Finite Group Databasebeta

← #199482

Subgroups — #199482

Order 512.

1109 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3113C211yesmaximal, characteristic
2#34059C211yesmaximal, characteristic
3#21844C211yesmaximal, characteristic
4#17545C211yesmaximal, characteristic
5#17449C211yesmaximal, characteristic
6#9328C218yesmaximal, complemented
7#8071C211yesmaximal, characteristic
8#7949C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11#3113—22no—
12#3113—22no—
13#3113C414yescomplemented
14#3113C2214yes—
15C2×#483—21no—
16C2×#475C412yescomplemented
17C2×#462C412yescomplemented
18C2×#451—21no—
19C2×#441—22no—
20#2920C2211yescharacteristic
21#2920C2211yescharacteristic
22C2×#432C2211yescharacteristic
23#2320—22no—
24#2320C2214yes—
25#2316—21no—
26#2313—22no—
27#2310—21no—
28#2295—22no—
29#2295—22no—
30#2283—22no—
31C4×#117C2211yescharacteristic
32C4×#117C2211yescharacteristic
33#1994—22no—
34#1986C2214yes—
35#1826—22no—
36#1508—22no—
37#1506C2214yes—
38#1503—22no—
39#1491C2214yes—
40#1487—22no—
41#1479—22no—
42#1300C2214yes—
43#1285C2214yes—
44C23×Q8D411yescharacteristic
45C22×Q16—22no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533C2×C412yes—
53#533C2312yes—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8C2×C412yes—
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8D412yes—
70C2×C4×Q8D412yes—
71C2×C4×Q8D412yes—
72C2×C4×Q8C2312yes—
73C2×#117—21no—
74C2×#117—21no—
75C2×#117—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80#483—24no—
81#483—24no—
82#475—24no—
83#475C2×C418yes—
84#462—24no—
85#462C2×C418yes—
86#451—24no—
87#451—24no—
88#441—28no—
89#441—28no—
90#432—21no—
91#432—21no—
92#432—21no—
93#432—21no—
94#432—21no—
95#432C2×C412yes—
96#432C2312yes—
97#432C2312yes—
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102C2×#106—22no—
103#423—21no—
104#423—21no—
105C2×C4⋊C8C2×C412yes—
106C2×#103—22no—
107C2×#103—22no—
108#407—21no—
109#407—21no—
110C2×C22⋊C8—21no—
111C4×M16—21no—
112C4×M16C2312yes—
113C2×#97—21no—
114#391—24no—
115#391—24no—
116#389—24no—
117#389—24no—
118#387—28no—
119#387—28no—
120#383—24no—
121#382—24no—
122#381—24no—
123#380—24no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4C2312yes—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4D412yes—
132C4×C22⋊C4D412yes—
133#376—24no—
134#376—21no—
135#376C2312yes—
136C43—21no—
137C43D412yes—
138#339—28no—
139#328—28no—
140Q8⋊C8—28no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—44no—
147C22×Q8C22⋊C412yes—
148C22×Q8C22⋊C412yes—
149C22×Q8C2×D412yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C2×Q16—28no—
156C2×Q16—48no—
157C2×M16—22no—
158C2×M16—41no—
159C22×C8—22no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8C22×C412yes—
167C4⋊Q8C22×C412yes—
168#125—22no—
169#125—22no—
170#125—41no—
171#125—41no—
172C22⋊Q8—22no—
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—41no—
177C22⋊Q8—41no—
178C22⋊Q8—41no—
179C22⋊Q8—41no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×Q8C22⋊C414yes—
201C4×Q8C22⋊C414yes—
202C4×Q8C22×C414yes—
203C4×Q8C2×D414yes—
204#117—22no—
205#117—22no—
206#117—44no—
207#117—44no—
208#117—41no—
209#117—41no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22⋊C412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—41no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—41no—
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C2×D414yes—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42#4111yescharacteristic
255C2×C42#4111yescharacteristic
256C2×C42#4111yescharacteristic
257C2×C42#4111yescharacteristic
258C2×C42C2411yescharacteristic
