Finite Group Databasebeta

← #7605695

Subgroups — #7605695

Order 512.

1358 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59871C211yesmaximal, characteristic
2#59870C211yesmaximal, characteristic
3#32553C214yesmaximal
4#29287C211yesmaximal, characteristic
5#28255C211yesmaximal, characteristic
6#28198C214yesmaximal
7#27738C212yesmaximal
8#27737C212yesmaximal
9#26113C212yesmaximal
10#26108C212yesmaximal
11#25324C212yesmaximal
12#25323C212yesmaximal
13#23004C212yesmaximal
14#23004C212yesmaximal
15#21712C211yesmaximal, characteristic
16C4×#487C211yesmaximal, characteristic
17C4×#487C211yesmaximal, characteristic
18C2×#487C2211yescharacteristic
19C2×#487C2211yescharacteristic
20C2×#462C2212yes—
21C2×#462C2212yes—
22C2×#446C2212yes—
23C2×#441C2214yes—
24#2806—22no—
25#2805—22no—
26#2584—22no—
27#2582C2214yes—
28#2581C2214yes—
29#2571C2212yes—
30#2568C2212yes—
31#2567—21no—
32#2567C2212yes—
33#2552—21no—
34#2552C2212yes—
35#2511—22no—
36#2509—22no—
37#2507—22no—
38#2369—21no—
39#2365C2212yes—
40#2365C2212yes—
41#2355—21no—
42Q8×C4⋊C4C2212yes—
43Q8×C4⋊C4C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49#2101C2214yes—
50#2099C2214yes—
51#2088C2214yes—
52#2088C2214yes—
53#2074C2214yes—
54#2072C2214yes—
55#2056C2214yes—
56#2056C2214yes—
57#1990C2211yescharacteristic
58#1990C2211yescharacteristic
59#1988C2212yes—
60#1987C2212yes—
61#1984C2212yes—
62#1956C2211yescharacteristic
63#1956C2211yescharacteristic
64#1948C2212yes—
65#1948C2212yes—
66#1939C2212yes—
67#1939C2212yes—
68#1935C2214yes—
69#1935C2214yes—
70#1933C2212yes—
71#1933C2212yes—
72#1933C2212yes—
73#1933C2212yes—
74#1878C2214yes—
75#1878C2214yes—
76#1865C2214yes—
77#1865C2214yes—
78#1842C2211yescharacteristic
79#1836C2214yes—
80#1834C2211yescharacteristic
81#1834C2211yescharacteristic
82#1805C2214yes—
83#1803C2212yes—
84#1803C2214yes—
85#1803C2212yes—
86#1770C2212yes—
87#1770C2212yes—
88C4×#103C2212yes—
89C4×#103C2212yes—
90C4×#103C2212yes—
91C4×#103C2212yes—
92C42×C8C2211yescharacteristic
93C22×Q16C2314yes—
94C2×C4⋊Q8D411yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×#125—22no—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2314yes—
105C2×C4×Q8C2314yes—
106C2×C4×Q8C2312yes—
107#487—24no—
108#487—24no—
109#462—28no—
110#462—28no—
111#446—44no—
112#441—48no—
113C2×#106C2314yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2314yes—
118C2×#103C2312yes—
119C2×#103C2312yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2312yes—
123C2×#103C2312yes—
124C2×#103C2312yes—
125C2×#103C2312yes—
126C2×#103C2312yes—
127C2×#103C2312yes—
128C2×#97C2314yes—
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135#400—24no—
136#400—24no—
137#400—24no—
138#398—24no—
139#398—24no—
140#398—24no—
141#395—24no—
142#395—24no—
143#391—24no—
144#391—22no—
145#391—22no—
146#391—21no—
147#391—21no—
148#391D412yes—
149#391C2312yes—
150#391C2312yes—
151#391C2314yes—
152#391C2314yes—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—24no—
158#389—22no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—24no—
167#389—24no—
168#389D412yes—
169#389D412yes—
170#389C2312yes—
171#389C2312yes—
172#389C2314yes—
173#389C2314yes—
174#387—24no—
175#387—24no—
176#387—24no—
177#387—24no—
178#384—22no—
179#384—22no—
180#384D411yescharacteristic
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#384C2314yes—
186#382—22no—
187#382C2314yes—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C43C2311yescharacteristic
203#340C2314yes—
204#340C2314yes—
205#340C2312yes—
206#340C2314yes—
207#340C2312yes—
208#340C2314yes—
209#340C2312yes—
210#340C2312yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#336C2314yes—
216#336C2314yes—
217C22×Q8—21no—
218C22×Q8—21no—
219C22×Q8C2×D412yes—
220C22×Q8C2412yes—
