Finite Group Databasebeta

← #7603995

Subgroups — #7603995

Order 512.

1320 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59871C211yesmaximal, characteristic
2#59870C211yesmaximal, characteristic
3#32478C214yesmaximal
4#29290C211yesmaximal, characteristic
5#28638C212yesmaximal
6#28284C211yesmaximal, characteristic
7#28036C214yesmaximal
8#27729C212yesmaximal
9#26322C212yesmaximal
10#26282C212yesmaximal
11#26092C212yesmaximal
12#25455C212yesmaximal
13#23063C212yesmaximal
14#22740C212yesmaximal
15C4×#500C211yesmaximal, characteristic
16C8×C4⋊Q8C211yesmaximal, characteristic
17C4×#487C211yesmaximal, characteristic
18C2×#500C2211yescharacteristic
19C2×#487C2211yescharacteristic
20C2×#477C2212yes—
21C2×C8×Q8C2212yes—
22#2806—22no—
23#2805—22no—
24#2584—22no—
25#2582—22no—
26#2581—22no—
27#2571—21no—
28#2568C2212yes—
29#2567C2212yes—
30#2567C2212yes—
31#2552—21no—
32#2552C2212yes—
33#2511C2214yes—
34#2509—22no—
35#2507C2214yes—
36#2369C2212yes—
37#2365—21no—
38#2365—21no—
39#2355C2212yes—
40Q8×C4⋊C4C2212yes—
41Q8×C4⋊C4C2212yes—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47#2100C2214yes—
48#2095C2214yes—
49#2087C2214yes—
50#2083C2214yes—
51#2072C2214yes—
52#2068C2214yes—
53#2068C2214yes—
54#2068C2214yes—
55#1990C2211yescharacteristic
56#1990C2211yescharacteristic
57#1990C2211yescharacteristic
58#1987C2212yes—
59#1987C2212yes—
60#1984C2212yes—
61#1981C2212yes—
62#1956C2211yescharacteristic
63#1947C2212yes—
64#1947C2212yes—
65#1947C2214yes—
66#1947C2212yes—
67#1947C2214yes—
68#1942C2212yes—
69#1942C2212yes—
70#1939C2212yes—
71#1939C2212yes—
72#1933C2214yes—
73#1933C2214yes—
74#1933C2212yes—
75#1933C2212yes—
76#1928C2214yes—
77#1928C2214yes—
78#1921C2214yes—
79#1921C2214yes—
80#1845C2211yescharacteristic
81#1842C2211yescharacteristic
82#1834C2214yes—
83#1834C2211yescharacteristic
84#1803C2212yes—
85C4×#107C2214yes—
86C4×#107C2212yes—
87C4×#107C2212yes—
88C8×C4⋊C4C2212yes—
89C8×C4⋊C4C2212yes—
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—22no—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106#500—24no—
107#487—24no—
108#477—28no—
109C8×Q8—28no—
110C2×#107C2314yes—
111C2×#107C2312yes—
112C2×#107C2312yes—
113C2×#107C2312yes—
114C2×#107C2312yes—
115C2×#107C2314yes—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2314yes—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2312yes—
120C2×#103C2314yes—
121C2×#103C2312yes—
122C2×#103C2312yes—
123C2×#103C2312yes—
124C2×#103C2314yes—
125C2×#103C2312yes—
126C2×C4×C8C2314yes—
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#400—24no—
134#400—24no—
135#400—24no—
136#398—24no—
137#398—24no—
138#398—24no—
139#395—24no—
140#395—24no—
141#391—22no—
142#391—22no—
143#391—21no—
144#391—21no—
145#391—21no—
146#391—24no—
147#391D412yes—
148#391C2312yes—
149#391C2314yes—
150#391C2314yes—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—24no—
156#389—22no—
157#389—22no—
158#389—24no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—24no—
165#389D412yes—
166#389D412yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2314yes—
171#389C2314yes—
172#387—24no—
173#387—24no—
174#387—24no—
175#387—24no—
176#384—22no—
177#384D411yescharacteristic
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2314yes—
183#384C2314yes—
184#382—22no—
185#382—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—22no—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8Q1618yes—
233C4⋊Q8#4114yes—
234#125—84no—
235C4×Q8—24no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×Q8—24no—
239C4×Q8—82no—
240C4×Q8—82no—
241C4×Q8—82no—
242C4×Q8—84no—
243C4×Q8—84no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
