Finite Group Databasebeta

← #195773

Subgroups — #195773

Order 512.

1027 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3160C211yesmaximal, characteristic
2#60250C211yesmaximal, characteristic
3#17648C211yesmaximal, characteristic
4#17501C211yesmaximal, characteristic
5#17464C211yesmaximal, characteristic
6#9068C218yesmaximal, complemented
7#7885C211yesmaximal, characteristic
8C2×#1278C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#533C2211yescharacteristic
11#3160—22no—
12#3160—22no—
13#3160C414yes—
14#3160C2214yes—
15C2×#500C411yescharacteristic
16C2×#500C411yescharacteristic
17C2×#499—22no—
18C2×#498C411yescharacteristic
19C2×#498C411yescharacteristic
20#2986C2214yes—
21#2985—22no—
22#2984—22no—
23#2983—21no—
24#2982—21no—
25#2962—22no—
26#2960—22no—
27#2924—22no—
28#2920C2211yescharacteristic
29#2920C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#427—22no—
32C2×C4×M16C2211yescharacteristic
33C22×C4×C8C2211yescharacteristic
34#1531—22no—
35#1528—22no—
36#1526C2214yes—
37#1493—22no—
38#1491—22no—
39#1485C2214yes—
40#1485C2214yes—
41#1481—22no—
42#1479—21no—
43#1479—21no—
44#1278C2214yes—
45#1278C2214yes—
46C22×M16—21no—
47C23×C8—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533C2×C412yes—
59#533C2×C412yes—
60#533D412yes—
61#533D412yes—
62#533C2312yes—
63#533C2312yes—
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×C4⋊Q8C2×C411yescharacteristic
66C2×C4⋊Q8C2×C411yescharacteristic
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×#125—21no—
69C2×#125D412yes—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×#117—21no—
79C2×#117D412yes—
80C22×C42C2311yescharacteristic
81#500—22no—
82#500—22no—
83#500C2×C414yes—
84#500C2×C414yes—
85#499—28no—
86#499—28no—
87#498—22no—
88#498—22no—
89#498C2×C414yes—
90#498C2×C414yes—
91#446—24no—
92#446—28no—
93#446—24no—
94#446—24no—
95C8×Q8—24no—
96C8×Q8—24no—
97C8×Q8—24no—
98#433—28no—
99#432—24no—
100#432—21no—
101#432—21no—
102#432—21no—
103#432D412yes—
104#431—24no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431D411yescharacteristic
111#431D411yescharacteristic
112#431C2312yes—
113#431C2312yes—
114#427—28no—
115#427—28no—
116C2×#107—22no—
117C2×#107—22no—
118C2×#106—22no—
119C2×#106—22no—
120#424—24no—
121#423—21no—
122#423D412yes—
123C2×C4⋊C8D412yes—
124#407—21no—
125#407—21no—
126C2×C22⋊C8—21no—
127#405—24no—
128C4×M16—21no—
129C4×M16—21no—
130C4×M16—21no—
131C4×M16—21no—
132C4×M16C2312yes—
133C4×M16C2312yes—
134C2×#97—21no—
135C2×C4×C8—21no—
136C2×C4×C8—21no—
137C2×C4×C8—21no—
138C2×C4×C8—21no—
139C2×C4×C8—21no—
140C2×C4×C8C2×C411yescharacteristic
141C2×C4×C8C2×C411yescharacteristic
142C2×C4×C8C2×C412yes—
143C2×C4×C8C2312yes—
144#330—28no—
145Q8⋊C8—24no—
146Q8⋊C8—24no—
147Q8⋊C8—28no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C23×C4C2×D411yescharacteristic
153C23×C4#4111yescharacteristic
154C23×C4#4111yescharacteristic
155C2×M16—22no—
156C2×M16—24no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—42no—
160C2×M16—41no—
161C22×C8—28no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—41no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8C22⋊C412yes—
179C4⋊Q8C22⋊C412yes—
180C4⋊Q8C22×C412yes—
181C4⋊Q8C22×C412yes—
182C4⋊Q8C22×C412yes—
183C4⋊Q8C22×C412yes—
184#125—22no—
185#125—22no—
186#125—22no—
187#125—22no—
188#125—41no—
189#125—41no—
190#125C22⋊C414yes—
191#125C2×D414yes—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C22⋊Q8—41no—
207C22⋊Q8—41no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—41no—
221C4×Q8—41no—
222C4×Q8—41no—
223C4×Q8—41no—
224#117—22no—
225#117—22no—
226#117—22no—
227#117—22no—
228#117—41no—
229#117—41no—
230#117C22⋊C414yes—
231#117C2×D414yes—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4C22⋊C412yes—
243C2×C4⋊C4C22⋊C412yes—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—41no—
246C2×C42—22no—
247C2×C42—41no—
