Finite Group Databasebeta

← #205161

Subgroups — #205161

Order 512.

884 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60943C211yesmaximal, characteristic
2#21370C211yesmaximal, characteristic
3#17664C211yesmaximal, characteristic
4#17648C211yesmaximal, characteristic
5#17464C211yesmaximal, characteristic
6C2×#1485C211yesmaximal, characteristic
7#17326C211yesmaximal, characteristic
8#9727C214yesmaximal, complemented
9#9725C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×#533C2211yescharacteristic
12#3249—22no—
13#3230C412yes—
14#3230C412yes—
15#3209C412yes—
16#3209C412yes—
17#3189—22no—
18#3170C2214yes—
19#3104—21no—
20#3104—21no—
21#3104—22no—
22#2920C2211yescharacteristic
23#2920C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27#1950—22no—
28#1855—22no—
29C4×#108C2214yes—
30#1537—22no—
31#1534—22no—
32#1531—22no—
33#1528—22no—
34#1526C2214yes—
35#1526C2214yes—
36#1485—22no—
37#1485C2212yes—
38#1485C2212yes—
39#1485C2212yes—
40#1485C2212yes—
41#1481—22no—
42#1479—22no—
43C2×Q8⋊C8—22no—
44#1454—22no—
45#1452—21no—
46#1451C2212yes—
47#1450—21no—
48C8×M16C2212yes—
49C22×M16—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#533—21no—
59#533—21no—
60#533—21no—
61#533—21no—
62#533C2×C412yes—
63#533C2×C412yes—
64#533C2312yes—
65#533C2312yes—
66C2×C4⋊Q8C2×C411yescharacteristic
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×#125—21no—
71C2×#125—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×#117—21no—
81C2×#117—21no—
82C22×C42C2311yescharacteristic
83#500C2×C414yes—
84#494C2×C414yes—
85#487C2×C414yes—
86#487C2×C414yes—
87#484—22no—
88#484—22no—
89#484—24no—
90#477—22no—
91#477—22no—
92#477—24no—
93#467—24no—
94#467—24no—
95#462—22no—
96#462—22no—
97#462—24no—
98#451—22no—
99#451—22no—
100#451—24no—
101C4×Q16—24no—
102C4×Q16—24no—
103#432—21no—
104#432—21no—
105#432D412yes—
106#432D412yes—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431C2×C412yes—
114#431C2×C412yes—
115#431D412yes—
116#431D411yescharacteristic
117#431D411yescharacteristic
118#431D412yes—
119#431C2311yescharacteristic
120#431C2311yescharacteristic
121C2×#108—24no—
122#423—21no—
123#423—21no—
124C2×C4⋊C8D412yes—
125C2×C4⋊C8D412yes—
126#407—21no—
127#407—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8D412yes—
130C4×M16—21no—
131C4×M16—21no—
132C4×M16—21no—
133C4×M16C2312yes—
134C4×M16C2312yes—
135C4×M16C2312yes—
136C2×#97—21no—
137C2×C4×C8—21no—
138C2×C4×C8—21no—
139C2×C4×C8C2312yes—
140#341—24no—
141#330—28no—
142#330—28no—
143Q8⋊C8—24no—
144Q8⋊C8—24no—
145Q8⋊C8—24no—
146Q8⋊C8—24no—
147#322—22no—
148#322—22no—
149#322—22no—
150C82—22no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C23×C4C2×D411yescharacteristic
156C23×C4#4111yescharacteristic
157C23×C4#4111yescharacteristic
158C2×Q16—24no—
159C2×Q16—24no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—44no—
164C2×M16—41no—
165C22×C8—22no—
166C22×C8—41no—
167C22×C8C2×D414yes—
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⋊Q8—21no—
179C4⋊Q8—21no—
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#125—41no—
191#125—41no—
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—41no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C22⋊Q8—41no—
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—41no—
217C4×Q8—41no—
218C4×Q8—41no—
219C4×Q8—41no—
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#117—41no—
231#117—41no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
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⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C42—41no—
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#108—48no—
261#107—24no—
