Finite Group Databasebeta

← #212751

Subgroups — #212751

Order 512.

565 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#23818C211yesmaximal, characteristic
2#22301C211yesmaximal, characteristic, complemented
3#22210C211yesmaximal, characteristic
4#9872C211yesmaximal, characteristic, complemented
5#9640C211yesmaximal, characteristic
6#9558C211yesmaximal, characteristic
7#9557C211yesmaximal, characteristic, complemented
8#9556C211yesmaximal, characteristic
9#9555C211yesmaximal, characteristic, complemented
10#8787C211yesmaximal, characteristic, complemented
11#8212C211yesmaximal, characteristic
12#8206C211yesmaximal, characteristic
13#8205C211yesmaximal, characteristic, complemented
14C8×#103C211yesmaximal, characteristic, complemented
15#7845C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C4×#125C2211yescharacteristic
19C4×#124C2211yescharacteristic
20#2101C411yescharacteristic
21#2101C411yescharacteristic
22#2074C411yescharacteristic
23#2073C411yescharacteristic
24#2042C411yescharacteristic
25#2042C411yescharacteristic
26#2026C411yescharacteristic
27#2025C411yescharacteristic
28#1984C2211yescharacteristic
29#1954C2211yescharacteristic
30#1940—21no—
31#1939—21no—
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1923—21no—
35#1923—21no—
36#1917—21no—
37#1916—21no—
38#1851—21no—
39#1851—21no—
40#1842C2211yescharacteristic
41#1842C2211yescharacteristic
42#1770C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1744C2211yescharacteristic
49#1744C2211yescharacteristic
50#1744C2211yescharacteristic
51C2×Q8⋊C8C2211yescharacteristic
52C2×Q8⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57C2×#322C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C2×#125C2×C411yescharacteristic
69C2×#124C2×C411yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C8×Q8—22no—
73C8×Q8—22no—
74#436—22no—
75#436—22no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×#103C2×C411yescharacteristic
88C2×#103C2×C411yescharacteristic
89C2×#103C2×C411yescharacteristic
90C2×#103C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×#102C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×#97—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#391C2×C411yescharacteristic
108#390C2×C411yescharacteristic
109#389C2×C411yescharacteristic
110#389C2×C411yescharacteristic
111#388—21no—
112#388—21no—
113#387—21no—
114#387—21no—
115#382C2×C411yescharacteristic
116#382C2×C411yescharacteristic
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C43C2311yescharacteristic
124#340C2×C411yescharacteristic
125#340C2×C411yescharacteristic
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#336—21no—
133#336—21no—
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336C2×C411yescharacteristic
141#336C2×C411yescharacteristic
142#336C2×C411yescharacteristic
143#336C2×C411yescharacteristic
144#336D411yescharacteristic
145#336D411yescharacteristic
146#336D411yescharacteristic
147#336D411yescharacteristic
148Q8⋊C8—22no—
149Q8⋊C8—22no—
150D4⋊C8—22no—
151D4⋊C8—22no—
152#322—22no—
153#322—22no—
154#322—22no—
155C82—22no—
156C22×Q8C22×C411yescharacteristic
157C22×D4C22×C411yescharacteristic
158C23×C4—21no—
159C2×M16—42no—
160C2×M16—81no—
161C2×M16—81no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8C22×C411yescharacteristic
169C22×C8C22×C411yescharacteristic
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176#125—22no—
177#125—41no—
178#124—22no—
179#124—41no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42#4111yescharacteristic
213C2×C42C2411yescharacteristic
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—42no—
223C4⋊C8—81no—
224C4⋊C8—81no—
225C4⋊C8—81no—
226C4⋊C8—81no—
227C4⋊C8—81no—
228C4⋊C8—81no—
229#103—22no—
230#103—22no—
231#103—22no—
232#103—41no—
233#102—22no—
234#102—41no—
235#102—41no—
236#102—41no—
237C22⋊C8—42no—
238C22⋊C8—81no—
239C22⋊C8—81no—
240#97—81no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—42no—
250C4×C8—81no—
251C4×C8#4112yes—
252C4×C8#4112yes—
253C4×C8#4112yes—
254C4×C8#4112yes—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265C24—21no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8#13112yes—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4C4×D411yescharacteristic
292C22×C4C4×D411yescharacteristic
293C22×C4C4×D411yescharacteristic
294C22×C4C4×D411yescharacteristic
295C22×C4C23×C411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4111yescharacteristic
298M16—84no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8#12412yes—
340C2×C8#12412yes—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4#13112yes—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42C2×#4111yescharacteristic
382C42C2×#4111yescharacteristic
383C42C2×#4111yescharacteristic
384C42C2×#4111yescharacteristic
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23C2×C4×D411yescharacteristic
393Q8—22no—
394Q8—22no—
395Q8—41no—
396Q8—41no—
397D4—41no—
398D4—41no—
399D4—41no—
400D4—41no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
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—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4C4×#4111yescharacteristic
441C2×C4C4×#4111yescharacteristic
442C2×C4C2×#12411yescharacteristic
443C2×C4#52911yescharacteristic
444C2×C4#52911yescharacteristic
445C2×C4#53211yescharacteristic
446C2×C4C2×#13111yescharacteristic
447C2×C4C2×#13111yescharacteristic
448C2×C4#56811yescharacteristic
449C2×C4#56811yescharacteristic
450C2×C4#57611yescharacteristic
451C2×C4#57711yescharacteristic
452C2×C4#57711yescharacteristic
453C2×C4#57811yescharacteristic
454C8—22no—
455C8—22no—
456C8—42no—
457C8—42no—
458C8—41no—
459C8—41no—
460C8—41no—
461C8—41no—
462C8—41no—
463C8—41no—
464C8—81no—
465C8—81no—
466C8—81no—
467C8—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22C4×#12411yescharacteristic
476C22#293311yescharacteristic
477C22#293311yescharacteristic
478C22#295111yescharacteristic
479C22#295211yescharacteristic
480C22#295511yescharacteristic
481C22#295511yescharacteristic
482C4—21no—
483C4—21no—
484C4—41no—
485C4—42no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—42no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4#296611yescharacteristic
496C4#296711yescharacteristic
497C4#296811yescharacteristic
498C4#296911yescharacteristic
499C4#313611yescharacteristic
500C4#313711yescharacteristic
501C4#317811yescharacteristic
502C4#317911yescharacteristic
503C2—161no—
504C2#1802911yescharacteristic
505C2#1803111yescharacteristic
506C2#1803311yescharacteristic
507C2#1803311yescharacteristic
508C2#2163711yescharacteristic
509C2#2164311yescharacteristic
510C2#2166111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22301C2semidirect
4. #9872C2semidirect
7. #9557C2semidirect
9. #9555C2semidirect
10. #8787C2semidirect
13. #8205C2semidirect
14. C8×#103C2semidirect
15. #7845C2semidirect

Maximal subgroups

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

subgroupindex
#238182
#223012
#222102
#98722
#96402
#95582
#95572
#95562
#95552
#87872
#82122
#82062
#82052
C8×#1032
#78452