Finite Group Databasebeta

← #192529

Subgroups — #192529

Order 512.

811 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60288C211yesmaximal, characteristic
2#21975C211yesmaximal, characteristic, complemented
3#17822C211yesmaximal, characteristic, complemented
4#17554C211yesmaximal, characteristic
5#17492C211yesmaximal, characteristic
6#8877C214yesmaximal, complemented
7#8876C214yesmaximal, complemented
8#8069C211yesmaximal, characteristic, complemented
9#7936C211yesmaximal, characteristic, complemented
10C2×#529C2211yescharacteristic
11#2994—21no—
12#2994—21no—
13#2992C2212yes—
14#2992C2212yes—
15#2984—21no—
16#2974—21no—
17#2973—21no—
18#2972—21no—
19#2968—22no—
20#2966—21no—
21#2966—21no—
22#2924—22no—
23#2920C2211yescharacteristic
24#2920C2211yescharacteristic
25C2×#432C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#423C2211yescharacteristic
28#2283C2211yescharacteristic
29#1939—21no—
30#1939—21no—
31#1851—21no—
32#1851—21no—
33#1843C2212yescomplemented
34#1832C2212yescomplemented
35#1573C2214yescomplemented
36#1571—21no—
37#1570—21no—
38#1569—22no—
39#1512C414yes—
40#1511C412yes—
41#1510C412yes—
42#1500C2214yes—
43#1495—22no—
44#1487—21no—
45#1486—21no—
46#1483C2214yes—
47#1300C2214yescomplemented
48#1283C2212yescomplemented
49#1283C2212yescomplemented
50C22×M16—21no—
51#529—21no—
52#529—21no—
53#529—21no—
54#529—21no—
55#529C2×C412yes—
56#529D412yes—
57#529Q812yes—
58#529C2312yes—
59C2×#126—21no—
60C2×#125Q811yescharacteristic
61C2×#124Q811yescharacteristic
62C2×#123—21no—
63C2×C22⋊Q8C2×C411yescharacteristic
64C2×#121C2×C411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×D4—21no—
67C2×C4×D4D411yescharacteristic
68C2×#117—21no—
69C22×C42C2311yescharacteristic
70#446—21no—
71#446—21no—
72#446—22no—
73#446—22no—
74C8×Q8—21no—
75C8×Q8—21no—
76#436—21no—
77#436—21no—
78#435—22no—
79#435—22no—
80#435—22no—
81#435—22no—
82#435—22no—
83#435—22no—
84C8×D4—21no—
85C8×D4—21no—
86C8×D4—22no—
87C8×D4—22no—
88C8×D4—22no—
89C8×D4—22no—
90#433—22no—
91#433—22no—
92#433—22no—
93#433—22no—
94#432—22no—
95#432—21no—
96#432—21no—
97#432—21no—
98#432—21no—
99#432—21no—
100#432C2×C412yes—
101#432D412yes—
102#432C2312yes—
103#431—22no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431C2312yes—
112#424—22no—
113#424—22no—
114#423—21no—
115#423—21no—
116#423—21no—
117#423—21no—
118#423—21no—
119#423—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—22no—
122C2×C4⋊C8—22no—
123C2×C4⋊C8D411yescharacteristic
124C2×C4⋊C8D411yescharacteristic
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128#407—22no—
129#407—21no—
130#407C2×C412yes—
131C2×C22⋊C8—22no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8D411yescharacteristic
134C2×C22⋊C8D411yescharacteristic
135#405—22no—
136#405—22no—
137C4×M16—21no—
138C4×M16C2312yes—
139C2×#97—21no—
140C2×C4×C8D411yescharacteristic
141C2×C4×C8D411yescharacteristic
142#383—21no—
143#382C2×C412yes—
144#381C2×C412yes—
145#380—21no—
146C4×C4⋊C4—21no—
147C4×C22⋊C4—21no—
148#376C2312yes—
149#336—22no—
150#336—22no—
151#335—24no—
152#334—24no—
153#328C2×C418yes—
154#327C2×C418yes—
155Q8⋊C8—24no—
156D4⋊C8—24no—
157C22×Q8C4⋊C411yescharacteristic
158C22×D4—21no—
159C22×D4C4⋊C411yescharacteristic
160C23×C4—21no—
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4#4111yescharacteristic
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—41no—
169C2×M16—41no—
170C2×M16—82no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180#126—22no—
181#126—22no—
182#126—22no—
183#126—22no—
184#125—21no—
185#125—21no—
186#125C4⋊C412yes—
187#125C2×Q812yes—
188#124—21no—
189#124—21no—
190#124C4⋊C412yes—
191#124C2×Q812yes—
192#123—22no—
193#123—22no—
194#123—22no—
195#123—22no—
196C22⋊Q8—21no—
197C22⋊Q8—21no—
198C22⋊Q8C4⋊C412yes—
199C22⋊Q8C22×C412yes—
200#121—21no—
201#121—21no—
202#121C4⋊C412yes—
203#121C22×C412yes—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8C4⋊C412yes—
207C4×Q8C2×D412yes—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4C4⋊C412yes—
215C4×D4C2×D412yes—
216#117—22no—
217#117—22no—
218#117—22no—
219#117—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4C4⋊C411yescharacteristic
226C2×C4⋊C4C4⋊C411yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42C22×C411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×Q811yescharacteristic
242C2×C42C2×Q811yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42C2411yescharacteristic