259#106—28no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—41no—
263C4⋊C8—41no—
264C4⋊C8—41no—
265C4⋊C8C22×C414yes—
266#103—28no—
267#103—28no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8—41no—
272C22⋊C8—41no—
273C22⋊C8—41no—
274#97—22no—
275#97—22no—
276#97—22no—
277#97—22no—
278C4×C8—22no—
279C4×C8—41no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95C2×D414yes—
287C24#12011yescharacteristic
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—44no—
296C2×Q8—44no—
297C2×Q8—44no—
298C2×Q8—44no—
299C2×Q8—44no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—44no—
303C2×Q8—82no—
304C2×Q8C2×C22⋊C414yes—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—82no—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C4×D411yescharacteristic
323C22×C4C4×D411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4C4×D411yescharacteristic
326C22×C4#12011yescharacteristic
327C22×C4#12011yescharacteristic
328C22×C4#12011yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4#12111yescharacteristic
331C22×C4#12111yescharacteristic
332C22×C4#12111yescharacteristic
333C22×C4#12311yescharacteristic
334C22×C4#12311yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338Q16—416no—
339M16—44no—
340M16—82no—
341C2×C8—24no—
342C2×C8—44no—
343C2×C8—42no—
344C2×C8—42no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—88no—
367C4⋊C4C2×C22⋊C414yes—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C42—24no—
373C42—24no—
374C42—24no—
375C42—24no—
376C42—44no—
377C42—44no—
378C42—44no—
379C42—44no—
380C42—44no—
381C42—42no—
382C42—42no—
383C42—82no—
384C42C2×C22⋊C411yescharacteristic
385C42C2×C22⋊C411yescharacteristic
386C42C4×D411yescharacteristic
387C42C4×D411yescharacteristic
388C42C4×D411yescharacteristic
389C42C4×D411yescharacteristic
390C42#12111yescharacteristic
391C42#12111yescharacteristic
392C42#12111yescharacteristic
393C42#12111yescharacteristic
394C42#12311yescharacteristic
395C42#12311yescharacteristic
396C42C23×C411yescharacteristic
397C42C22×D411yescharacteristic
398C42C2×#4111yescharacteristic
399C42C2×#4111yescharacteristic
400C23—22no—
401C23—22no—
402C23—41no—
403C23#38611yescharacteristic
404C23#38611yescharacteristic
405C23C2×#12011yescharacteristic
406Q8—24no—
407Q8—24no—
408Q8—44no—
409Q8—44no—
410Q8—44no—
411Q8—44no—
412Q8—82no—
413Q8—82no—
414Q8—82no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4#38611yescharacteristic
451C2×C4#38611yescharacteristic
452C2×C4#38611yescharacteristic
453C2×C4#38611yescharacteristic
454C2×C4#38611yescharacteristic
455C2×C4#38611yescharacteristic
456C2×C4#42114yes—
457C2×C4C22×C22⋊C411yescharacteristic
458C2×C4C2×C4×D411yescharacteristic
459C2×C4C2×C4×D411yescharacteristic
460C2×C4C2×#12111yescharacteristic
461C2×C4C2×#12111yescharacteristic
462C2×C4C2×#12311yescharacteristic
463C8—44no—
464C8—82no—
465C8—82no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—44no—
474C22#189712yes—
475C22C2×#38611yescharacteristic
476C22#302212yes—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—44no—
487C4—44no—
488C4—84no—
489C4—84no—
490C4—82no—
491C4—82no—
492C4#189712yes—
493C4#190512yes—
494C4C2×#42112yes—
495C2—22no—
496C2—22no—
497C2—41no—
498C2#2094411yescharacteristic
499C2C2×#189711yescharacteristic
500C2C2×#190511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ 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. #9328C2semidirect
13. #3113C4semidirect
16. C2×#475C4semidirect
17. C2×#462C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31132
#340592
#218442
#175452
#174492
#93282
#80712
#79492