Finite Group Databasebeta

← #203191

Subgroups — #203191

Order 512.

714 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22008C211yesmaximal, characteristic
2#20569C211yesmaximal, characteristic, complemented
3C4×#440C211yesmaximal, characteristic
4#9542C211yesmaximal, characteristic
5#9541C211yesmaximal, characteristic
6#9536C211yesmaximal, characteristic, complemented
7#9533C211yesmaximal, characteristic, complemented
8#8948C211yesmaximal, characteristic
9#8948C211yesmaximal, characteristic
10#8846C211yesmaximal, characteristic, complemented
11#8844C211yesmaximal, characteristic, complemented
12#8275C211yesmaximal, characteristic, complemented
13C4×Q8⋊C8C211yesmaximal, characteristic, complemented
14C4×D4⋊C8C211yesmaximal, characteristic
15#8132C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×#440C411yescharacteristic
19C2×#440C411yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C42×Q8C2211yescharacteristic
22C42×D4C2211yescharacteristic
23#1954C2211yescharacteristic
24#1954C2211yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1923—21no—
28#1923—21no—
29#1817—21no—
30#1797—21no—
31#1776C411yescharacteristic
32#1776C411yescharacteristic
33C4×#107C2211yescharacteristic
34C8×C4⋊C4—21no—
35C8×C4⋊C4—21no—
36C8×C4⋊C4C2211yescharacteristic
37C8×C4⋊C4C2211yescharacteristic
38C4×C4⋊C8—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1763C411yescharacteristic
42#1763C411yescharacteristic
43#1762C411yescharacteristic
44#1762C411yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#102C2211yescharacteristic
47#1744—21no—
48#1744C2211yescharacteristic
49C4×#97C2211yescharacteristic
50C42×C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
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
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C22×C42—21no—
75C8×Q8—22no—
76C8×Q8—22no—
77C8×Q8—22no—
78C8×Q8—22no—
79#440—24no—
80#440C818yescomplemented
81C4×SD16—22no—
82C4×SD16—22no—
83C2×#107C2×C411yescharacteristic
84C2×#107C2×C411yescharacteristic
85C2×#106C2×C411yescharacteristic
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×#103C2×C411yescharacteristic
95C2×#103C2×C411yescharacteristic
96C2×#103C2×C411yescharacteristic
97C2×#102C2×C411yescharacteristic
98C2×#102C2×C411yescharacteristic
99C2×#102C2×C411yescharacteristic
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×#97C2×C411yescharacteristic
105C2×#97C2×C411yescharacteristic
106C2×#97C2×C411yescharacteristic
107C2×#97C2×C411yescharacteristic
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C43—21no—
135C43C2311yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336D411yescharacteristic
149#336D411yescharacteristic
150#336D411yescharacteristic
151#336D411yescharacteristic
152#335—22no—
153#334—22no—
154Q8⋊C8—22no—
155Q8⋊C8—22no—
156D4⋊C8—22no—
157D4⋊C8—22no—
158C22×Q8C22×C411yescharacteristic
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C2×SD16—22no—
164C2×SD16—22no—
165C2×SD16C2×C814yes—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8C2×C814yes—
185C4×Q8M1614yes—
186C4×Q8#4112yes—
187C4×Q8#4112yes—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4C2×C814yes—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—81no—
218C2×C42—81no—
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42#4111yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42C2411yescharacteristic
226#107—22no—
227#107C2×C814yes—
228#106—22no—
229C4⋊C8—24no—
230C4⋊C8—24no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—81no—
238C4⋊C8—81no—
239C4⋊C8#4112yes—
240C4⋊C8#4112yes—
241#103—22no—
242#103—22no—
243#103C2×C814yes—
244#102—22no—
245#102—22no—
246#102C2×C814yes—
247C22⋊C8—81no—
248C22⋊C8—81no—
249C22⋊C8—81no—
250C22⋊C8—81no—
251#97—22no—
252#97—22no—
253#97—22no—
254#97C2×C814yes—
