Finite Group Databasebeta

← #7604007

Subgroups — #7604007

Order 512.

1282 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59875C211yesmaximal, characteristic
2#59871C211yesmaximal, characteristic
3#32481C214yesmaximal
4#29287C211yesmaximal, characteristic
5#28647C212yesmaximal
6#28284C211yesmaximal, characteristic
7#28059C214yesmaximal
8#27737C212yesmaximal
9#26320C212yesmaximal
10#26279C212yesmaximal
11#26088C212yesmaximal
12#25454C212yesmaximal
13#23069C212yesmaximal
14#22742C212yesmaximal
15C4×#498C211yesmaximal, characteristic
16#21712C211yesmaximal, characteristic
17C4×#487C211yesmaximal, characteristic
18C2×#498C2211yescharacteristic
19C2×#487C2211yescharacteristic
20C2×#477C2212yes—
21C2×#446C2212yes—
22#2810—22no—
23#2806—22no—
24#2584—22no—
25#2582—22no—
26#2568—21no—
27#2568C2212yes—
28#2568C2212yes—
29#2567C2212yes—
30#2552—21no—
31#2552C2212yes—
32#2514—22no—
33#2514—22no—
34#2514C2214yes—
35#2507C2214yes—
36#2369C2212yes—
37#2365—21no—
38#2355—21no—
39#2355C2212yes—
40#2355C2212yes—
41Q8×C4⋊C4C2212yes—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47#2101C2214yes—
48#2096C2214yes—
49#2088C2214yes—
50#2085C2214yes—
51#2072C2214yes—
52#2068C2214yes—
53#2066C2214yes—
54#2066C2214yes—
55#1990C2211yescharacteristic
56#1990C2211yescharacteristic
57#1990C2211yescharacteristic
58#1988C2212yes—
59#1987C2212yes—
60#1984C2212yes—
61#1981C2212yes—
62#1956C2211yescharacteristic
63#1949C2214yes—
64#1948C2212yes—
65#1948C2212yes—
66#1948C2214yes—
67#1947C2212yes—
68#1941C2212yes—
69#1941C2212yes—
70#1939C2212yes—
71#1939C2212yes—
72#1935C2214yes—
73#1935C2214yes—
74#1933C2212yes—
75#1933C2212yes—
76#1927C2214yes—
77#1927C2214yes—
78#1919C2214yes—
79#1918C2214yes—
80#1844C2211yescharacteristic
81#1842C2211yescharacteristic
82#1836C2214yes—
83#1834C2211yescharacteristic
84#1803C2212yes—
85#1776C2214yes—
86C4×#106C2212yes—
87C4×#106C2212yes—
88#1770C2212yes—
89#1770C2212yes—
90C4×#103C2212yes—
91C4×#103C2212yes—
92C42×C8C2211yescharacteristic
93C2×C4⋊Q8D411yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×#125—22no—
99C2×C4×Q8—22no—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105#498—24no—
106#487—24no—
107#477—28no—
108#446—28no—
109C2×#107C2314yes—
110C2×#106C2312yes—
111C2×#106C2312yes—
112C2×#106C2312yes—
113C2×#106C2312yes—
114C2×#106C2314yes—
115C2×C4⋊C8C2312yes—
116C2×C4⋊C8C2314yes—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2312yes—
119C2×#103C2314yes—
120C2×#103C2312yes—
121C2×#103C2312yes—
122C2×#103C2312yes—
123C2×#103C2314yes—
124C2×#103C2312yes—
125C2×#97C2314yes—
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132#401—24no—
133#400—24no—
134#400—24no—
135#400—24no—
136#398—24no—
137#398—24no—
138#398—24no—
139#395—24no—
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—24no—
155#389—22no—
156#389—22no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—24no—
163#389D412yes—
164#389D412yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2314yes—
169#389C2314yes—
170#387—24no—
171#387—24no—
172#387—24no—
173#387—24no—
174#384—22no—
175#384D411yescharacteristic
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#384C2314yes—
181#383—24no—
182#382—22no—
183#382—22no—
184#382C2314yes—
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#4114yes—
234#125—84no—
235C4×Q8—24no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×Q8—24no—
239C4×Q8—44no—
240C4×Q8—82no—
241C4×Q8—82no—
242C4×Q8—84no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