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—21no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2414yes—
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294#107—24no—
295#107—28no—
296#107—24no—
297#107—24no—
298#107—42no—
299#107—44no—
300C4⋊C8—28no—
301C4⋊C8—24no—
302C4⋊C8—24no—
303C4⋊C8—42no—
304#103—24no—
305#103—24no—
306#103—24no—
307#103—28no—
308#103—44no—
309#103—42no—
310C4×C8—28no—
311C4×C8—22no—
312C4×C8—22no—
313C4×C8—22no—
314C4×C8—22no—
315C4×C8—41no—
316C4×C8#4114yes—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—22no—
326#95—24no—
327#95—22no—
328#95—22no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95C2×D414yes—
334#95C2×D414yes—
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—81no—
347C2×Q8—81no—
348C2×Q8—81no—
349C2×Q8—81no—
350C2×Q8—81no—
351C2×Q8—81no—
352C2×Q8C2×Q1618yes—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4C22×D414yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D414yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C2511yescharacteristic
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—24no—
382C2×C8C2×#4112yes—
383C2×C8C2×#4112yes—
384C2×C82−1+412yes—
385C4⋊C4—28no—
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—42no—
411C4⋊C4—42no—
412C4⋊C4—81no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C4⋊C4—81no—
416C4⋊C4—81no—
417C4⋊C4—84no—
418C4⋊C4—84no—
419C4⋊C4—81no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4C2×Q1618yes—
426C4⋊C4C2×Q1618yes—
427C4⋊C4C2×#4114yes—
428C42—22no—
429C42—22no—
430C42—24no—
431C42—21no—
432C42—24no—
433C42—21no—
434C42—41no—
435C42—44no—
436C42—44no—
437C42—44no—
438C42—44no—
439C42—84no—
440C42—84no—
441C42C2×Q1614yes—
442C42C2×#4112yes—
443C42C2×#4112yes—
444C422−1+412yes—
445C422−1+412yes—
446C23C23×D411yescharacteristic
447Q8—216no—
448Q8—42no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—84no—
453Q8—84no—
454Q8—84no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—24no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—24no—
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—24no—
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—42no—
492C2×C4—42no—
493C2×C4#54712yes—
494C2×C4#54712yes—
495C2×C4#54812yes—
496C2×C4#54812yes—
497C2×C4C22×Q1612yes—
498C2×C4C22×Q1618yes—
499C2×C4C22×Q1612yes—
500C2×C4C22×Q1612yes—
501C2×C4#57612yes—
502C2×C4#57712yes—
503C2×C4C22×#4111yescharacteristic
504C2×C4C22×#4111yescharacteristic
505C2×C4C2×2−1+411yescharacteristic
506C2×C4C2×2−1+411yescharacteristic
507C2×C4C2×2−1+411yescharacteristic
508C2×C4#58511yescharacteristic
509C8—22no—
510C8—22no—
511C8—22no—
512C8—44no—
513C8—41no—
514C22C2×#54711yescharacteristic
515C22C2×#54811yescharacteristic
516C22#349711yescharacteristic
517C22C23×Q1611yescharacteristic
518C22C2×#57611yescharacteristic
519C22C2×#57711yescharacteristic
520C22#358411yescharacteristic
521C4—28no—
522C4—22no—
523C4—22no—
524C4—22no—
525C4—44no—
526C4—44no—
527C4—41no—
528C4—82no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4—82no—
533C4—82no—
534C4—82no—
535C4#333814yes—
536C4#336314yes—
537C4#348011yescharacteristic
538C4#350911yescharacteristic
539C4#352311yescharacteristic
540C4#355511yescharacteristic
541C2#5965311yescharacteristic
542C2C2×#333811yescharacteristic
543C2#6128511yescharacteristic
544C2C2×#336311yescharacteristic
545C2#6132511yescharacteristic
546C2#6236011yescharacteristic
547C2#6238111yescharacteristic
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
#324782
#292902
#286382
#282842
#280362
#277292
#263222
#262822
#260922
#254552
#230632
#227402
C4×#5002
C8×C4⋊Q82
C4×#4872