Finite Group Databasebeta

← #10723910

Subgroups — #10723910

Order 768.

1398 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#391C211yesmaximal, characteristic
2C2×#5229C211yesmaximal, characteristic
3C2×#5229C211yesmaximal, characteristic
4#71568C212yesmaximal
5#71536C212yesmaximal
6#69420C212yesmaximal
7#69411C212yesmaximal
8#69406C212yesmaximal
9Dic3⋊#95C212yesmaximal
10#14467—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C42C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×Dic3⋊C4C2212yes—
17C22×Dic3⋊C4C2212yes—
18C22×Dic3⋊C4C2212yes—
19C22×C4×Dic3C2211yescharacteristic
20C22×C4×Dic3C2211yescharacteristic
21C3×#391—24no—
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24#5229—24no—
25#5229C418yescomplemented
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×#391—31no—
36C2×#391—31no—
37C2×#391D311yescharacteristic
38#1444—32no—
39#1444—32no—
40#1441—32no—
41#1436—32no—
42#1436—32no—
43C4⋊#95—32no—
44C22×C3×Q8—41no—
45C22×C3×Q8—41no—
46C22×C3×Q8—41no—
47C22×C3×Q8—41no—
48C22×C3×Q8C2×C414yes—
49C22×C3×Q8D414yes—
50C22×C3×Q8Q814yes—
51C23×C3×C4C2312yes—
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2312yes—
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—41no—
62C2×C3×C4⋊C4—41no—
63C2×C3×C42—22no—
64C2×C3×C42—41no—
65C2×C3×C42—41no—
66C2×C4⋊Dic3—22no—
67C2×C4⋊Dic3—41no—
68C2×C4⋊Dic3—41no—
69C2×C4⋊Dic3—41no—
70C2×C4⋊Dic3—41no—
71C2×C4⋊Dic3C2×C414yes—
72C2×Dic3⋊C4—24no—
73C2×Dic3⋊C4—42no—
74C2×Dic3⋊C4—42no—
75C2×Dic3⋊C4—42no—
76C2×Dic3⋊C4—42no—
77C2×Dic3⋊C4—42no—
78C2×Dic3⋊C4—42no—
79C2×Dic3⋊C4—42no—
80C2×Dic3⋊C4Q818yescomplemented
81C2×C4×Dic3—22no—
82C2×C4×Dic3—41no—
83C2×C4×Dic3—41no—
84C2×C4×Dic3—41no—
85C2×C4×Dic3—41no—
86C2×C4×Dic3C2×C414yes—
87C3×#95—24no—
88C3×#95—24no—
89C3⋊#95—24no—
90C3⋊#95—24no—
91C3⋊#95—24no—
92C3⋊#95—24no—
93C3⋊#95—42no—
94C3⋊#95—42no—
95C3⋊#95—42no—
96C3⋊#95C2×C418yes—
97C3⋊#95C2×C418yes—
98C23×Q8C2×D311yescharacteristic
99C22×C4⋊C4—32no—
100C22×C4⋊C4—31no—
101C22×C4⋊C4—32no—
102C22×C4⋊C4—31no—
103C22×C4⋊C4—32no—
104C22×C4⋊C4C2×D311yescharacteristic
105C22×C42—31no—
106C22×C42—31no—
107C22×C42C2×D311yescharacteristic
108#391—24no—
109#391—38no—
110#391—64no—
111C2×#95—32no—
112C2×#95—32no—
113C2×#95—32no—
114C2×#95—32no—
115C2×#95—32no—
116C2×#95—32no—
117C2×#95—32no—
118C2×#95—32no—
119C2×#95—32no—
120C2×#95C2×D312yes—
121C2×#95C2×D312yes—
122C24×C3C2411yescharacteristic
123C2×C3×Q8—44no—
124C2×C3×Q8—44no—
125C2×C3×Q8—44no—
126C2×C3×Q8—44no—
127C2×C3×Q8—44no—
128C2×C3×Q8—44no—
129C2×C3×Q8C4⋊C4116yescomplemented
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4C22×C412yes—
156C22×C3×C4C22×C412yes—
157C22×C3×C4C22×C412yes—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×Q812yes—
162C22×C3×C4C2×Q812yes—
163C22×C3×C4C2×Q812yes—
164C22×C3×C4C2×Q814yes—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3C22×C414yes—
189C22×Dic3C22×C414yes—
190C22×Dic3C2×Q814yes—
