Finite Group Databasebeta

← #7615157

Subgroups — #7615157

Order 512.

1334 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
1#59875C211yesmaximal, characteristic
2#59870C211yesmaximal, characteristic
3#32530C214yesmaximal
4#28744C212yesmaximal
5#28542C212yesmaximal
6#28284C211yesmaximal, characteristic
7#28283C211yesmaximal, characteristic
8#28028C214yesmaximal
9#26320C212yesmaximal
10#26319C212yesmaximal
11#24683C212yesmaximal
12#24682C212yesmaximal
13#22814C212yesmaximal
14#22728C212yesmaximal
15C4×#498C211yesmaximal, characteristic
16#21712C211yesmaximal, characteristic
17C4×#494C211yesmaximal, characteristic
18C2×#498C2211yescharacteristic
19C2×#494C2211yescharacteristic
20C2×#477C2212yes—
21C2×#475C2212yes—
22C2×#446C2212yes—
23C2×C4×Q16C2212yes—
24C2×C4×Q16C2212yes—
25#2810—22no—
26#2805—22no—
27#2581—22no—
28#2571—21no—
29#2568—21no—
30#2568C2212yes—
31#2568C2212yes—
32#2568C2212yes—
33#2567C2212yes—
34#2514—22no—
35#2514—22no—
36#2514C2214yes—
37#2511C2214yes—
38#2509—22no—
39#2365—21no—
40#2355—21no—
41#2355C2212yes—
42#2355C2212yes—
43#2355C2212yes—
44Q8×C4⋊C4C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50#2097C2214yes—
51#2096C2214yes—
52#2074C2214yes—
53#2072C2214yes—
54#2068C2214yes—
55#2066C2214yes—
56#2056C2214yes—
57#2056C2214yes—
58#1990C2211yescharacteristic
59#1990C2211yescharacteristic
60#1984C2212yes—
61#1981C2212yes—
62#1981C2212yes—
63#1971C2214yes—
64#1971C2214yes—
65#1969C2212yes—
66#1969C2212yes—
67#1956C2211yescharacteristic
68#1956C2211yescharacteristic
69#1949C2214yes—
70#1948C2212yes—
71#1947C2214yes—
72#1947C2212yes—
73#1941C2212yes—
74#1941C2212yes—
75#1939C2212yes—
76#1939C2212yes—
77#1928C2214yes—
78#1927C2214yes—
79#1927C2214yes—
80#1927C2214yes—
81#1847C2214yes—
82#1845C2211yescharacteristic
83#1844C2211yescharacteristic
84#1842C2211yescharacteristic
85#1803C2212yes—
86#1803C2212yes—
87#1776C2214yes—
88C4×#106C2212yes—
89C4×#106C2212yes—
90#1770C2212yes—
91#1770C2212yes—
92C42×C8C2211yescharacteristic
93C22×Q16C2312yes—
94C22×Q16C2312yes—
95C2×C4⋊Q8D411yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4×Q8—22no—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106#498—24no—
107#494—24no—
108#477—44no—
109#475—44no—
110#446—28no—
111C4×Q16—28no—
112C4×Q16—44no—
113C2×#107C2312yes—
114C2×#107C2314yes—
115C2×#107C2312yes—
116C2×#106C2312yes—
117C2×#106C2312yes—
118C2×#106C2312yes—
119C2×#106C2314yes—
120C2×#106C2312yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2314yes—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2312yes—
125C2×#103C2312yes—
126C2×#103C2312yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#97C2314yes—
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140#401—24no—
141#400—24no—
142#400—24no—
143#400—24no—
144#400—24no—
145#400—24no—
146#400—24no—
147#395—24no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391—22no—
153#391—21no—
154#391—21no—
155#391D412yes—
156#391C2312yes—
157#391C2312yes—
158#391C2314yes—
159#391C2314yes—
160#391C2314yes—
161#391C2314yes—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—24no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389D412yes—
173#389D412yes—
174#389C2312yes—
175#389C2312yes—
176#387—24no—
177#387—24no—
178#387—24no—
179#387—24no—
180#387—24no—
181#387—24no—
182#384D411yescharacteristic
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2314yes—
188#383—24no—
189#382—22no—
190#382—22no—
191#382—22no—
192#382—22no—
193#382C2314yes—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C43C2311yescharacteristic
210#340C2312yes—
211#340C2312yes—
212#340C2314yes—
213#340C2314yes—
214#340C2312yes—
215#340C2312yes—
216#336C2314yes—
217#336C2314yes—
218C22×Q8—21no—
219C22×Q8—21no—
220C22×Q8C2×D412yes—
221C22×Q8C2412yes—
222C22×Q8C2412yes—
223C2×Q16—24no—
224C2×Q16—24no—
225C2×Q16—24no—
226C2×Q16—24no—
227C2×Q16—42no—
228C2×Q16—42no—
229C22×C8C2411yescharacteristic
