Finite Group Databasebeta

← #381153

Subgroups — #381153

Order 512.

1316 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2640C211yesmaximal, characteristic
2C2×#1768C211yesmaximal, characteristic, complemented
3#14350C211yesmaximal, characteristic, complemented
4#11384C212yesmaximal, complemented
5#11077C212yesmaximal, complemented
6#11076C212yesmaximal
7#11009C212yesmaximal
8#10928C211yesmaximal, characteristic, complemented
9#10927C211yesmaximal, characteristic
10#10926C211yesmaximal, characteristic, complemented
11#10925C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×#97C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2640—24no—
22#2640C418yescomplemented
23C2×#397—22no—
24C2×#393—22no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#1768—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36#1443—22no—
37#1443—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C23×Q8—21no—
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—42no—
49C2×C4⋊Q8—42no—
50C2×C4⋊Q8C2×C414yes—
51C2×#124—22no—
52C2×#124—42no—
53C2×#124—42no—
54C2×#124C2×C414yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×C4⋊C8—24no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—24no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#103—42no—
75C2×#103—42no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—42no—
79C2×#102—42no—
80C2×#102—42no—
81C2×#102—42no—
82C2×#97—22no—
83C2×#97—42no—
84C2×C4×C8—22no—
85C2×C4×C8—42no—
86#397—216no—
87#393—216no—
88#391—22no—
89#391—22no—
90#391C2×C414yes—
91#391C2×C414yes—
92#390—22no—
93#390—22no—
94#390C2×C414yes—
95#390C2×C414yes—
96#376C2×C414yes—
97#376D414yes—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D412yes—
107C2×#95D412yes—
108#340—42no—
109#340—42no—
110#340—42no—
111#340—42no—
112#336—24no—
113C25—21no—
114C22×Q8—24no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—82no—
125C22×Q8—82no—
126C22×Q8C22×C412yes—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4C22×C412yes—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2411yescharacteristic
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8#4114yes—
158C22×C8#4114yes—
159C4⋊Q8—48no—
160C4⋊Q8—48no—
161#124—48no—
162#124—48no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—82no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4—82no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42C22×C412yes—
205C2×C42C22×C412yes—
206C2×C42C22×C412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C4⋊C8—216no—
211#103—48no—
212#103—48no—
213#102—48no—
214#102—48no—
215#97—44no—
216C4×C8—44no—
217#95—28no—
218#95—28no—
219#95—82no—
220#95—82no—
221#95—82no—
222#95—82no—
223#95—82no—
224#95—82no—
225#95C22⋊C418yes—
226#95C22⋊C418yes—
227C24—22no—
228C24—42no—
229C24—41no—
230C24—41no—
231C24—42no—
232C24—42no—
233C24—42no—
234C24—42no—
235C24—42no—
236C24C22×D411yescharacteristic
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—48no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—88no—
255C2×Q8—84no—
256C2×Q8—84no—
257C2×Q8—88no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4C2×C22⋊C414yes—
324C22×C4C2×C22⋊C414yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C23×C411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C2×#4112yes—
330C22×C4C2×#4112yes—
331C22×C4H(5,𝔽2)12yes—
332C22×C4H(5,𝔽2)12yes—
333C22×C4H(5,𝔽2)12yes—
334C22×C42−1+412yes—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—48no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C22⋊C4—48no—
363C22⋊C4—48no—
364C22⋊C4—88no—
365C22⋊C4—88no—
366C42—22no—
367C42—22no—
368C42—24no—
369C42—24no—
370C42—24no—
371C42—22no—
372C42—24no—
373C42—22no—
374C23—42no—
375C23—42no—
376C23—44no—
377C23—44no—
378C23—42no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23—82no—
395C23C22×C22⋊C411yescharacteristic
396C23#53212yes—
397C23#53511yescharacteristic
398C23#53511yescharacteristic
399C23#53712yes—
400C23#57612yes—
401C23#57712yes—
402C23#57712yes—
403C23#57812yes—
404Q8—44no—
405Q8—44no—
406Q8—816no—
407Q8—82no—
408Q8—82no—
409D4—44no—
410D4—44no—
411D4—82no—
412D4—82no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4C4×#4111yescharacteristic
467C2×C4C4×#4111yescharacteristic
468C2×C4#51812yes—
469C2×C4#52012yes—
470C2×C4#52512yes—
471C2×C4#53511yescharacteristic
472C2×C4#53512yes—
473C2×C4#53611yescharacteristic
474C2×C4#53812yes—
475C2×C4#53812yes—
476C2×C4#53812yes—
477C2×C4#53812yes—
478C2×C4#54212yes—
479C2×C4#54212yes—
480C2×C4#54212yes—
481C2×C4#54212yes—
482C8—44no—
483C8—44no—
484C22—82no—
485C22—82no—
486C22—84no—
487C22—84no—
488C22—82no—
489C22—162no—
490C22—162no—
491C22—162no—
492C22—162no—
493C22#229711yescharacteristic
494C22#230911yescharacteristic
495C22#231111yescharacteristic
496C22#266212yes—
497C22#266212yes—
498C22#289712yes—
499C22#289812yes—
500C22#317312yes—
501C22#317512yes—
502C22#317812yes—
503C22#317912yes—
504C22#322011yescharacteristic
505C22#322111yescharacteristic
506C22#322211yescharacteristic
507C22#322311yescharacteristic
508C22#322412yes—
509C22#322512yes—
510C22#325712yes—
511C22#325812yes—
512C22#326212yes—
513C22#326312yes—
514C4—42no—
515C4—42no—
516C4—44no—
517C4—44no—
518C4—82no—
519C4—82no—
520C4—82no—
521C4—162no—
522C4—162no—
523C4#233712yes—
524C4#233712yes—
525C4#263112yes—
526C4#263212yes—
527C2—162no—
528C2#1428811yescharacteristic
529C2#2141212yes—
530C2#2181311yescharacteristic
531C2#2181411yescharacteristic
532C2#2189612yes—
533C2#3034812yes—
534C2#3035012yes—
535C2#3035212yes—
536C2#3035412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1768C2semidirect
3. #14350C2semidirect
4. #11384C2semidirect
5. #11077C2semidirect
8. #10928C2semidirect
10. #10926C2semidirect
22. #2640C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26402
C2×#17682
#143502
#113842
#110772
#110762
#110092
#109282
#109272
#109262
#109252