Finite Group Databasebeta

← #826003

Subgroups — #826003

Order 512.

2202 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2370C212yesmaximal
2#16563C212yesmaximal
3#16088C212yesmaximal
4#15367C214yesmaximal
5#15363C212yesmaximal
6#15196C211yesmaximal, characteristic
7#15133C212yesmaximal
8#15105C214yesmaximal
9#14151C212yesmaximal
10#14054C212yesmaximal
11#14054C212yesmaximal
12#14046C212yesmaximal
13#14005C212yesmaximal
14#13910C211yesmaximal, characteristic
15C4×#398C211yesmaximal, characteristic
16C22×#125C2212yes—
17C2×#398C2211yescharacteristic
18C2×#398C2214yes—
19#2370—216no—
20C2×#389C2214yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2214yes—
26C2×#389C2214yes—
27C2×#389C2212yes—
28C2×#387C2214yes—
29C2×#387C2214yes—
30C2×#384C2212yes—
31C2×#384C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×C4×C4⋊C4C2214yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40#1445C2214yes—
41#1445C2214yes—
42#1445C2211yescharacteristic
43#1444C2214yes—
44#1444C2211yescharacteristic
45#1444C2212yes—
46#1444C2212yes—
47#1444C2212yes—
48#1444C2212yes—
49#1442C2214yes—
50#1442C2212yes—
51#1441C2212yes—
52#1441C2212yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2211yescharacteristic
60#1441C2212yes—
61#1441C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2211yescharacteristic
74C4×#95C2212yes—
75C4×#95C2211yescharacteristic
76C2×#125—28no—
77C2×#125—28no—
78C2×#125Q8116yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2314yes—
107C22×C42C2314yes—
108#398—24no—
109#398—48no—
110#389—28no—
111#389—28no—
112#389—28no—
113#389—28no—
114#389—216no—
115#389—48no—
116#389—48no—
117#389—44no—
118#387—216no—
119#387—216no—
120#384—28no—
121#384—42no—
122#384—42no—
123C4×C4⋊C4—28no—
124C4×C4⋊C4—28no—
125C4×C4⋊C4—28no—
126C4×C4⋊C4—216no—
127C4×C4⋊C4—42no—
128C4×C4⋊C4—42no—
129C4×C4⋊C4D418yes—
130C2×#95C2312yes—
131C2×#95C2311yescharacteristic
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2311yescharacteristic
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C23×C4C2414yes—
161C23×C4C2414yes—
162C23×C4C2412yes—
163C23×C4C2411yescharacteristic
164C23×C4C2411yescharacteristic
165C23×C4C2412yes—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2411yescharacteristic
169C23×C4C2411yescharacteristic
170C23×C4C2412yes—
171C23×C4C2411yescharacteristic
172C23×C4C2414yes—
173C23×C4C2411yescharacteristic
174C23×C4C2411yescharacteristic
175#125—264no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4C2×D418yes—
229C2×C4⋊C4C2×Q8116yes—
230C2×C4⋊C4C2×Q8116yes—
231C2×C4⋊C4C2×Q818yes—
232C2×C4⋊C4C2×Q818yes—
233C2×C4⋊C4#4114yes—
234C2×C4⋊C4#4118yes—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—28no—
249C2×C42—28no—
250C2×C42—44no—
251C2×C42—44no—
252C2×C42—44no—
253C2×C42—44no—
254C2×C42C2×D418yes—
255C2×C42C2×D414yes—
256C2×C42C2×Q818yes—
257#95—24no—
258#95—24no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—22no—
266#95—24no—
267#95—24no—
268#95—28no—
269#95—24no—
270#95—24no—
271#95—42no—
272#95—41no—
273#95—44no—
274#95—44no—
275#95—42no—
276#95—42no—
277#95—44no—
278#95—41no—
279#95—41no—
