Finite Group Databasebeta

← #202084

Subgroups — #202084

Order 512.

1057 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3113C211yesmaximal, characteristic
2#60250C211yesmaximal, characteristic
3#17749C211yesmaximal, characteristic
4C2×#1487C211yesmaximal, characteristic
5#17365C211yesmaximal, characteristic
6#9528C218yesmaximal, complemented
7#8121C211yesmaximal, characteristic
8#7966C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11#3113—22no—
12#3113—22no—
13#3113C414yes—
14#3113C2214yes—
15C2×#483C412yes—
16C2×#475—21no—
17C2×#462—21no—
18C2×#451C412yes—
19#2986—22no—
20#2985C2214yes—
21#2984—22no—
22#2983—21no—
23#2982—21no—
24#2962—22no—
25#2960—22no—
26C2×#441—22no—
27#2924—22no—
28#2920C2211yescharacteristic
29#2920C2211yescharacteristic
30C2×#432C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C2×#423C2211yescharacteristic
33#1559C2214yes—
34#1550—22no—
35#1548—21no—
36#1546—21no—
37#1487—21no—
38#1487—21no—
39#1487C2214yes—
40C2×Q8⋊C8—22no—
41#1467C2214yes—
42#1463—22no—
43#1461—22no—
44#1307C2214yes—
45#1287C2214yes—
46C23×Q8C2×C411yescharacteristic
47C22×Q16—22no—
48C22×M16—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533C2×C412yes—
54#533D412yes—
55#533D412yes—
56#533C2312yes—
57C2×C4⋊Q8D411yescharacteristic
58C2×C4⋊Q8D411yescharacteristic
59C2×#125—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8C2×C412yes—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8C2×C412yes—
74C2×C4×Q8D412yes—
75C2×C4×Q8C2312yes—
76C2×#117—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#483—24no—
82#483C2×C418yes—
83#475—24no—
84#475—24no—
85#462—24no—
86#462—24no—
87#451—24no—
88#451C2×C418yes—
89#446—24no—
90#446—24no—
91#446—28no—
92#446—24no—
93C8×Q8—24no—
94C8×Q8—24no—
95C8×Q8—24no—
96#441—28no—
97#441—28no—
98#433—28no—
99#432—24no—
100#432—21no—
101#432—21no—
102#432—21no—
103#432—21no—
104#432C2×C412yes—
105#432C2312yes—
106#432C2312yes—
107#432C2312yes—
108#431—24no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431D412yes—
115#431D411yescharacteristic
116#431D411yescharacteristic
117#431C2312yes—
118C2×#106—22no—
119#424—24no—
120#423—21no—
121#423—21no—
122#423—21no—
123#423—21no—
124#423—21no—
125#423C2312yes—
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8—21no—
130C2×#103—22no—
131C2×#103—22no—
132#407—21no—
133#407D412yes—
134C2×C22⋊C8—21no—
135C2×C22⋊C8C2×C412yes—
136#405—24no—
137C4×M16—21no—
138C4×M16—21no—
139C2×#97—21no—
140C2×C4×C8D412yes—
141#333—24no—
142#332—24no—
143Q8⋊C8—24no—
144Q8⋊C8—24no—
145#324—28no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—44no—
153C22×Q8C22⋊C412yes—
154C22×Q8C22×C412yes—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4C22×C411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4#4111yescharacteristic
160C2×Q16—28no—
161C2×Q16—48no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—41no—
165C2×M16—41no—
166C2×M16—82no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—41no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8C22⋊C412yes—
175C4⋊Q8C22⋊C412yes—
176C4⋊Q8C2×D412yes—
177C4⋊Q8C2×D412yes—
178#125—22no—
179#125—22no—
180#125—22no—
181#125—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8C22⋊C414yes—
189C22⋊Q8C22×C414yes—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—41no—
205C4×Q8—41no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8C22⋊C414yes—
213C4×Q8C2×D414yes—
214#117—22no—
215#117—22no—
216#117—22no—
217#117—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4C22⋊C412yes—
235C2×C4⋊C4C22×C412yes—
236C2×C22⋊C4—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42—41no—
243C2×C42—41no—
244C2×C42C22×C411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42#4111yescharacteristic
253C2×C42#4111yescharacteristic
254C2×C42#4111yescharacteristic
255C2×C42C2411yescharacteristic
