Finite Group Databasebeta

← #190663

Subgroups — #190663

Order 512.

1068 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34208C211yesmaximal, characteristic
2#21435C212yesmaximal, complemented
3#8718C211yesmaximal, characteristic, complemented
4#8718C211yesmaximal, characteristic, complemented
5#8583C211yesmaximal, characteristic, complemented
6#8581C212yesmaximal, complemented
7#8579C211yesmaximal, characteristic, complemented
8#8337C212yesmaximal
9#8226C212yesmaximal, complemented
10#8224C211yesmaximal, characteristic, complemented
11C4×#329C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#520C411yescharacteristic
14C2×#520C411yescharacteristic
15#2364—22no—
16#2364—22no—
17#2352—22no—
18#2352—22no—
19#2328C411yescharacteristic
20#2328C411yescharacteristic
21#2327C411yescharacteristic
22#2327C411yescharacteristic
23#2311C411yescharacteristic
24#2311C411yescharacteristic
25C4×#124C2211yescharacteristic
26C4×#124C2211yescharacteristic
27C4×#124C2211yescharacteristic
28C4×#124C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1924—22no—
32#1923—22no—
33#1917—22no—
34#1916—22no—
35#1767C2212yes—
36#1767C2212yes—
37#1749C2212yes—
38#1749C2212yes—
39#1745C2212yes—
40C4×#97C2212yes—
41C2×#329C2212yes—
42C2×#329C2212yes—
43C2×#329C2212yes—
44C2×#329C2212yes—
45#1295C2211yescharacteristic
46#1295C2211yescharacteristic
47#1295C2211yescharacteristic
48#1295C2211yescharacteristic
49#1295C2212yes—
50#1295C2212yes—
51C22×#41C2×C411yescharacteristic
52C22×#41C2×C411yescharacteristic
53#532—22no—
54#532—22no—
55#532—24no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#127C2×C411yescharacteristic
58C2×#124C2311yescharacteristic
59C2×#124C2311yescharacteristic
60C2×#121—22no—
61#520—44no—
62#520—44no—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—22no—
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×#117—21no—
77C2×#117—21no—
78C2×#117C2×C411yescharacteristic
79C2×#117C2×C411yescharacteristic
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C42C2311yescharacteristic
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×#97—22no—
90C2×#97C2312yes—
91C2×#97C2312yes—
92C2×#97C2312yes—
93C2×#97C2312yes—
94C2×C4×C8—22no—
95#391C2×C411yescharacteristic
96#391C2×C411yescharacteristic
97#391C2×C411yescharacteristic
98#391C2×C411yescharacteristic
99#390C2×C411yescharacteristic
100#390C2×C411yescharacteristic
101#390C2×C411yescharacteristic
102#390C2×C411yescharacteristic
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#382—22no—
118#382C2×C411yescharacteristic
119#382C2×C411yescharacteristic
120#382C2311yescharacteristic
121#382C2311yescharacteristic
122#380—22no—
123#380—22no—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D412yes—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4D412yes—
131C4×C22⋊C4D412yes—
132#376C2×C411yescharacteristic
133#376C2×C411yescharacteristic
134C2×#95—22no—
135C43C2311yescharacteristic
136C43C2311yescharacteristic
137#336—22no—
138#336—22no—
139#336—22no—
140#336—22no—
141#336—22no—
142#336—22no—
143#329—24no—
144#329—24no—
145#329—24no—
146#329—24no—
147C2×#41—22no—
148C2×#41—22no—
149C2×#41—22no—
150C2×#41—22no—
151C2×#41—42no—
152C2×#41—42no—
153C2×#41—42no—
154C2×#41—42no—
155C22×Q8C22×C411yescharacteristic
156C22×Q8C22×C411yescharacteristic
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—22no—
160C22×D4C22×C411yescharacteristic
161C22×D4C22×C411yescharacteristic
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D412yes—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178#127—22no—
179#127—22no—
180#124—21no—
181#124—21no—
182#124—22no—
183#124#4114yes—
184#124#4112yes—
185#124#4112yes—
186#121—28no—
