Finite Group Databasebeta

← #369005

Subgroups — #369005

Order 512.

1356 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×M16C211yesmaximal, characteristic
2#34953C211yesmaximal, characteristic
3C2×#1995C211yesmaximal, characteristic
4C2×#1986C211yesmaximal, characteristic
5#13850C211yesmaximal, characteristic
6#11209C214yesmaximal
7#10886C214yesmaximal
8#10852C211yesmaximal, characteristic
9#10701C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12Q8×M16—24no—
13Q8×M16—24no—
14C2×#446—22no—
15C2×#446—21no—
16C2×C8×Q8—21no—
17C2×#423C2212yes—
18C2×#423C2211yescharacteristic
19C2×C4×M16C2212yes—
20C2×C4×M16C2211yescharacteristic
21#2584C412yescomplemented
22#2584C412yescomplemented
23#2583—22no—
24#2567C412yescomplemented
25#2546C412yescomplemented
26#2502—21no—
27#2434—21no—
28#2399—21no—
29#2399—21no—
30C4×C4⋊Q8—22no—
31#2296C2214yes—
32C4×#117C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×#376C2211yescharacteristic
35#1995—24no—
36#1986—24no—
37C2×#339C2212yes—
38C2×#339C2212yes—
39C2×#339C2214yes—
40C2×#339C2214yes—
41C2×#339C2212yes—
42C2×#339C2212yes—
43#1446—21no—
44#1445—21no—
45#1441—22no—
46C4×#95C2214yes—
47C23×Q8D411yescharacteristic
48C22×M16C2312yes—
49C22×M16C2311yescharacteristic
50C22×M16C2311yescharacteristic
51#533—22no—
52#533—22no—
53#533—22no—
54#533—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#125C2×C412yes—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8C2×C412yes—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8D412yes—
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×#117C2312yes—
74C2×#117C2312yes—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D412yes—
77C22×C4⋊C4D411yescharacteristic
78C22×C42C2312yes—
79C22×C42C2312yes—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82#446—216no—
83#446—28no—
84C8×Q8—28no—
85#423—24no—
86#423—24no—
87#423—22no—
88#423D414yes—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—21no—
93C4×M16—24no—
94C4×M16—24no—
95C4×M16—22no—
96C4×M16D414yes—
97C2×#97—22no—
98C2×#97—21no—
99C2×C4×C8—22no—
100C2×C4×C8—21no—
101#398—22no—
102#398—22no—
103#398—22no—
104#398—22no—
105#395—22no—
106#395—22no—
107#395—22no—
108#395—22no—
109#391—22no—
110#391—22no—
111#389—24no—
112#389—24no—
113#389C2×C414yes—
114#389C2×C414yes—
115#389C2×C414yes—
116#389C2×C414yes—
117#385—24no—
118#385—24no—
119#384—22no—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C412yes—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128#376—21no—
129#376—22no—
130#376—22no—
131C2×#95—24no—
132C2×#95—24no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C43—21no—
137#339—24no—
138#339—24no—
139#339—28no—
140#339—24no—
141#339—28no—
142#339Q818yescomplemented
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—44no—
148C22×Q8C22⋊C411yescharacteristic
149C22×Q8C22⋊C412yes—
150C22×Q8C22⋊C411yescharacteristic
151C23×C4—22no—
152C23×C4—22no—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C2×M16—21no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—21no—
162C2×M16—22no—
163C2×M16—24no—
164C2×M16—22no—
165C2×M16—24no—
166C2×M16C2×D414yes—
167C2×M16C2×D412yes—
168C2×M16C2×D412yes—
169C2×M16C2×Q814yes—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—21no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—48no—
178#125—24no—
179#125—24no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183C22⋊Q8—42no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—24no—
189C4×Q8—24no—
190C4×Q8—22no—
191C4×Q8—24no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—24no—
195C4×Q8—22no—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8—42no—
200#117—22no—
201#117—22no—
202#117—22no—
203#117—22no—
204#117—22no—
205#117—22no—
206#117C2×Q814yes—
207#117#4114yes—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C22⋊C412yes—
223C2×C4⋊C4C22⋊C412yes—
224C2×C4⋊C4C22⋊C412yes—
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22⋊C412yes—
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C414yes—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
