Finite Group Databasebeta

← #368994

Subgroups — #368994

Order 512.

1141 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60663C211yesmaximal, characteristic
2#60192C211yesmaximal, characteristic
3#21329C211yesmaximal, characteristic
4#21311C211yesmaximal, characteristic
5#21302C211yesmaximal, characteristic
6#11218C214yesmaximal
7#10900C214yesmaximal
8#10853C211yesmaximal, characteristic
9C4×#344C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#533C2211yescharacteristic
12#3166—22no—
13#3166C414yescomplemented
14#3165—21no—
15#3165—21no—
16#3165C412yescomplemented
17#3165C412yescomplemented
18#3162—22no—
19#3160—22no—
20Q8×M16—22no—
21Q8×M16—22no—
22#2962—22no—
23#2962—22no—
24#2961—22no—
25#2960—22no—
26#2911C2214yes—
27#2905C2212yes—
28C2×#423C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30#2873C2212yes—
31C2×C4×M16C2211yescharacteristic
32C2×C4×M16C2211yescharacteristic
33C4×#117C2211yescharacteristic
34#1995—22no—
35#1994—22no—
36#1989—22no—
37#1988—21no—
38#1987—21no—
39#1985—22no—
40#1984—22no—
41C2×#344C2212yes—
42C2×#344C2212yes—
43C2×#341C2214yes—
44#1735C2214yes—
45#1733C2214yes—
46#1729C2212yes—
47#1729C2212yes—
48C22×M16C2312yes—
49C22×M16C2312yes—
50C22×M16C2311yescharacteristic
51C22×M16C2311yescharacteristic
52C23×C8C2312yes—
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#533—21no—
62#533—21no—
63#533—21no—
64#533—21no—
65#533C2×C412yes—
66#533C2×C412yes—
67#533D412yes—
68#533D412yes—
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×C4⋊Q8C2×C411yescharacteristic
71C2×C4⋊Q8D411yescharacteristic
72C2×C4⋊Q8D411yescharacteristic
73C2×#125C2×C412yes—
74C2×#125D412yes—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8—21no—
83C2×#117C2312yes—
84C2×#117C2312yes—
85C2×#117C2312yes—
86C22×C42C2311yescharacteristic
87#501—24no—
88#501—24no—
89#501—24no—
90#501—24no—
91#500—22no—
92#500—22no—
93#499—28no—
94#498—22no—
95#498—22no—
96#446—24no—
97#446—24no—
98#446—28no—
99#446—28no—
100C8×Q8—24no—
101C8×Q8—24no—
102#428—24no—
103#428—24no—
104#428C2×C418yes—
105#428C2×C418yes—
106#427—28no—
107#424—24no—
108#424—24no—
109#423—24no—
110#423—21no—
111#423—21no—
112#423D412yes—
113#423D412yes—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122#405—24no—
123#405—24no—
124C4×M16—24no—
125C4×M16—21no—
126C4×M16—21no—
127C4×M16—21no—
128C4×M16—21no—
129C4×M16—21no—
130C4×M16—21no—
131C4×M16C2×C412yes—
132C4×M16C2×C412yes—
133C2×#97—21no—
134C2×#97D412yes—
135C2×C4×C8—21no—
136C2×C4×C8D412yes—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141#376—21no—
142C43—21no—
143#344—24no—
144#344Q818yescomplemented
145#341—28no—
146#340—24no—
147#340—24no—
148#339—28no—
149#336—24no—
150#336—24no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C23×C4#4111yescharacteristic
156C23×C4#4111yescharacteristic
157C23×C4C2411yescharacteristic
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—24no—
170C2×M16—24no—
171C2×M16C22×C414yes—
172C2×M16C2×D414yes—
173C2×M16C2×Q814yes—
174C2×M16C2×Q814yes—
175C2×M16#4114yes—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—24no—
181C22×C8—22no—
182C22×C8—24no—
183C22×C8C2×D412yes—
184C22×C8C2×D412yes—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8C22⋊C412yes—
198C4⋊Q8C22⋊C412yes—
199C4⋊Q8C22⋊C412yes—
200C4⋊Q8C22⋊C412yes—
201#125—22no—
202#125—22no—
203#125—22no—
204#125—22no—
205#125—22no—
206#125—22no—
207#125C22⋊C414yes—
208#125C22⋊C414yes—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—41no—
218C22⋊Q8—41no—
219C22⋊Q8—41no—
220C22⋊Q8—41no—
221C22⋊Q8—41no—
222C22⋊Q8—41no—
223C22⋊Q8—41no—
224C22⋊Q8—41no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—41no—
234C4×Q8—41no—
235C4×Q8—41no—
236C4×Q8—41no—
237C4×Q8—41no—
238C4×Q8—41no—
239C4×Q8—41no—
240C4×Q8—41no—
241#117—22no—
242#117—22no—
243#117—22no—
244#117—22no—
245#117—22no—
246#117—22no—
247#117—22no—
248#117—22no—
249#117—22no—
250#117C22×C414yes—
251#117C2×D414yes—
252#117C2×Q814yes—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4C22×C412yes—
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—41no—
273C2×C42—41no—
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42#4111yescharacteristic
285C2×C42#4111yescharacteristic
286#107—28no—
287#107—28no—
