Finite Group Databasebeta

← #240942

Subgroups — #240942

Order 512.

871 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60663C211yesmaximal, characteristic
2C2×#1950C211yesmaximal, characteristic
3#17708C211yesmaximal, characteristic
4#17708C211yesmaximal, characteristic
5C2×#1542C211yesmaximal, characteristic
6C2×#1481C211yesmaximal, characteristic
7#10556C214yesmaximal, complemented
8#10555C214yesmaximal, complemented
9#7932C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#533C2211yescharacteristic
12#3166—22no—
13#3166—22no—
14#3165—21no—
15#3165—21no—
16#3165C2212yes—
17#3165C2212yes—
18#3162—22no—
19#3160—22no—
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C22×#108—21no—
23#2911—21no—
24#2911—21no—
25C2×#423C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28#1950—22no—
29#1950C2214yes—
30C2×#341—21no—
31#1550—22no—
32#1550C2214yes—
33#1549—21no—
34#1549C2212yes—
35#1548—21no—
36#1548C2212yes—
37#1542—22no—
38#1542—22no—
39#1542—22no—
40#1542—22no—
41C2×#333C2212yes—
42C2×#332C2212yes—
43#1481—22no—
44#1481C2214yes—
45C2×Q8⋊C8—22no—
46#1283C2214yes—
47C22×M16—21no—
48C22×M16D411yescharacteristic
49C22×M16D411yescharacteristic
50C23×C8—21no—
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#533—21no—
62#533—21no—
63#533D412yes—
64#533D412yes—
65#533D412yes—
66#533C2312yes—
67C2×C4⋊Q8D411yescharacteristic
68C2×C4⋊Q8D411yescharacteristic
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⋊Q8—21no—
77C2×C4×Q8—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8—21no—
81C2×#117—21no—
82C2×#117—21no—
83C22×C42C2311yescharacteristic
84#501—24no—
85#501—24no—
86#501—24no—
87#501—24no—
88#500—22no—
89#500—22no—
90#499—24no—
91#499—24no—
92#498—22no—
93#498—22no—
94#431—21no—
95#431—21no—
96#431—21no—
97#431—21no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431C2312yes—
102C2×#108—24no—
103C2×#108—44no—
104#428—22no—
105#428—22no—
106#428—22no—
107#428—22no—
108#428—22no—
109#428—22no—
110#428—22no—
111#428—22no—
112#427—24no—
113#427—24no—
114#423—21no—
115#423—21no—
116#423—21no—
117#423C2312yes—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8D412yes—
125C2×C4⋊C8D412yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130C4×M16—21no—
131C4×M16—21no—
132C4×M16D412yes—
133C4×M16C2312yes—
134C2×#97—21no—
135C2×C4×C8—21no—
136C2×C4×C8—21no—
137C2×C4×C8—21no—
138#341—24no—
139#333—24no—
140#333—24no—
141#333—24no—
142#332—24no—
143#332—24no—
144#332—24no—
145Q8⋊C8—28no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C23×C4C2×D411yescharacteristic
151C23×C4#4111yescharacteristic
152C23×C4#4111yescharacteristic
153C2×M16—21no—
154C2×M16—21no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—41no—
159C2×M16—42no—
160C2×M16—42no—
161C2×M16SD1614yes—
162C2×M16Q1614yes—
163C2×M16C2×D412yes—
164C2×M16C2×D412yes—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8—41no—
174C22×C8—41no—
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⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8C2×D412yes—
188C4⋊Q8C2×D412yes—
189C4⋊Q8C2×D412yes—
190C4⋊Q8C2×D412yes—
191#125—22no—
192#125—22no—
193#125—22no—
194#125—22no—
195#125—41no—
196#125—41no—
197#125—41no—
198#125—41no—
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—41no—
208C22⋊Q8—41no—
209C22⋊Q8—41no—
210C22⋊Q8—41no—
211C22⋊Q8—41no—
212C22⋊Q8—41no—
213C22⋊Q8—41no—
214C22⋊Q8—41no—
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—41no—
224C4×Q8—41no—
225C4×Q8—41no—
226C4×Q8—41no—
227C4×Q8—41no—
228C4×Q8—41no—
229C4×Q8—41no—
230C4×Q8—41no—
231#117—22no—
232#117—22no—
233#117—22no—
234#117—22no—
235#117—41no—
236#117—41no—
237#117—41no—
238#117—41no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42#4111yescharacteristic
264C2×C42#4111yescharacteristic
265C2×C42#4111yescharacteristic
266C2×C42C2411yescharacteristic
267#108—48no—
268#107—24no—
269#107—24no—
270#107—24no—
271#107—24no—
272#106—24no—
273#106—24no—
274#106—24no—
275#106—24no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—41no—
