Finite Group Databasebeta

← #198932

Subgroups — #198932

Order 512.

1034 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60943C211yesmaximal, characteristic
2C2×#2961C211yesmaximal, characteristic
3#21321C211yesmaximal, characteristic
4#17708C211yesmaximal, characteristic
5#17575C211yesmaximal, characteristic
6#9290C214yesmaximal, complemented
7#9288C214yesmaximal, complemented
8#7957C211yesmaximal, characteristic
9C2×#1275C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#533C2211yescharacteristic
12#3249—22no—
13#3230C412yescomplemented
14#3230C2212yes—
15#3209C412yescomplemented
16#3209C2212yes—
17#3189—22no—
18#3170C414yescomplemented
19#3104—22no—
20#3104—22no—
21#2961—22no—
22#2961—22no—
23#2961—22no—
24#2961C2214yes—
25C2×#446—22no—
26C2×#446—22no—
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#424—21no—
30C2×#423C2211yescharacteristic
31C2×#405—21no—
32C22×#97C2211yescharacteristic
33#2285C2211yescharacteristic
34#1993C2212yes—
35#1993C2212yes—
36#1992—22no—
37#1735—21no—
38#1735—21no—
39#1550—22no—
40#1549C2212yes—
41#1548C2212yes—
42#1542—22no—
43#1520—22no—
44#1518C2214yes—
45#1503—22no—
46#1286C2214yes—
47#1275C2214yes—
48C22×M16—21no—
49C23×C8D411yescharacteristic
50C23×C8D411yescharacteristic
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#533C2×C412yes—
62#533C2×C412yes—
63#533C2×C412yes—
64#533D412yes—
65#533D412yes—
66#533C2312yes—
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×#125—21no—
72C2×#125—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8D412yes—
77C2×C4×Q8—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8D412yes—
81C2×#117—21no—
82C2×#117—21no—
83C2×#117—21no—
84C22×C42C2311yescharacteristic
85#500C2×C414yes—
86#494C2×C414yes—
87#487C2×C414yes—
88#487C2×C414yes—
89#484—24no—
90#484—24no—
91#477—24no—
92#477—24no—
93#467—28no—
94#462—24no—
95#462—24no—
96#451—24no—
97#451—24no—
98#446—28no—
99#446—28no—
100#446—28no—
101#446—28no—
102C4×Q16—28no—
103#431—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431C2×C412yes—
109#431C2312yes—
110#431C2312yes—
111#424—24no—
112#424—24no—
113#423—21no—
114#423—21no—
115#423D412yes—
116#423C2312yes—
117C2×C4⋊C8—24no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8D412yes—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124#405—24no—
125#405—24no—
126C2×#97—22no—
127C2×#97—21no—
128C2×#97—21no—
129C2×#97—21no—
130C2×#97—21no—
131C2×#97D412yes—
132C2×#97C2312yes—
133C2×C4×C8—22no—
134C2×C4×C8—21no—
135C2×C4×C8C2×C412yes—
136#384—21no—
137#382C2312yes—
138#380—21no—
139#380—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146#333—24no—
147#332—24no—
148#328—28no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8C22⋊C412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4#4111yescharacteristic
156C2×Q16—28no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—42no—
160C2×M16—42no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—41no—
171C22×C8Q1614yescomplemented
172C22×C8C2×D412yes—
173C22×C8C2×D412yes—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8C22⋊C412yes—
185C4⋊Q8C22⋊C412yes—
186C4⋊Q8C22×C412yes—
187C4⋊Q8C22×C412yes—
188C4⋊Q8C22×C412yes—
189C4⋊Q8C22×C412yes—
190#125—22no—
191#125—22no—
192#125—22no—
193#125—22no—
194#125—22no—
195#125—22no—
196#125—41no—
197#125—41no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—41no—
209C22⋊Q8—41no—
210C22⋊Q8—41no—
211C22⋊Q8—41no—
212C22⋊Q8C22⋊C414yes—
213C22⋊Q8C2×D414yes—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—41no—
225C4×Q8—41no—
226C4×Q8—41no—
227C4×Q8—41no—
228C4×Q8C22⋊C414yes—
229C4×Q8C2×D414yes—
230#117—22no—
231#117—22no—
232#117—22no—
233#117—22no—
234#117—22no—
235#117—22no—
236#117—41no—
237#117—41no—
238#117—42no—
239#117—42no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4C22⋊C412yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C42—22no—
259C2×C42—41no—
260C2×C42—41no—
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42#4111yescharacteristic
270C2×C42#4111yescharacteristic
271C2×C42#4111yescharacteristic
272C2×C42C2411yescharacteristic
273#107—28no—
274C4⋊C8—216no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—41no—
