Finite Group Databasebeta

← #10690568

Subgroups — #10690568

Order 768.

964 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊C8C211yesmaximal, characteristic
2C2×#5210C211yesmaximal, characteristic
3#72126C211yesmaximal, characteristic
4C2×#4469C212yesmaximal, complemented
5#67674C212yesmaximal, complemented
6#9081—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C23×C3×C8—21no—
9C22×#217C411yescharacteristic
10C22×#217C411yescharacteristic
11C22×Dic3⋊C4C411yescharacteristic
12C22×Dic3⋊C4C2211yescharacteristic
13C2×C3×C22⋊C8—21no—
14C2×C3×C22⋊C8—24no—
15C2×C3×C22⋊C8C2212yes—
16#5210—21no—
17#5210—21no—
18#5210—21no—
19#5210—21no—
20C2×C3⋊#95C411yescharacteristic
21C2×C3⋊#95C2211yescharacteristic
22C2×#830—21no—
23C2×#830—21no—
24C2×#830C2212yes—
25#4469—24no—
26C22×C22⋊C8D311yescharacteristic
27C2×#385—31no—
28#1779—31no—
29C2×#324—32no—
30#1280—32no—
31C25×C3C2×C411yescharacteristic
32C23×C3×C4—22no—
33C23×C3×C4—22no—
34C23×C3×C4—42no—
35C23×C3×C4—42no—
36C23×C3×C4C2×C411yescharacteristic
37C23×C3×C4D411yescharacteristic
38C23×C3×C4D411yescharacteristic
39C23×C3×C4D411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C23×Dic3C2×C411yescharacteristic
43C23×Dic3C2×C411yescharacteristic
44C23×Dic3C2×C411yescharacteristic
45C23×Dic3C2×C411yescharacteristic
46C22×C3×C8—24no—
47C22×C3×C8—22no—
48C22×C3×C8—42no—
49C22×C3×C8—42no—
50C2×#217—21no—
51C2×#217—21no—
52C2×#217—22no—
53C2×#217—21no—
54C2×#217—21no—
55C2×#217—42no—
56C2×#217C814yescomplemented
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—22no—
62C2×Dic3⋊C4—22no—
63C2×Dic3⋊C4—42no—
64C22×C3⋊C8—22no—
65C3×C22⋊C8—24no—
66C3×C22⋊C8—22no—
67C3×C22⋊C8D414yes—
68C3⋊#95—21no—
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72#830—22no—
73#830—22no—
74#830—42no—
75#830—42no—
76C24×C4C2×D311yescharacteristic
77C23×C8—21no—
78C22×C4⋊C4—31no—
79C22×C4⋊C4—31no—
80C22×C22⋊C4—31no—
81C22×C22⋊C4—31no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—32no—
85C2×C22⋊C8—61no—
86C2×C22⋊C8—61no—
87C2×C22⋊C8C2×D312yes—
88#385—61no—
89#385—61no—
90#385—61no—
91#385—61no—
92C2×#95—31no—
93C2×#95—31no—
94#324—64no—
95C24×C3—22no—
96C24×C3—22no—
97C24×C3—22no—
98C24×C3—22no—
99C24×C3—41no—
100C24×C3—41no—
101C24×C3C22⋊C411yescharacteristic
102C24×C3C22⋊C411yescharacteristic
103C24×C3C2×C812yes—
104C24×C3M1612yes—
105C24×C3C22×C411yescharacteristic
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—21no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—48no—
126C22×C3×C4—48no—
127C22×C3×C4C22⋊C411yescharacteristic
128C22×C3×C4C22⋊C411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D412yes—
132C22×C3×C4#4112yes—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—22no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—22no—
139C22×Dic3—21no—
140C22×Dic3—21no—
141C22×Dic3—22no—
142C22×Dic3—22no—
143C22×Dic3—41no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3—41no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C22×Dic3C2×C812yescomplemented
150C22×Dic3C2×C812yescomplemented
151C2×C3×C8—24no—
152C2×C3×C8—48no—
153C2×C3×C8—42no—
154C2×C3×C8—42no—
155#217—24no—
156#217—24no—
157#217—42no—
158#217—42no—
159Dic3⋊C4—42no—
160Dic3⋊C4—42no—
