Finite Group Databasebeta

← #188519

Subgroups — #188519

Order 512.

1012 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34221C211yesmaximal, characteristic
2#20814C211yesmaximal, characteristic
3C2×#1518C211yesmaximal, characteristic
4C2×#1506C211yesmaximal, characteristic
5#13963C211yesmaximal, characteristic
6#8578C212yesmaximal
7#8576C212yesmaximal
8#8574C212yesmaximal
9#8572C212yesmaximal
10#8017C211yesmaximal, characteristic
11#8013C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2371—22no—
16#2368—22no—
17#2356—21no—
18#2356—21no—
19#2343—21no—
20#2343—21no—
21#2331C412yescomplemented
22#2327C412yescomplemented
23#2320C411yescharacteristic, complemented
24#2320C411yescharacteristic, complemented
25C4×C22⋊Q8C2212yes—
26C4×C22⋊Q8C2212yes—
27#2283C2211yescharacteristic
28C2×#382C2211yescharacteristic
29#2276C411yescharacteristic, complemented
30#2276C411yescharacteristic, complemented
31C2×#376C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1843—21no—
34#1832—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1768—22no—
38#1518—24no—
39#1506—24no—
40C2×#328C2211yescharacteristic
41C2×#328C2212yes—
42C2×#328C2211yescharacteristic
43C2×#328C2212yes—
44C2×#328C2211yescharacteristic
45C2×#328C2211yescharacteristic
46#1444—21no—
47#1442—21no—
48#1436—22no—
49C4⋊#95C2212yes—
50#1434C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54#1294C2212yes—
55#533—22no—
56#533—22no—
57#533D414yes—
58#533Q814yes—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8C2311yescharacteristic
64C2×C22⋊Q8C2311yescharacteristic
65C2×C4×Q8—21no—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8C2×C412yes—
70C2×#117—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C42—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76#432—22no—
77#432—22no—
78#431—22no—
79#431—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C22⋊C8C2312yes—
85C2×C22⋊C8C2311yescharacteristic
86C2×C22⋊C8C2311yescharacteristic
87C2×#97—21no—
88C2×C4×C8—21no—
89#391—22no—
90#391—22no—
91#391C2×C412yes—
92#391C2×C412yes—
93#389C2×C412yes—
94#389C2×C412yes—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#385—22no—
106#385—22no—
107#383—21no—
108#383—21no—
109#383—21no—
110#382—21no—
111#382—21no—
112#382—22no—
113#382—22no—
114#382C2×C412yes—
115#381—21no—
116#380C2312yes—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4C2×C412yes—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—21no—
130#376—21no—
131#376—21no—
132#376—21no—
133#376—22no—
134#376—22no—
135#376C2×C411yescharacteristic
136#376C2×C412yes—
137#376C2×C411yescharacteristic
138#376D411yescharacteristic
139#376D411yescharacteristic
140#376D412yes—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—22no—
146C2×#95D411yescharacteristic
147C2×#95D411yescharacteristic
148C2×#95D412yes—
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152#336—22no—
153#336—22no—
154#328—22no—
155#328—22no—
156#328—22no—
157#328—22no—
158#328—24no—
159#328—24no—
160C22×Q8—21no—
161C22×Q8C22×C411yescharacteristic
162C22×Q8C22×C411yescharacteristic
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2411yescharacteristic
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8C4⋊C414yescomplemented
177C4⋊Q8C4⋊C414yescomplemented
178#125—24no—
179#125—24no—
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C22⋊Q8C2×D412yes—
187C22⋊Q8C2×D412yes—
188C22⋊Q8C2×Q812yes—
189C22⋊Q8C2×Q812yes—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—44no—
195C4×Q8—44no—
196C4×Q8C4⋊C414yescomplemented
197C4×Q8C4⋊C414yescomplemented
198#117—24no—
199#117—24no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C412yes—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—44no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42C22×C412yes—
