Finite Group Databasebeta

← #199066

Subgroups — #199066

Order 512.

1182 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#21329C211yesmaximal, characteristic
4C2×#1548C211yesmaximal, characteristic
5#17575C211yesmaximal, characteristic
6#9297C218yesmaximal, complemented
7#7957C211yesmaximal, characteristic
8#7854C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3209—22no—
12#3209—22no—
13#3209C414yescomplemented
14#3209C2214yes—
15C2×#494C412yescomplemented
16C2×#487C412yescomplemented
17C2×#462—22no—
18C2×#451—22no—
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22#2369—22no—
23#2365—22no—
24#2345—22no—
25Q8×C22⋊C4—22no—
26#2331C2214yes—
27#2316—21no—
28C4×C4⋊Q8—22no—
29#2285C2211yescharacteristic
30#2274—21no—
31C4×#117C2211yescharacteristic
32#1995C2214yes—
33#1994—22no—
34#1733—21no—
35#1733—21no—
36#1548—22no—
37#1548C2214yes—
38C2×#333—22no—
39#1520—22no—
40#1518C2214yes—
41#1503—22no—
42#1286C2214yes—
43#1274C2214yes—
44C23×Q8D412yes—
45C22×Q16—24no—
46C22×M16—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533C2×C412yes—
54#533C2312yes—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C412yes—
64C2×C4⋊Q8C2×C412yes—
65C2×C4⋊Q8D412yes—
66C2×C4⋊Q8D412yes—
67C2×C4⋊Q8D412yes—
68C2×C4⋊Q8C2312yes—
69C2×#125—21no—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8—21no—
72C2×C4×Q8—24no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×#117—21no—
76C2×#117—21no—
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C42C2311yescharacteristic
82#494—24no—
83#494C2×C418yes—
84#487—24no—
85#487C2×C418yes—
86#462—28no—
87#462—28no—
88#451—28no—
89#451—28no—
90#431—21no—
91#431—21no—
92#431—21no—
93#431—21no—
94#431—21no—
95#431C2×C412yes—
96#431C2312yes—
97#431C2312yes—
98#423—21no—
99#423—21no—
100#423—21no—
101#423C2312yes—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×#103—24no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2×C412yes—
111#391—24no—
112#391—28no—
113#389—24no—
114#389—24no—
115#389—24no—
116#387—28no—
117#384—21no—
118#382C2312yes—
119#380—24no—
120#380—21no—
121#380D412yes—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4D412yes—
129C4×C22⋊C4—24no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132#376C2312yes—
133C43D412yes—
134#339—24no—
135#339—24no—
136#333—28no—
137#328—28no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—84no—
144C22×Q8C22⋊C414yes—
145C22×Q8C22⋊C414yes—
146C22×Q8C2×D414yes—
147C23×C4—22no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C2×Q16—216no—
152C2×Q16—88no—
153C2×M16—22no—
154C2×M16—41no—
155C2×M16—42no—
156C2×M16—42no—
157C22×C8—22no—
158C22×C8—41no—
159C22×C8—41no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8C22⋊C414yes—
179C4⋊Q8C22⋊C414yes—
180C4⋊Q8C22×C412yes—
181C4⋊Q8C22×C412yes—
182C4⋊Q8C22×C414yes—
183C4⋊Q8C2×D414yes—
184#125—22no—
185#125—22no—
186#125—41no—
187#125—41no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194C22⋊Q8—41no—
195C22⋊Q8—41no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—41no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—88no—
205#117—22no—
206#117—22no—
207#117—22no—
208#117—22no—
209#117—22no—
210#117—41no—
211#117—41no—
212#117—42no—
213#117—42no—
214#117Q1614yescomplemented
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—41no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—28no—
247C2×C42—41no—
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42#4111yescharacteristic
258C2×C42#4111yescharacteristic
259C2×C42#4111yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42C2411yescharacteristic
262C4⋊C8—22no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—41no—
268C4⋊C8—41no—
269C4⋊C8—41no—
270#103—216no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—41no—
275C4×C8—22no—
276C4×C8—22no—
277C4×C8—41no—
278C4×C8C22×C414yes—
279#95—28no—