161Dic3⋊C4—41no—
162Dic3⋊C4—42no—
163Dic3⋊C4—41no—
164C2×C3⋊C8—42no—
165C2×C3⋊C8—42no—
166C2×C3⋊C8—42no—
167C25C4×D311yescharacteristic
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—31no—
171C23×C4—31no—
172C23×C4—31no—
173C23×C4—31no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4C4×D311yescharacteristic
177C23×C4D1211yescharacteristic
178C23×C4D1211yescharacteristic
179C23×C4#6711yescharacteristic
180C23×C4#6711yescharacteristic
181C23×C4C22×D311yescharacteristic
182C22×C8—24no—
183C22×C8—22no—
184C22×C8—42no—
185C22×C8—42no—
186C22×C8—62no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—62no—
191C2×C4⋊C4—62no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—122no—
194C2×C22⋊C4—34no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—122no—
201C22⋊C8—24no—
202C22⋊C8—22no—
203C22⋊C8—62no—
204C22⋊C8—62no—
205C22⋊C8—122no—
206C22⋊C8—122no—
207C22⋊C8#6714yes—
208#95—61no—
209#95—61no—
210#95—61no—
211#95—61no—
212C23×C3—21no—
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—21no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—24no—
221C23×C3—24no—
222C23×C3—21no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—24no—
229C23×C3—24no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3—44no—
233C23×C3—44no—
234C23×C3—44no—
235C23×C3—44no—
236C23×C3C22⋊C812yes—
237C23×C3C22⋊C812yes—
238C23×C3#9912yes—
239C23×C3#10112yes—
240C23×C3C2×C22⋊C411yescharacteristic
241C23×C3C4×D412yes—
242C23×C3C22×C811yescharacteristic
243C23×C3C2×M1611yescharacteristic
244C23×C3#13112yes—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—44no—
259C2×C3×C4—44no—
260C2×C3×C4—44no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4#9912yes—
264C2×C3×C4#12011yescharacteristic
265C2×C3×C4#12112yes—
266C2×C3×C4#12311yescharacteristic
267C2×Dic3—22no—
268C2×Dic3—22no—
269C2×Dic3—24no—
270C2×Dic3—42no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C2×Dic3—42no—
276C2×Dic3—42no—
277C2×Dic3—41no—
278C2×Dic3—41no—
279C2×Dic3—41no—
280C2×Dic3—41no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—82no—
284C2×Dic3—82no—
285C3×C8—44no—
286C3⋊C8—82no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—41no—
292C24—41no—
293C24D3×C812yes—
294C24ΓL(1,𝔽25)12yes—
295C24(C2×D3)⋊C411yescharacteristic
296C24(C2×D3)⋊C411yescharacteristic
297C24C2×C4×D311yescharacteristic
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—32no—
311C22×C4—32no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—48no—
320C22×C4—48no—
321C22×C4—61no—
322C22×C4—61no—
323C22×C4—62no—
324C22×C4—61no—
325C22×C4—61no—
326C22×C4—62no—
327C22×C4—61no—
328C22×C4—61no—
329C22×C4—62no—
330C22×C4—62no—
331C22×C4—121no—
332C22×C4—121no—
333C22×C4—121no—
334C22×C4—121no—
335C22×C4—121no—
336C22×C4—121no—
337C22×C4(C2×D3)⋊C411yescharacteristic
338C22×C4(C2×D3)⋊C411yescharacteristic
339C22×C4C2×D1211yescharacteristic
340C22×C4#23512yes—
341C22×C4C2×#6711yescharacteristic
342C22×C4C2×#6712yes—
343C2×C8—24no—
344C2×C8—48no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—122no—
348C2×C8—122no—
349C2×C8—122no—
350C4⋊C4—122no—
351C4⋊C4—122no—
352C4⋊C4—121no—
353C4⋊C4—122no—
354C4⋊C4—121no—
355C22⋊C4—64no—
356C22⋊C4—64no—
357C22⋊C4—122no—