187#121—28no—
188C4×Q8—42no—
189C4×Q8—42no—
190C4×Q8—42no—
191C4×Q8—42no—
192C4×D4—28no—
193C4×D4—28no—
194C4×D4—42no—
195C4×D4—42no—
196C4×D4—42no—
197C4×D4—42no—
198C4×D4—44no—
199C4×D4—44no—
200C4×D4—44no—
201C4×D4—44no—
202#117—24no—
203#117—42no—
204#117—44no—
205#117—44no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C22⋊C411yescharacteristic
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4C22⋊C411yescharacteristic
235C2×C22⋊C4C22⋊C412yes—
236C2×C22⋊C4C22⋊C411yescharacteristic
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2411yescharacteristic
255C4⋊C8—44no—
256C4⋊C8—44no—
257C22⋊C8—44no—
258C22⋊C8—44no—
259#97—22no—
260#97—22no—
261#97—22no—
262#97—22no—
263#97—22no—
264#97—22no—
265#97—22no—
266#97—22no—
267#97—44no—
268C4×C8—44no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95C22⋊C412yes—
278#95C22⋊C412yes—
279#95C22⋊C412yes—
280#95C22⋊C412yes—
281C24—21no—
282C24—21no—
283C24C2×C22⋊C411yescharacteristic
284#41—48no—
285#41—48no—
286#41—48no—
287#41—48no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8#11712yes—
295C2×Q8#11712yes—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—28no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—82no—
310C2×D4—82no—
311C2×D4#11712yes—
312C2×D4#11712yes—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D412yes—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C4⋊C4—28no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C22⋊C4—22no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4#10014yes—
396C42—21no—
397C42—21no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—42no—
403C42—42no—
404C42—42no—
405C42—42no—
406C42—41no—
407C42—41no—
408C42—42no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42C2×#4112yes—
413C42C2×#4111yescharacteristic
414C42C2×#4111yescharacteristic
415C23—21no—
416C23—22no—
417C23—22no—
418C23—21no—
419C23—22no—
420C23—22no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23C2×#10012yes—
430C23C22×C22⋊C411yescharacteristic
431Q8—42no—
432Q8—42no—
433Q8—42no—
434Q8—42no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—42no—
439D4—48no—
440D4—48no—
441D4—88no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—48no—
468C2×C4—48no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—162no—
484C2×C4—162no—
485C2×C4—162no—
486C2×C4—162no—
487C2×C4—162no—
488C2×C4—162no—
489C2×C4C2×#10012yes—
490C2×C4C2×#10012yes—
491C2×C4#41312yes—
492C2×C4C2×#11711yescharacteristic
493C2×C4C2×#11711yescharacteristic
494C2×C4#52512yes—
495C2×C4#52512yes—
496C8—44no—
497C8—44no—
498C8—82no—
499C8—82no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—44no—
507C22—42no—
508C22—42no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—82no—
514C22—82no—
515C22#228011yescharacteristic
516C22C22×#10011yescharacteristic
517C22C2×#41311yescharacteristic
518C22#290112yes—
519C22#290112yes—
520C4—22no—
521C4—22no—
522C4—22no—
523C4—22no—
524C4—22no—
525C4—21no—
526C4—21no—
527C4—42no—
528C4—41no—
529C4—41no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—42no—
534C4—81no—
535C4—81no—
536C4—81no—
537C4—81no—
538C2—42no—
539C2—82no—
540C2—82no—
541C2#1768511yescharacteristic
542C2#1768511yescharacteristic
543C2#1768712yes—
544C2#2072611yescharacteristic
545C2#2074912yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21435C2semidirect
3. #8718C2semidirect
4. #8718C2semidirect
5. #8583C2semidirect
6. #8581C2semidirect
7. #8579C2semidirect
9. #8226C2semidirect
10. #8224C2semidirect
11. C4×#329C2semidirect

Maximal subgroups

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

subgroupindex
#342082
#214352
#87182
#85832
#85812
#85792
#83372
#82262
#82242
C4×#3292