Finite Group Databasebeta

← #205033

Subgroups — #205033

Order 512.

1099 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35241C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic, complemented
3C2×#1506C211yesmaximal, characteristic, complemented
4C2×#1509C211yesmaximal, characteristic
5C2×#1503C211yesmaximal, characteristic, complemented
6C2×#1500C211yesmaximal, characteristic
7#13797C211yesmaximal, characteristic, complemented
8#9716C214yesmaximal
9#9714C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×#529C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2857C2212yes—
15#2850C2212yes—
16#2664—22no—
17#2568C411yescharacteristic, complemented
18#2565C412yescomplemented
19#2547C411yescharacteristic, complemented
20#2546C411yescharacteristic, complemented
21#2545C412yescomplemented
22#2542C411yescharacteristic, complemented
23#2498—22no—
24#2417—22no—
25#2323—22no—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×#376C2211yescharacteristic
29#1940—22no—
30#1939—22no—
31#1797C2214yes—
32#1509—24no—
33#1506—24no—
34#1503—24no—
35#1500—24no—
36C2×#328C2212yes—
37C2×#328C2214yes—
38C2×#328C2212yes—
39C2×#328C2214yes—
40C2×#327C2212yes—
41C2×#327C2212yes—
42C4⋊#95—22no—
43#1434—22no—
44#1434C2214yes—
45#533—22no—
46#533—22no—
47#533—22no—
48#533—22no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2×C411yescharacteristic
52#529—22no—
53#529—22no—
54#529—22no—
55#529—22no—
56C2×#126—21no—
57C2×#125—21no—
58C2×#125C2×C411yescharacteristic
59C2×#124C2×C411yescharacteristic
60C2×#124C2×C412yes—
61C2×#123—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8C2312yes—
65C2×C22⋊Q8C2311yescharacteristic
66C2×#121C2311yescharacteristic
67C2×C4×Q8—21no—
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×#117—21no—
73C2×#117—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D412yes—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80#432—22no—
81#432—22no—
82#431—22no—
83#431—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—24no—
87C2×C22⋊C8C2312yes—
88C2×C22⋊C8C2312yes—
89C2×#97—21no—
90C2×C4×C8—21no—
91#400—22no—
92#399—22no—
93#398—24no—
94#398—22no—
95#397—22no—
96#396—24no—
97#391—22no—
98#391C2×C412yes—
99#391C2×C412yes—
100#391C2×C412yes—
101#390C2×C412yes—
102#389—22no—
103#389—24no—
104#388—24no—
105#388—24no—
106#387—24no—
107#386—24no—
108#384C2×C412yes—
109#382C2×C412yes—
110#380—24no—
111#380—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4D412yes—
117C4×C4⋊C4D412yes—
118C4×C4⋊C4D414yes—
119#376—22no—
120#376D414yes—
121C2×#95—21no—
122C2×#95—24no—
123C2×#95—24no—
124C2×#95D412yes—
125C2×#95C2312yes—
126C2×#95C2312yes—
127#336—24no—
128#328—24no—
129#328—28no—
130#328—28no—
131#328—24no—
132#327—24no—
133#327—24no—
134C22×Q8—21no—
135C22×Q8C22×C411yescharacteristic
136C22×Q8C22×C412yes—
137C22×D4—21no—
138C22×D4C22×C411yescharacteristic
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D414yes—
145C23×C4C2411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—82no—
153#126—42no—
154#126—42no—
155#125—22no—
156#125—22no—
157#125—42no—
158#125—42no—
159#124—22no—
160#124—22no—
161#124—82no—
162#123—42no—
163#123—42no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166C22⋊Q8—22no—
167C22⋊Q8—22no—
168C22⋊Q8—21no—
169C22⋊Q8—21no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—48no—
173C22⋊Q8—42no—
174C22⋊Q8—42no—
175#121—21no—
176#121—21no—
177#121—21no—
178#121—21no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—24no—
182C4×Q8—24no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—42no—
188C4×D4—42no—
189#117—42no—
190#117—42no—
191#117—42no—
192#117—42no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C414yes—
