Finite Group Databasebeta

← #206261

Subgroups — #206261

Order 512.

988 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
1C2×#3231C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic
3#21864C211yesmaximal, characteristic
4#21860C211yesmaximal, characteristic
5#21409C211yesmaximal, characteristic
6#21322C211yesmaximal, characteristic
7C2×#1453C211yesmaximal, characteristic
8#9840C211yesmaximal, characteristic, complemented
9#9836C211yesmaximal, characteristic, complemented
10#9834C211yesmaximal, characteristic, complemented
11#9828C211yesmaximal, characteristic, complemented
12#9826C211yesmaximal, characteristic, complemented
13#9826C211yesmaximal, characteristic, complemented
14#9821C211yesmaximal, characteristic, complemented
15#9819C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#533C2211yescharacteristic
18#3231—24no—
19#3231C418yescomplemented
20C2×#500C2211yescharacteristic
21C2×#498C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#485—22no—
25C2×#479—22no—
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29#2285C2211yescharacteristic
30C2×#384C2211yescharacteristic
31#1992—22no—
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1990C2211yescharacteristic
39#1990C2211yescharacteristic
40#1990C2211yescharacteristic
41#1990C2211yescharacteristic
42#1956C2211yescharacteristic
43#1956C2211yescharacteristic
44#1956C2211yescharacteristic
45#1956C2211yescharacteristic
46#1956C2211yescharacteristic
47#1956C2211yescharacteristic
48#1948—22no—
49#1894—22no—
50#1873—22no—
51#1872—22no—
52#1853—22no—
53#1844C2211yescharacteristic
54#1844C2211yescharacteristic
55#1834C2211yescharacteristic
56#1834C2211yescharacteristic
57#1805—22no—
58#1803—22no—
59#1763—22no—
60#1736—22no—
61#1453—24no—
62C2×#322C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×#322C2211yescharacteristic
65C2×C82C2211yescharacteristic
66C22×M16D412yes—
67#533—22no—
68#533—22no—
69#533—22no—
70#533—22no—
71#533—22no—
72#533—22no—
73#533C2×C414yes—
74#533C2×C414yes—
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×C4⋊Q8C2311yescharacteristic
78C2×C4⋊Q8C2311yescharacteristic
79C2×#125—21no—
80C2×#125—21no—
81C2×C22⋊Q8—21no—
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—21no—
85C2×C4×Q8—21no—
86C2×C4×Q8—21no—
87C2×C4×Q8—21no—
88C2×C4×Q8—21no—
89C2×#117—21no—
90C2×#117—21no—
91C2×#117—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C42C2311yescharacteristic
96#500—22no—
97#500C2×C414yescomplemented
98#498—22no—
99#498C2×C414yescomplemented
100#487—22no—
101#487—22no—
102#487C2×C414yescomplemented
103#487C2×C414yescomplemented
104#485—216no—
105#479—216no—
106#431—22no—
107#431—22no—
108#431C2×C414yes—
109#431D414yes—
110C2×#107—22no—
111C2×#106—22no—
112C2×#106—22no—
113C2×#106—22no—
114C2×C4⋊C8D412yes—
115C2×C4⋊C8D412yes—
116C2×#103—22no—
117C2×#103—22no—
118C2×#103—22no—
119C2×#103—22no—
120C2×#103—22no—
121C2×#103—22no—
122C2×#103—22no—
123C2×#103—22no—
124C2×C22⋊C8D412yes—
125C2×C22⋊C8D412yes—
126C4×M16—22no—
127C4×M16—22no—
128C2×#97D412yes—
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135#384—21no—
136#384—21no—
137#384—21no—
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#384C2311yescharacteristic
141#384C2311yescharacteristic
142#382—21no—
143#380—21no—
144#380—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147#340—22no—
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152#340—22no—
153#340—22no—
154#340—22no—
155#340—22no—
156#340—22no—
157#340—22no—
158#340—22no—
159#322—22no—
160#322—22no—
161#322—22no—
162C82—22no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—21no—
167C23×C4—41no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C2×M16—24no—
172C2×M16—24no—
173C2×M16—42no—
174C22×C8C2×D412yes—
175C22×C8C2×D412yes—
176C22×C8C2×D412yes—
177C22×C8C2×D412yes—
178C22×C8C2×D412yes—
179C22×C8C2×D412yes—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8C22×C412yes—
193C4⋊Q8C22×C412yes—
194C4⋊Q8C22×C412yes—
