Finite Group Databasebeta

← #192932

Subgroups — #192932

Order 512.

922 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2925C211yesmaximal, characteristic
2#20747C211yesmaximal, characteristic
3#17817C211yesmaximal, characteristic, complemented
4#17807C211yesmaximal, characteristic, complemented
5#17443C211yesmaximal, characteristic
6#8905C214yesmaximal, complemented
7#8903C214yesmaximal, complemented
8#8097C211yesmaximal, characteristic, complemented
9#7891C211yesmaximal, characteristic, complemented
10C2×C4×#41C2211yescharacteristic
11#2925—22no—
12#2925—22no—
13#2925—22no—
14#2925C2214yes—
15#2920C2211yescharacteristic
16#2920C2211yescharacteristic
17C2×#433—22no—
18C2×#432—21no—
19C2×#431—21no—
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C2×#408—21no—
24C2×#405—21no—
25C4×#117C2211yescharacteristic
26#1826—21no—
27#1806C2212yes—
28#1806C2212yes—
29#1789—21no—
30C4×#104—22no—
31#1574—22no—
32#1573—22no—
33#1573—22no—
34#1571C2212yescomplemented
35#1570C2212yescomplemented
36#1567C2212yescomplemented
37#1566C2212yescomplemented
38#1565—21no—
39#1565—21no—
40#1488C414yescomplemented
41#1487C411yescharacteristic, complemented
42#1487C411yescharacteristic, complemented
43#1486C411yescharacteristic, complemented
44#1486C411yescharacteristic, complemented
45#1476C2214yes—
46#1303C2214yescomplemented
47#1279C2214yescomplemented
48C22×#41D411yescharacteristic
49C22×M16—21no—
50C23×C8—21no—
51C4×#41—21no—
52C4×#41—21no—
53C4×#41—21no—
54C4×#41—21no—
55C4×#41C2×C412yes—
56C4×#41D412yes—
57C4×#41Q812yes—
58C4×#41C2312yes—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8Q811yescharacteristic
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4Q811yescharacteristic
65C2×#117—21no—
66C2×#117—21no—
67C2×#117D411yescharacteristic
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70#433—28no—
71#433—28no—
72#432—24no—
73#432—21no—
74#432—24no—
75#432—21no—
76#432—21no—
77#432C2×C412yes—
78#431—24no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431—24no—
83#431—21no—
84#431—21no—
85#431C2×C411yescharacteristic
86#431C2×C411yescharacteristic
87#431C2×C412yes—
88#431C2312yes—
89#423—21no—
90#423—21no—
91#423C2×C411yescharacteristic
92#423C2×C411yescharacteristic
93#423D412yes—
94#423C2311yescharacteristic
95#423C2311yescharacteristic
96#423C2312yes—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8D411yescharacteristic
103C2×C4⋊C8D411yescharacteristic
104C2×C4⋊C8D412yes—
105C2×C4⋊C8D412yes—
106C2×C4⋊C8C2312yes—
107C2×#104—22no—
108C2×#104—22no—
109#408—24no—
110#408—24no—
111#407—21no—
112#407—21no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—21no—
116#405—24no—
117#405—24no—
118C4×M16—21no—
119C4×M16—21no—
120C2×#97—22no—
121C2×C4×C8—22no—
122C2×C4×C8C2312yes—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2312yes—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127#376—21no—
128C43—21no—
129#339—22no—
130#339—22no—
131#335—22no—
132#335—22no—
133#335—24no—
134#334—22no—
135#334—22no—
136#334—24no—
137Q8⋊C8C2×C414yes—
138Q8⋊C8C2×C414yes—
139D4⋊C8C2×C414yes—
140D4⋊C8C2×C414yes—
141C2×#41—21no—
142C2×#41—21no—
143C2×#41—42no—
144C2×#41—42no—
145C2×#41C4⋊C412yes—
146C2×#41C2×D412yes—
147C22×Q8C4⋊C411yescharacteristic
148C22×D4—21no—
149C22×D4C4⋊C411yescharacteristic
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4C2×D411yescharacteristic
154C23×C4#4111yescharacteristic
155C23×C4#4111yescharacteristic
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—44no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—44no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8C4⋊C412yes—
175C4×Q8C4⋊C412yes—
176C4×Q8C22×C412yes—
177C4×Q8C2×Q812yes—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4C4⋊C412yes—
191C4×D4C4⋊C412yes—
192C4×D4C22×C412yes—
193C4×D4C2×Q812yes—
194#117—21no—
195#117—21no—
196#117—22no—
197#117—22no—
198#117—22no—
199#117—22no—
200#117—24no—
201#117—42no—
202#117C4⋊C412yes—
203#117C2×D412yes—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4C4⋊C411yescharacteristic
208C2×C4⋊C4C4⋊C411yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—41no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—41no—
222C2×C42—41no—
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×Q811yescharacteristic
231C2×C42C2×Q811yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42C2411yescharacteristic
235C4⋊C8—22no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8C22×C412yes—
