Finite Group Databasebeta

← #7603988

Subgroups — #7603988

Order 512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59875C211yesmaximal, characteristic
2#59869C211yesmaximal, characteristic
3#32479C214yesmaximal
4#29286C211yesmaximal, characteristic
5#28639C212yesmaximal
6#28284C211yesmaximal, characteristic
7#28033C214yesmaximal
8#27730C212yesmaximal
9#26319C212yesmaximal
10#26281C212yesmaximal
11#26089C212yesmaximal
12#25454C212yesmaximal
13#23063C212yesmaximal
14#22739C212yesmaximal
15C4×#498C211yesmaximal, characteristic
16C8×C4⋊Q8C211yesmaximal, characteristic
17C4×#487C211yesmaximal, characteristic
18C2×#498C2211yescharacteristic
19C2×#487C2211yescharacteristic
20C2×#477C2212yes—
21C2×C8×Q8C2212yes—
22#2810—22no—
23#2809—22no—
24#2571C2212yes—
25#2568—22no—
26#2568—21no—
27#2568C2212yes—
28#2568C2212yes—
29#2546—21no—
30#2546C2212yes—
31#2514—22no—
32#2514—22no—
33#2514C2214yes—
34#2508—22no—
35#2508C2214yes—
36#2369C2212yes—
37#2355—21no—
38#2355—21no—
39#2355C2212yes—
40#2355C2212yes—
41#2355C2212yes—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47#2101C2214yes—
48#2097C2214yes—
49#2086C2214yes—
50#2085C2214yes—
51#2074C2214yes—
52#2068C2214yes—
53#2066C2214yes—
54#2066C2214yes—
55#1990C2211yescharacteristic
56#1990C2211yescharacteristic
57#1990C2211yescharacteristic
58#1988C2212yes—
59#1988C2212yes—
60#1984C2212yes—
61#1981C2212yes—
62#1956C2211yescharacteristic
63#1949C2214yes—
64#1949C2214yes—
65#1947C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1941C2212yes—
69#1941C2212yes—
70#1939C2212yes—
71#1939C2212yes—
72#1933C2214yes—
73#1933C2214yes—
74#1933C2212yes—
75#1933C2212yes—
76#1928C2214yes—
77#1927C2214yes—
78#1921C2214yes—
79#1920C2214yes—
80#1844C2211yescharacteristic
81#1842C2211yescharacteristic
82#1835C2214yes—
83#1834C2211yescharacteristic
84#1803C2212yes—
85C4×#107C2214yes—
86C4×#106C2212yes—
87C4×#106C2212yes—
88C8×C4⋊C4C2212yes—
89C8×C4⋊C4C2212yes—
90C4×#103C2212yes—
91C4×#103C2212yes—
92C42×C8C2211yescharacteristic
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8D411yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4×Q8—21no—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2312yes—
104#498—24no—
105#487—24no—
106#477—28no—
107C8×Q8—28no—
108C2×#107C2314yes—
109C2×#107C2314yes—
110C2×#106C2312yes—
111C2×#106C2312yes—
112C2×#106C2312yes—
113C2×#106C2312yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2314yes—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2312yes—
118C2×#103C2314yes—
119C2×#103C2312yes—
120C2×#103C2312yes—
121C2×#103C2312yes—
122C2×#103C2314yes—
123C2×#103C2312yes—
124C2×C4×C8C2314yes—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131#401—24no—
132#400—24no—
133#400—24no—
134#400—24no—
135#400—24no—
136#400—24no—
137#398—24no—
138#398—24no—
139#391—22no—
140#391—22no—
141#391—22no—
142#391—22no—
143#391—22no—
144#391—22no—
145#391—21no—
146#391—21no—
147#391—21no—
148#391—24no—
149#391D412yes—
150#391C2312yes—
151#391C2314yes—
152#391C2314yes—
153#389—22no—
154#389—22no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389D412yes—
161#389D412yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2314yes—
166#389C2314yes—
167#387—24no—
168#387—24no—
169#387—24no—
170#387—24no—
171#387—24no—
172#384D411yescharacteristic
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#383—24no—
178#382—22no—
179#382—22no—
180#382—22no—
181#382—24no—
182#382C2314yes—
183#382C2314yes—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2314yes—
197C4×C4⋊C4C2314yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C43C2311yescharacteristic
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2312yes—
205#340C2314yes—
206#340C2312yes—
207#340C2312yes—
208#340C2312yes—
