Finite Group Databasebeta

← #10702899

Subgroups — #10702899

Order 768.

1406 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2308C211yesmaximal, characteristic
2C2×#5220C211yesmaximal, characteristic
3C22×#834C211yesmaximal, characteristic
4#71544C212yesmaximal, complemented
5C2×#4526C212yesmaximal, complemented
6#11174—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C24×Dic3C2211yescharacteristic
9C22×C3×C22⋊C4C2212yes—
10C22×#217—21no—
11C22×#217C2211yescharacteristic
12C22×#217C2212yes—
13C3×#386—24no—
14C3×#381—21no—
15C3×#379—21no—
16#5220—22no—
17#5220C414yescomplemented
18C2×#834—22no—
19C2×#834—24no—
20C2×C3⋊#95—21no—
21C2×C3⋊#95—24no—
22#4526—24no—
23C22×#99—31no—
24#2308D311yescharacteristic
25C2×#386—31no—
26C2×#342—32no—
27C22⋊#95—32no—
28C25×C3D411yescharacteristic
29C25×C3C2311yescharacteristic
30C22×C3×D4—22no—
31C22×C3×D4—22no—
32C22×C3×D4—22no—
33C22×C3×D4—22no—
34C22×C3×D4—22no—
35C22×C3×D4—22no—
36C22×C3×D4C2×C412yes—
37C22×C3×D4D412yes—
38C23×C3×C4D411yescharacteristic
39C23×C3×C4D414yes—
40C23×Dic3—22no—
41C23×Dic3—21no—
42C23×Dic3—21no—
43C23×Dic3—22no—
44C23×Dic3—21no—
45C23×Dic3—22no—
46C23×Dic3—44no—
47C23×Dic3C2×C412yes—
48C2×C3×C22⋊C4—22no—
49C2×C3×C22⋊C4—22no—
50C2×C3×C22⋊C4—22no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C22⋊C4—22no—
55C2×C3×C22⋊C4—22no—
56C2×#217—22no—
57C2×#217—22no—
58C2×#217—22no—
59C2×#217—22no—
60C2×#217—22no—
61C2×#217—22no—
62C2×#217—24no—
63C2×#217—21no—
64C2×#217—22no—
65C2×#217—44no—
66C2×#217C2×C412yes—
67C2×#217D414yes—
68C3×#95—24no—
69#834—28no—
70C3⋊#95—24no—
71C3⋊#95—48no—
72C23×D4C2×D311yescharacteristic
73C24×C4—31no—
74C22×C22⋊C4—32no—
75C22×C22⋊C4—31no—
76C22×C22⋊C4—61no—
77C22×C22⋊C4C2×D312yes—
78C2×#99—64no—
79C2×#99—62no—
80#386—24no—
81#386—34no—
82#386—62no—
83#381—21no—
84#379—21no—
85C2×#95—61no—
86C2×#95—64no—
87#342—64no—
88C24×C3—21no—
89C24×C3—22no—
90C24×C3—22no—
91C24×C3—21no—
92C24×C3—42no—
93C24×C3—42no—
94C24×C3—42no—
95C24×C3—42no—
96C24×C3C22⋊C412yes—
97C24×C3C22×C411yescharacteristic
98C24×C3C2×D411yescharacteristic
99C24×C3C2×D412yes—
100C24×C3C2×D411yescharacteristic
101C24×C3C2×D412yes—
102C24×C3#4112yes—
103C24×C3#4112yes—
104C24×C3#4112yes—
105C24×C3#4112yes—
106C2×C3×D4—22no—
107C2×C3×D4—22no—
108C2×C3×D4—24no—
109C2×C3×D4—24no—
110C2×C3×D4—22no—
111C2×C3×D4—24no—
112C2×C3×D4—24no—
113C2×C3×D4—22no—
114C2×C3×D4—42no—
115C2×C3×D4—42no—
116C2×C3×D4—42no—
117C2×C3×D4—42no—
118C2×C3×D4—42no—
119C2×C3×D4—42no—
120C2×C3×D4—42no—
121C2×C3×D4—42no—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—24no—
127C22×C3×C4—24no—
128C22×C3×C4—24no—
129C22×C3×C4—44no—
130C22×C3×C4—44no—
131C22×C3×C4C22⋊C412yes—
132C22×Dic3—28no—
133C22×Dic3—24no—
134C22×Dic3—22no—
135C22×Dic3—24no—
136C22×Dic3—22no—
137C22×Dic3—24no—
138C22×Dic3—44no—
139C22×Dic3—44no—
140C22×Dic3—48no—
141C22×Dic3—44no—
142C22×Dic3—42no—
143C22×Dic3—41no—
144C22×Dic3—42no—
145C22×Dic3—41no—
146C22×Dic3—44no—
147C22×Dic3—42no—
148C22×Dic3—44no—
149C22×Dic3—42no—
150C22×Dic3—42no—
151C22×Dic3—42no—
152C22×Dic3—42no—
153C3×C22⋊C4—42no—
154C3×C22⋊C4—42no—
155C3×C22⋊C4—42no—
156C3×C22⋊C4—42no—
157C3×C22⋊C4—82no—
158C3×C22⋊C4—82no—
159#217—24no—
160#217—24no—
161#217—24no—
162#217—48no—
163#217—41no—
164#217—41no—
165#217—42no—
166#217—42no—
167C25#6711yescharacteristic
168C25C22×D311yescharacteristic
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4C4×D312yes—
176C22×D4D1212yes—
