Finite Group Databasebeta

← #400013

Subgroups — #400013

Order 512.

1492 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2327C211yesmaximal, characteristic
2C2×#2074C211yesmaximal, characteristic
3C2×#2016C211yesmaximal, characteristic
4C2×#1878C212yesmaximal
5#14261C211yesmaximal, characteristic
6#11358C211yesmaximal, characteristic
7#11067C212yesmaximal
8#11024C212yesmaximal
9#11010C211yesmaximal, characteristic
10#10796C211yesmaximal, characteristic
11#10789C212yesmaximal
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19#2327—24no—
20#2327—24no—
21C2×#391C2212yes—
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#387—22no—
27C2×#387—22no—
28C2×#382C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#2074—24no—
31#2016—24no—
32#1878—28no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40#1443—22no—
41#1442—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C23×Q8C2311yescharacteristic
45C23×Q8C2312yes—
46C22×Q16—24no—
47C22×Q16—24no—
48C22×Q16—24no—
49C22×Q16—24no—
50C22×Q16—24no—
51C22×Q16—42no—
52C22×Q16—42no—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8D414yes—
59C2×C4×Q8—24no—
60C2×C4×Q8—24no—
61C2×C4×Q8—24no—
62C2×C4×Q8—24no—
63C2×C4×Q8—42no—
64C2×C4×Q8—42no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C42—22no—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88#391—24no—
89#391—24no—
90#391—22no—
91#391—22no—
92#391—22no—
93#391—42no—
94#391D414yes—
95#389—42no—
96#387—216no—
97#387—216no—
98#382—22no—
99#382D414yes—
100#376—22no—
101#376—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D412yes—
113C2×#95D412yes—
114#340—24no—
115#340—24no—
116#340—42no—
117#336—24no—
118#336—22no—
119#336—24no—
120#336—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—21no—
141C22×Q8C2×D412yes—
142C22×Q8C2×D414yes—
143C22×Q8#4114yes—
144C22×Q8#4114yes—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2411yescharacteristic
156C2×Q16—216no—
157C2×Q16—48no—
158C2×Q16—48no—
159C2×Q16—48no—
160C2×Q16—48no—
161C2×Q16—48no—
162C2×Q16—48no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8#4114yes—
177C4⋊Q8—216no—
178C4⋊Q8—216no—
179C4×Q8—48no—
180C4×Q8—48no—
181C4×Q8—48no—
182C4×Q8—48no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4C2×D414yes—
208C2×C4⋊C4C2×D412yes—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—28no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—44no—
221C2×C42—44no—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42#4112yes—
226#103—48no—
227#103—48no—
228#103—48no—
229#103—48no—
230#95—28no—
231#95—28no—
232#95—28no—
233#95—24no—
234#95—24no—
235#95—28no—
236#95—42no—
237#95—42no—
238#95—42no—
239#95—44no—
240#95—82no—
241#95—82no—
242C24C22×D411yescharacteristic
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8#12318yes—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4#12014yes—
346C22×C4#12114yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C2×#4112yes—
351C22×C4C2×#4112yes—
352C22×C4C2×#4112yes—
353C22×C4C2×#4111yescharacteristic
354C22×C4H(5,𝔽2)11yescharacteristic
355C22×C4H(5,𝔽2)11yescharacteristic
356C22×C4H(5,𝔽2)12yes—
357Q16—432no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C4⋊C4—232no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—416no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—84no—
385C42—24no—
386C42—24no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—416no—
392C42—42no—
393C42—42no—
394C23C2×#12011yescharacteristic
395C23C2×#12111yescharacteristic
396C23#52511yescharacteristic
397C23#52511yescharacteristic
398C23#53211yescharacteristic
399C23#53511yescharacteristic
400C23#53511yescharacteristic
401C23#57712yes—
402C23#57812yes—
403C23#57812yes—
404C23#57812yes—
405Q8—28no—
406Q8—28no—
407Q8—44no—
408Q8—44no—
409Q8—44no—
410Q8—44no—
411Q8—44no—
412Q8—44no—
413Q8—44no—
414Q8—44no—
415Q8—82no—
416Q8—82no—
417C2×C4—24no—
418C2×C4—28no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4C2×#12312yes—
478C2×C4#52512yes—
479C2×C4#52512yes—
480C2×C4#52912yes—
481C2×C4#53812yes—
482C2×C4#53812yes—
483C2×C4D4211yescharacteristic
484C2×C4D4211yescharacteristic
485C2×C4#54612yes—
486C2×C4#54612yes—
487C2×C4#54611yescharacteristic
488C2×C4#54611yescharacteristic
489C2×C4#55012yes—
490C2×C4#55112yes—
491C2×C4#55312yes—
492C2×C4#55312yes—
493C8—44no—
494C8—82no—
495C22#239211yescharacteristic
496C22#240811yescharacteristic
497C22#243111yescharacteristic
498C22#253511yescharacteristic
499C22#260311yescharacteristic
500C22#263111yescharacteristic
501C22#284911yescharacteristic
502C22#301112yes—
503C22#301312yes—
504C22#304412yes—
505C22#304612yes—
506C22#312112yes—
507C22#312112yes—
508C22#312712yes—
509C22#312712yes—
510C22#317712yes—
511C22#317812yes—
512C22#322211yescharacteristic
513C22#322311yescharacteristic
514C22#322311yescharacteristic
515C22#322311yescharacteristic
516C22#322312yes—
517C22#322512yes—
518C4—44no—
519C4—48no—
520C4—44no—
521C4—81no—
522C4—81no—
523C4—82no—
524C4—84no—
525C4—82no—
526C4—161no—
527C4—161no—
528C4#241812yes—
529C4#244412yes—
530C4#245612yes—
531C4#246712yes—
532C2#1479511yescharacteristic
533C2#2369112yes—
534C2#2422112yes—
535C2#2560911yescharacteristic
536C2#2561211yescharacteristic
537C2#2817011yescharacteristic
538C2#2817311yescharacteristic
539C2#2950912yes—
540C2#3005011yescharacteristic
541C2#3005211yescharacteristic
542C2#3301912yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#23272
C2×#20742
C2×#20162
C2×#18782
#142612
#113582
#110672
#110242
#110102
#107962
#107892