288#106—28no—
289#106—28no—
290C4⋊C8—28no—
291C4⋊C8—28no—
292C4⋊C8—22no—
293C4⋊C8—22no—
294C4⋊C8—22no—
295C4⋊C8—22no—
296C4⋊C8—41no—
297C4⋊C8—41no—
298C4⋊C8—41no—
299C4⋊C8—41no—
300#97—28no—
301#97—22no—
302#97—22no—
303#97—22no—
304#97—22no—
305C4×C8—28no—
306C4×C8—22no—
307C4×C8—22no—
308C4×C8—22no—
309C4×C8—22no—
310#95—41no—
311#95—41no—
312C24C2×#4111yescharacteristic
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—44no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—82no—
323C2×Q8—82no—
324C2×Q8—82no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—81no—
332C22×C4C4×D411yescharacteristic
333C22×C4C4×D411yescharacteristic
334C22×C4C4×D411yescharacteristic
335C22×C4C4×D411yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4#12111yescharacteristic
339C22×C4#12111yescharacteristic
340C22×C4#12311yescharacteristic
341C22×C4#12311yescharacteristic
342C22×C4#12311yescharacteristic
343C22×C4#12311yescharacteristic
344C22×C4C23×C411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×Q812yes—
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4112yes—
352C22×C42−1+412yes—
353C22×C42−1+412yes—
354M16—24no—
355M16—24no—
356M16—24no—
357M16—24no—
358M16—24no—
359M16—28no—
360M16—42no—
361M16—42no—
362M16—42no—
363M16—42no—
364M16C4×Q818yes—
365C2×C8—24no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—44no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—44no—
379C2×C8—44no—
380C2×C8C4⋊Q814yes—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4C2×C22⋊C414yes—
406C4⋊C4C2×C22⋊C414yes—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C42—24no—
414C42—24no—
415C42—24no—
416C42—42no—
417C42—42no—
418C42—42no—
419C42—82no—
420C42—82no—
421C42C2×C22⋊C411yescharacteristic
422C42C2×C22⋊C411yescharacteristic
423C42C2×C22⋊C411yescharacteristic
424C42C2×C22⋊C411yescharacteristic
425C42C4×D411yescharacteristic
426C42C4×D411yescharacteristic
427C42C4×D411yescharacteristic
428C42C4×D411yescharacteristic
429C42#12011yescharacteristic
430C42#12011yescharacteristic
431C42#12011yescharacteristic
432C42#12011yescharacteristic
433C42#12111yescharacteristic
434C42#12111yescharacteristic
435C42#12111yescharacteristic
436C42#12111yescharacteristic
437C23—41no—
438C23—41no—
439C23C2×C4×D411yescharacteristic
440C23C2×#12111yescharacteristic
441C23C2×#12311yescharacteristic
442C23#53312yes—
443C23#54112yes—
444Q8—44no—
445Q8—44no—
446Q8—82no—
447Q8—82no—
448Q8—82no—
449Q8—82no—
450Q8—82no—
451Q8—82no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4#38611yescharacteristic
476C2×C4#38611yescharacteristic
477C2×C4#38611yescharacteristic
478C2×C4#38611yescharacteristic
479C2×C4#38611yescharacteristic
480C2×C4#38611yescharacteristic
481C2×C4#38611yescharacteristic
482C2×C4#38611yescharacteristic
483C2×C4C22×C22⋊C411yescharacteristic
484C2×C4C2×C4×D411yescharacteristic
485C2×C4C2×C4×Q812yes—
486C2×C4#51912yes—
487C2×C4C2×#12011yescharacteristic
488C2×C4C2×#12111yescharacteristic
489C2×C4C2×C4⋊Q811yescharacteristic
490C2×C4#53211yescharacteristic
491C2×C4#53611yescharacteristic
492C2×C4#53611yescharacteristic
493C2×C4#54712yes—
494C2×C4#54812yes—
495C2×C4#54812yes—
496C2×C4D4×Q812yes—
497C2×C4D4×Q812yes—
498C2×C4D4×Q812yes—
499C2×C4#55612yes—
500C2×C4#55712yes—
501C8—28no—
502C8—24no—
503C8—42no—
504C8—44no—
505C8—82no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—42no—
513C22—42no—
514C22C2×#38611yescharacteristic
515C22C4×C4⋊Q811yescharacteristic
516C22#232711yescharacteristic
517C22#242912yes—
518C22#244612yes—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—44no—
527C4—44no—
528C4—84no—
529C4—82no—
530C4—82no—
531C4#188112yes—
532C4Q8×C22⋊C412yes—
533C4#236512yes—
534C4#243211yescharacteristic
535C4#243411yescharacteristic
536C4#254511yescharacteristic
537C4#254611yescharacteristic
538C2—22no—
539C2—21no—
540C2—21no—
541C2—22no—
542C2#1402411yescharacteristic
543C2C2×#188111yescharacteristic
544C2#2093911yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #3166C4semidirect
16. #3165C4semidirect
17. #3165C4semidirect
144. #344Q8semidirect

Maximal subgroups

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

subgroupindex
#606632
#601922
#213292
#213112
#213022
#112182
#109002
#108532
C4×#3442