177C23×C4—32no—
178C23×C4—61no—
179C23×C4—62no—
180C23×C4—61no—
181C23×C4—62no—
182C23×C4—61no—
183C23×C4—62no—
184C23×C4—124no—
185C23×C4#6711yescharacteristic
186C23×C4#6714yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—32no—
196C2×C22⋊C4—34no—
197C2×C22⋊C4—62no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4—62no—
200C2×C22⋊C4—62no—
201C2×C22⋊C4—62no—
202C2×C22⋊C4—62no—
203C2×C22⋊C4—64no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—62no—
206C2×C22⋊C4—124no—
207#99—68no—
208#95—24no—
209#95—64no—
210#95—128no—
211C23×C3—24no—
212C23×C3—22no—
213C23×C3—24no—
214C23×C3—22no—
215C23×C3—22no—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—42no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—44no—
224C23×C3—42no—
225C23×C3—44no—
226C23×C3—44no—
227C23×C3—44no—
228C23×C3—42no—
229C23×C3—42no—
230C23×C3—48no—
231C23×C3—42no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—81no—
236C23×C3—81no—
237C23×C3—84no—
238C23×C3C2×C22⋊C411yescharacteristic
239C23×C3#11711yescharacteristic
240C23×C3#11711yescharacteristic
241C23×C3C4×D412yes—
242C23×C3C4×D412yes—
243C23×C3#12011yescharacteristic
244C23×C3#12011yescharacteristic
245C23×C3#12112yes—
246C23×C3#12112yes—
247C23×C3#12112yes—
248C23×C3#12112yes—
249C23×C3#12312yes—
250C23×C3#12312yes—
251C23×C3#12311yescharacteristic
252C23×C3#12311yescharacteristic
253C23×C3#12312yes—
254C23×C3#12312yes—
255C23×C3#12412yes—
256C23×C3#12412yes—
257C23×C3#12612yes—
258C23×C3#12612yes—
259C3×D4—44no—
260C3×D4—44no—
261C3×D4—44no—
262C3×D4—44no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—48no—
269C2×C3×C4—44no—
270C2×C3×C4—44no—
271C2×C3×C4—44no—
272C2×C3×C4—84no—
273C2×Dic3—24no—
274C2×Dic3—28no—
275C2×Dic3—42no—
276C2×Dic3—44no—
277C2×Dic3—48no—
278C2×Dic3—44no—
279C2×Dic3—42no—
280C2×Dic3—44no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—48no—
284C2×Dic3—42no—
285C2×Dic3—44no—
286C2×Dic3—82no—
287C2×Dic3—82no—
288C2×Dic3—84no—
289C2×Dic3—84no—
290C24—22no—
291C24—21no—
292C24—21no—
293C24—22no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24(C2×D3)⋊C412yes—
299C24C2×C4×D311yescharacteristic
300C24C2×D1211yescharacteristic
301C24#23512yes—
302C24D3×D412yes—
303C24#23712yes—
304C24#23712yes—
305C24#23712yes—
306C24C2×#6711yescharacteristic
307C24C2×#6712yes—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—24no—
311C2×D4—22no—
312C2×D4—24no—
313C2×D4—22no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—68no—
334C22×C4—64no—
335C22×C4—62no—
336C22×C4—64no—
337C22×C4—62no—
338C22×C4—64no—
339C22×C4—124no—
340C22×C4—124no—
341C22×C4—128no—
342C22×C4—124no—
343C22×C4—122no—
344C22×C4—121no—
345C22×C4—122no—
346C22×C4—121no—
347C22×C4—124no—
348C22×C4—122no—
349C22×C4—124no—
350C22×C4—122no—
351C22×C4—122no—
352C22×C4—122no—
353C22×C4—122no—
354C22×C4(C2×D3)⋊C412yes—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—64no—
360C22⋊C4—64no—
361C22⋊C4—64no—
362C22⋊C4—82no—
363C22⋊C4—82no—
364C22⋊C4—128no—
365C22⋊C4—121no—
366C22⋊C4—122no—
367C22⋊C4—121no—
368C22⋊C4—122no—
369C22×C3—21no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—22no—
375C22×C3—21no—
376C22×C3—22no—
377C22×C3—24no—
378C22×C3—24no—
379C22×C3—24no—
380C22×C3—24no—
381C22×C3—24no—
382C22×C3—24no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3—42no—
386C22×C3—41no—
387C22×C3—42no—
388C22×C3—41no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3—44no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—82no—
