Finite Group Databasebeta

← #224436

Subgroups — #224436

Order 512.

837 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36552C211yesmaximal, characteristic
2#22625C211yesmaximal, characteristic
3#13936C211yesmaximal, characteristic
4#9913C212yesmaximal
5#9909C211yesmaximal, characteristic
6#9886C212yesmaximal
7#9885C211yesmaximal, characteristic
8#8576C212yesmaximal
9#8572C211yesmaximal, characteristic
10#8221C212yesmaximal
11C4×#328C211yesmaximal, characteristic
12#2857C2211yescharacteristic
13#2851C2212yes—
14#2590—22no—
15#2568C411yescharacteristic, complemented
16#2552C411yescharacteristic, complemented
17#2511—21no—
18#2508C412yescomplemented
19#2505—22no—
20#2502—21no—
21#2453C412yescomplemented
22#2365C411yescharacteristic, complemented
23#2360—21no—
24#2355—21no—
25Q8×C4⋊C4C411yescharacteristic, complemented
26C4×C22⋊Q8C2211yescharacteristic
27C4×C22⋊Q8C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1939—21no—
30#1916—22no—
31#1797C2211yescharacteristic
32#1797C2212yes—
33C8×C4⋊C4—21no—
34C4×C22⋊C8C2211yescharacteristic
35C4×C22⋊C8C2212yes—
36C2×#328C2211yescharacteristic
37C2×#328C2212yes—
38C2×#328C2211yescharacteristic
39#1443—22no—
40C4⋊#95—21no—
41C4⋊#95—21no—
42C4⋊#95C2211yescharacteristic
43#1434C2212yes—
44#1434C2212yes—
45C4×#95C2211yescharacteristic
46#1294C2211yescharacteristic
47#1294C2211yescharacteristic
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125C2×C412yes—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8C2311yescharacteristic
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C412yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65C22×C42C2312yes—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—22no—
68C2×C22⋊C8C2312yes—
69C2×C22⋊C8C2311yescharacteristic
70C2×C22⋊C8C2311yescharacteristic
71C2×C4×C8—21no—
72C2×C4×C8—22no—
73#400—22no—
74#400—22no—
75#400—22no—
76#398—22no—
77#398—22no—
78#398—22no—
79#391—21no—
80#391—21no—
81#391—21no—
82#391—21no—
83#391C2×C412yes—
84#391C2×C411yescharacteristic
85#389—22no—
86#389—21no—
87#389—21no—
88#389—22no—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C411yescharacteristic
93#389C2×C412yes—
94#387—22no—
95#387—21no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#385—22no—
102#385—22no—
103#385—22no—
104#385—21no—
105#384—21no—
106#384—21no—
107#384C2×C411yescharacteristic
108#382—22no—
109#382C2×C412yes—
110#380—21no—
111#380C2311yescharacteristic
112#380C2312yes—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C22⋊C4—21no—
125C4×C22⋊C4—22no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—21no—
132C2×#95D412yes—
133C2×#95C2311yescharacteristic
134C2×#95C2312yes—
135C2×#95C2311yescharacteristic
136#336—21no—
137#336—22no—
138#336—21no—
139#336—22no—
140#328—22no—
141#328—24no—
142#328Q814yescomplemented
143C22×Q8—21no—
144C22×Q8C22×C411yescharacteristic
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2411yescharacteristic
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C4⋊Q8—42no—
157#125—82no—
158C22⋊Q8—21no—
159C22⋊Q8—21no—
160C22⋊Q8—42no—
161C22⋊Q8—42no—
162C22⋊Q8C2×Q812yes—
163C22⋊Q8#4112yes—
164C4×Q8—42no—
165C4×Q8—44no—
166C4×Q8—82no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C412yes—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—82no—
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C412yes—
211C2×C42C22×C412yes—
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C4⋊C8—42no—
218C4⋊C8—82no—
219C22⋊C8—21no—
220C22⋊C8—21no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—21no—
224C22⋊C8C2×Q812yes—
225C4×C8—42no—
226C4×C8—82no—
227#95—21no—
228#95—22no—
229#95—21no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—22no—
234#95—21no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—21no—
241#95—22no—
242#95—21no—
243#95—21no—
244#95—22no—
245#95—21no—
246#95—22no—
247#95—42no—
248#95—44no—
249#95—82no—
250#95—81no—
251#95—82no—
252#95—82no—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256#95C22⋊C412yes—
257#95C2×Q812yes—
258C24C22×D411yescharacteristic
259C2×Q8—21no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8C4×Q812yes—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4#13512yes—
318C22×C4#13512yes—
319C22×C4C23×C411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×Q811yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C42−1+411yescharacteristic
324C22×C42−1+411yescharacteristic
325C2×C8—22no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—42no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—44no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—44no—
352C4⋊C4—81no—
353C4⋊C4—82no—
354C4⋊C4—81no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—84no—
360C4⋊C4C4×Q812yes—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C42—22no—
368C42—22no—
369C42—42no—
370C42—42no—
371C42—41no—
372C42—41no—
373C42—42no—
374C42—42no—
375C42—44no—
376C42—42no—
377C42—42no—
378C42—82no—
379C42—84no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23C22×C22⋊C411yescharacteristic
385C23#54811yescharacteristic
386C23D4×Q811yescharacteristic
387C23C2×#13511yescharacteristic
388C23C2×#13511yescharacteristic
389C23#57711yescharacteristic
390C23#57811yescharacteristic
391Q8—42no—
392Q8—42no—
393Q8—84no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—82no—
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—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4#41012yes—
470C2×C4#41612yes—
471C2×C4#41612yes—
472C2×C4#42112yes—
473C2×C4C2×C4×Q811yescharacteristic
474C2×C4#51911yescharacteristic
475C2×C4#54811yescharacteristic
476C2×C4D4×Q811yescharacteristic
477C8—42no—
478C8—81no—
479C8—82no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22Q8×C22⋊C411yescharacteristic
493C22C2×#41011yescharacteristic
494C22#288311yescharacteristic
495C22C2×#41611yescharacteristic
496C22#289811yescharacteristic
497C22C2×#42111yescharacteristic
498C22#290111yescharacteristic
499C22#336411yescharacteristic
500C22#336511yescharacteristic
501C22#338111yescharacteristic
502C22#338311yescharacteristic
503C4—22no—
504C4—21no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—41no—
512C4—41no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—82no—
517C4—82no—
518C4—82no—
519C4—81no—
520C4—81no—
521C4—82no—
522C4—162no—
523C4#152114yes—
524C2—21no—
525C2—21no—
526C2—21no—
527C2—21no—
528C2C2×#152111yescharacteristic
529C2#1760011yescharacteristic
530C2#1760511yescharacteristic
531C2#1761711yescharacteristic
532C2#2251311yescharacteristic
533C2#2252111yescharacteristic
534C2#2252911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2568C4semidirect
16. #2552C4semidirect
18. #2508C4semidirect
21. #2453C4semidirect
22. #2365C4semidirect
25. Q8×C4⋊C4C4semidirect
142. #328Q8semidirect

Maximal subgroups

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

subgroupindex
#365522
#226252
#139362
#99132
#99092
#98862
#98852
#85762
#85722
#82212
C4×#3282