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—22no—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42C22×C412yes—
252C2×C42C22×C412yes—
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×Q814yes—
262C2×C42#4112yes—
263C2×C42#4112yes—
264C4⋊C8—28no—
265C4⋊C8—28no—
266C4⋊C8—24no—
267C4⋊C8—24no—
268#97—28no—
269#97—24no—
270C4×C8—28no—
271C4×C8—24no—
272#95—22no—
273#95—22no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—48no—
279#95—48no—
280#95—42no—
281#95—42no—
282#95—42no—
283C24C22×D411yescharacteristic
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8—22no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—44no—
292C2×Q8—48no—
293C2×Q8—48no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—82no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—42no—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C412yes—
327C22×C4#12112yes—
328C22×C4#12112yes—
329C22×C4#12112yes—
330C22×C4#12112yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×Q812yes—
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4112yes—
339C22×C42−1+412yes—
340C22×C42−1+412yes—
341M16—22no—
342M16—24no—
343M16—22no—
344M16—22no—
345M16—28no—
346M16—42no—
347M16—42no—
348M16—42no—
349M16C4⋊Q814yescomplemented
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8—42no—
355C2×C8—44no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—48no—
369C4⋊C4—48no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—48no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C4⋊C4C4×Q818yes—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—24no—
391C42—24no—
392C42—24no—
393C42—24no—
394C42—44no—
395C42—42no—
396C42—42no—
397C42—42no—
398C42—42no—
399C42—42no—
400C42—42no—
401C42—44no—
402C42—82no—
403C42C4×D412yes—
404C42C4×D412yes—
405C42C4×D412yes—
406C42C4×D412yes—
407C42#12012yes—
408C42#12012yes—
409C42#12312yes—
410C42#12312yes—
411C23—22no—
412C23—22no—
413C23C22×C22⋊C411yescharacteristic
414C23C2×#12111yescharacteristic
415C23C2×#12111yescharacteristic
416C23#54812yes—
417C23D4×Q812yes—
418Q8—28no—
419Q8—48no—
420Q8—44no—
421Q8—84no—
422Q8—84no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—28no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—48no—
447C2×C4—48no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—84no—
457C2×C4—84no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4#38612yes—
463C2×C4#38612yes—
464C2×C4#38612yes—
465C2×C4#38612yes—
466C2×C4#42114yes—
467C2×C4C2×C4×D411yescharacteristic
468C2×C4C2×C4×D411yescharacteristic
469C2×C4C2×C4×Q812yes—
470C2×C4#51912yes—
471C2×C4C2×#12011yescharacteristic
472C2×C4C2×#12311yescharacteristic
473C2×C4C2×C4⋊Q811yescharacteristic
474C2×C4#53211yescharacteristic
475C2×C4#53312yes—
476C2×C4#53611yescharacteristic
477C2×C4#53611yescharacteristic
478C2×C4#54112yes—
479C2×C4#54712yes—
480C2×C4#54812yes—
481C2×C4D4×Q812yes—
482C2×C4D4×Q812yes—
483C2×C4#55612yes—
484C2×C4#55712yes—
485C8—24no—
486C8—42no—
487C8—44no—
488C22—24no—
489C22—24no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22C2×#38611yescharacteristic
495C22Q8×C22⋊C412yes—
496C22#242912yes—
497C22#254511yescharacteristic
498C22#254611yescharacteristic
499C22C2×#42112yes—
500C22#290112yes—
501C4—22no—
502C4—22no—
503C4—24no—
504C4—22no—
505C4—22no—
506C4—48no—
507C4—42no—
508C4—42no—
509C4—42no—
510C4—84no—
511C4—84no—
512C4—82no—
513C4—82no—
514C4C4×C4⋊Q811yescharacteristic
515C4#232711yescharacteristic
516C4#236512yes—
517C4#243211yescharacteristic
518C4#243411yescharacteristic
519C4#244612yes—
520C2—21no—
521C2—22no—
522C2—21no—
523C2#1402411yescharacteristic
524C2#2090211yescharacteristic
525C2#2090311yescharacteristic
526C2#2252914yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #2584C4semidirect
22. #2584C4semidirect
24. #2567C4semidirect
25. #2546C4semidirect
142. #339Q8semidirect
349. M16C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×M162
#349532
C2×#19952
C2×#19862
#138502
#112092
#108862
#108522
#107012