Finite Group Databasebeta

← #230533

Subgroups — #230533

Order 512.

595 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#23248C211yesmaximal, characteristic, complemented
2#22835C211yesmaximal, characteristic
3C4×#474C211yesmaximal, characteristic
4#9877C211yesmaximal, characteristic, complemented
5#9606C211yesmaximal, characteristic, complemented
6#9558C211yesmaximal, characteristic
7#9556C211yesmaximal, characteristic
8#9546C211yesmaximal, characteristic
9#9541C211yesmaximal, characteristic
10#9536C211yesmaximal, characteristic, complemented
11#9533C211yesmaximal, characteristic, complemented
12#8291C211yesmaximal, characteristic, complemented
13#8284C211yesmaximal, characteristic, complemented
14#8263C211yesmaximal, characteristic, complemented
15#8172C211yesmaximal, characteristic
16C2×#474C411yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18C42×D4C2211yescharacteristic
19#1988C411yescharacteristic
20#1985—21no—
21#1984—21no—
22#1954C2211yescharacteristic
23#1954C2211yescharacteristic
24#1947C411yescharacteristic
25#1940—21no—
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1923—21no—
29#1923—21no—
30#1918C411yescharacteristic
31#1918C411yescharacteristic
32#1862C411yescharacteristic
33#1862C411yescharacteristic
34#1831C2211yescharacteristic
35#1817—21no—
36#1802C411yescharacteristic
37C4×#107C2211yescharacteristic
38C4×#107C2211yescharacteristic
39#1771—21no—
40#1770C2211yescharacteristic
41C8×C4⋊C4C2211yescharacteristic
42C8×C4⋊C4C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44#1767C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×C22⋊C8—21no—
50C4×#97C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×C4×Q8—21no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C22×C42—21no—
73#474—24no—
74C2×#107C2×C411yescharacteristic
75C2×#107C2×C411yescharacteristic
76C2×#107C2×C411yescharacteristic
77C2×#107C2×C411yescharacteristic
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8C2×C411yescharacteristic
81C2×C4⋊C8C2×C411yescharacteristic
82C2×C4⋊C8D411yescharacteristic
83C2×C4⋊C8D411yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×#102C2×C411yescharacteristic
89C2×#102C2×C411yescharacteristic
90C2×#102C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×#97—21no—
97C2×#97—21no—
98C2×#97—21no—
99C2×#97—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109#391—21no—
110#389—21no—
111#389C2×C411yescharacteristic
112#389C2×C411yescharacteristic
113#384C2×C411yescharacteristic
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2311yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C22⋊C4—21no—
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#336—21no—
141#336C2×C411yescharacteristic
142#336C2×C411yescharacteristic
143#336D411yescharacteristic
144#336D411yescharacteristic
145#335—22no—
146#334—22no—
147D4⋊C8—22no—
148D4⋊C8—22no—
149#322—22no—
150#322—22no—
151C22×Q8—21no—
152C22×D4C22×C411yescharacteristic
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8C22×C411yescharacteristic
163C22×C8C22×C411yescharacteristic
164C22×C8C22×C411yescharacteristic
165C22×C8C22×C411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C4⋊Q8—22no—
171C4⋊Q8—41no—
172C4×Q8—81no—
173C4×Q8—81no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—41no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—81no—
203C2×C42—81no—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42#4111yescharacteristic
209C2×C42#4111yescharacteristic
210C2×C42C2411yescharacteristic
211#107—22no—
212#107—22no—
213#107—22no—
214#107M1614yes—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—41no—
225C4⋊C8—41no—
226C4⋊C8—81no—
227C4⋊C8—81no—
228C4⋊C8#4112yes—
229#102—22no—
230#102—22no—
231#102—22no—
232#102—22no—
233C22⋊C8—42no—
234C22⋊C8—42no—
235C22⋊C8—81no—
236C22⋊C8—81no—
237#97—42no—
238#97—42no—
239#97—42no—
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—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—81no—
253C4×C8—81no—
254C4×C8—81no—
255C4×C8#4112yes—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263C24—21no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—81no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4#13112yes—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—24no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4—81no—
292C22×C4C4×D411yescharacteristic
293C22×C4C4×D411yescharacteristic
294C22×C4C4×D411yescharacteristic
295C22×C4C4×D411yescharacteristic
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—22no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—22no—
311C2×C8—22no—
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—21no—
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×C8C2×M1612yes—
340C2×C8#13712yes—
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—81no—
354C4⋊C4—81no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4C2×M1612yes—
358C4⋊C4C2×M1612yes—
359C4⋊C4#13112yes—
360C22⋊C4—24no—
361C22⋊C4—42no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—22no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—81no—
381C42—81no—
382C42C2×#4111yescharacteristic
383C42C2×#4111yescharacteristic
384C422−1+411yescharacteristic
385C422−1+411yescharacteristic
386C23—22no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23C2×C4×D411yescharacteristic
394Q8—84no—
395D4—22no—
396D4—22no—
397D4—41no—
398D4—41no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
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—41no—
416C2×C4—41no—
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—84no—
434C2×C4—84no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4#44412yes—
440C2×C4C4×#4111yescharacteristic
441C2×C4#51911yescharacteristic
442C2×C4#54711yescharacteristic
443C2×C4#54811yescharacteristic
444C2×C4#55211yescharacteristic
445C2×C4#55611yescharacteristic
446C2×C4C22×M1611yescharacteristic
447C2×C4C2×#13111yescharacteristic
448C2×C4C2×#13111yescharacteristic
449C2×C4#56811yescharacteristic
450C2×C4C2×#13711yescharacteristic
451C2×C4#57511yescharacteristic
452C2×C4#57611yescharacteristic
453C2×C4#57611yescharacteristic
454C8—22no—
455C8—22no—
456C8—42no—
457C8—42no—
458C8—41no—
459C8—41no—
460C8—41no—
461C8—42no—
462C8—41no—
463C8—41no—
464C8—41no—
465C8—81no—
466C8—81no—
467C8—81no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22#236211yescharacteristic
476C22#292911yescharacteristic
477C22D4×M1611yescharacteristic
478C22C4×#13711yescharacteristic
479C22#294811yescharacteristic
480C22C2×#44411yescharacteristic
481C22#295611yescharacteristic
482C4—21no—
483C4—21no—
484C4—22no—
485C4—41no—
486C4—42no—
487C4—41no—
488C4—42no—
489C4—42no—
490C4—41no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4#296511yescharacteristic
496C4#296811yescharacteristic
497C4#298411yescharacteristic
498C4#298511yescharacteristic
499C4#334611yescharacteristic
500C4#334711yescharacteristic
501C4#337011yescharacteristic
502C4#337511yescharacteristic
503C2—82no—
504C2#1791611yescharacteristic
505C2#1792211yescharacteristic
506C2#1804911yescharacteristic
507C2#1806411yescharacteristic
508C2#2305211yescharacteristic
509C2#2307911yescharacteristic
510C2#2308111yescharacteristic
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 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23248C2semidirect
4. #9877C2semidirect
5. #9606C2semidirect
10. #9536C2semidirect
11. #9533C2semidirect
12. #8291C2semidirect
13. #8284C2semidirect
14. #8263C2semidirect

Maximal subgroups

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

subgroupindex
#232482
#228352
C4×#4742
#98772
#96062
#95582
#95562
#95462
#95412
#95362
#95332
#82912
#82842
#82632
#81722