195C4⋊Q8C22×C412yes—
196#125—24no—
197#125—24no—
198#125—42no—
199#125—42no—
200C22⋊Q8—24no—
201C22⋊Q8—24no—
202C22⋊Q8—24no—
203C22⋊Q8—24no—
204C22⋊Q8—42no—
205C22⋊Q8—42no—
206C22⋊Q8—42no—
207C22⋊Q8—42no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—42no—
213C4×Q8—42no—
214C4×Q8—42no—
215C4×Q8—42no—
216#117—24no—
217#117—24no—
218#117—42no—
219#117—42no—
220#117—82no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—82no—
250C2×C4⋊C4—82no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42—41no—
257C2×C42C22×C412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D412yes—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42#4112yes—
263C2×C42#4112yes—
264C2×C42C2411yescharacteristic
265#107—28no—
266#106—28no—
267#106—82no—
268#106—82no—
269C4⋊C8—24no—
270C4⋊C8C22⋊C418yes—
271#103—28no—
272#103—28no—
273#103—44no—
274#103—44no—
275#103—82no—
276#103—82no—
277#103—82no—
278#103—82no—
279C22⋊C8—24no—
280C22⋊C8C22⋊C418yes—
281#97—24no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8C22×C412yes—
291C4×C8C22×C412yes—
292C4×C8C2×D412yes—
293C4×C8C2×D412yes—
294#95—41no—
295#95—41no—
296C24#12711yescharacteristic
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4C2×C22⋊C412yes—
331C22×C4C4×D412yes—
332C22×C4C4×D412yes—
333C22×C4#12112yes—
334C22×C4#12112yes—
335C22×C4#12412yes—
336C22×C4#12711yescharacteristic
337C22×C4#12711yescharacteristic
338C22×C4#12711yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342M16—48no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8C2×C22⋊C414yes—
359C2×C8C2×C22⋊C414yes—
360C2×C8#13714yes—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—82no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—82no—
399C42—44no—
400C42—44no—
401C42—82no—
402C42C2×C22⋊C412yes—
403C42C4×D412yes—
404C42C4×D412yes—
405C42#12112yes—
406C42#12112yes—
407C42#12412yes—
408C42C23×C411yescharacteristic
409C42C22×D411yescharacteristic
410C42C2×#4111yescharacteristic
411C42C2×#4111yescharacteristic
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23#39012yes—
419C23C2×#12711yescharacteristic
420Q8—44no—
421Q8—44no—
422Q8—44no—
423Q8—44no—
424Q8—82no—
425Q8—82no—
426Q8—82no—
427Q8—82no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—164no—
465C2×C4#39012yes—
466C2×C4#39012yes—
467C2×C4#39012yes—
468C2×C4C22×C22⋊C411yescharacteristic
469C2×C4C2×C4×D411yescharacteristic
470C2×C4C2×C4×D411yescharacteristic
471C2×C4C2×#12111yescharacteristic
472C2×C4C2×#12111yescharacteristic
473C2×C4C2×#12411yescharacteristic
474C2×C4C2×#13712yes—
475C2×C4#57512yes—
476C2×C4#57712yes—
477C2×C4#57712yes—
478C2×C4#57712yes—
479C2×C4#57812yes—
480C8—24no—
481C8—24no—
482C8—24no—
483C8—42no—
484C8—42no—
485C8—42no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22C2×#39011yescharacteristic
494C22#315211yescharacteristic
495C22#315311yescharacteristic
496C22#315411yescharacteristic
497C22#315411yescharacteristic
498C22#315411yescharacteristic
499C22#315611yescharacteristic
500C4—22no—
501C4—22no—
502C4—22no—
503C4—82no—
504C4—82no—
505C4—82no—
506C4—82no—
507C4—161no—
508C4—161no—
509C4—161no—
510C4—161no—
511C4#289812yes—
512C4C4×#13712yes—
513C4#295212yes—
514C4#305612yes—
515C4#306012yes—
516C4#313712yes—
517C2—42no—
518C2#1971411yescharacteristic
519C2#1971711yescharacteristic
520C2#1972011yescharacteristic
521C2#1972311yescharacteristic
522C2#2120511yescharacteristic
523C2#2122911yescharacteristic
524C2#2123711yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#127
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #9840C2semidirect
9. #9836C2semidirect
10. #9834C2semidirect
11. #9828C2semidirect
12. #9826C2semidirect
13. #9826C2semidirect
14. #9821C2semidirect
15. #9819C2semidirect
19. #3231C4semidirect
97. #500C2×C4semidirect
99. #498C2×C4semidirect
102. #487C2×C4semidirect
103. #487C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32312
#219812
#218642
#218602
#214092
#213222
C2×#14532
#98402
#98362
#98342
#98282
#98262
#98212
#98192