230C22×C8C2414yes—
231C22×C8C2411yescharacteristic
232C22×C8C2411yescharacteristic
233C22×C8C2411yescharacteristic
234C4⋊Q8—24no—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238C4⋊Q8—22no—
239C4⋊Q8—22no—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243C4×Q8—24no—
244C4×Q8—24no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—24no—
248C4×Q8—24no—
249C4×Q8—44no—
250C4×Q8—82no—
251C4×Q8—84no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
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—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2414yes—
295C2×C42C2411yescharacteristic
296C2×C42C2411yescharacteristic
297C2×C42C2411yescharacteristic
298#107—24no—
299#107—42no—
300#107—44no—
301#106—24no—
302#106—24no—
303#106—24no—
304#106—42no—
305#106—44no—
306C4⋊C8—28no—
307C4⋊C8—24no—
308C4⋊C8—24no—
309C4⋊C8—42no—
310#103—24no—
311#103—24no—
312#103—44no—
313#103—44no—
314#103—42no—
315#103—42no—
316#103—44no—
317#103—44no—
318#97—28no—
319C4×C8—22no—
320C4×C8—22no—
321C4×C8—22no—
322C4×C8—41no—
323C4×C8#4114yes—
324C4×C8#4114yes—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95C2×D414yes—
346#95C2×D414yes—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—81no—
367C2×Q8—81no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D414yes—
378C22×C4C22×D414yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C2511yescharacteristic
385Q16—48no—
386Q16—48no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—24no—
392C2×C8—21no—
393C2×C8—24no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—24no—
397C2×C8C2×#4112yes—
398C2×C8C2×#4112yes—
399C2×C8C2×#4112yes—
400C2×C8C2×#4112yes—
401C2×C82−1+412yes—
402C4⋊C4—24no—
403C4⋊C4—28no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
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—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C4⋊C4—81no—
436C4⋊C4—84no—
437C4⋊C4—81no—
438C4⋊C4—82no—
439C4⋊C4—84no—
440C42—22no—
441C42—22no—
442C42—24no—
443C42—21no—
444C42—24no—
445C42—21no—
446C42—41no—
447C42—44no—
448C42—44no—
449C42—44no—
450C42—44no—
451C42—84no—
452C42—84no—
453C42#13714yes—
454C42C2×#4112yes—
455C42C2×#4112yes—
456C42H(5,𝔽2)12yes—
457C422−1+412yes—
458C23C23×D411yescharacteristic
459Q8—48no—
460Q8—42no—
461Q8—42no—
462Q8—42no—
463Q8—42no—
464Q8—42no—
465Q8—42no—
466Q8—42no—
467Q8—42no—
468Q8—84no—
469Q8—84no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
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—42no—
508C2×C4—42no—
509C2×C4#54712yes—
510C2×C4#54712yes—
511C2×C4#55012yes—
512C2×C4#55012yes—
513C2×C4C2×#13712yes—
514C2×C4C2×#13712yes—
515C2×C4C2×#13712yes—
516C2×C4#57712yes—
517C2×C4#57812yes—
518C2×C4C22×#4111yescharacteristic
519C2×C4C22×#4111yescharacteristic
520C2×C4C2×H(5,𝔽2)11yescharacteristic
521C2×C4C2×2−1+411yescharacteristic
522C2×C4C2×2−1+411yescharacteristic
523C2×C4#58511yescharacteristic
524C8—22no—
525C8—22no—
526C8—44no—
527C8—41no—
528C8#55614yes—
529C22C2×#54711yescharacteristic
530C22C2×#55011yescharacteristic
531C22#349711yescharacteristic
532C22C22×#13711yescharacteristic
533C22C2×#57711yescharacteristic
534C22C2×#57811yescharacteristic
535C22#358411yescharacteristic
536C4—22no—
537C4—22no—
538C4—22no—
539C4—44no—
540C4—44no—
541C4—41no—
542C4—82no—
543C4—82no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4—82no—
550C4#335614yes—
551C4#339814yes—
552C4C2×#55611yescharacteristic
553C4#350611yescharacteristic
554C4#350711yescharacteristic
555C4#355011yescharacteristic
556C2#5982111yescharacteristic
557C2C2×#335611yescharacteristic
558C2#6128611yescharacteristic
559C2C2×#339811yescharacteristic
560C2#6140711yescharacteristic
561C2#6236611yescharacteristic
562C2#6238211yescharacteristic
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
#598752
#598702
#325302
#287442
#285422
#282842
#282832
#280282
#263202
#263192
#246832
#246822
#228142
#227282
C4×#4982
#217122
C4×#4942