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C412yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—24no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—44no—
226C2×C42—82no—
227C2×C42—82no—
228C2×C42C22×C412yes—
229C2×C42C22×C412yes—
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D414yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42#4112yes—
238C2×C42#4112yes—
239C4⋊C8—42no—
240C4⋊C8—42no—
241C4⋊C8—84no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246#97—42no—
247C4×C8—42no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—22no—
254#95—22no—
255#95—84no—
256#95—84no—
257#95C22⋊C414yes—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260#95C2×D414yes—
261#95C2×D414yes—
262C24—21no—
263C24C22×D411yescharacteristic
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—42no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—44no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—42no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4C2×C22⋊C414yes—
311C22×C4C2×C22⋊C414yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4#12112yes—
317C22×C4#12112yes—
318C22×C4#12112yes—
319C22×C4#12112yes—
320C22×C4C23×C411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—44no—
341C4⋊C4—48no—
342C4⋊C4—48no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—44no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—84no—
355C22⋊C4—48no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—84no—
366C42—44no—
367C42—42no—
368C42—42no—
369C42—48no—
370C42—42no—
371C42—44no—
372C42—84no—
373C42C4×D412yes—
374C42C4×D412yes—
375C42C4×D412yes—
376C42C4×D412yes—
377C42#12412yes—
378C42#12412yes—
379C42#12712yes—
380C42#12712yes—
381C23—21no—
382C23—21no—
383C23—22no—
384C23—41no—
385C23—41no—
386C23—42no—
387C23—42no—
388C23—41no—
389C23C22×C22⋊C411yescharacteristic
390C23C2×#12111yescharacteristic
391C23C2×#12111yescharacteristic
392C23#57712yes—
393C23#57812yes—
394Q8—81no—
395Q8—81no—
396Q8—82no—
397Q8—82no—
398Q8—84no—
399D4—81no—
400D4—81no—
401D4—81no—
402D4—81no—
403D4—82no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—84no—
445C2×C4—84no—
446C2×C4—82no—
447C2×C4—84no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—84no—
451C2×C4—82no—
452C2×C4—84no—
453C2×C4#39012yes—
454C2×C4#39012yes—
455C2×C4#39012yes—
456C2×C4#39012yes—
457C2×C4#42114yes—
458C2×C4C2×C4×D411yescharacteristic
459C2×C4C2×C4×D411yescharacteristic
460C2×C4C2×#12411yescharacteristic
461C2×C4C2×#12711yescharacteristic
462C8—82no—
463C8—82no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—81no—
473C22—82no—
474C22—82no—
475C22—81no—
476C22—81no—
477C22C2×#39011yescharacteristic
478C22#288311yescharacteristic
479C22#288411yescharacteristic
480C22#289812yes—
481C22C2×#42112yes—
482C22#304211yescharacteristic
483C22#304411yescharacteristic
484C22#304612yes—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—44no—
493C4—44no—
494C4—84no—
495C4—84no—
496C4—81no—
497C4—81no—
498C4—82no—
499C4—82no—
500C4—81no—
501C4#196314yes—
502C2—21no—
503C2—22no—
504C2—21no—
505C2—161no—
506C2#1760012yes—
507C2#1760112yes—
508C2C2×#196311yescharacteristic
509C2#2121411yescharacteristic
510C2#2126211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
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
2. #21953C2semidirect
3. C2×#1506C2semidirect
5. C2×#1503C2semidirect
7. #13797C2semidirect
9. #9714C2semidirect
17. #2568C4semidirect
18. #2565C4semidirect
19. #2547C4semidirect
20. #2546C4semidirect
21. #2545C4semidirect
22. #2542C4semidirect

Maximal subgroups

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

subgroupindex
#352412
#219532
C2×#15062
C2×#15092
C2×#15032
C2×#15002
#137972
#97162
#97142