280#95—28no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95C2×D414yes—
285C24#12011yescharacteristic
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—48no—
292C2×Q8—48no—
293C2×Q8—44no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—84no—
297C2×Q8—82no—
298C2×Q8—84no—
299C2×Q8—84no—
300C2×Q8—84no—
301C2×Q8C2×C22⋊C418yes—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—84no—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4C4×D411yescharacteristic
319C22×C4#12011yescharacteristic
320C22×C4#12011yescharacteristic
321C22×C4#12011yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12111yescharacteristic
326C22×C4#12311yescharacteristic
327C22×C4#12311yescharacteristic
328C22×C4C2×Q1612yes—
329C22×C4#13512yes—
330C22×C4#13712yes—
331C22×C4#13712yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335Q16—816no—
336M16—42no—
337M16—42no—
338M16—82no—
339C2×C8—24no—
340C2×C8—44no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—82no—
344C2×C8—82no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—88no—
372C22⋊C4—24no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—44no—
381C42—44no—
382C42—42no—
383C42—42no—
384C42—44no—
385C42—88no—
386C42C2×C22⋊C411yescharacteristic
387C42C2×C22⋊C411yescharacteristic
388C42C4×D411yescharacteristic
389C42C4×D411yescharacteristic
390C42C4×D411yescharacteristic
391C42C4×D411yescharacteristic
392C42#12111yescharacteristic
393C42#12111yescharacteristic
394C42#12111yescharacteristic
395C42#12111yescharacteristic
396C42#12311yescharacteristic
397C42#12311yescharacteristic
398C42C2×Q1614yescomplemented
399C42C23×C411yescharacteristic
400C42C22×D411yescharacteristic
401C42C2×#4111yescharacteristic
402C42C2×#4111yescharacteristic
403C23—41no—
404C23—41no—
405C23#38611yescharacteristic
406C23#38611yescharacteristic
407C23#45112yes—
408C23#45212yes—
409C23C2×#12011yescharacteristic
410Q8—48no—
411Q8—48no—
412Q8—44no—
413Q8—48no—
414Q8—84no—
415Q8—82no—
416Q8—82no—
417Q8—82no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—28no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—48no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4#38611yescharacteristic
448C2×C4#38611yescharacteristic
449C2×C4#38611yescharacteristic
450C2×C4#38611yescharacteristic
451C2×C4#38611yescharacteristic
452C2×C4#38611yescharacteristic
453C2×C4C4×Q1612yes—
454C2×C4C4×Q1612yes—
455C2×C4#44112yes—
456C2×C4#44112yes—
457C2×C4#45112yes—
458C2×C4#45212yes—
459C2×C4#46212yes—
460C2×C4#46212yes—
461C2×C4#46412yes—
462C2×C4#46412yes—
463C2×C4C22×C22⋊C411yescharacteristic
464C2×C4C2×C4×D411yescharacteristic
465C2×C4C2×C4×D411yescharacteristic
466C2×C4C2×#12111yescharacteristic
467C2×C4C2×#12111yescharacteristic
468C2×C4C2×#12311yescharacteristic
469C2×C4C22×Q1611yescharacteristic
470C2×C4C2×#13511yescharacteristic
471C2×C4C2×#13711yescharacteristic
472C2×C4C2×#13711yescharacteristic
473C8—44no—
474C8—82no—
475C8—82no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—42no—
483C22—42no—
484C22#187812yes—
485C22#187812yes—
486C22C2×#38611yescharacteristic
487C22C2×#45111yescharacteristic
488C22C2×#45211yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—44no—
496C4—48no—
497C4—82no—
498C4—84no—
499C4—82no—
500C4—82no—
501C4#187812yes—
502C4#187812yes—
503C4#190512yes—
504C4#191112yes—
505C4C2×C4×Q1611yescharacteristic
506C4C2×#44111yescharacteristic
507C4C2×#46211yescharacteristic
508C4C2×#46411yescharacteristic
509C2—21no—
510C2—21no—
511C2—22no—
512C2—22no—
513C2C2×#187811yescharacteristic
514C2C2×#190511yescharacteristic
515C2C2×#191111yescharacteristic
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. #9297C2semidirect
13. #3209C4semidirect
15. C2×#494C4semidirect
16. C2×#487C4semidirect
214. #117Q16semidirect
398. C42C2×Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#342162
#213292
C2×#15482
#175752
#92972
#79572
#78542