255C4×C8—24no—
256C4×C8—24no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—42no—
270C4×C8—42no—
271C4×C8—81no—
272C4×C8—81no—
273C4×C8—81no—
274C4×C8—81no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281C24—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8#11712yes—
286C2×Q8C22×C812yes—
287C2×Q8C2×M1612yes—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4C22×C812yes—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—81no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4—81no—
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C2×#4111yescharacteristic
320SD16—28no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8C22×C812yes—
362C2×C8C22×C812yes—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4#11712yes—
376C4⋊C4C22×C812yes—
377C4⋊C4C22×C812yes—
378C4⋊C4C2×M1612yes—
379C22⋊C4—24no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—42no—
383C42—22no—
384C42—22no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—24no—
390C42—24no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—81no—
398C42—81no—
399C42—81no—
400C42—81no—
401C42C22×C812yes—
402C42C2×M1612yes—
403C42#13112yes—
404C42#13112yes—
405C42C2×#4111yescharacteristic
406C42C2×#4111yescharacteristic
407C42C2×#4111yescharacteristic
408C42C2×#4111yescharacteristic
409C23—22no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23C2×C4×D411yescharacteristic
417Q8—22no—
418Q8—22no—
419Q8—22no—
420Q8#43114yes—
421D4—22no—
422D4—22no—
423D4—41no—
424D4—41no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
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×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—84no—
457C2×C4—84no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4C8×D412yes—
463C2×C4C8×D412yes—
464C2×C4#43512yes—
465C2×C4#43512yes—
466C2×C4C2×#11711yescharacteristic
467C2×C4C4×#4111yescharacteristic
468C2×C4#52511yescharacteristic
469C2×C4#52511yescharacteristic
470C2×C4#52911yescharacteristic
471C2×C4#52911yescharacteristic
472C2×C4C23×C811yescharacteristic
473C2×C4C22×M1611yescharacteristic
474C2×C4C2×#13111yescharacteristic
475C2×C4C2×#13111yescharacteristic
476C2×C4#57611yescharacteristic
477C2×C4#57711yescharacteristic
478C2×C4#57711yescharacteristic
479C2×C4#57811yescharacteristic
480C8—22no—
481C8—22no—
482C8—42no—
483C8—42no—
484C8—41no—
485C8—42no—
486C8—41no—
487C8—41no—
488C8—42no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—81no—
493C8—81no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22#229011yescharacteristic
502C22C2×C8×D411yescharacteristic
503C22C2×#43511yescharacteristic
504C22#294611yescharacteristic
505C22#294711yescharacteristic
506C22#295611yescharacteristic
507C22#295611yescharacteristic
508C4—24no—
509C4—21no—
510C4—21no—
511C4—22no—
512C4—22no—
513C4—22no—
514C4—22no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—81no—
520C4—81no—
521C4C2×#43111yescharacteristic
522C4#292211yescharacteristic
523C4C8×#4111yescharacteristic
524C4#296511yescharacteristic
525C4#311911yescharacteristic
526C4#312111yescharacteristic
527C4#312411yescharacteristic
528C4#312611yescharacteristic
529C2—82no—
530C2#1785211yescharacteristic
531C2#1785311yescharacteristic
532C2#1787411yescharacteristic
533C2#1787511yescharacteristic
534C2#2110111yescharacteristic
535C2#2112811yescharacteristic
536C2#2113911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20569C2semidirect
6. #9536C2semidirect
7. #9533C2semidirect
10. #8846C2semidirect
11. #8844C2semidirect
12. #8275C2semidirect
13. C4×Q8⋊C8C2semidirect
15. #8132C2semidirect
80. #440C8semidirect

Maximal subgroups

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

subgroupindex
#220082
#205692
C4×#4402
#95422
#95412
#95362
#95332
#89482
#88462
#88442
#82752
C4×Q8⋊C82
C4×D4⋊C82
#81322