Finite Group Databasebeta

← #188395

Subgroups — #188395

Order 512.

1020 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34060C211yesmaximal, characteristic
2#21978C211yesmaximal, characteristic, complemented
3#21350C211yesmaximal, characteristic, complemented
4C2×#1489C211yesmaximal, characteristic
5C2×#1488C211yesmaximal, characteristic
6#8570C212yesmaximal, complemented
7#8569C212yesmaximal, complemented
8#8568C212yesmaximal, complemented
9#8567C212yesmaximal, complemented
10#7904C211yesmaximal, characteristic, complemented
11#7901C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2320C411yescharacteristic
18#2320C411yescharacteristic
19#2319C411yescharacteristic
20#2319C411yescharacteristic
21#2318—21no—
22#2318—21no—
23#2317C412yes—
24#2314—22no—
25#2311C412yes—
26#2295C2212yes—
27#2293C2212yes—
28#2290—22no—
29#2285—21no—
30#2283—21no—
31C4×#117C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1940—22no—
34#1940—21no—
35#1940—21no—
36#1851—21no—
37#1851—21no—
38#1851—22no—
39#1842C2212yes—
40#1831C2212yes—
41C4×C4⋊C8C2212yes—
42C4×C4⋊C8C2212yes—
43#1489—24no—
44#1488—24no—
45C2×Q8⋊C8C2211yescharacteristic
46C2×Q8⋊C8C2211yescharacteristic
47C2×Q8⋊C8C2212yes—
48C2×D4⋊C8C2211yescharacteristic
49C2×D4⋊C8C2211yescharacteristic
50C2×D4⋊C8C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55C22×#41C2×C411yescharacteristic
56C22×M16—21no—
57C22×M16—21no—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41D414yes—
61C4×#41Q814yes—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×#117—21no—
69C2×#117—22no—
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C412yes—
72C22×C4⋊C4C2×C412yes—
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#432—22no—
77#432—22no—
78#431—22no—
79#431—22no—
80#423—22no—
81#423—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2311yescharacteristic
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C4×M16—22no—
94C4×M16—22no—
95C2×#97—21no—
96C2×#97—21no—
97C2×C4×C8C2312yes—
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#391C2×C412yes—
101#391C2×C412yes—
102#390C2×C412yes—
103#390C2×C412yes—
104#389C2×C412yes—
105#389C2×C412yes—
106#389C2×C412yes—
107#389C2×C412yes—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#387—22no—
116#386—22no—
117#386—22no—
118#385—22no—
119#385—22no—
120#384—22no—
121#383—21no—
122#383—21no—
123#382—22no—
124#382—21no—
125#382—21no—
126#381—21no—
127#381—21no—
128#380—22no—
129#380—22no—
130#380—21no—
131#380—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4D412yes—
138C4×C4⋊C4D411yescharacteristic
139C4×C4⋊C4D411yescharacteristic
140C4×C4⋊C4C2312yes—
141C4×C22⋊C4—22no—
142C4×C22⋊C4D411yescharacteristic
143C4×C22⋊C4D411yescharacteristic
144C4×C22⋊C4D412yes—
145#376—22no—
146#376C2×C411yescharacteristic
147#376C2×C411yescharacteristic
148C2×#95—21no—
149C43C2312yes—
150#336—22no—
151#336—22no—
152#336—22no—
153#336—22no—
154Q8⋊C8—22no—
155Q8⋊C8—22no—
156Q8⋊C8—24no—
157D4⋊C8—22no—
158D4⋊C8—22no—
159D4⋊C8—24no—
160C2×#41—22no—
161C2×#41—44no—
162C2×#41C4⋊C414yes—
163C22×Q8C22×C411yescharacteristic
164C22×D4—21no—
165C22×D4C22×C411yescharacteristic
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—22no—
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C2×M16—44no—
174C2×M16—42no—
175C2×M16—42no—
176C2×M16—81no—
177C2×M16—81no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—21no—
184C4×Q8—22no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8C4⋊C414yes—
188C4×Q8C2×D412yes—
189C4×Q8C2×Q812yes—
190C4×D4—24no—
191C4×D4—24no—
192C4×D4—24no—
193C4×D4—24no—
194C4×D4—22no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4C4⋊C414yes—
198C4×D4C2×D412yes—
199C4×D4C2×Q812yes—
200#117—22no—
201#117—24no—
202#117—24no—
203#117—44no—
204#117—82no—
205#117—82no—
206#117C4⋊C414yes—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4C22⋊C412yes—
229C2×C22⋊C4C22⋊C412yes—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—81no—
239C2×C42—81no—
240C2×C42C22×C412yes—
241C2×C42C22×C412yes—