398C22×C3—84no—
399C22×C3—84no—
400C22×C3#38111yescharacteristic
401C22×C3#38611yescharacteristic
402C22×C3#38611yescharacteristic
403C22×C3#38812yes—
404C22×C3#38812yes—
405C22×C3#39412yes—
406C22×C3#39412yes—
407C22×C3#39712yes—
408C22×C3#39712yes—
409C22×C3B(4,𝔽2)12yes—
410C22×C3#45812yes—
411C3×C4—42no—
412C3×C4—84no—
413Dic3—44no—
414Dic3—82no—
415Dic3—84no—
416C23—24no—
417C23—22no—
418C23—24no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—41no—
424C23—41no—
425C23—42no—
426C23—41no—
427C23—41no—
428C23—42no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—42no—
434C23—42no—
435C23—48no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—81no—
441C23—81no—
442C23—84no—
443C23#87711yescharacteristic
444C23#87711yescharacteristic
445C23#87912yes—
446C23#87912yes—
447C23#67⋊C412yes—
448C23#88211yescharacteristic
449C23#88312yes—
450C23#88512yes—
451C23#88612yes—
452C23#88611yescharacteristic
453C23C2×(C2×D3)⋊C411yescharacteristic
454C23C4×#6712yes—
455C23#92911yescharacteristic
456C23#93012yes—
457C23#93512yes—
458C23#93512yes—
459C23#93612yes—
460C23#93812yes—
461C23#93912yes—
462C23#93912yes—
463C23#95311yescharacteristic
464D4—44no—
465D4—44no—
466D4—44no—
467D4—44no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—48no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—64no—
478C2×C4—68no—
479C2×C4—84no—
480C2×C4—122no—
481C2×C4—124no—
482C2×C4—128no—
483C2×C4—124no—
484C2×C4—122no—
485C2×C4—124no—
486C2×C4—122no—
487C2×C4—128no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—124no—
491C2×C4—244no—
492C2×C4—244no—
493C2×C4—242no—
494C2×C4—242no—
495C2×C3—21no—
496C2×C3—22no—
497C2×C3—21no—
498C2×C3—22no—
499C2×C3—22no—
500C2×C3—22no—
501C2×C3—22no—
502C2×C3—82no—
503C2×C3#144011yescharacteristic
504C2×C3#188911yescharacteristic
505C2×C3#189011yescharacteristic
506C2×C3#204414yes—
507C22—24no—
508C22—24no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—21no—
514C22—22no—
515C22—22no—
516C22—21no—
517C22—21no—
518C22—22no—
519C22—21no—
520C22—22no—
521C22—41no—
522C22—41no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—41no—
527C22—42no—
528C22—42no—
529C22—41no—
530C22—42no—
531C22—44no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—82no—
536C22—84no—
537C22—84no—
538C22#494311yescharacteristic
539C22#494412yes—
540C22#494512yes—
541C22#495212yes—
542C22#495312yes—
543C22#495811yescharacteristic
544C22#496112yes—
545C22#521311yescharacteristic
546C22#522212yes—
547C22#524312yes—
548C22#524412yes—
549C4—42no—
550C4—84no—
551C4—124no—
552C4—242no—
553C4—244no—
554C3#1117411yescharacteristic, complemented, Sylow
555C2—21no—
556C2—22no—
557C2—21no—
558C2—22no—
559C2—22no—
560C2—22no—
561C2—22no—
562C2—82no—
563C2#7154911yescharacteristic
564C2#7165411yescharacteristic
565C2#7165511yescharacteristic
566C2#7185014yes—
subgroupG/H
centerC23#953
commutatorC24×C3C22×C4
FrattiniC25C22×D3
FittingC3×#2308C2
socleC23×C3#120
prime radicalC23×C3×D4C22
cosocleC25×C3C23
hypercenter#2308D3
hypocenterC3#11174
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C25 ◁ #2308

Complements

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

NH ≅ G/Ntype
4. #71544C2semidirect
5. C2×#4526C2semidirect
17. #5220C4semidirect
554. C3#11174semidirect

Maximal subgroups

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

subgroupindex
C3×#23082
C2×#52202
C22×#8342
#715442
C2×#45262
#111743