Finite Group Databasebeta

← #203372

Subgroups — #203372

Order 512.

658 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21339C211yesmaximal, characteristic, complemented
2#21115C211yesmaximal, characteristic
3#20485C211yesmaximal, characteristic
4#9547C211yesmaximal, characteristic, complemented
5#9543C211yesmaximal, characteristic, complemented
6#9539C211yesmaximal, characteristic, complemented
7#9535C211yesmaximal, characteristic, complemented
8#8949C211yesmaximal, characteristic
9#8945C211yesmaximal, characteristic
10#8849C211yesmaximal, characteristic
11#8845C211yesmaximal, characteristic
12#8277C211yesmaximal, characteristic, complemented
13#8183C211yesmaximal, characteristic
14#8175C211yesmaximal, characteristic, complemented
15#8133C211yesmaximal, characteristic, complemented
16C2×C4×SD16C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#405—21no—
20#2293C2211yescharacteristic
21C42×Q8C2211yescharacteristic
22#1984C2211yescharacteristic
23#1984C2211yescharacteristic
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1926C411yescharacteristic
27#1926C411yescharacteristic
28#1919C411yescharacteristic
29#1919C411yescharacteristic
30#1916—21no—
31#1916—21no—
32#1880C411yescharacteristic
33#1880C411yescharacteristic
34#1865C411yescharacteristic
35#1865C411yescharacteristic
36C4×#106C2211yescharacteristic
37#1770C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39#1767C2211yescharacteristic
40C4×C4⋊C8—21no—
41C4×C4⋊C8C2211yescharacteristic
42C4×#103C2211yescharacteristic
43C4×#102C2211yescharacteristic
44#1753—21no—
45C8×C22⋊C4—21no—
46#1750—21no—
47#1744—21no—
48#1744C2211yescharacteristic
49#1744C2211yescharacteristic
50C42×C8C2211yescharacteristic
51C2×#334C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C22×M16—21no—
69C23×C8—21no—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75C4×SD16—22no—
76C4×SD16—22no—
77#436—22no—
78#436—22no—
79C8×D4—22no—
80C8×D4—22no—
81C2×#106C2×C411yescharacteristic
82C2×#106C2×C411yescharacteristic
83C2×#106C2×C411yescharacteristic
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#103C2×C411yescharacteristic
91C2×#103C2×C411yescharacteristic
92C2×#103C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98#405—44no—
99C2×#97—21no—
100C2×#97—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8D411yescharacteristic
104C2×C4×C8D411yescharacteristic
105C2×C4×C8D411yescharacteristic
106C2×C4×C8D411yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#390C2×C411yescharacteristic
114#390C2×C411yescharacteristic
115#389C2×C411yescharacteristic
116#389C2×C411yescharacteristic
117#388—21no—
118#388—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—21no—
127C43—21no—
128C43C2311yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336C2×C411yescharacteristic
148#336C2×C411yescharacteristic
149#336C2×C411yescharacteristic
150#336C2×C411yescharacteristic
151#334—22no—
152#334—22no—
153Q8⋊C8—22no—
154Q8⋊C8—22no—
155D4⋊C8—22no—
156D4⋊C8—22no—
157C22×Q8C22×C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C2×SD16—22no—
161C2×SD16—22no—
162C2×SD16—41no—
163C2×M16—24no—
164C2×M16—42no—
165C2×M16—42no—
166C22×C8—24no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4#4112yes—
194C4×D4#4112yes—
195#117—42no—
196#117—42no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42C2411yescharacteristic
227#106—22no—
228#106—22no—
229#106—41no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—42no—
237C4⋊C8—42no—
238C4⋊C8#4112yes—
239C4⋊C8#4112yes—
240#103—22no—
241#103—22no—
242#103—41no—
243#102—22no—
244#102—22no—
245#102—41no—
246C22⋊C8—24no—
247C22⋊C8—24no—
248#97—42no—
249#97—42no—
250C4×C8—22no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—41no—
264C4×C8—41no—
265C4×C8—42no—
266C4×C8—42no—
267C4×C8SD1614yescomplemented
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274C24—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8#13112yes—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4#11712yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4C4×D411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4C23×C411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4111yescharacteristic
307SD16—44no—
308M16—48no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—48no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8C2×SD1612yes—
351C2×C8C2×SD1612yes—
352C4⋊C4—24no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4#11712yes—
369C4⋊C4#13112yes—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C42—24no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—41no—
384C42—41no—
385C42—42no—
386C42—42no—
387C42—42no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42C2×SD1612yes—
395C42#13512yes—
396C42C2×#4111yescharacteristic
397C42C2×#4111yescharacteristic
398C42C2×#4111yescharacteristic
399C42C2×#4111yescharacteristic
400C23—22no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23C2×C4×D411yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—41no—
411Q8—41no—
412D4—22no—
413D4—22no—
414D4—41no—
415D4—41no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4C4×SD1612yes—
454C2×C4C4×SD1612yes—
455C2×C4C2×#11711yescharacteristic
456C2×C4C4×#4111yescharacteristic
457C2×C4#52511yescharacteristic
458C2×C4#52511yescharacteristic
459C2×C4#52911yescharacteristic
460C2×C4#52911yescharacteristic
461C2×C4C2×#13111yescharacteristic
462C2×C4C2×#13111yescharacteristic
463C2×C4#56811yescharacteristic
464C2×C4#56811yescharacteristic
465C2×C4C22×SD1611yescharacteristic
466C2×C4C2×#13511yescharacteristic
467C2×C4#57511yescharacteristic
468C2×C4#57511yescharacteristic
469C8—22no—
470C8—22no—
471C8—42no—
472C8—42no—
473C8—41no—
474C8—41no—
475C8—41no—
476C8—41no—
477C8—41no—
478C8—41no—
479C8—81no—
480C8—81no—
481C8—81no—
482C8—81no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22#229011yescharacteristic
491C22#292911yescharacteristic
492C22#293011yescharacteristic
493C22C2×C4×SD1611yescharacteristic
494C22#294311yescharacteristic
495C22#295511yescharacteristic
496C22#295511yescharacteristic
497C4—22no—
498C4—21no—
499C4—21no—
500C4—22no—
501C4—41no—
502C4—42no—
503C4—42no—
504C4—41no—
505C4—42no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4#292411yescharacteristic
511C4#292511yescharacteristic
512C4#296811yescharacteristic
513C4#296911yescharacteristic
514C4#310111yescharacteristic
515C4#310111yescharacteristic
516C4#310211yescharacteristic
517C4#310211yescharacteristic
518C2—82no—
519C2#1790111yescharacteristic
520C2#1790911yescharacteristic
521C2#1792911yescharacteristic
522C2#1793711yescharacteristic
523C2#2110911yescharacteristic
524C2#2111711yescharacteristic
525C2#2113811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21339C2semidirect
4. #9547C2semidirect
5. #9543C2semidirect
6. #9539C2semidirect
7. #9535C2semidirect
12. #8277C2semidirect
14. #8175C2semidirect
15. #8133C2semidirect
267. C4×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#213392
#211152
#204852
#95472
#95432
#95392
#95352
#89492
#89452
#88492
#88452
#82772
#81832
#81752
#81332