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×Q812yes—
251C2×C42#4112yes—
252C2×C42#4112yes—
253C2×C42C2411yescharacteristic
254C4⋊C8—22no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—22no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—44no—
261C4⋊C8—44no—
262C4⋊C8—42no—
263C4⋊C8—42no—
264C4⋊C8—82no—
265C4⋊C8—82no—
266C22⋊C8—42no—
267C22⋊C8—42no—
268#97—42no—
269#97—42no—
270C4×C8—22no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—22no—
274C4×C8—21no—
275C4×C8—21no—
276#95—21no—
277#95—21no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95C22⋊C411yescharacteristic
288#95C22⋊C411yescharacteristic
289#95C22⋊C411yescharacteristic
290#95C22⋊C411yescharacteristic
291#95C22⋊C412yes—
292#95C22⋊C412yes—
293C24—21no—
294C24C2×C22⋊C411yescharacteristic
295#41—48no—
296#41—48no—
297C2×Q8—21no—
298C2×Q8—24no—
299C2×Q8C2×C4⋊C412yes—
300C2×D4—24no—
301C2×D4—21no—
302C2×D4—24no—
303C2×D4—44no—
304C2×D4C2×C4⋊C412yes—
305C22×C4—24no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—44no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C4⋊C412yes—
337C22×C4#11712yes—
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4C22⋊Q812yes—
341C22×C4C22⋊Q812yes—
342C22×C4C23×C411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345M16—48no—
346M16—84no—
347C2×C8—24no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—41no—
362C2×C8—41no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—44no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4C2×C4⋊C412yes—
378C4⋊C4C2×C4⋊C412yes—
379C22⋊C4—24no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—24no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C42—22no—
390C42—22no—
391C42—24no—
392C42—24no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—24no—
398C42—24no—
399C42—42no—
400C42—42no—
401C42—42no—
402C42—42no—
403C42—81no—
404C42—81no—
405C42C2×C4⋊C412yes—
406C42#11712yes—
407C42#12112yes—
408C42#12112yes—
409C42C22⋊Q812yes—
410C42C22⋊Q812yes—
411C42C22×D411yescharacteristic
412C42C22×Q811yescharacteristic
413C42C2×#4111yescharacteristic
414C42C2×#4111yescharacteristic
415C23—22no—
416C23—21no—
417C23—21no—
418C23—22no—
419C23—22no—
420C23—42no—
421C23—41no—
422C23—41no—
423C23—42no—
424C23#38012yes—
425C23C22×C22⋊C411yescharacteristic
426Q8—24no—
427Q8—42no—
428D4—24no—
429D4—48no—
430D4—42no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—48no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—84no—
475C2×C4—88no—
476C2×C4#38012yes—
477C2×C4#38012yes—
478C2×C4#38012yes—
479C2×C4C22×C4⋊C411yescharacteristic
480C2×C4C2×#11711yescharacteristic
481C2×C4C2×#12111yescharacteristic
482C2×C4C2×#12111yescharacteristic
483C2×C4C2×C22⋊Q811yescharacteristic
484C2×C4C2×C22⋊Q811yescharacteristic
485C2×C4#56814yes—
486C2×C4#57611yescharacteristic
487C2×C4#57711yescharacteristic
488C2×C4#57711yescharacteristic
489C2×C4#57811yescharacteristic
490C8—44no—
491C8—81no—
492C8—81no—
493C8—81no—
494C8—82no—
495C8—81no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—82no—
505C22—82no—
506C22C2×#38011yescharacteristic
507C22#287912yes—
508C22#289711yescharacteristic
509C22#289811yescharacteristic
510C22#290112yes—
511C4—22no—
512C4—22no—
513C4—21no—
514C4—22no—
515C4—21no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—82no—
524C4—82no—
525C4#290512yes—
526C4#292112yes—
527C4#304511yescharacteristic
528C4#304611yescharacteristic
529C4#308311yescharacteristic
530C4#308411yescharacteristic
531C2—42no—
532C2—82no—
533C2#1750512yes—
534C2#1750612yes—
535C2#2062711yescharacteristic
536C2#2065211yescharacteristic
537C2#2065511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21978C2semidirect
3. #21350C2semidirect
6. #8570C2semidirect
7. #8569C2semidirect
8. #8568C2semidirect
9. #8567C2semidirect
10. #7904C2semidirect
11. #7901C2semidirect

Maximal subgroups

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

subgroupindex
#340602
#219782
#213502
C2×#14892
C2×#14882
#85702
#85692
#85682
#85672
#79042
#79012