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—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2414yes—
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291#107—28no—
292#106—24no—
293#106—24no—
294#106—24no—
295#106—42no—
296#106—44no—
297C4⋊C8—28no—
298C4⋊C8—24no—
299C4⋊C8—24no—
300C4⋊C8—42no—
301#103—24no—
302#103—24no—
303#103—24no—
304#103—28no—
305#103—44no—
306#103—42no—
307#97—28no—
308C4×C8—22no—
309C4×C8—22no—
310C4×C8—22no—
311C4×C8—22no—
312C4×C8—41no—
313C4×C8#4114yes—
314#95—24no—
315#95—24no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—22no—
325#95—24no—
326#95—22no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95C2×D414yes—
333#95C2×D414yes—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—81no—
349C2×Q8—81no—
350C2×Q8—81no—
351C2×Q8—81no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4C22×D414yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D414yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C2511yescharacteristic
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—24no—
381C2×C8C2×#4112yes—
382C2×C8C2×#4112yes—
383C2×C82−1+412yes—
384C4⋊C4—24no—
385C4⋊C4—28no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
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—22no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—44no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—44no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—81no—
415C4⋊C4—81no—
416C4⋊C4—84no—
417C4⋊C4—81no—
418C4⋊C4—81no—
419C4⋊C4—82no—
420C4⋊C4—82no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C4⋊C4—84no—
424C4⋊C4C2×#4114yes—
425C42—22no—
426C42—22no—
427C42—24no—
428C42—21no—
429C42—24no—
430C42—21no—
431C42—41no—
432C42—44no—
433C42—44no—
434C42—44no—
435C42—44no—
436C42—84no—
437C42—84no—
438C42#13714yes—
439C42C2×#4112yes—
440C42C2×#4112yes—
441C422−1+412yes—
442C422−1+412yes—
443C23C23×D411yescharacteristic
444Q8—48no—
445Q8—42no—
446Q8—42no—
447Q8—42no—
448Q8—42no—
449Q8—84no—
450Q8—84no—
451Q8—84no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—24no—
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—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4#54712yes—
492C2×C4#54712yes—
493C2×C4#54812yes—
494C2×C4#54812yes—
495C2×C4C2×#13712yes—
496C2×C4C2×#13712yes—
497C2×C4C2×#13712yes—
498C2×C4#57712yes—
499C2×C4#57712yes—
500C2×C4C22×#4111yescharacteristic
501C2×C4C22×#4111yescharacteristic
502C2×C4C2×2−1+411yescharacteristic
503C2×C4C2×2−1+411yescharacteristic
504C2×C4C2×2−1+411yescharacteristic
505C2×C4#58511yescharacteristic
506C8—22no—
507C8—22no—
508C8—22no—
509C8—44no—
510C8—41no—
511C22C2×#54711yescharacteristic
512C22C2×#54811yescharacteristic
513C22#349711yescharacteristic
514C22C22×#13711yescharacteristic
515C22C2×#57711yescharacteristic
516C22C2×#57711yescharacteristic
517C22#358411yescharacteristic
518C4—22no—
519C4—22no—
520C4—22no—
521C4—44no—
522C4—44no—
523C4—44no—
524C4—41no—
525C4—82no—
526C4—82no—
527C4—82no—
528C4—82no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4#335614yes—
533C4#336914yes—
534C4#348011yescharacteristic
535C4#350911yescharacteristic
536C4#352311yescharacteristic
537C4#355511yescharacteristic
538C2#5965311yescharacteristic
539C2C2×#335611yescharacteristic
540C2#6128511yescharacteristic
541C2C2×#336911yescharacteristic
542C2#6132611yescharacteristic
543C2#6236611yescharacteristic
544C2#6237111yescharacteristic
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
#598712
#324812
#292872
#286472
#282842
#280592
#277372
#263202
#262792
#260882
#254542
#230692
#227422
C4×#4982
#217122
C4×#4872