221C22×Q8C2412yes—
222C2×Q16—28no—
223C2×Q16—28no—
224C2×Q16—44no—
225C22×C8C2414yes—
226C22×C8C2411yescharacteristic
227C22×C8C2411yescharacteristic
228C22×C8C2411yescharacteristic
229C22×C8C2411yescharacteristic
230C4⋊Q8—24no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233C4⋊Q8—22no—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237C4⋊Q8#4114yes—
238C4⋊Q8#4114yes—
239#125—84no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—24no—
243C4×Q8—24no—
244C4×Q8—42no—
245C4×Q8—42no—
246C4×Q8—44no—
247C4×Q8—44no—
248C4×Q8—44no—
249C4×Q8—44no—
250C4×Q8—44no—
251C4×Q8—82no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2414yes—
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302#106—44no—
303C4⋊C8—24no—
304C4⋊C8—24no—
305C4⋊C8—42no—
306C4⋊C8—44no—
307#103—28no—
308#103—28no—
309#103—24no—
310#103—24no—
311#103—24no—
312#103—24no—
313#103—24no—
314#103—24no—
315#103—42no—
316#103—42no—
317#97—28no—
318C4×C8—22no—
319C4×C8—22no—
320C4×C8—22no—
321C4×C8—22no—
322C4×C8—22no—
323C4×C8—41no—
324#95—24no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—22no—
331#95—24no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95C2×D414yes—
341#95C2×D414yes—
342C2×Q8—24no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—81no—
360C2×Q8—81no—
361C2×Q8C2×#4114yes—
362C2×Q8C2×#4114yes—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D414yes—
373C22×C4C22×D414yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C2511yescharacteristic
380Q16—416no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—21no—
390C2×C8—24no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8H(5,𝔽2)12yes—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4—81no—
435C4⋊C4—82no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C42—22no—
439C42—22no—
440C42—21no—
441C42—24no—
442C42—21no—
443C42—24no—
444C42—41no—
445C42—44no—
446C42—44no—
447C42—44no—
448C42—44no—
449C42—84no—
450C42—84no—
451C42#13714yes—
452C42C2×#4112yes—
453C42C2×#4112yes—
454C422−1+412yes—
455C422−1+412yes—
456C23C23×D411yescharacteristic
457Q8—24no—
458Q8—24no—
459Q8—24no—
460Q8—24no—
461Q8—42no—
462Q8—42no—
463Q8—42no—
464Q8—42no—
465Q8—84no—
466Q8—84no—
467Q8—84no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—21no—
480C2×C4—24no—
481C2×C4—21no—
482C2×C4—24no—
483C2×C4—21no—
484C2×C4—24no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4#54812yes—
508C2×C4#54812yes—
509C2×C4#54812yes—
510C2×C4#54812yes—
511C2×C4C2×#13712yes—
512C2×C4C2×#13712yes—
513C2×C4C2×#13712yes—
514C2×C4#57712yes—
515C2×C4#57812yes—
516C2×C4C22×#4111yescharacteristic
517C2×C4C22×#4111yescharacteristic
518C2×C4C2×H(5,𝔽2)11yescharacteristic
519C2×C4C2×2−1+411yescharacteristic
520C2×C4C2×2−1+411yescharacteristic
521C2×C4#58511yescharacteristic
522C8—22no—
523C8—41no—
524C8—44no—
525C8—41no—
526C8—41no—
527C22C2×#54811yescharacteristic
528C22C2×#54811yescharacteristic
529C22#349611yescharacteristic
530C22C22×#13711yescharacteristic
531C22C2×#57711yescharacteristic
532C22C2×#57811yescharacteristic
533C22#358411yescharacteristic
534C4—22no—
535C4—22no—
536C4—22no—
537C4—44no—
538C4—44no—
539C4—44no—
540C4—44no—
541C4—41no—
542C4—82no—
543C4—82no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4#336914yes—
549C4#337414yes—
550C4#348111yescharacteristic
551C4#351511yescharacteristic
552C4#351511yescharacteristic
553C4#354411yescharacteristic
554C2#5966011yescharacteristic
555C2C2×#337411yescharacteristic
556C2C2×#336911yescharacteristic
557C2#6132711yescharacteristic
558C2#6132811yescharacteristic
559C2#6226011yescharacteristic
560C2#6234311yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#598712
#598702
#325532
#292872
#282552
#281982
#277382
#277372
#261132
#261082
#253242
#253232
#230042
#217122
C4×#4872