Finite Group Databasebeta

← #364187

Subgroups — #364187

Order 512.

1516 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2930C211yesmaximal, characteristic
2#20910C211yesmaximal, characteristic
3#20909C211yesmaximal, characteristic
4C2×#1811C211yesmaximal, characteristic, complemented
5C2×#1736C211yesmaximal, characteristic, complemented
6#11242C212yesmaximal, complemented
7#11105C212yesmaximal, complemented
8#10827C212yesmaximal, complemented
9#10774C212yesmaximal, complemented
10#10728C211yesmaximal, characteristic, complemented
11#10727C211yesmaximal, characteristic, complemented
12C22×#131C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C2×#525C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2930—24no—
18#2930C418yescomplemented
19C2×#436—22no—
20C2×#435—22no—
21C22×#108C2212yes—
22C2×#424C2211yescharacteristic
23C2×#417C2212yes—
24C2×#417C2212yes—
25C2×#408C2211yescharacteristic
26C2×#407C2211yescharacteristic
27C2×#407C2211yescharacteristic
28C2×#407C2211yescharacteristic
29C2×#405C2211yescharacteristic
30#1875—22no—
31#1874—22no—
32#1873—22no—
33#1872—22no—
34#1811—24no—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#341C2212yes—
40#1736—22no—
41#1736—22no—
42#1736C2212yes—
43#1736C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C22×#41C2311yescharacteristic
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C2×#131—22no—
50C2×#131—42no—
51C2×#131—42no—
52C2×#131C2×C414yescomplemented
53C22×M16—21no—
54C22×M16—42no—
55C22×M16—42no—
56C22×M16C2×C414yes—
57C22×M16D414yes—
58C22×M16D412yes—
59C22×M16D412yes—
60C22×M16C2312yes—
61C22×M16C2312yes—
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C23×C8—21no—
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67#525—22no—
68#525—22no—
69#525C2×C414yes—
70#525Q814yes—
71C2×#123—21no—
72C2×C22⋊Q8D412yes—
73C2×#121D412yes—
74C2×#120—21no—
75C2×C4×D4—22no—
76C2×#117—22no—
77C2×#117—22no—
78C2×#117—24no—
79C2×#117—42no—
80C2×#117C2311yescharacteristic
81C22×C4⋊C4—22no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—21no—
89C22×C42—21no—
90#436—216no—
91#435—216no—
92C2×#108—24no—
93C2×#108—42no—
94C2×#108—42no—
95#424—22no—
96#424C2×C414yescomplemented
97C2×C4⋊C8—22no—
98#417—24no—
99#417—24no—
100#417—42no—
101#417—42no—
102C2×#103—22no—
103C2×#103—22no—
104C2×#102—22no—
105C2×#102—22no—
106#408—22no—
107#408C2×C414yescomplemented
108#407—22no—
109#407—22no—
110#407—22no—
111#407—22no—
112#407—22no—
113#407C2×C414yes—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8—22no—
117C2×C22⋊C8—22no—
118#405—22no—
119#405C2×C414yescomplemented
120C2×#97—21no—
121C2×C4×C8—21no—
122#343—24no—
123#343—24no—
124#343—42no—
125#343—42no—
126#341—24no—
127#340—22no—
128#340—22no—
129#340—24no—
130#340—24no—
131#340—24no—
132C25#4111yescharacteristic
133C2×#41—21no—
134C2×#41—21no—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—22no—
138C2×#41—22no—
139C2×#41C22×C412yes—
140C2×#41C2×Q812yes—
141C22×Q8C2×D411yescharacteristic
142C22×D4—21no—
143C22×D4—22no—
144C22×D4C2×D411yescharacteristic
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—44no—
153C23×C4—42no—
154C23×C4—84no—
155C23×C4C22×C412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×Q812yes—
160C23×C4#4111yescharacteristic
161C23×C4#4112yes—
162C23×C4C2411yescharacteristic
163#131—48no—
164#131—48no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—24no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—48no—
185C2×M16—48no—
186C2×M16—42no—
187C2×M16—42no—
188C2×M16—42no—
189C2×M16—42no—
190C2×M16—42no—
191C2×M16—42no—
192C2×M16C22⋊C418yes—
193C2×M16C22⋊C418yes—
194C2×M16C22×C412yes—
195C2×M16C22×C412yes—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—24no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—42no—
206C22×C8—42no—
207C22×C8C22×C412yes—
208C22×C8C22×C412yes—
209C22×C8C2×D411yescharacteristic
210C22×C8C2×D411yescharacteristic
211C22×C8C2×D412yes—
212C22×C8C2×D412yes—
213C22×C8C2×D412yes—
214#123—24no—
215#123—24no—
216C22⋊Q8—24no—
217C22⋊Q8C4⋊C418yes—
218#121—24no—
219#121C4⋊C418yes—
220#120—24no—
221#120—24no—
222C4×D4—28no—
223C4×D4—28no—
224#117—21no—
225#117—21no—
226#117—28no—
227#117—48no—
228#117—44no—
229#117C22×C412yes—
230#117C2×Q812yes—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—48no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4#4114yes—
249C2×C4⋊C4#4114yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—42no—
