Finite Group Databasebeta

← #223086

Subgroups — #223086

Order 512.

905 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36193C211yesmaximal, characteristic
2#22599C211yesmaximal, characteristic
3#13918C211yesmaximal, characteristic
4#9908C211yesmaximal, characteristic
5#9884C211yesmaximal, characteristic
6#9152C212yesmaximal
7#9112C212yesmaximal
8#8717C212yesmaximal
9#8572C211yesmaximal, characteristic
10C4×#328C211yesmaximal, characteristic
11#8026C212yesmaximal
12C2×#431C2212yes—
13#2857C2211yescharacteristic
14#2855C2212yes—
15#2567C411yescharacteristic, complemented
16#2552C411yescharacteristic, complemented
17#2520—22no—
18#2511C412yescomplemented
19#2508—21no—
20#2507C412yescomplemented
21#2442—21no—
22#2438—22no—
23#2370—21no—
24#2369C411yescharacteristic, complemented
25#2355C411yescharacteristic, complemented
26#2344—21no—
27C4×C22⋊Q8C2211yescharacteristic
28C2×#389—21no—
29#2285C2212yes—
30C2×#382C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32#1939—21no—
33#1916—22no—
34#1817C2212yes—
35#1797C2211yescharacteristic
36C8×C4⋊C4—21no—
37C4×C22⋊C8C2211yescharacteristic
38C2×#328C2211yescharacteristic
39C2×#328C2211yescharacteristic
40C2×#328C2212yes—
41#1442—22no—
42C4⋊#95—21no—
43C4⋊#95C2211yescharacteristic
44#1434C2212yes—
45C4×#95C2211yescharacteristic
46#1294C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2211yescharacteristic
51#1294C2212yes—
52#1294C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8C2311yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×#117—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C42—21no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66#431—24no—
67#431—42no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8C2312yes—
71C2×C22⋊C8C2311yescharacteristic
72C2×C22⋊C8C2311yescharacteristic
73C2×C4×C8—22no—
74C2×C4×C8—21no—
75#400—22no—
76#400—22no—
77#398—22no—
78#398—22no—
79#398—22no—
80#395—22no—
81#391—21no—
82#391—22no—
83#391C2×C412yes—
84#391C2×C412yes—
85#391C2×C411yescharacteristic
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—22no—
92#389—44no—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C411yescharacteristic
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—21no—
102#387—22no—
103#387—22no—
104#385—22no—
105#385—21no—
106#384C2×C412yes—
107#384C2×C411yescharacteristic
108#384C2×C412yes—
109#382—21no—
110#382—22no—
111#382—24no—
112#382—42no—
113#381—22no—
114#380—21no—
115#380—22no—
116#380C2312yes—
117#380C2311yescharacteristic
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4D412yes—
130C4×C4⋊C4D411yescharacteristic
131C4×C22⋊C4—21no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—21no—
137C2×#95D412yes—
138C2×#95C2311yescharacteristic
139C2×#95C2312yes—
140C2×#95C2311yescharacteristic
141#336—21no—
142#336—22no—
143#336—22no—
144#336—21no—
145#328—22no—
146#328—22no—
147#328—42no—
148C22×Q8—21no—
149C22×Q8C22×C411yescharacteristic
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C4⋊Q8—42no—
162#125—44no—
163C22⋊Q8—21no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167C22⋊Q8—81no—
168C22⋊Q8—81no—
169C4×Q8—42no—
170#117—82no—
171#117—82no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4SD1614yes—
199C2×C4⋊C4Q1614yes—
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C412yes—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C412yes—
222C2×C42C22×C412yes—
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C4⋊C8—44no—
229C4⋊C8—42no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234C22⋊C8—21no—
235C22⋊C8#4112yes—
236C4×C8—44no—
237C4×C8—42no—
238#95—21no—
239#95—21no—
240#95—22no—
241#95—22no—
242#95—21no—
243#95—22no—
244#95—22no—
245#95—21no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—22no—
253#95—21no—
254#95—21no—
255#95—22no—
256#95—22no—
257#95—21no—
258#95—44no—
259#95—42no—
260#95—44no—
261#95—44no—
262#95—81no—
263#95C22⋊C412yes—
264#95C22⋊C412yes—
265#95C22⋊C412yes—
266#95C22⋊C412yes—
267#95#4112yes—
268C24C22×D411yescharacteristic
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—42no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—81no—
275C2×Q8—81no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×SD1614yes—
328C22×C4C2×SD1612yes—
329C22×C4C2×Q1614yes—
330C22×C4C2×Q1612yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4H(5,𝔽2)11yescharacteristic
336C22×C42−1+411yescharacteristic
337C2×C8—22no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—41no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—81no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—81no—
369C4⋊C4—88no—
370C4⋊C4—84no—
371C4⋊C4#10318yes—
372C4⋊C4#10314yes—
373C4⋊C4#10314yes—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—42no—
378C22⋊C4—41no—
379C22⋊C4—42no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—41no—
391C42—41no—
392C42—42no—
393C42—42no—
394C42—82no—
395C42—82no—
396C42—84no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23C22×C22⋊C411yescharacteristic
402C23#54611yescharacteristic
403C23#54711yescharacteristic
404C23C22×SD1611yescharacteristic
405C23C22×Q1611yescharacteristic
406C23#57611yescharacteristic
407C23#57711yescharacteristic
408Q8—81no—
409Q8—81no—
410Q8—162no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4#41012yes—
487C2×C4C2×#10312yes—
488C2×C4C2×#10314yes—
489C2×C4C2×#10314yes—
490C2×C4C2×#10312yes—
491C2×C4#42112yes—
492C2×C4C4×#4111yescharacteristic
493C2×C4#52011yescharacteristic
494C2×C4#54611yescharacteristic
495C2×C4#54711yescharacteristic
496C8—42no—
497C8—82no—
498C8—81no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22#233911yescharacteristic
512C22C2×#41011yescharacteristic
513C22#288411yescharacteristic
514C22C22×#10311yescharacteristic
515C22#289711yescharacteristic
516C22C2×#42111yescharacteristic
517C22#290111yescharacteristic
518C22#329511yescharacteristic
519C22#329911yescharacteristic
520C22#333511yescharacteristic
521C22#333811yescharacteristic
522C4—22no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—24no—
528C4—24no—
529C4—41no—
530C4—41no—
531C4—42no—
532C4—42no—
533C4—82no—
534C4—82no—
535C4—81no—
536C4—82no—
537C4—81no—
538C4—82no—
539C4—81no—
540C4—81no—
541C4—162no—
542C4#151814yes—
543C2—21no—
544C2—21no—
545C2—21no—
546C2—21no—
547C2C2×#151811yescharacteristic
548C2#1760211yescharacteristic
549C2#1761011yescharacteristic
550C2#1761911yescharacteristic
551C2#2247811yescharacteristic
552C2#2248711yescharacteristic
553C2#2250711yescharacteristic
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. #2567C4semidirect
16. #2552C4semidirect
18. #2511C4semidirect
20. #2507C4semidirect
24. #2369C4semidirect
25. #2355C4semidirect

Maximal subgroups

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

subgroupindex
#361932
#225992
#139182
#99082
#98842
#91522
#91122
#87172
#85722
C4×#3282
#80262