282C4⋊C8C2×D414yes—
283#103—28no—
284#103—28no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8—41no—
289#97—28no—
290#97—22no—
291#97—22no—
292#97—22no—
293#97—22no—
294C4×C8—28no—
295C4×C8—22no—
296C4×C8—22no—
297C4×C8—41no—
298C4×C8C22×C414yes—
299#95—22no—
300#95—22no—
301C24#12111yescharacteristic
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—82no—
310C2×Q8—82no—
311C2×Q8—82no—
312C2×Q8—82no—
313C2×Q8C2×C22⋊C414yes—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C4×D411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4C4×D411yescharacteristic
326C22×C4C4×D411yescharacteristic
327C22×C4#12011yescharacteristic
328C22×C4#12011yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4#12111yescharacteristic
331C22×C4#12111yescharacteristic
332C22×C4#12111yescharacteristic
333C22×C4#12111yescharacteristic
334C22×C4#12311yescharacteristic
335C22×C4#12311yescharacteristic
336C22×C4C2×Q1612yes—
337C22×C4#13512yes—
338C22×C4#13712yes—
339C22×C4#13712yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C2×#4111yescharacteristic
343Q16—88no—
344M16—42no—
345M16—42no—
346M16—44no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—44no—
359C2×C8—42no—
360C2×C8—82no—
361C2×C8—82no—
362C2×C8C2×Q1614yes—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C4⋊C4C2×C22⋊C414yes—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—82no—
396C42—24no—
397C42—24no—
398C42—42no—
399C42—42no—
400C42—82no—
401C42C2×C22⋊C411yescharacteristic
402C42C2×C22⋊C411yescharacteristic
403C42C4×D411yescharacteristic
404C42C4×D411yescharacteristic
405C42C4×D411yescharacteristic
406C42C4×D411yescharacteristic
407C42#12011yescharacteristic
408C42#12011yescharacteristic
409C42#12111yescharacteristic
410C42#12111yescharacteristic
411C42#12311yescharacteristic
412C42#12311yescharacteristic
413C42C23×C411yescharacteristic
414C42C22×D411yescharacteristic
415C42C22×D411yescharacteristic
416C42C2×#4111yescharacteristic
417C23—41no—
418C23—41no—
419C23#38611yescharacteristic
420C23#38611yescharacteristic
421C23#46212yes—
422C23#46412yes—
423C23C2×#12111yescharacteristic
424Q8—44no—
425Q8—44no—
426Q8—82no—
427Q8—82no—
428Q8—82no—
429Q8—82no—
430Q8—82no—
431Q8—82no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
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—82no—
455C2×C4#38611yescharacteristic
456C2×C4#38611yescharacteristic
457C2×C4#38611yescharacteristic
458C2×C4#38611yescharacteristic
459C2×C4#38611yescharacteristic
460C2×C4#38611yescharacteristic
461C2×C4C4×Q1612yes—
462C2×C4C4×Q1612yes—
463C2×C4#44112yes—
464C2×C4#44112yes—
465C2×C4#45112yes—
466C2×C4#45112yes—
467C2×C4#45212yes—
468C2×C4#45212yes—
469C2×C4#46212yes—
470C2×C4#46412yes—
471C2×C4C22×C22⋊C411yescharacteristic
472C2×C4C2×C4×D411yescharacteristic
473C2×C4C2×C4×D411yescharacteristic
474C2×C4C2×#12011yescharacteristic
475C2×C4C2×#12111yescharacteristic
476C2×C4C2×#12311yescharacteristic
477C2×C4C22×Q1611yescharacteristic
478C2×C4C2×#13511yescharacteristic
479C2×C4C2×#13711yescharacteristic
480C2×C4C2×#13711yescharacteristic
481C8—24no—
482C8—44no—
483C8—42no—
484C8—44no—
485C8—82no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—42no—
493C22—42no—
494C22#187812yes—
495C22#187812yes—
496C22C2×#38611yescharacteristic
497C22C2×#46211yescharacteristic
498C22C2×#46411yescharacteristic
499C4—21no—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—21no—
507C4—44no—
508C4—44no—
509C4—82no—
510C4—82no—
511C4—162no—
512C4#187812yes—
513C4#187812yes—
514C4C2×C4×Q1611yescharacteristic
515C4C2×#44111yescharacteristic
516C4C2×#45111yescharacteristic
517C4C2×#45211yescharacteristic
518C2—21no—
519C2—22no—
520C2—22no—
521C2—21no—
522C2C2×#187811yescharacteristic
523C2#2099511yescharacteristic
524C2#2100911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC24#121
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
6. #9290C2semidirect
7. #9288C2semidirect
13. #3230C4semidirect
15. #3209C4semidirect
18. #3170C4semidirect
171. C22×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#609432
C2×#29612
#213212
#177082
#175752
#92902
#92882
#79572
C2×#12752