280#95—42no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95C2×D418yes—
287C24C2511yescharacteristic
288C22×C4—24no—
289C22×C4—21no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—24no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4C22×D412yes—
363C22×C4C22×D418yes—
364C22×C4C22×D412yes—
365C22×C4C22×D412yes—
366C22×C4C22×Q814yes—
367C22×C4C22×Q814yes—
368C22×C4C22×Q818yes—
369C22×C4C22×Q818yes—
370C22×C4C22×Q814yes—
371C22×C4C2×#4112yes—
372C22×C4C2×#4112yes—
373C22×C4C2×#4118yes—
374C22×C4C2×#4112yes—
375C22×C4C2×#4114yes—
376C22×C4H(5,𝔽2)14yes—
377C22×C4H(5,𝔽2)12yes—
378C22×C42−1+412yes—
379C22×C42−1+412yes—
380C22×C42−1+412yes—
381C4⋊C4—232no—
382C4⋊C4—232no—
383C4⋊C4—216no—
384C4⋊C4—44no—
385C4⋊C4—416no—
386C4⋊C4—416no—
387C4⋊C4—416no—
388C4⋊C4—44no—
389C4⋊C4—48no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—84no—
396C4⋊C4—88no—
397C4⋊C4—88no—
398C4⋊C4—84no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4C22⋊Q8132yes—
402C42—216no—
403C42—28no—
404C42—216no—
405C42—48no—
406C42—44no—
407C42—44no—
408C42—88no—
409C42—88no—
410C23C23×D411yescharacteristic
411C23C23×Q812yes—
412C23C22×#4112yes—
413C23C22×#4111yescharacteristic
414C23C2×H(5,𝔽2)11yescharacteristic
415C23C2×H(5,𝔽2)12yes—
416C23C2×2−1+411yescharacteristic
417C23C2×2−1+411yescharacteristic
418C23C2×2−1+411yescharacteristic
419C23#58512yes—
420C23#58511yescharacteristic
421C2×C4—22no—
422C2×C4—28no—
423C2×C4—28no—
424C2×C4—28no—
425C2×C4—28no—
426C2×C4—22no—
427C2×C4—28no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—28no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—28no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—28no—
445C2×C4—24no—
446C2×C4—28no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—41no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4C2×C22⋊Q8116yes—
508C2×C4C2×C22⋊Q818yes—
509C2×C4#53714yes—
510C2×C4#54714yes—
511C2×C4#55718yes—
512C2×C4Q8214yes—
513C2×C4#55914yes—
514C22C22×C22⋊Q812yes—
515C22C2×#53711yescharacteristic
516C22C2×#54711yescharacteristic
517C22C2×#55712yes—
518C22C2×Q8211yescharacteristic
519C22Q8×#4112yes—
520C22C2×#55911yescharacteristic
521C22#348112yes—
522C22#348812yes—
523C22#349711yescharacteristic
524C22#350312yes—
525C22#350312yes—
526C22#350412yes—
527C22#350511yescharacteristic
528C22#350911yescharacteristic
529C22#351512yes—
530C22#351612yes—
531C22#351712yes—
532C22#353711yescharacteristic
533C22#353811yescharacteristic
534C22#355111yescharacteristic
535C22#355912yes—
536C22#356311yescharacteristic
537C4—24no—
538C4—216no—
539C4—42no—
540C4—44no—
541C4—42no—
542C4—44no—
543C4—44no—
544C4—48no—
545C4—42no—
546C4—42no—
547C4—82no—
548C4—81no—
549C4—84no—
550C4—84no—
551C4#258218yes—
552C2C2×#258211yescharacteristic
553C2#3581612yes—
554C2#3627512yes—
555C2#3948711yescharacteristic
556C2#4295211yescharacteristic
557C2#4522312yes—
558C2#4522611yescharacteristic
559C2#4543511yescharacteristic
560C2#5963211yescharacteristic
561C2#5967012yes—
562C2#5992511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#23702
#165632
#160882
#153672
#153632
#151962
#151332
#151052
#141512
#140542
#140462
#140052
#139102
C4×#3982