Finite Group Databasebeta

← #192925

Subgroups — #192925

Order 512.

893 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60268C211yesmaximal, characteristic
2#21856C211yesmaximal, characteristic
3#17809C211yesmaximal, characteristic, complemented
4C2×#1566C211yesmaximal, characteristic, complemented
5#17473C211yesmaximal, characteristic
6#8905C214yesmaximal, complemented
7#8904C214yesmaximal, complemented
8#8097C211yesmaximal, characteristic, complemented
9#7890C211yesmaximal, characteristic, complemented
10#3444C2211yescharacteristic
11#2994—22no—
12#2993—22no—
13#2991—21no—
14#2990—21no—
15#2989—21no—
16#2988—21no—
17#2960—21no—
18#2933—22no—
19#2931—21no—
20#2925C2212yes—
21#2920C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2906C2212yes—
24C2×#423C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C4×#117C2211yescharacteristic
28#1994—21no—
29#1962—21no—
30#1871—22no—
31#1806C2214yes—
32#1574—21no—
33#1574—21no—
34#1571C2212yes—
35#1570C2212yes—
36#1566—22no—
37#1566C2214yes—
38#1565—21no—
39#1565—21no—
40C2×#334—22no—
41#1494C414yescomplemented
42#1493C411yescharacteristic, complemented
43#1493C411yescharacteristic, complemented
44#1492C411yescharacteristic, complemented
45#1492C411yescharacteristic, complemented
46#1476C2214yes—
47#1303C2214yes—
48#1279C2212yes—
49#1279C2212yes—
50C2×#131—22no—
51C22×M16—21no—
52C22×M16—21no—
53C23×C8—21no—
54#533C2×C411yescharacteristic
55#533D411yescharacteristic
56#533Q811yescharacteristic
57#532C2×C411yescharacteristic
58#532D411yescharacteristic
59#532Q811yescharacteristic
60#529—21no—
61#529—21no—
62#529—21no—
63#529—21no—
64#529C2×C412yes—
65#529Q812yes—
66#525—21no—
67#525—21no—
68#525D412yes—
69C4×#41—21no—
70C4×#41—21no—
71C4×#41C2312yes—
72C2×#117—21no—
73C22×C42C2311yescharacteristic
74#446—22no—
75#446—22no—
76C8×Q8—22no—
77C8×Q8—22no—
78#436—22no—
79#436—22no—
80#436—24no—
81#435—24no—
82#435—24no—
83C8×D4—22no—
84C8×D4—22no—
85C8×D4—24no—
86#433—24no—
87#432—21no—
88#432—22no—
89#432C2×C412yes—
90#431—21no—
91#431—21no—
92#431—21no—
93#431—22no—
94#431D411yescharacteristic
95#431D411yescharacteristic
96#431D411yescharacteristic
97#431D411yescharacteristic
98#431C2312yes—
99#424—22no—
100#424—24no—
101#423—21no—
102#423—21no—
103#423—22no—
104#423—21no—
105#423—21no—
106#423—21no—
107#423C2×C411yescharacteristic
108#423C2×C411yescharacteristic
109#423D412yes—
110#423C2312yes—
111#423C2312yes—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—22no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8D412yes—
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2311yescharacteristic
119C2×#104—24no—
120#408—22no—
121#407—21no—
122C2×C22⋊C8—21no—
123#405—22no—
124C4×M16—21no—
125C2×C4×C8—21no—
126C2×C4×C8—21no—
127C2×C4×C8—21no—
128C2×C4×C8—21no—
129C2×C4×C8C2×C411yescharacteristic
130C2×C4×C8C2×C411yescharacteristic
131C2×C4×C8C2×C412yes—
132C2×C4×C8C2312yes—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2312yes—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137#376—21no—
138C43—21no—
139#339—24no—
140#335—22no—
141#335—22no—
142#334—28no—
143#334—22no—
144#334—22no—
145Q8⋊C8C2×C414yes—
146Q8⋊C8C2×C414yes—
147D4⋊C8C2×C414yes—
148D4⋊C8C2×C414yes—
149C2×#41—21no—
150C2×#41—21no—
151C2×#41C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4#4111yescharacteristic
154C23×C4#4111yescharacteristic
155#131—84no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—22no—
159C2×M16—24no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—41no—
163C2×M16—82no—
164C2×M16—82no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—24no—
172C22×C8—22no—
173C22×C8—44no—
174C22×C8—41no—
175C4⋊Q8C4⋊C411yescharacteristic
176C4⋊Q8C4⋊C411yescharacteristic
177C4⋊Q8C4⋊C411yescharacteristic
178#127C4⋊C411yescharacteristic
179#126—22no—
180#126—22no—
181#125—21no—
182#125—21no—
183#125C4⋊C412yes—
184#124—21no—
185#124—21no—
186#124C4⋊C412yes—
187#123—22no—
188#123—22no—
189#123—22no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8C4⋊C412yes—
195C22⋊Q8C4⋊C412yes—
196#121—21no—
197#121—21no—
198#121—21no—
199#121—21no—
200#121C4⋊C412yes—
201#121C4⋊C412yes—
202#120—22no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8C22×C412yes—
208C4×Q8C2×Q812yes—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4C22×C412yes—
220C4×D4C2×Q812yes—
221#117—21no—
222#117—21no—
223#117—22no—
224#117—22no—
225#117—22no—
226#117—24no—
227#117—42no—
228#117C2×D412yes—
229C2×C4⋊C4—22no—
230C2×C22⋊C4—41no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—41no—
234C2×C42—41no—
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×Q811yescharacteristic