255C2×C42—21no—
256C2×C42—44no—
257C2×C42—42no—
258#108—48no—
259C4⋊C8—28no—
260#103—44no—
261#103—44no—
262#102—44no—
263#102—44no—
264C22⋊C8—28no—
265C22⋊C8—28no—
266C22⋊C8—44no—
267C22⋊C8—44no—
268#97—24no—
269C4×C8—24no—
270C24—22no—
271C24—22no—
272C24—21no—
273C24—42no—
274C24—42no—
275C24#11712yes—
276C24#12112yes—
277C24C22⋊Q812yes—
278C24C2×#4111yescharacteristic
279#41—24no—
280#41—24no—
281#41—24no—
282#41—24no—
283#41—42no—
284#41—42no—
285#41—42no—
286#41—42no—
287C2×Q8—21no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8C2×C4⋊C412yes—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—28no—
294C2×D4—21no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—82no—
300C2×D4—82no—
301C2×D4C2×C4⋊C412yes—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—48no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—84no—
334C22×C4—84no—
335C22×C4—816no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—84no—
339C22×C4C2×C22⋊C414yes—
340C22×C4C2×C4⋊C414yes—
341C22×C4#11712yes—
342C22×C4#12114yes—
343C22×C4#12112yes—
344C22×C4C22⋊Q814yes—
345C22×C4C22⋊Q812yes—
346C22×C4C23×C411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×Q811yescharacteristic
351C22×C4C2×#4112yes—
352C22×C4C2×#4112yes—
353C22×C4C2×#4111yescharacteristic
354C22×C4H(5,𝔽2)12yes—
355C22×C4H(5,𝔽2)12yes—
356M16—24no—
357M16—24no—
358M16—48no—
359M16—48no—
360M16—48no—
361M16—42no—
362M16—42no—
363M16—42no—
364M16—42no—
365M16—42no—
366M16—42no—
367M16—42no—
368M16—42no—
369M16—42no—
370M16—42no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—24no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—48no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8C2×C22⋊C412yes—
395C2×C8C2×C22⋊C412yes—
396C2×C8C2×C22⋊C414yes—
397C4⋊C4—22no—
398C4⋊C4—28no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—48no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4C2×C4⋊C414yes—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—28no—
412C22⋊C4—28no—
413C22⋊C4—48no—
414C42—22no—
415C42—22no—
416C42—48no—
417C23—22no—
418C23—22no—
419C23—24no—
420C23—24no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23—42no—
438C23#38014yes—
439C23C2×#11711yescharacteristic
440C23C2×#12111yescharacteristic
441C23C2×C22⋊Q811yescharacteristic
442C23#55112yes—
443C23#55112yes—
444Q8—24no—
445Q8—42no—
446D4—24no—
447D4—44no—
448D4—44no—
449D4—42no—
450D4—88no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—84no—
474C2×C4—84no—
475C2×C4—816no—
476C2×C4—82no—
477C2×C4—84no—
478C2×C4—82no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4#38014yes—
482C2×C4C22×C22⋊C411yescharacteristic
483C2×C4C22×C4⋊C411yescharacteristic
484C2×C4C4×#4112yes—
485C2×C4#51812yes—
486C2×C4C2×#12111yescharacteristic
487C2×C4C2×C22⋊Q811yescharacteristic
488C2×C4#52511yescharacteristic
489C2×C4#52511yescharacteristic
490C2×C4#53212yes—
491C2×C4#53411yescharacteristic
492C2×C4#53411yescharacteristic
493C2×C4#53412yes—
494C2×C4#54612yes—
495C2×C4#54612yes—
496C2×C4#55312yes—
497C2×C4#55312yes—
498C2×C4#55412yes—
499C2×C4#55412yes—
500C8—22no—
501C8—24no—
502C8—22no—
503C8—42no—
504C8—42no—
505C8—82no—
506C8—82no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—24no—
514C22—24no—
515C22—24no—
516C22—48no—
517C22—48no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—44no—
524C22—81no—
525C22—81no—
526C22—82no—
527C22C2×#38011yescharacteristic
528C22#235312yes—
529C22#253411yescharacteristic
530C22#253411yescharacteristic
531C22#256412yes—
532C22#306812yes—
533C22#306912yes—
534C4—24no—
535C4—42no—
536C4—44no—
537C4—84no—
538C4—84no—
539C4#229711yescharacteristic
540C4#230811yescharacteristic
541C4#234912yes—
542C4#254312yes—
543C4#257911yescharacteristic
544C4#257911yescharacteristic
545C2—22no—
546C2—22no—
547C2—44no—
548C2—82no—
549C2#1396011yescharacteristic
550C2#2067912yes—
551C2#2813711yescharacteristic
552C2#2813811yescharacteristic
553C2#2813912yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1811C2semidirect
5. C2×#1736C2semidirect
6. #11242C2semidirect
7. #11105C2semidirect
8. #10827C2semidirect
9. #10774C2semidirect
10. #10728C2semidirect
11. #10727C2semidirect
18. #2930C4semidirect
52. C2×#131C2×C4semidirect
96. #424C2×C4semidirect
107. #408C2×C4semidirect
119. #405C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29302
#209102
#209092
C2×#18112
C2×#17362
#112422
#111052
#108272
#107742
#107282
#107272