358C22⋊C4—122no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—22no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—48no—
377C22×C3—48no—
378C22×C3—41no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—41no—
382C22×C3#32414yescomplemented
383C22×C3#38611yescharacteristic
384C22×C3C2×C22⋊C811yescharacteristic
385C22×C3#40811yescharacteristic
386C22×C3C2×#9911yescharacteristic
387C22×C3C2×#9911yescharacteristic
388C22×C3C2×#10111yescharacteristic
389C22×C3#41311yescharacteristic
390C22×C3C8×D412yes—
391C22×C3#43512yes—
392C3×C4—22no—
393C3×C4—21no—
394C3×C4—22no—
395C3×C4—21no—
396C3×C4—41no—
397Dic3—42no—
398Dic3—42no—
399Dic3—81no—
400Dic3—81no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—21no—
418C23—24no—
419C23—42no—
420C23—42no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23#80612yes—
426C23(C2×D3)⋊C812yes—
427C23(C2×D3)⋊C812yes—
428C23#82412yes—
429C23C2×D3×C811yescharacteristic
430C23C2×ΓL(1,𝔽25)11yescharacteristic
431C23#90112yes—
432C23C2×(C2×D3)⋊C411yescharacteristic
433C23C4×#6712yes—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—64no—
455C2×C4—122no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4—242no—
469C2×C4—242no—
470C2×C4#80612yes—
471C2×C4#92911yescharacteristic
472C2×C4#93012yes—
473C2×C4#95311yescharacteristic
474C8—44no—
475C8—242no—
476C2×C3—24no—
477C2×C3—22no—
478C2×C3—22no—
479C2×C3—42no—
480C2×C3C2×#32411yescharacteristic
481C2×C3#145911yescharacteristic
482C2×C3#146811yescharacteristic
483C2×C3#147211yescharacteristic
484C2×C3#186911yescharacteristic
485C2×C3#189811yescharacteristic
486C2×C3#190111yescharacteristic
487C22—21no—
488C22—21no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—21no—
493C22—21no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—48no—
505C22—48no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22#446914yescomplemented
511C22C2×#80611yescharacteristic
512C22C2×#80611yescharacteristic
513C22C2×(C2×D3)⋊C811yescharacteristic
514C22C8×#6712yes—
515C22#511011yescharacteristic
516C22#511112yes—
517C22#513211yescharacteristic
518C22C2×#82411yescharacteristic
519C22#521311yescharacteristic
520C4—21no—
521C4—22no—
522C4—22no—
523C4—21no—
524C4—41no—
525C4—122no—
526C4—122no—
527C4—241no—
528C4—241no—
529C3#908111yescharacteristic, complemented, Sylow
530C2—24no—
531C2—22no—
532C2—22no—
533C2—42no—
534C2#6991911yescharacteristic
535C2#6992011yescharacteristic
536C2C2×#446911yescharacteristic
537C2#6992311yescharacteristic
538C2#7154311yescharacteristic
539C2#7197511yescharacteristic
540C2#7200311yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC23×C3C22×C8
FrattiniC23×C4C22×D3
FittingC22×C3×C22⋊C8C2
socleC23×C3C2×C22⋊C4
prime radicalC25×C3C2×C4
cosocleC23×C3×C4C23
hypercenterC22×C22⋊C8D3
hypocenterC3#9081
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊C8

Complements

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

NH ≅ G/Ntype
4. C2×#4469C2semidirect
5. #67674C2semidirect
56. C2×#217C8semidirect
149. C22×Dic3C2×C8semidirect
150. C22×Dic3C2×C8semidirect
382. C22×C3#324semidirect
510. C22#4469semidirect
529. C3#9081semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊C82
C2×#52102
#721262
C2×#44692
#676742
#90813