243C2×C42C2×Q811yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42C2411yescharacteristic
247C4⋊C8—24no—
248C4⋊C8—24no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—41no—
259C4⋊C8—41no—
260C4⋊C8C22×C412yes—
261C4⋊C8C22×C412yes—
262C4⋊C8C2×D411yescharacteristic
263C4⋊C8C2×D414yes—
264C4⋊C8C2×D411yescharacteristic
265C4⋊C8C2×D411yescharacteristic
266C4⋊C8C2×D411yescharacteristic
267#104—84no—
268C22⋊C8—24no—
269C22⋊C8—24no—
270C22⋊C8—22no—
271C22⋊C8—21no—
272C22⋊C8—21no—
273C22⋊C8—21no—
274C22⋊C8—21no—
275C22⋊C8—22no—
276#97—24no—
277#97—22no—
278C4×C8—24no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—22no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—22no—
287C4×C8—22no—
288C4×C8C22×C412yes—
289C4×C8C22×C412yes—
290#95—22no—
291#95—41no—
292C24#12311yescharacteristic
293#41—24no—
294#41—24no—
295#41—42no—
296#41—42no—
297#41—82no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8C2×C4⋊C412yes—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4C2×C4⋊C412yes—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4C2×C4⋊C411yescharacteristic
317C22×C4C2×C4⋊C411yescharacteristic
318C22×C4C4×D411yescharacteristic
319C22×C4C4×D411yescharacteristic
320C22×C4C4×D411yescharacteristic
321C22×C4C4×D411yescharacteristic
322C22×C4#12011yescharacteristic
323C22×C4#12011yescharacteristic
324C22×C4C22⋊Q811yescharacteristic
325C22×C4C22⋊Q811yescharacteristic
326C22×C4C22⋊Q811yescharacteristic
327C22×C4C22⋊Q811yescharacteristic
328C22×C4#12311yescharacteristic
329C22×C4#12311yescharacteristic
330C22×C4#12311yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334M16—44no—
335M16—42no—
336M16—42no—
337M16—44no—
338M16—82no—
339M16—84no—
340M16—82no—
341C2×C8—24no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—24no—
351C2×C8—44no—
352C2×C8—44no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—82no—
358C2×C8—84no—
359C2×C8—82no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—42no—
372C4⋊C4C2×C4⋊C412yes—
373C4⋊C4C2×C4⋊C412yes—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—44no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—44no—
388C42—42no—
389C42—42no—
390C42—82no—
391C42C2×C4⋊C411yescharacteristic
392C42C2×C4⋊C411yescharacteristic
393C42C4×D411yescharacteristic
394C42C4×D411yescharacteristic
395C42C4×D411yescharacteristic
396C42C4×D411yescharacteristic
397C42#12011yescharacteristic
398C42#12011yescharacteristic
399C42C22⋊Q811yescharacteristic
400C42C22⋊Q811yescharacteristic
401C42C22⋊Q811yescharacteristic
402C42C22⋊Q811yescharacteristic
403C42C23×C411yescharacteristic
404C42C22×D411yescharacteristic
405C42C22×D411yescharacteristic
406C42C22×Q811yescharacteristic
407C23—22no—
408C23—22no—
409C23—22no—
410C23—41no—
411C23#38511yescharacteristic
412C23#38511yescharacteristic
413C23C2×#12311yescharacteristic
414Q8—24no—
415Q8—42no—
416Q8—42no—
417D4—24no—
418D4—44no—
419D4—44no—
420D4—42no—
421D4—42no—
422D4—82no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4#38511yescharacteristic
455C2×C4#38511yescharacteristic
456C2×C4#38511yescharacteristic
457C2×C4#38511yescharacteristic
458C2×C4#38511yescharacteristic
459C2×C4#38511yescharacteristic
460C2×C4#44312yes—
461C2×C4#44412yes—
462C2×C4C22×C4⋊C411yescharacteristic
463C2×C4C2×C4×D411yescharacteristic
464C2×C4C2×C4×D411yescharacteristic
465C2×C4C2×#12011yescharacteristic
466C2×C4C2×C22⋊Q811yescharacteristic
467C2×C4C2×C22⋊Q811yescharacteristic
468C2×C4#57712yes—
469C2×C4#57712yes—
470C8—24no—
471C8—42no—
472C8—42no—
473C8—42no—
474C8—41no—
475C8—41no—
476C8—44no—
477C8—41no—
478C8—41no—
479C8—82no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—44no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22#186612yes—
492C22C2×#38511yescharacteristic
493C22#309912yes—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—84no—
505C4#186112yes—
506C4#186612yes—
507C4#291212yes—
508C4C2×#44311yescharacteristic
509C4C2×#44411yescharacteristic
510C4#300911yescharacteristic
511C4#301011yescharacteristic
512C2—22no—
513C2—22no—
514C2—41no—
515C2—82no—
516C2#2084811yescharacteristic
517C2C2×#186111yescharacteristic
518C2C2×#186611yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radical#3444C22
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. #17809C2semidirect
4. C2×#1566C2semidirect
6. #8905C2semidirect
7. #8904C2semidirect
8. #8097C2semidirect
9. #7890C2semidirect
41. #1494C4semidirect
42. #1493C4semidirect
43. #1493C4semidirect
44. #1492C4semidirect
45. #1492C4semidirect

Maximal subgroups

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

subgroupindex
#602682
#218562
#178092
C2×#15662
#174732
#89052
#89042
#80972
#78902