262#107—24no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—41no—
267C4⋊C8—41no—
268C4⋊C8C22⋊C412yes—
269C4⋊C8C22⋊C412yes—
270C4⋊C8C22⋊C414yes—
271C4⋊C8C2×D412yes—
272C4⋊C8C2×D412yes—
273#103—24no—
274#103—24no—
275#103—24no—
276#103—24no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—22no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C22⋊C8—41no—
283C22⋊C8—41no—
284C22⋊C8C22⋊C412yes—
285C22⋊C8C22⋊C412yes—
286C22⋊C8C22⋊C414yes—
287#97—22no—
288#97—22no—
289#97—41no—
290#97C2×D414yes—
291C4×C8—22no—
292C4×C8—21no—
293C4×C8—22no—
294C4×C8—21no—
295C4×C8—22no—
296C4×C8—22no—
297C4×C8—41no—
298C4×C8C22×C414yes—
299C4×C8C2×D412yes—
300C4×C8C2×D412yes—
301C24#12411yescharacteristic
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—82no—
313C2×Q8—82no—
314C22×C4—24no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C4×D411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4C4×D411yescharacteristic
326C22×C4C4×D411yescharacteristic
327C22×C4#12111yescharacteristic
328C22×C4#12111yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4#12111yescharacteristic
331C22×C4#12411yescharacteristic
332C22×C4#12411yescharacteristic
333C22×C4#12411yescharacteristic
334C22×C4#12711yescharacteristic
335C22×C4#12711yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339Q16—84no—
340Q16—84no—
341M16—24no—
342M16—44no—
343M16—42no—
344M16—44no—
345M16—44no—
346C2×C8—24no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—44no—
350C2×C8—44no—
351C2×C8—42no—
352C2×C8—44no—
353C2×C8—42no—
354C2×C8C2×C22⋊C414yes—
355C2×C8C2×C22⋊C414yes—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C42—42no—
385C42—42no—
386C42—42no—
387C42—42no—
388C42C2×C22⋊C411yescharacteristic
389C42C2×C22⋊C411yescharacteristic
390C42C4×D411yescharacteristic
391C42C4×D411yescharacteristic
392C42C4×D411yescharacteristic
393C42C4×D411yescharacteristic
394C42#12111yescharacteristic
395C42#12111yescharacteristic
396C42#12111yescharacteristic
397C42#12111yescharacteristic
398C42#12711yescharacteristic
399C42#12711yescharacteristic
400C42C23×C411yescharacteristic
401C42C22×D411yescharacteristic
402C42C22×D411yescharacteristic
403C42C22×D411yescharacteristic
404C23—22no—
405C23—41no—
406C23—41no—
407C23#39011yescharacteristic
408C23#39011yescharacteristic
409C23C2×#12411yescharacteristic
410Q8—42no—
411Q8—42no—
412Q8—44no—
413Q8—44no—
414Q8—44no—
415Q8—82no—
416Q8—82no—
417Q8—82no—
418Q8—82no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4#39011yescharacteristic
445C2×C4#39011yescharacteristic
446C2×C4#39011yescharacteristic
447C2×C4#39011yescharacteristic
448C2×C4#39011yescharacteristic
449C2×C4#39011yescharacteristic
450C2×C4#47314yes—
451C2×C4C22×C22⋊C411yescharacteristic
452C2×C4C2×C4×D411yescharacteristic
453C2×C4C2×C4×D411yescharacteristic
454C2×C4C2×#12111yescharacteristic
455C2×C4C2×#12111yescharacteristic
456C2×C4C2×#12711yescharacteristic
457C8—24no—
458C8—24no—
459C8—44no—
460C8—44no—
461C8—42no—
462C8—42no—
463C8—82no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—82no—
473C22#197912yes—
474C22C2×#39011yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—82no—
484C4—82no—
485C4—82no—
486C4—82no—
487C4#197912yes—
488C4#290112yes—
489C4#295512yes—
490C4C2×#47312yes—
491C2—22no—
492C2—41no—
493C2—41no—
494C2#2124211yescharacteristic
495C2C2×#197911yescharacteristic
496C2#2125311yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radicalC24#124
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
8. #9727C2semidirect
9. #9725C2semidirect

Maximal subgroups

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

subgroupindex
#609432
#213702
#176642
#176482
#174642
C2×#14852
#173262
#97272
#97252