191C22×Dic3C2×Q814yes—
192C22×Dic3#4114yes—
193C3×C4⋊C4—44no—
194C3×C42—44no—
195C4⋊Dic3—44no—
196C4⋊Dic3—44no—
197Dic3⋊C4—48no—
198Dic3⋊C4—48no—
199Dic3⋊C4—48no—
200C4×Dic3—44no—
201C4×Dic3—44no—
202C22×Q8—41no—
203C22×Q8—41no—
204C22×Q8—41no—
205C22×Q8—41no—
206C22×Q8Dic614yes—
207C22×Q8C4×D314yes—
208C22×Q8#6714yes—
209C23×C4—32no—
210C23×C4—32no—
211C23×C4—32no—
212C23×C4—32no—
213C23×C4C22×D312yes—
214C23×C4C22×D312yes—
215C23×C4C22×D311yescharacteristic
216C23×C4C22×D311yescharacteristic
217C23×C4C22×D311yescharacteristic
218C2×C4⋊C4—22no—
219C2×C4⋊C4—34no—
220C2×C4⋊C4—38no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—64no—
224C2×C4⋊C4—62no—
225C2×C4⋊C4—122no—
226C2×C4⋊C4—122no—
227C2×C4⋊C4—122no—
228C2×C4⋊C4—122no—
229C2×C4⋊C4—122no—
230C2×C4⋊C4—122no—
231C2×C4⋊C4—122no—
232C2×C4⋊C4—121no—
233C2×C4⋊C4—121no—
234C2×C4⋊C4—121no—
235C2×C4⋊C4—121no—
236C2×C42—22no—
237C2×C42—34no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42—62no—
241C2×C42—121no—
242C2×C42—121no—
243C2×C42—121no—
244C2×C42—121no—
245#95—24no—
246#95—24no—
247#95—38no—
248#95—38no—
249#95—64no—
250#95—64no—
251#95—64no—
252#95—64no—
253#95—122no—
254#95—122no—
255#95—122no—
256C23×C3C23×C411yescharacteristic
257C23×C3C22×D411yescharacteristic
258C23×C3C22×Q811yescharacteristic
259C23×C3C22×Q812yes—
260C23×C3C2×#4112yes—
261C23×C3H(5,𝔽2)11yescharacteristic
262C23×C3H(5,𝔽2)11yescharacteristic
263C23×C3H(5,𝔽2)11yescharacteristic
264C23×C3H(5,𝔽2)11yescharacteristic
265C23×C3H(5,𝔽2)12yes—
266C23×C32−1+412yes—
267C3×Q8—416no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4C2×C4⋊C414yes—
301C2×C3×C4C2×C4⋊C414yes—
302C2×C3×C4C2×C4⋊C414yes—
303C2×Dic3—24no—
304C2×Dic3—24no—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3—24no—
309C2×Dic3—24no—
310C2×Dic3—24no—
311C2×Dic3—24no—
312C2×Dic3—24no—
313C2×Dic3—24no—
314C2×Dic3—24no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3C4×Q818yes—
331C24C23×D311yescharacteristic
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8Dic3⋊C4116yescomplemented
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—34no—
365C22×C4—34no—
366C22×C4—34no—
367C22×C4—34no—
368C22×C4—34no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4C2×Dic612yes—
393C22×C4C2×Dic612yes—
394C22×C4C2×Dic612yes—
395C22×C4C2×C4×D312yes—
396C22×C4C2×C4×D312yes—
397C22×C4C2×C4×D312yes—
398C22×C4D3×Q814yes—
399C22×C4C2×#6712yes—
400C22×C4C2×#6712yes—
401C22×C4C2×#6712yes—
402C4⋊C4—44no—
403C4⋊C4—128no—
404C4⋊C4—124no—
405C4⋊C4—124no—
406C4⋊C4—128no—
407C4⋊C4—128no—
408C42—44no—
409C42—124no—
410C42—124no—
411C22×C3C22×C4⋊C411yescharacteristic
412C22×C3C2×C4×Q812yes—
413C22×C3#51811yescharacteristic
414C22×C3#51811yescharacteristic
415C22×C3#52012yes—
416C22×C3#53411yescharacteristic
417C22×C3#53411yescharacteristic
418C22×C3#53812yes—
419C22×C3#53912yes—
420C22×C3#54312yes—
421C22×C3#54311yescharacteristic
422C22×C3#54311yescharacteristic