209#340C2312yes—
210#340C2312yes—
211#340C2312yes—
212#340C2312yes—
213#336C2314yes—
214#336C2314yes—
215C22×Q8—21no—
216C22×Q8—21no—
217C22×Q8—21no—
218C22×Q8C2×D412yes—
219C22×Q8C2412yes—
220C22×C8C2411yescharacteristic
221C22×C8C2414yes—
222C22×C8C2411yescharacteristic
223C22×C8C2411yescharacteristic
224C22×C8C2411yescharacteristic
225C4⋊Q8—24no—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233C4⋊Q8—84no—
234C4⋊Q8#4114yes—
235C4×Q8—24no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×Q8—24no—
239C4×Q8—44no—
240C4×Q8—44no—
241C4×Q8—82no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2414yes—
285C2×C42C2411yescharacteristic
286C2×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288#107—28no—
289#107—44no—
290#106—24no—
291#106—24no—
292#106—24no—
293#106—42no—
294C4⋊C8—28no—
295C4⋊C8—24no—
296C4⋊C8—24no—
297C4⋊C8—42no—
298#103—24no—
299#103—24no—
300#103—24no—
301#103—28no—
302#103—44no—
303#103—42no—
304C4×C8—28no—
305C4×C8—22no—
306C4×C8—22no—
307C4×C8—22no—
308C4×C8—22no—
309C4×C8—41no—
310C4×C8#4114yes—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—24no—
320#95—24no—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—22no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95C2×D414yes—
332#95C2×D414yes—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8—81no—
350C2×Q8—81no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D414yes—
361C22×C4C22×D414yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C2511yescharacteristic
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—24no—
380C2×C8C2×#4112yes—
381C2×C8C2×#4112yes—
382C2×C82−1+412yes—
383C4⋊C4—24no—
384C4⋊C4—28no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—81no—
415C4⋊C4—81no—
416C4⋊C4—81no—
417C4⋊C4—81no—
418C4⋊C4—82no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4C2×#4114yes—
422C42—22no—
423C42—22no—
424C42—24no—
425C42—21no—
426C42—24no—
427C42—21no—
428C42—41no—
429C42—44no—
430C42—44no—
431C42—44no—
432C42—44no—
433C42—84no—
434C42—84no—
435C42#13514yes—
436C42C2×#4112yes—
437C42C2×#4112yes—
438C422−1+412yes—
439C422−1+412yes—
440C23C23×D411yescharacteristic
441Q8—48no—
442Q8—42no—
443Q8—42no—
444Q8—42no—
445Q8—42no—
446Q8—84no—
447Q8—84no—
448Q8—84no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—21no—
464C2×C4—24no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4#54712yes—
489C2×C4#54712yes—
490C2×C4#54812yes—
491C2×C4#54812yes—
492C2×C4C2×#13512yes—
493C2×C4C2×#13512yes—
494C2×C4C2×#13512yes—
495C2×C4#57712yes—
496C2×C4#57812yes—
497C2×C4C22×#4111yescharacteristic
498C2×C4C22×#4111yescharacteristic
499C2×C4C2×2−1+411yescharacteristic
500C2×C4C2×2−1+411yescharacteristic
501C2×C4C2×2−1+411yescharacteristic
502C2×C4#58511yescharacteristic
503C8—22no—
504C8—22no—
505C8—22no—
506C8—44no—
507C8—41no—
508C22C2×#54711yescharacteristic
509C22C2×#54811yescharacteristic
510C22#349711yescharacteristic
511C22C22×#13511yescharacteristic
512C22C2×#57711yescharacteristic
513C22C2×#57811yescharacteristic
514C22#358411yescharacteristic
515C4—22no—
516C4—22no—
517C4—22no—
518C4—44no—
519C4—44no—
520C4—44no—
521C4—41no—
522C4—82no—
523C4—82no—
524C4—82no—
525C4—82no—
526C4—82no—
527C4—82no—
528C4—82no—
529C4#334014yes—
530C4#336414yes—
531C4#348011yescharacteristic
532C4#350911yescharacteristic
533C4#352311yescharacteristic
534C4#355511yescharacteristic
535C2#5965311yescharacteristic
536C2C2×#334011yescharacteristic
537C2#6128611yescharacteristic
538C2C2×#336411yescharacteristic
539C2#6132611yescharacteristic
540C2#6235811yescharacteristic
541C2#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
#598692
#324792
#292862
#286392
#282842
#280332
#277302
#263192
#262812
#260892
#254542
#230632
#227392
C4×#4982
C8×C4⋊Q82
C4×#4872