252C4⋊C8C22×C412yes—
253C4⋊C8C2×D412yes—
254C4⋊C8C2×D412yes—
255C4⋊C8C2×D414yes—
256#104—44no—
257#104—44no—
258C22⋊C8—28no—
259C22⋊C8—28no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—22no—
265C22⋊C8—41no—
266C22⋊C8—41no—
267#97—28no—
268#97—22no—
269#97—41no—
270C4×C8—28no—
271C4×C8—22no—
272C4×C8—22no—
273C4×C8—41no—
274C4×C8C22×C412yes—
275C4×C8C22×C412yes—
276#95—22no—
277#95—41no—
278C24—21no—
279C24#12311yescharacteristic
280#41—22no—
281#41—22no—
282#41—22no—
283#41—22no—
284#41—42no—
285#41—42no—
286#41—44no—
287#41—42no—
288#41—42no—
289C2×Q8—21no—
290C2×Q8—42no—
291C2×Q8C2×C4⋊C412yes—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—42no—
296C2×D4—44no—
297C2×D4C2×C4⋊C412yes—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—81no—
317C22×C4—82no—
318C22×C4C2×C4⋊C411yescharacteristic
319C22×C4C2×C4⋊C411yescharacteristic
320C22×C4C4×D411yescharacteristic
321C22×C4C4×D411yescharacteristic
322C22×C4C4×D411yescharacteristic
323C22×C4C4×D411yescharacteristic
324C22×C4#12011yescharacteristic
325C22×C4#12011yescharacteristic
326C22×C4C22⋊Q811yescharacteristic
327C22×C4C22⋊Q811yescharacteristic
328C22×C4C22⋊Q811yescharacteristic
329C22×C4C22⋊Q811yescharacteristic
330C22×C4#12311yescharacteristic
331C22×C4#12311yescharacteristic
332C22×C4#12311yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336M16—44no—
337M16—44no—
338M16—42no—
339M16—42no—
340M16—42no—
341M16—42no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—22no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—44no—
352C2×C8—44no—
353C2×C8—44no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—82no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—24no—
364C4⋊C4—42no—
365C4⋊C4C2×C4⋊C412yes—
366C4⋊C4C2×C4⋊C412yes—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—44no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—44no—
381C42—42no—
382C42—42no—
383C42—82no—
384C42C2×C4⋊C411yescharacteristic
385C42C2×C4⋊C411yescharacteristic
386C42C4×D411yescharacteristic
387C42C4×D411yescharacteristic
388C42C4×D411yescharacteristic
389C42C4×D411yescharacteristic
390C42#12011yescharacteristic
391C42#12011yescharacteristic
392C42C22⋊Q811yescharacteristic
393C42C22⋊Q811yescharacteristic
394C42C22⋊Q811yescharacteristic
395C42C22⋊Q811yescharacteristic
396C42C23×C411yescharacteristic
397C42C22×D411yescharacteristic
398C42C22×D411yescharacteristic
399C42C22×Q811yescharacteristic
400C23—22no—
401C23—22no—
402C23—22no—
403C23—41no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23#38511yescharacteristic
408C23#38511yescharacteristic
409C23C2×#12311yescharacteristic
410Q8—22no—
411Q8—22no—
412Q8—42no—
413D4—22no—
414D4—22no—
415D4—42no—
416D4—44no—
417D4—42no—
418D4—42no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4#38511yescharacteristic
457C2×C4#38511yescharacteristic
458C2×C4#38511yescharacteristic
459C2×C4#38511yescharacteristic
460C2×C4#38511yescharacteristic
461C2×C4#38511yescharacteristic
462C2×C4#44312yes—
463C2×C4#44412yes—
464C2×C4C22×C4⋊C411yescharacteristic
465C2×C4C2×C4×D411yescharacteristic
466C2×C4C2×C4×D411yescharacteristic
467C2×C4C2×#12011yescharacteristic
468C2×C4C2×C22⋊Q811yescharacteristic
469C2×C4C2×C22⋊Q811yescharacteristic
470C2×C4#57512yes—
471C2×C4#57512yes—
472C8—42no—
473C8—42no—
474C8—42no—
475C8—42no—
476C8—42no—
477C8—42no—
478C8—44no—
479C8—81no—
480C8—81no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—44no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—84no—
493C22#186612yes—
494C22C2×#38511yescharacteristic
495C22#309312yes—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—84no—
508C4#186612yes—
509C4#291112yes—
510C4C2×#44311yescharacteristic
511C4C2×#44411yescharacteristic
512C4#300511yescharacteristic
513C4#300511yescharacteristic
514C2—22no—
515C2—22no—
516C2—41no—
517C2—82no—
518C2#2084511yescharacteristic
519C2#2086711yescharacteristic
520C2C2×#186611yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17817C2semidirect
4. #17807C2semidirect
6. #8905C2semidirect
7. #8903C2semidirect
8. #8097C2semidirect
9. #7891C2semidirect
34. #1571C22semidirect
35. #1570C22semidirect
36. #1567C22semidirect
37. #1566C22semidirect
40. #1488C4semidirect
41. #1487C4semidirect
42. #1487C4semidirect
43. #1486C4semidirect
44. #1486C4semidirect
46. #1303C22semidirect
47. #1279C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29252
#207472
#178172
#178072
#174432
#89052
#89032
#80972
#78912