Finite Group Databasebeta

← #219794

Subgroups — #219794

Order 512.

1026 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2496C211yesmaximal, characteristic
2#22601C212yesmaximal, complemented
3C2×#1473C212yesmaximal
4#9654C212yesmaximal, complemented
5#9086C212yesmaximal, complemented
6#9082C212yesmaximal, complemented
7#8559C211yesmaximal, characteristic
8#8309C212yesmaximal
9#8152C211yesmaximal, characteristic
10C22×#126C2212yes—
11C2×#432C2212yes—
12C2×#431C2212yes—
13#2496—22no—
14#2496—22no—
15#2496—22no—
16#2496—22no—
17C2×#400C412yes—
18C2×#397C412yes—
19C2×#389C411yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#387C2212yes—
22C2×#384C411yescharacteristic
23C2×#380C411yescharacteristic
24C2×#380C411yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27#1940—22no—
28#1917—22no—
29#1916—22no—
30#1817C2212yes—
31#1817C2212yes—
32#1797C2212yes—
33#1771—22no—
34#1749C2212yes—
35#1473—28no—
36C2×#324C2212yes—
37C2×#324C2212yes—
38C2×#324C2212yes—
39C2×#324C2212yes—
40C2×#323C2212yes—
41C2×#323C2211yescharacteristic
42C2×#323C2211yescharacteristic
43#1280C2211yescharacteristic
44#1280C2211yescharacteristic
45#1280C2212yes—
46C24×C4C2×C411yescharacteristic
47C2×#126—24no—
48C2×#126—24no—
49C2×#126—22no—
50C2×#126—22no—
51C2×#126—22no—
52C2×#126—22no—
53C2×#126—42no—
54C2×#126—42no—
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2311yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70#432—24no—
71#432—42no—
72#431—24no—
73#431—42no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C22⋊C8C2312yes—
78C2×C22⋊C8C2312yes—
79C2×C22⋊C8C2312yes—
80C2×C22⋊C8C2312yes—
81C2×#97—22no—
82C2×#97—22no—
83C2×C4×C8—22no—
84#400—22no—
85#400—22no—
86#400—22no—
87#400—22no—
88#397—22no—
89#397—22no—
90#397—22no—
91#397—22no—
92#389—22no—
93#389—21no—
94#389—21no—
95#389—22no—
96#389—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#384—22no—
102#384—21no—
103#384—21no—
104#380—22no—
105#380—21no—
106#380—21no—
107#380—21no—
108#380—21no—
109#380—22no—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4D412yes—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4D412yes—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C2×#95C2×C412yes—
117C2×#95C2×C412yes—
118C2×#95C2×C411yescharacteristic
119C2×#95C2×C412yes—
120C2×#95C2×C412yes—
121C2×#95C2×C412yes—
122C2×#95C2311yescharacteristic
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#324—24no—
130#324—24no—
131#324—42no—
132#324—42no—
133#323—24no—
134#323—22no—
135#323D414yes—
136C25C22×C411yescharacteristic
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—41no—
140C23×C4—41no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4C22×C412yes—
144C23×C4C22×C412yes—
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C412yes—
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C412yes—
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157#126—44no—
158#126—44no—
159#126—44no—
160#126—44no—
161#126—44no—
162#126—44no—
163#126—44no—
164#126—44no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4C2×D412yes—
213C2×C22⋊C4#4112yes—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C4⋊C8—44no—
229C4⋊C8—44no—
230C4⋊C8—44no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8C2×D414yes—
239#97—44no—
240#97—44no—
241C4×C8—44no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24C4×D412yes—
267C24C23×C411yescharacteristic
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—84no—
320C22×C4—84no—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C4×D412yes—
329C22×C4#13112yes—
330C22×C4C22×D412yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4H(5,𝔽2)12yes—
334C2×C8—24no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C4⋊C4—24no—
347C4⋊C4—42no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—24no—
382C42—42no—
383C42—42no—
384C42—42no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23C22×C22⋊C411yescharacteristic
413C23C2×C4×D411yescharacteristic
414C23#51811yescharacteristic
415C23C2×#13111yescharacteristic
416C23#56811yescharacteristic
417C23#56811yescharacteristic
418C23#56811yescharacteristic
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—84no—
473C2×C4—84no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4#41012yes—
483C2×C4#41312yes—
484C2×C4D4212yes—
485C2×C4#54612yes—
486C8—44no—
487C8—82no—
488C8—82no—
489C8—82no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22D4×C22⋊C411yescharacteristic
514C22#287911yescharacteristic
515C22#288011yescharacteristic
516C22C2×#41011yescharacteristic
517C22#288411yescharacteristic
518C22C2×#41311yescharacteristic
519C22#288911yescharacteristic
520C22#293211yescharacteristic
521C22#293511yescharacteristic
522C22#298011yescharacteristic
523C22#298111yescharacteristic
524C4—22no—
525C4—22no—
526C4—22no—
527C4—22no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—82no—
536C4—82no—
537C4—81no—
538C2—21no—
539C2—21no—
540C2—21no—
541C2—21no—
542C2—82no—
543C2#1739811yescharacteristic
544C2#1741011yescharacteristic
545C2#1742011yescharacteristic
546C2#1742311yescharacteristic
547C2#2240311yescharacteristic
548C2#2240611yescharacteristic
549C2#2241111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22601C2semidirect
4. #9654C2semidirect
5. #9086C2semidirect
6. #9082C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24962
#226012
C2×#14732
#96542
#90862
#90822
#85592
#83092
#81522