Finite Group Databasebeta

← #189909

Subgroups — #189909

Order 512.

1008 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34193C211yesmaximal, characteristic
2#21380C211yesmaximal, characteristic
3#20375C211yesmaximal, characteristic
4#8722C212yesmaximal, complemented
5#8721C212yesmaximal, complemented
6#8720C212yesmaximal, complemented
7#8720C212yesmaximal, complemented
8#8035C211yesmaximal, characteristic, complemented
9#8034C211yesmaximal, characteristic, complemented
10#8029C211yesmaximal, characteristic, complemented
11#8028C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#408—21no—
15C2×C4×M16C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17C22×#97C2211yescharacteristic
18#2354C411yescharacteristic
19#2354C411yescharacteristic
20#2354C412yes—
21#2354C412yes—
22Q8×C4⋊C4C411yescharacteristic
23Q8×C4⋊C4C411yescharacteristic
24#2347—21no—
25#2347—22no—
26#2341—21no—
27#2319—22no—
28#2319—21no—
29#2285C2212yes—
30#2285C2211yescharacteristic
31#2283C2212yes—
32#2283C2211yescharacteristic
33#2276—21no—
34#1984—21no—
35#1954—21no—
36#1870—22no—
37#1750—22no—
38#1749C2212yes—
39#1749C2212yes—
40#1749C2212yes—
41#1749C2212yes—
42#1729—21no—
43#1295C2211yescharacteristic
44#1295C2211yescharacteristic
45#1295C2212yes—
46#1295C2212yes—
47#1295C2212yes—
48#1295C2212yes—
49#1295C2212yes—
50#1295C2211yescharacteristic
51#1295C2212yes—
52#1295C2211yescharacteristic
53C22×#41D411yescharacteristic
54C22×M16—21no—
55C22×M16—21no—
56C23×C8—21no—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41—22no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×D4C2×C412yes—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D412yes—
68C2×C4×D4D411yescharacteristic
69C2×#117C2×C412yes—
70C2×#117C2×C412yes—
71C2×#117C2×C412yes—
72C2×#117D411yescharacteristic
73C2×#117D412yes—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76#432—22no—
77#432—22no—
78C2×C4⋊C8—21no—
79#408—28no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C4×M16—22no—
86C4×M16—22no—
87C4×M16—22no—
88C4×M16—22no—
89C2×#97—22no—
90C2×#97—22no—
91C2×#97—21no—
92C2×#97C2311yescharacteristic
93C2×#97C2311yescharacteristic
94C2×#97C2311yescharacteristic
95C2×#97C2311yescharacteristic
96C2×#97C2312yes—
97C2×#97C2312yes—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100#390—21no—
101#390—21no—
102#390—22no—
103#390—22no—
104#389—21no—
105#389—21no—
106#389—22no—
107#389—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#384C2×C412yes—
115#384C2×C412yes—
116#384C2×C411yescharacteristic
117#384C2×C411yescharacteristic
118#383—22no—
119#383—21no—
120#382C2311yescharacteristic
121#382C2312yes—
122#382C2311yescharacteristic
123#381—22no—
124#381—21no—
125#380—22no—
126#380—21no—
127#380—22no—
128#380—21no—
129#380C2×C412yes—
130#380C2×C412yes—
131#380C2×C411yescharacteristic
132#380C2×C411yescharacteristic
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4C2×C412yes—
142C4×C4⋊C4C2×C412yes—
143C4×C4⋊C4C2×C411yescharacteristic
144C4×C4⋊C4C2×C411yescharacteristic
145C4×C22⋊C4—22no—
146C4×C22⋊C4—21no—
147#376—22no—
148#376—21no—
149#336—22no—
150#336—22no—
151#336—22no—
152#336—22no—
153C2×#41—24no—
154C2×#41—22no—
155C2×#41—22no—
156C2×#41—42no—
157C22×Q8C22⋊C411yescharacteristic
158C22×D4C22⋊C411yescharacteristic
159C22×D4C22⋊C412yes—
160C23×C4C22×C412yes—
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C2×M16—24no—
167C2×M16—42no—
168C2×M16—42no—
169C2×M16—42no—
170C2×M16—42no—
171C2×M16—42no—
172C22×C8—24no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—42no—
180C22×C8—42no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—44no—
186C4×Q8—44no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—22no—
195#117—22no—
196#117—22no—
197#117—24no—
198#117—24no—