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42#4111yescharacteristic
258C2×C42#4111yescharacteristic
259C2×C42C2411yescharacteristic
260#107—28no—
261#107—28no—
262#106—28no—
263#106—28no—
264C4⋊C8—28no—
265C4⋊C8—28no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—41no—
269C4⋊C8—41no—
270C4⋊C8C2×D412yes—
271C4⋊C8C2×D412yes—
272C4⋊C8C2×D414yes—
273C22⋊C8—21no—
274C22⋊C8—21no—
275C22⋊C8—22no—
276C22⋊C8—22no—
277C22⋊C8—22no—
278C22⋊C8—41no—
279C22⋊C8—41no—
280#97—28no—
281#97—22no—
282#97—22no—
283#97—22no—
284#97—41no—
285C4×C8—28no—
286C4×C8—21no—
287C4×C8—22no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—22no—
292C4×C8—22no—
293C4×C8—22no—
294C4×C8—22no—
295C4×C8—22no—
296C4×C8—41no—
297C4×C8C22×C412yes—
298C4×C8C22×C412yes—
299C24#12311yescharacteristic
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—82no—
313C2×Q8—82no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4C4×D411yescharacteristic
326C22×C4C4×D411yescharacteristic
327C22×C4C4×D411yescharacteristic
328C22×C4#12011yescharacteristic
329C22×C4#12011yescharacteristic
330C22×C4#12111yescharacteristic
331C22×C4#12111yescharacteristic
332C22×C4#12111yescharacteristic
333C22×C4#12111yescharacteristic
334C22×C4#12311yescharacteristic
335C22×C4#12311yescharacteristic
336C22×C4#12311yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340M16—24no—
341M16—44no—
342M16—42no—
343M16—44no—
344M16—44no—
345C2×C8—28no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—28no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8—44no—
355C2×C8—42no—
356C2×C8—44no—
357C2×C8—42no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4C2×C22⋊C414yes—
381C4⋊C4C2×C22⋊C414yes—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C42—24no—
387C42—24no—
388C42—42no—
389C42—42no—
390C42C2×C22⋊C411yescharacteristic
391C42C2×C22⋊C411yescharacteristic
392C42C4×D411yescharacteristic
393C42C4×D411yescharacteristic
394C42C4×D411yescharacteristic
395C42C4×D411yescharacteristic
396C42#12011yescharacteristic
397C42#12011yescharacteristic
398C42#12111yescharacteristic
399C42#12111yescharacteristic
400C42#12111yescharacteristic
401C42#12111yescharacteristic
402C42C23×C411yescharacteristic
403C42C22×D411yescharacteristic
404C42C22×D411yescharacteristic
405C42C22×D411yescharacteristic
406C23—22no—
407C23—22no—
408C23—41no—
409C23#38611yescharacteristic
410C23#38611yescharacteristic
411C23C2×#12311yescharacteristic
412Q8—44no—
413Q8—42no—
414Q8—42no—
415Q8—44no—
416Q8—42no—
417Q8—42no—
418Q8—44no—
419Q8—82no—
420Q8—82no—
421Q8—82no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4#38611yescharacteristic
449C2×C4#38611yescharacteristic
450C2×C4#38611yescharacteristic
451C2×C4#38611yescharacteristic
452C2×C4#38611yescharacteristic
453C2×C4#38611yescharacteristic
454C2×C4#44312yes—
455C2×C4#44412yes—
456C2×C4#45614yes—
457C2×C4C22×C22⋊C411yescharacteristic
458C2×C4C2×C4×D411yescharacteristic
459C2×C4C2×C4×D411yescharacteristic
460C2×C4C2×#12011yescharacteristic
461C2×C4C2×#12111yescharacteristic
462C2×C4C2×#12111yescharacteristic
463C8—24no—
464C8—24no—
465C8—44no—
466C8—42no—
467C8—42no—
468C8—44no—
469C8—82no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—44no—
478C22#187612yes—
479C22#188612yes—
480C22C2×#38611yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—44no—
489C4—44no—
490C4—82no—
491C4—82no—
492C4#187612yes—
493C4#188612yes—
494C4C2×#44311yescharacteristic
495C4C2×#44411yescharacteristic
496C4#295712yes—
497C4C2×#45612yes—
498C2—22no—
499C2—22no—
500C2—41no—
501C2C2×#187611yescharacteristic
502C2#2091911yescharacteristic
503C2C2×#188611yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC24#123
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
6. #9068C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31602
#602502
#176482
#175012
#174642
#90682
#78852
C2×#12782