285C4⋊C8—41no—
286C4⋊C8—41no—
287C4⋊C8C2×D414yes—
288C22⋊C8—22no—
289C22⋊C8—41no—
290C22⋊C8—41no—
291C22⋊C8—41no—
292#97—22no—
293#97—22no—
294C4×C8—22no—
295C4×C8—22no—
296C4×C8—22no—
297C4×C8—41no—
298C4×C8—41no—
299C4×C8—41no—
300C24#12411yescharacteristic
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—44no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—82no—
311C2×Q8—82no—
312C2×Q8—82no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—81no—
319C22×C4#12011yescharacteristic
320C22×C4#12011yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12111yescharacteristic
326C22×C4#12111yescharacteristic
327C22×C4#12311yescharacteristic
328C22×C4#12311yescharacteristic
329C22×C4#12311yescharacteristic
330C22×C4#12311yescharacteristic
331C22×C4#12411yescharacteristic
332C22×C4#12411yescharacteristic
333C22×C4#12411yescharacteristic
334C22×C4C2×SD1612yes—
335C22×C4C2×Q1612yes—
336C22×C4#13712yes—
337C22×C4#13712yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341M16—22no—
342M16—22no—
343M16—24no—
344M16—24no—
345M16—42no—
346M16—42no—
347M16—82no—
348M16C2×SD1614yes—
349M16C2×Q1614yes—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—44no—
358C2×C8—44no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—82no—
363C2×C8—82no—
364C2×C8—82no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C42—42no—
394C42—42no—
395C42—42no—
396C42—42no—
397C42#12011yescharacteristic
398C42#12011yescharacteristic
399C42#12111yescharacteristic
400C42#12111yescharacteristic
401C42#12111yescharacteristic
402C42#12111yescharacteristic
403C42#12111yescharacteristic
404C42#12111yescharacteristic
405C42#12311yescharacteristic
406C42#12311yescharacteristic
407C42#12311yescharacteristic
408C42#12311yescharacteristic
409C42C22×D411yescharacteristic
410C42C22×D411yescharacteristic
411C42C22×D411yescharacteristic
412C42C2×#4111yescharacteristic
413C23—41no—
414C23—41no—
415C23#39411yescharacteristic
416C23#39411yescharacteristic
417C23#48712yes—
418C23#49112yes—
419C23C2×#12411yescharacteristic
420Q8—44no—
421Q8—82no—
422Q8—82no—
423Q8—82no—
424Q8—82no—
425Q8—82no—
426Q8—82no—
427Q8—82no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4#39411yescharacteristic
449C2×C4#39411yescharacteristic
450C2×C4#39411yescharacteristic
451C2×C4#39411yescharacteristic
452C2×C4#39411yescharacteristic
453C2×C4#39411yescharacteristic
454C2×C4#44812yes—
455C2×C4#44812yes—
456C2×C4#45112yes—
457C2×C4#45112yes—
458C2×C4#46112yes—
459C2×C4#46112yes—
460C2×C4#46212yes—
461C2×C4#46212yes—
462C2×C4#48712yes—
463C2×C4#49112yes—
464C2×C4C2×#12011yescharacteristic
465C2×C4C2×#12111yescharacteristic
466C2×C4C2×#12111yescharacteristic
467C2×C4C2×#12111yescharacteristic
468C2×C4C2×#12311yescharacteristic
469C2×C4C2×#12311yescharacteristic
470C2×C4C22×SD1611yescharacteristic
471C2×C4C22×Q1611yescharacteristic
472C2×C4C2×#13711yescharacteristic
473C2×C4C2×#13711yescharacteristic
474C8—24no—
475C8—24no—
476C8—44no—
477C8—82no—
478C8—82no—
479C8—82no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—42no—
487C22—42no—
488C22#204312yes—
489C22#204312yes—
490C22C2×#39411yescharacteristic
491C22C2×#48711yescharacteristic
492C22C2×#49111yescharacteristic
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—82no—
501C4—82no—
502C4—82no—
503C4—82no—
504C4#204312yes—
505C4#204312yes—
506C4C2×#44811yescharacteristic
507C4C2×#45111yescharacteristic
508C4C2×#46111yescharacteristic
509C4C2×#46211yescharacteristic
510C4#306712yes—
511C2—21no—
512C2—21no—
513C2—22no—
514C2—22no—
515C2C2×#204311yescharacteristic
516C2#2352511yescharacteristic
517C2#2353011yescharacteristic
subgroupG/H
centerC2×C4#394
commutator, Frattini, cosocleC2×C42C24
socleC22C2×#394
prime radicalC24#124
LCSG ▷ C2×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
7. #10556C2semidirect
8. #10555C2semidirect

Maximal subgroups

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

subgroupindex
#606632
C2×#19502
#177082
C2×#15422
C2×#14812
#105562
#105552
#79322