236C2×C42C22×C412yes—
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×Q812yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247C4⋊C8—44no—
248C4⋊C8—44no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C22⋊C8—22no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—22no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257#97—42no—
258C4×C8—42no—
259#95—21no—
260#95—21no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—21no—
273#95—21no—
274#95—22no—
275#95—21no—
276#95—21no—
277#95—22no—
278#95—44no—
279#95—44no—
280#95—42no—
281#95—42no—
282#95—82no—
283#95C22⋊C411yescharacteristic
284#95C22⋊C411yescharacteristic
285#95C22⋊C412yes—
286#95C22⋊C411yescharacteristic
287#95C22⋊C411yescharacteristic
288#95C22⋊C412yes—
289#95C22⋊C412yes—
290#95C22⋊C412yes—
291#95C22⋊C412yes—
292#95C22⋊C412yes—
293C24C22×D411yescharacteristic
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—24no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—81no—
300C2×Q8—81no—
301C2×Q8C2×C4⋊C412yes—
302C2×Q8C2×C4⋊C412yes—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
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×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C412yes—
347C22×C4#12112yes—
348C22×C4#12112yes—
349C22×C4C22⋊Q812yes—
350C22×C4C22⋊Q812yes—
351C22×C4C23×C411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×Q811yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—41no—
363C2×C8—41no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4—84no—
382C4⋊C4—88no—
383C4⋊C4C2×C4⋊C412yes—
384C4⋊C4C2×C4⋊C412yes—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C42—24no—
392C42—24no—
393C42—24no—
394C42—24no—
395C42—42no—
396C42—42no—
397C42—44no—
398C42—48no—
399C42—44no—
400C42—82no—
401C42—81no—
402C42—81no—
403C42—82no—
404C42C2×C4⋊C412yes—
405C42C2×C4⋊C412yes—
406C42#11712yes—
407C42#11712yes—
408C42#12112yes—
409C42#12112yes—
410C42C22⋊Q812yes—
411C42C22⋊Q812yes—
412C23—22no—
413C23—21no—
414C23—21no—
415C23C22×C22⋊C411yescharacteristic
416C23C2×#12111yescharacteristic
417C23C2×C22⋊Q811yescharacteristic
418C23#57712yes—
419C23#57812yes—
420Q8—42no—
421Q8—42no—
422Q8—42no—
423Q8—42no—
424Q8—84no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
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—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4#38012yes—
491C2×C4#38012yes—
492C2×C4#38012yes—
493C2×C4#38012yes—
494C2×C4#41014yes—
495C2×C4C22×C4⋊C411yescharacteristic
496C2×C4C2×#11711yescharacteristic
497C2×C4C2×#12111yescharacteristic
498C2×C4C2×C22⋊Q811yescharacteristic
499C8—82no—
500C8—81no—
501C8—81no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22C2×#38011yescharacteristic
511C22C2×#41012yes—
512C22#289812yes—
513C22#290111yescharacteristic
514C22#290111yescharacteristic
515C22#304311yescharacteristic
516C22#304411yescharacteristic
517C22#308212yes—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—22no—
523C4—22no—
524C4—22no—
525C4—22no—
526C4—24no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—42no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—162no—
539C2—21no—
540C2—21no—
541C2—22no—
542C2#1760512yes—
543C2#1760912yes—
544C2#2063211yescharacteristic
545C2#2065111yescharacteristic
546C2#2065511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #2331C4semidirect
22. #2327C4semidirect
23. #2320C4semidirect
24. #2320C4semidirect
29. #2276C4semidirect
30. #2276C4semidirect
176. C4⋊Q8C4⋊C4semidirect
177. C4⋊Q8C4⋊C4semidirect
196. C4×Q8C4⋊C4semidirect
197. C4×Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342212
#208142
C2×#15182
C2×#15062
#139632
#85782
#85762
#85742
#85722
#80172
#80132