Finite Group Databasebeta

← #198948

Subgroups — #198948

Order 512.

1114 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#60192C211yesmaximal, characteristic
3#21324C211yesmaximal, characteristic
4C2×#1549C211yesmaximal, characteristic
5#17575C211yesmaximal, characteristic
6#9290C218yesmaximal, complemented
7#7957C211yesmaximal, characteristic
8#7860C211yesmaximal, 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—
19Q8×M16—22no—
20Q8×M16—22no—
21#2962—22no—
22#2962—22no—
23#2961C2214yes—
24#2960—22no—
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27#2905—21no—
28C2×#423C2211yescharacteristic
29#2873—21no—
30C2×C4×M16C2211yescharacteristic
31#2285C2211yescharacteristic
32#1993C2214yes—
33#1990—22no—
34#1736—21no—
35#1736—21no—
36#1549—22no—
37#1549C2214yes—
38C2×#332—22no—
39#1520—22no—
40#1518C2214yes—
41#1503—22no—
42#1286C2214yes—
43#1275C2214yes—
44C23×Q8—21no—
45C22×Q16—24no—
46C22×M16—21no—
47C22×M16D411yescharacteristic
48C22×M16D411yescharacteristic
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533C2×C412yes—
54#533D412yes—
55#533D412yes—
56#533C2312yes—
57C2×C4⋊Q8—21no—
58C2×C4⋊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⋊Q8C2×C412yes—
67C2×C4⋊Q8C2×C412yes—
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×C4⋊Q8C2312yes—
71C2×#125—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8D412yes—
74C2×C4×Q8—21no—
75C2×C4×Q8D412yes—
76C2×#117—21no—
77C2×#117—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C42C2311yescharacteristic
81#494—24no—
82#494C2×C418yes—
83#487—24no—
84#487C2×C418yes—
85#462—28no—
86#462—28no—
87#451—28no—
88#451—28no—
89#446—24no—
90#446—24no—
91#446—28no—
92#446—28no—
93C8×Q8—24no—
94C8×Q8—24no—
95#431—21no—
96#431—21no—
97#431—21no—
98#431—21no—
99#431—21no—
100#431C2×C412yes—
101#431C2312yes—
102#431C2312yes—
103#424—24no—
104#424—24no—
105#423—24no—
106#423—21no—
107#423—21no—
108#423D412yes—
109#423C2312yes—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8D412yes—
114C2×#103—24no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117#405—24no—
118#405—24no—
119C4×M16—24no—
120C4×M16—21no—
121C4×M16—21no—
122C4×M16—21no—
123C4×M16—21no—
124C2×#97C2312yes—
125C2×C4×C8—21no—
126C2×C4×C8C2×C412yes—
127C2×C4×C8D412yes—
128#384—21no—
129#382C2312yes—
130#380—21no—
131#380—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134#340—24no—
135#340—24no—
136#332—28no—
137#328—28no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—41no—
143C22×Q8—41no—
144C22×Q8—41no—
145C22×Q8—84no—
146C22×Q8C22⋊C412yes—
147C23×C4—41no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4#4111yescharacteristic
151C2×Q16—216no—
152C2×Q16—88no—
153C2×M16—21no—
154C2×M16—21no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—42no—
163C2×M16—42no—
164C2×M16Q1614yescomplemented
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—41no—
168C22×C8C2×D412yes—
169C22×C8C2×D412yes—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—22no—
182C4⋊Q8—22no—
183C4⋊Q8—41no—
184C4⋊Q8—41no—
185C4⋊Q8—41no—
186C4⋊Q8—41no—
187C4⋊Q8—41no—
188C4⋊Q8—41no—
189C4⋊Q8C22⋊C412yes—
190C4⋊Q8C22⋊C412yes—
191C4⋊Q8C22×C414yes—
192C4⋊Q8C22×C412yes—
193C4⋊Q8C22×C412yes—
194#125—22no—
195#125—22no—
196#125—22no—
197#125—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C22⋊Q8C22⋊C414yes—
205C22⋊Q8C2×D414yes—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8C22⋊C414yes—
213C4×Q8C2×D414yes—
214#117—22no—
215#117—22no—
216#117—22no—
217#117—22no—
218#117—42no—
219#117—42no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4C22⋊C412yes—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—41no—
243C2×C42—22no—
244C2×C42—41no—
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42#4111yescharacteristic
254C2×C42#4111yescharacteristic
255C2×C42#4111yescharacteristic
256C2×C42C2411yescharacteristic
257C4⋊C8—28no—
258C4⋊C8—28no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—22no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—41no—
266C4⋊C8C2×D414yes—