256#106—28no—
257C4⋊C8—28no—
258C4⋊C8—28no—
259C4⋊C8—22no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—22no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8—41no—
270C4⋊C8—41no—
271C4⋊C8—41no—
272#103—28no—
273#103—28no—
274C22⋊C8—22no—
275C22⋊C8—21no—
276C22⋊C8—21no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—22no—
280C22⋊C8—41no—
281C22⋊C8C22×C414yes—
282C22⋊C8C2×D414yes—
283#97—28no—
284#97—22no—
285#97—22no—
286#97—22no—
287#97—22no—
288C4×C8—28no—
289C4×C8—22no—
290C4×C8—22no—
291C4×C8—41no—
292C4×C8C2×D412yes—
293C4×C8C2×D412yes—
294C24C4×D411yescharacteristic
295C2×Q8—24no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—44no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8C2×C22⋊C414yes—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—82no—
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4C4×D411yescharacteristic
333C22×C4C4×D411yescharacteristic
334C22×C4C4×D411yescharacteristic
335C22×C4C4×D411yescharacteristic
336C22×C4#12011yescharacteristic
337C22×C4#12011yescharacteristic
338C22×C4#12111yescharacteristic
339C22×C4#12111yescharacteristic
340C22×C4#12111yescharacteristic
341C22×C4#12111yescharacteristic
342C22×C4#12311yescharacteristic
343C22×C4#12311yescharacteristic
344C22×C4C23×C411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C2×#4111yescharacteristic
347Q16—416no—
348M16—44no—
349M16—82no—
350M16—82no—
351M16—82no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8—24no—
355C2×C8—22no—
356C2×C8—44no—
357C2×C8—44no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—82no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4C2×C22⋊C414yes—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C42—24no—
384C42—24no—
385C42—42no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42C2×C22⋊C411yescharacteristic
390C42C2×C22⋊C411yescharacteristic
391C42C4×D411yescharacteristic
392C42C4×D411yescharacteristic
393C42#12011yescharacteristic
394C42#12011yescharacteristic
395C42#12111yescharacteristic
396C42#12111yescharacteristic
397C42#12111yescharacteristic
398C42#12111yescharacteristic
399C42#12311yescharacteristic
400C42#12311yescharacteristic
401C42C22×D411yescharacteristic
402C42C22×D411yescharacteristic
403C42C22×D411yescharacteristic
404C42C2×#4111yescharacteristic
405C23—22no—
406C23—22no—
407C23—41no—
408C23#38611yescharacteristic
409C23#38611yescharacteristic
410C23C2×C4×D411yescharacteristic
411Q8—24no—
412Q8—24no—
413Q8—44no—
414Q8—42no—
415Q8—42no—
416Q8—44no—
417Q8—44no—
418Q8—44no—
419Q8—44no—
420Q8—42no—
421Q8—42no—
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—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4#38611yescharacteristic
455C2×C4#38611yescharacteristic
456C2×C4#38611yescharacteristic
457C2×C4#38611yescharacteristic
458C2×C4#38611yescharacteristic
459C2×C4#38611yescharacteristic
460C2×C4#47112yes—
461C2×C4#47312yes—
462C2×C4C22×C22⋊C411yescharacteristic
463C2×C4C2×C4×D411yescharacteristic
464C2×C4C2×#12011yescharacteristic
465C2×C4C2×#12111yescharacteristic
466C2×C4C2×#12111yescharacteristic
467C2×C4C2×#12311yescharacteristic
468C8—42no—
469C8—42no—
470C8—44no—
471C8—44no—
472C8—82no—
473C8—82no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—44no—
482C22#190512yes—
483C22#190812yes—
484C22C2×#38611yescharacteristic
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—44no—
495C4—44no—
496C4—44no—
497C4—82no—
498C4#189112yes—
499C4#190812yes—
500C4C2×#47111yescharacteristic
501C4C2×#47311yescharacteristic
502C2—22no—
503C2—22no—
504C2—41no—
505C2C2×#189111yescharacteristic
506C2C2×#190511yescharacteristic
507C2C2×#190811yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9528C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31132
#602502
#177492
C2×#14872
#173652
#95282
#81212
#79662