199#117—82no—
200#117—82no—
201#117—82no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4C22⋊C412yes—
228C2×C22⋊C4C22⋊C412yes—
229C2×C22⋊C4C22⋊C412yes—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42C22×C412yes—
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42#4112yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251C2×C42C2411yescharacteristic
252C4⋊C8—42no—
253C22⋊C8—28no—
254C22⋊C8—28no—
255C22⋊C8—44no—
256C22⋊C8—44no—
257C22⋊C8—42no—
258#97—21no—
259#97—21no—
260#97—21no—
261#97—21no—
262#97—21no—
263#97—21no—
264#97—21no—
265#97—21no—
266#97—22no—
267#97—22no—
268#97—22no—
269#97—22no—
270#97—42no—
271C4×C8—42no—
272C4×C8—42no—
273#95—22no—
274#95—22no—
275#95—21no—
276#95—21no—
277#95—22no—
278#95—22no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—22no—
283#95—21no—
284#95—21no—
285#95—22no—
286#95—21no—
287C24C2×C22⋊C412yes—
288C24C2×C22⋊C411yescharacteristic
289#41—48no—
290#41—48no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8#10414yes—
294C2×D4—24no—
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—42no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4#11712yes—
331C22×C4#11712yes—
332C22×C4#12312yes—
333C22×C4#12312yes—
334C22×C4#12312yes—
335C22×C4#12312yes—
336C22×C4C23×C411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339M16—48no—
340M16—48no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—24no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—48no—
349C2×C8—42no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—41no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—81no—
375C4⋊C4—81no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—81no—
382C22⋊C4—81no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—44no—
395C42—44no—
396C42—42no—
397C42—42no—
398C42—82no—
399C42—82no—
400C42—82no—
401C42#11712yes—
402C42#11712yes—
403C42#12312yes—
404C42#12312yes—
405C42#12312yes—
406C42#12312yes—
407C42C2×#4111yescharacteristic
408C42C2×#4111yescharacteristic
409C42C2×#4111yescharacteristic
410C42C2×#4111yescharacteristic
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23#38112yes—
425C23C22×C22⋊C411yescharacteristic
426Q8—28no—
427D4—48no—
428D4—44no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4#38112yes—
478C2×C4#38112yes—
479C2×C4#38112yes—
480C2×C4C2×#10414yes—
481C2×C4C2×#10412yes—
482C2×C4#42112yes—
483C2×C4C2×#11711yescharacteristic
484C2×C4C2×#11711yescharacteristic
485C2×C4C2×#12311yescharacteristic
486C2×C4C2×#12311yescharacteristic
487C2×C4C2×#12311yescharacteristic
488C2×C4C2×#12311yescharacteristic
489C8—42no—
490C8—42no—
491C8—44no—
492C8—42no—
493C8—42no—
494C8—82no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22C2×#38111yescharacteristic
510C22#288811yescharacteristic
511C22#288911yescharacteristic
512C22C22×#10411yescharacteristic
513C22C2×#42111yescharacteristic
514C22#290111yescharacteristic
515C22#290111yescharacteristic
516C4—21no—
517C4—21no—
518C4—22no—
519C4—21no—
520C4—21no—
521C4—24no—
522C4—21no—
523C4—21no—
524C4—22no—
525C4—42no—
526C4—42no—
527C4—81no—
528C4—82no—
529C4—81no—
530C4—81no—
531C4—82no—
532C4—81no—
533C4#182614yes—
534C2—42no—
535C2—82no—
536C2#1765411yescharacteristic
537C2#1765511yescharacteristic
538C2#1768311yescharacteristic
539C2#1768411yescharacteristic
540C2#2069511yescharacteristic
541C2C2×#182611yescharacteristic
542C2#2070611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8722C2semidirect
5. #8721C2semidirect
6. #8720C2semidirect
7. #8720C2semidirect
8. #8035C2semidirect
9. #8034C2semidirect
10. #8029C2semidirect
11. #8028C2semidirect

Maximal subgroups

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

subgroupindex
#341932
#213802
#203752
#87222
#87212
#87202
#80352
#80342
#80292
#80282