267#103—216no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8—41no—
272#97—28no—
273#97—22no—
274#97—22no—
275C4×C8—28no—
276C4×C8—22no—
277C4×C8—22no—
278C4×C8—22no—
279C4×C8—22no—
280C4×C8—41no—
281C4×C8C22×C414yes—
282#95—22no—
283#95—22no—
284C24#12111yescharacteristic
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—24no—
289C2×Q8—44no—
290C2×Q8—48no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—82no—
295C2×Q8—84no—
296C2×Q8—84no—
297C2×Q8—84no—
298C2×Q8—84no—
299C2×Q8—84no—
300C2×Q8C2×C22⋊C414yes—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—84no—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12311yescharacteristic
324C22×C4#12311yescharacteristic
325C22×C4C2×Q1612yes—
326C22×C4#13512yes—
327C22×C4#13712yes—
328C22×C4#13712yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4111yescharacteristic
332Q16—816no—
333M16—24no—
334M16—24no—
335M16—42no—
336M16—42no—
337M16—42no—
338M16—42no—
339M16—44no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—24no—
343C2×C8—22no—
344C2×C8—24no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—44no—
348C2×C8—44no—
349C2×C8—42no—
350C2×C8—82no—
351C2×C8C2×Q1614yescomplemented
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4C2×C22⋊C414yes—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—82no—
379C42—24no—
380C42—24no—
381C42—82no—
382C42C2×C22⋊C411yescharacteristic
383C42C2×C22⋊C411yescharacteristic
384C42C4×D411yescharacteristic
385C42C4×D411yescharacteristic
386C42C4×D411yescharacteristic
387C42C4×D411yescharacteristic
388C42#12011yescharacteristic
389C42#12011yescharacteristic
390C42#12111yescharacteristic
391C42#12111yescharacteristic
392C42#12311yescharacteristic
393C42#12311yescharacteristic
394C42C23×C411yescharacteristic
395C42C22×D411yescharacteristic
396C42C22×D411yescharacteristic
397C42C2×#4111yescharacteristic
398C23—41no—
399C23—41no—
400C23#38611yescharacteristic
401C23#38611yescharacteristic
402C23#46212yes—
403C23#46412yes—
404C23C2×#12111yescharacteristic
405Q8—44no—
406Q8—48no—
407Q8—44no—
408Q8—44no—
409Q8—82no—
410Q8—84no—
411Q8—84no—
412Q8—84no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—44no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4#38611yescharacteristic
437C2×C4#38611yescharacteristic
438C2×C4#38611yescharacteristic
439C2×C4#38611yescharacteristic
440C2×C4#38611yescharacteristic
441C2×C4#38611yescharacteristic
442C2×C4C4×Q1612yes—
443C2×C4C4×Q1612yes—
444C2×C4#44112yes—
445C2×C4#44112yes—
446C2×C4#45112yes—
447C2×C4#45112yes—
448C2×C4#45212yes—
449C2×C4#45212yes—
450C2×C4#46212yes—
451C2×C4#46412yes—
452C2×C4C22×C22⋊C411yescharacteristic
453C2×C4C2×C4×D411yescharacteristic
454C2×C4C2×C4×D411yescharacteristic
455C2×C4C2×#12011yescharacteristic
456C2×C4C2×#12111yescharacteristic
457C2×C4C2×#12311yescharacteristic
458C2×C4C22×Q1611yescharacteristic
459C2×C4C2×#13511yescharacteristic
460C2×C4C2×#13711yescharacteristic
461C2×C4C2×#13711yescharacteristic
462C8—24no—
463C8—44no—
464C8—42no—
465C8—44no—
466C8—82no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—42no—
474C22—42no—
475C22#187812yes—
476C22#187812yes—
477C22C2×#38611yescharacteristic
478C22C2×#46211yescharacteristic
479C22C2×#46411yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—44no—
487C4—44no—
488C4—84no—
489C4—162no—
490C4#187812yes—
491C4#187812yes—
492C4#190212yes—
493C4#191012yes—
494C4C2×C4×Q1611yescharacteristic
495C4C2×#44111yescharacteristic
496C4C2×#45111yescharacteristic
497C4C2×#45211yescharacteristic
498C2—21no—
499C2—22no—
500C2—22no—
501C2—21no—
502C2C2×#187811yescharacteristic
503C2C2×#190211yescharacteristic
504C2C2×#191011yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
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. #9290C2semidirect
13. #3209C4semidirect
15. C2×#494C4semidirect
16. C2×#487C4semidirect
164. C2×M16Q16semidirect
351. C2×C8C2×Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#601922
#213242
C2×#15492
#175752
#92902
#79572
#78602