Finite Group Databasebeta

← #413883

Subgroups — #413883

Order 512.

1492 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2327C211yesmaximal, characteristic
2C2×#2043C211yesmaximal, characteristic
3C2×#2016C211yesmaximal, characteristic
4C2×#1878C212yesmaximal
5#14261C211yesmaximal, characteristic
6#11208C211yesmaximal, characteristic
7#11190C212yesmaximal
8#11052C211yesmaximal, characteristic
9#11024C212yesmaximal
10#10796C211yesmaximal, characteristic
11#10789C212yesmaximal
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19#2327—28no—
20C2×#391C2211yescharacteristic
21C2×#391C2212yes—
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#387—22no—
26C2×#387—22no—
27C2×#382C2211yescharacteristic
28C2×#376C2211yescharacteristic
29#2043—24no—
30#2016—24no—
31#1878—28no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1443—22no—
40#1442—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×Q8C2312yes—
45C22×Q16—24no—
46C22×Q16—24no—
47C22×Q16—24no—
48C22×Q16—24no—
49C22×Q16—24no—
50C22×Q16—42no—
51C22×Q16—42no—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8D414yes—
58C2×C4×Q8—24no—
59C2×C4×Q8—42no—
60C2×C4×Q8—42no—
61C2×C4×Q8—42no—
62C2×C4×Q8—42no—
63C2×C4×Q8D418yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
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—24no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88#391—24no—
89#391—22no—
90#391—24no—
91#391—22no—
92#391—24no—
93#391—22no—
94#391—22no—
95#389—22no—
96#389—22no—
97#387—216no—
98#387—216no—
99#382—24no—
100#376—24no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—24no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95D412yes—
112C2×#95D412yes—
113#340—24no—
114#340—42no—
115#340—42no—
116#336—24no—
117#336—24no—
118#336—22no—
119#336—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—21no—
140C22×Q8C2×D414yes—
141C22×Q8C2×D414yes—
142C22×Q8#4112yes—
143C22×Q8#4114yes—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C2×Q16—216no—
156C2×Q16—48no—
157C2×Q16—48no—
158C2×Q16—48no—
159C2×Q16—48no—
160C2×Q16—48no—
161C2×Q16—48no—
162C22×C8—22no—
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#4114yes—
176C4⋊Q8—216no—
177C4⋊Q8—216no—
178C4×Q8—48no—
179C4×Q8—48no—
180C4×Q8—48no—
181C4×Q8—48no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4#4112yes—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—28no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—44no—
224C2×C42—44no—
225C2×C42C2×D414yes—
226C2×C42C2×D412yes—
227C2×C42#4112yes—
228#103—48no—
229#103—48no—
230#103—48no—
231#103—48no—
232#95—24no—
233#95—24no—
234#95—216no—
235#95—28no—
236#95—24no—
237#95—24no—
238#95—44no—
239#95—42no—
240#95—42no—
241#95—42no—
242#95—82no—
243#95—82no—
244C24C22×D411yescharacteristic
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—22no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—22no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—41no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—41no—
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—42no—
287C2×Q8—42no—
288C2×Q8#12118yes—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4#12014yes—
347C22×C4#12112yes—
348C22×C4#12112yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D412yes—
351C22×C4C2×#4112yes—
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4112yes—
355C22×C4C2×#4111yescharacteristic
356C22×C4H(5,𝔽2)11yescharacteristic
357C22×C4H(5,𝔽2)12yes—
358C22×C4H(5,𝔽2)11yescharacteristic
359Q16—432no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
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—
372C2×C8—42no—
373C2×C8—42no—
374C4⋊C4—216no—
375C4⋊C4—216no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—416no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—84no—
390C42—24no—
391C42—24no—
392C42—24no—
393C42—24no—
394C42—416no—
395C42—42no—
396C42—42no—
397C23C2×#12011yescharacteristic
398C23C2×#12111yescharacteristic
399C23#52511yescharacteristic
400C23#52511yescharacteristic
401C23#53211yescharacteristic
402C23#53511yescharacteristic
403C23#53511yescharacteristic
404C23#57712yes—
405C23#57812yes—
406C23#57812yes—
407C23#57812yes—
408Q8—28no—
409Q8—28no—
410Q8—44no—
411Q8—44no—
412Q8—44no—
413Q8—44no—
414Q8—44no—
415Q8—44no—
416Q8—44no—
417Q8—44no—
418Q8—82no—
419Q8—82no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—28no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4C2×#12112yes—
481C2×C4#52512yes—
482C2×C4#52512yes—
483C2×C4#53212yes—
484C2×C4#53412yes—
485C2×C4#53412yes—
486C2×C4#54611yescharacteristic
487C2×C4#54612yes—
488C2×C4#54611yescharacteristic
489C2×C4#55112yes—
490C2×C4#55112yes—
491C2×C4#55311yescharacteristic
492C2×C4#55312yes—
493C2×C4#55311yescharacteristic
494C2×C4#55312yes—
495C2×C4#55912yes—
496C8—44no—
497C8—82no—
498C22#238811yescharacteristic
499C22#253511yescharacteristic
500C22#254411yescharacteristic
501C22#259811yescharacteristic
502C22#260311yescharacteristic
503C22#261311yescharacteristic
504C22#284711yescharacteristic
505C22#301112yes—
506C22#301312yes—
507C22#304311yescharacteristic
508C22#304411yescharacteristic
509C22#304411yescharacteristic
510C22#304411yescharacteristic
511C22#312012yes—
512C22#312112yes—
513C22#312712yes—
514C22#312712yes—
515C22#317712yes—
516C22#317812yes—
517C22#322312yes—
518C22#322312yes—
519C22#322512yes—
520C22#322512yes—
521C4—48no—
522C4—42no—
523C4—42no—
524C4—44no—
525C4—82no—
526C4—82no—
527C4—82no—
528C4—84no—
529C4—161no—
530C4—161no—
531C4#253412yes—
532C4#254312yes—
533C4#260012yes—
534C4#260212yes—
535C2#1515911yescharacteristic
536C2#2358612yes—
537C2#2816811yescharacteristic
538C2#2817011yescharacteristic
539C2#2862511yescharacteristic
540C2#2862811yescharacteristic
541C2#2937912yes—
542C2#2950912yes—
543C2#2963312yes—
544C2#3291511yescharacteristic
545C2#3291811yescharacteristic
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×#20432
C2×#20162
C2×#18782
#142612
#112082
#111902
#110522
#110242
#107962
#107892