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—41no—
262C4⋊C8—41no—
263C4⋊C8—41no—
264C4⋊C8—41no—
265C4⋊C8—82no—
266C4⋊C8—82no—
267C4⋊C8C2×D412yes—
268C4⋊C8C2×D412yes—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—21no—
275C22⋊C8—22no—
276C22⋊C8—21no—
277C22⋊C8—21no—
278C22⋊C8—21no—
279C22⋊C8—22no—
280C22⋊C8—41no—
281C22⋊C8—41no—
282C22⋊C8C22×C414yes—
283#97—22no—
284#97—22no—
285#97—22no—
286#97—22no—
287#97—22no—
288#97—22no—
289C4×C8—22no—
290C4×C8—22no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—22no—
296C4×C8—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95C22×C414yes—
301C24—21no—
302C24C4×D411yescharacteristic
303C2×Q8—21no—
304C2×Q8—81no—
305C2×Q8C2×C4⋊C412yes—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—21no—
309C2×D4—44no—
310C2×D4—81no—
311C2×D4C2×C4⋊C412yes—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—81no—
322C22×C4—81no—
323C22×C4—81no—
324C22×C4C2×C4⋊C411yescharacteristic
325C22×C4C2×C4⋊C411yescharacteristic
326C22×C4C4×D411yescharacteristic
327C22×C4C4×D411yescharacteristic
328C22×C4C4×D411yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4#12011yescharacteristic
332C22×C4#12011yescharacteristic
333C22×C4C22⋊Q811yescharacteristic
334C22×C4C22⋊Q811yescharacteristic
335C22×C4C22⋊Q811yescharacteristic
336C22×C4C22⋊Q811yescharacteristic
337C22×C4#12311yescharacteristic
338C22×C4#12311yescharacteristic
339C22×C4C23×C411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C2×#4111yescharacteristic
342M16—44no—
343M16—82no—
344M16—82no—
345M16—82no—
346M16—82no—
347M16—82no—
348C2×C8—24no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—81no—
364C2×C8—81no—
365C2×C8—82no—
366C2×C8—82no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—44no—
378C4⋊C4C2×C4⋊C412yes—
379C4⋊C4C2×C4⋊C412yes—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—82no—
391C42—22no—
392C42—22no—
393C42—42no—
394C42—42no—
395C42C2×C4⋊C411yescharacteristic
396C42C2×C4⋊C411yescharacteristic
397C42C4×D411yescharacteristic
398C42C4×D411yescharacteristic
399C42#12011yescharacteristic
400C42#12011yescharacteristic
401C42C22⋊Q811yescharacteristic
402C42C22⋊Q811yescharacteristic
403C42C22⋊Q811yescharacteristic
404C42C22⋊Q811yescharacteristic
405C42#12311yescharacteristic
406C42#12311yescharacteristic
407C42C22×D411yescharacteristic
408C42C22×D411yescharacteristic
409C42C22×Q811yescharacteristic
410C42C2×#4111yescharacteristic
411C23—22no—
412C23—22no—
413C23—41no—
414C23—41no—
415C23—42no—
416C23—81no—
417C23—81no—
418C23#38511yescharacteristic
419C23#38511yescharacteristic
420C23C2×C4×D411yescharacteristic
421Q8—42no—
422Q8—42no—
423D4—42no—
424D4—42no—
425D4—44no—
426D4—44no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
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—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—82no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4#38511yescharacteristic
461C2×C4#38511yescharacteristic
462C2×C4#38511yescharacteristic
463C2×C4#38511yescharacteristic
464C2×C4#38511yescharacteristic
465C2×C4#38511yescharacteristic
466C2×C4C22×C4⋊C411yescharacteristic
467C2×C4C2×C4×D411yescharacteristic
468C2×C4C2×#12011yescharacteristic
469C2×C4C2×C22⋊Q811yescharacteristic
470C2×C4C2×C22⋊Q811yescharacteristic
471C2×C4C2×#12311yescharacteristic
472C8—44no—
473C8—41no—
474C8—41no—
475C8—42no—
476C8—41no—
477C8—41no—
478C8—42no—
479C8—82no—
480C8—82no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—82no—
493C22—162no—
494C22C2×#38511yescharacteristic
495C22#295512yes—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—21no—
507C4—42no—
508C4—42no—
509C4—42no—
510C4—84no—
511C4#302111yescharacteristic
512C4#302211yescharacteristic
513C2—22no—
514C2—41no—
515C2—41no—
516C2—82no—
517C2#2086511yescharacteristic
518C2#2088411yescharacteristic
519C2#2089211yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ 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
2. #21975C2semidirect
3. #17822C2semidirect
6. #8877C2semidirect
7. #8876C2semidirect
8. #8069C2semidirect
9. #7936C2semidirect
33. #1843C22semidirect
34. #1832C22semidirect
35. #1573C22semidirect
47. #1300C22semidirect
48. #1283C22semidirect
49. #1283C22semidirect

Maximal subgroups

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

subgroupindex
#602882
#219752
#178222
#175542
#174922
#88772
#88762
#80692
#79362