—22no—
425C2×D3—22no—
426C2×D3—22no—
427C2×D3—22no—
428C2×D3#44714yes—
429C3×C4—22no—
430C3×C4—22no—
431C3×C4—44no—
432C3×C4—81no—
433C3×C4#53612yes—
434C3×C4#53612yes—
435Dic3—22no—
436Dic3—22no—
437Dic3—22no—
438Dic3—44no—
439Dic3—81no—
440Dic3—81no—
441C23—32no—
442C23—32no—
443C23—32no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—61no—
452C23—61no—
453C23—61no—
454C23—61no—
455C23C2×D3×D411yescharacteristic
456Q8—48no—
457Q8—128no—
458Q8—1216no—
459D4—42no—
460D4—42no—
461D4—244no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—32no—
468C2×C4—32no—
469C2×C4—32no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—68no—
476C2×C4—68no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—61no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—64no—
484C2×C4—64no—
485C2×C4—64no—
486C2×C4—1216no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4D3×D812yes—
496C2×C4#91512yes—
497C2×C4C2×D3×D412yes—
498C2×C4#100711yescharacteristic
499C2×C4#101011yescharacteristic
500C2×C4#101012yes—
501C8—42no—
502C8—122no—
503C2×C3—161no—
504C2×C3#261411yescharacteristic
505C2×C3C2×#44711yescharacteristic
506C2×C3#300411yescharacteristic
507C2×C3#322611yescharacteristic
508C2×C3#322611yescharacteristic
509C2×C3#323811yescharacteristic
510C2×C3#323811yescharacteristic
511D3—24no—
512C22—34no—
513C22—62no—
514C22—62no—
515C22—62no—
516C22—62no—
517C22—62no—
518C22—62no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22D3×#12011yescharacteristic
527C22#562811yescharacteristic
528C22#567511yescharacteristic
529C22C2×D3×D811yescharacteristic
530C22C2×#91511yescharacteristic
531C22D3×AGL(1,ℤ/8)11yescharacteristic
532C22D3×#13711yescharacteristic
533C4—22no—
534C4—22no—
535C4—44no—
536C4—62no—
537C4—62no—
538C4—62no—
539C4—81no—
540C4—124no—
541C4—241no—
542C4—241no—
543C4#562812yes—
544C4#567512yes—
545C3#2963811yescharacteristic, complemented, Sylow
546C2—64no—
547C2—161no—
548C2#7687011yescharacteristic
549C2D3×#44711yescharacteristic
550C2#8003911yescharacteristic
551C2#8096911yescharacteristic
552C2#8106711yescharacteristic
553C2#8157111yescharacteristic
554C2#8169011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2023C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#2023D3
hypocenterC3#29638
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. C2×D3×C4⋊Q8C2semidirect
4. C3×#2023C2semidirect
5. #71782C2semidirect
6. #70163C2semidirect
7. #69638C2semidirect
8. #68871C2semidirect
9. #68826C2semidirect
10. #68664C2semidirect
11. #68632C2semidirect
54. #2023D3semidirect
545. C3#29638semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊Q82
C2×#47742
#764752
C3×#20232
#717822
#701632
#696382
#688712
#688262
#686642
#686322
#296383