423C22×C3#54412yes—
424C22×C3D4×Q812yes—
425C22×C3#55012yes—
426C22×C3#55412yes—
427C22×C3#55511yescharacteristic
428C22×C3#55712yes—
429C22×C3Q8211yescharacteristic
430C22×C3Q8212yes—
431C22×C3H(3,𝔽4)11yescharacteristic
432C22×C3#56212yes—
433C22×C3#56311yescharacteristic
434C3×C4—24no—
435C3×C4—24no—
436C3×C4—24no—
437C3×C4—44no—
438C3×C4—82no—
439Dic3—28no—
440Dic3—44no—
441Dic3—44no—
442Dic3—82no—
443C23C22×Dic611yescharacteristic
444C23C22×C4×D311yescharacteristic
445C23#100411yescharacteristic
446C23#100411yescharacteristic
447C23#100411yescharacteristic
448C23#100411yescharacteristic
449C23C2×D3×Q812yes—
450C23#100712yes—
451C23D3×#4112yes—
452C23#100912yes—
453C23C22×#6711yescharacteristic
454Q8—416no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—38no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—64no—
489C2×C4—64no—
490C2×C4—64no—
491C2×C4—64no—
492C2×C4—64no—
493C2×C4—64no—
494C2×C4—64no—
495C2×C4—64no—
496C2×C4—64no—
497C2×C4—64no—
498C2×C4—64no—
499C2×C4—64no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4C2×Dic3⋊C414yes—
516C2×C4C2×Dic3⋊C414yes—
517C2×C4C2×Dic3⋊C414yes—
518C2×C3#231511yescharacteristic
519C2×C3#232912yes—
520C2×C3Q8×C4⋊C412yes—
521C2×C3#236911yescharacteristic
522C2×C3#237911yescharacteristic
523C2×C3#273212yes—
524C2×C3#273311yescharacteristic
525C2×C3#273411yescharacteristic
526C2×C3#280612yes—
527C2×C3#280911yescharacteristic
528C2×C3#281111yescharacteristic
529C22#548311yescharacteristic
530C22#548311yescharacteristic
531C22#548611yescharacteristic
532C22#548611yescharacteristic
533C22Q8×Dic612yes—
534C22#556912yes—
535C22C4×D3×Q812yes—
536C22#557412yes—
537C22#563412yes—
538C22#563612yes—
539C22#563812yes—
540C22#564712yes—
541C22#567011yescharacteristic
542C22#567812yes—
543C22#568311yescharacteristic
544C22#572211yescharacteristic
545C22#572912yes—
546C22#573511yescharacteristic
547C22C22×Dic3⋊C411yescharacteristic
548C22#580411yescharacteristic
549C22#580411yescharacteristic
550C22Q8×#6712yes—
551C22#583412yes—
552C4—24no—
553C4—24no—
554C4—24no—
555C4—44no—
556C4—68no—
557C4—82no—
558C4—124no—
559C4—124no—
560C4—242no—
561C3#1446711yescharacteristic, complemented, Sylow
562C2#7649411yescharacteristic
563C2#7649611yescharacteristic
564C2#7660012yes—
565C2#7660311yescharacteristic
566C2#7660411yescharacteristic
567C2#7661212yes—
568C2#8209511yescharacteristic
569C2Q8×Dic3⋊C412yes—
570C2#8280712yes—
571C2#8299111yescharacteristic
572C2#8325011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#391C2
socle, prime radical, cosocleC24×C3C24
hypercenterC2×#391D3
hypocenterC3#14467
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
25. #5229C4semidirect
80. C2×Dic3⋊C4Q8semidirect
129. C2×C3×Q8C4⋊C4semidirect
338. C2×Q8Dic3⋊C4semidirect
561. C3#14467semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3912
C2×#52292
#715682
#715362
#694202
#694112
#694062
Dic3⋊#952
#144673