Finite Group Databasebeta

← #230502

Subgroups — #230502

Order 512.

630 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23151C211yesmaximal, characteristic, complemented
2#22795C211yesmaximal, characteristic
3C4×#474C211yesmaximal, characteristic
4#9874C211yesmaximal, characteristic, complemented
5#9604C211yesmaximal, characteristic, complemented
6#9558C211yesmaximal, characteristic
7#9554C211yesmaximal, characteristic
8#9545C211yesmaximal, characteristic
9#9541C211yesmaximal, characteristic
10#9533C211yesmaximal, characteristic, complemented
11#9533C211yesmaximal, characteristic, complemented
12#8291C211yesmaximal, characteristic, complemented
13C8×#107C211yesmaximal, characteristic, complemented
14#8264C211yesmaximal, characteristic, complemented
15#8174C211yesmaximal, characteristic
16C2×#474C411yescharacteristic
17C2×#446—21no—
18C2×#431—21no—
19C4×C4⋊Q8C2211yescharacteristic
20C42×D4C2211yescharacteristic
21#1988C411yescharacteristic
22#1985—21no—
23#1954C2211yescharacteristic
24#1954C2211yescharacteristic
25#1947C411yescharacteristic
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1923—21no—
29#1923—21no—
30#1918C411yescharacteristic
31#1918C411yescharacteristic
32#1917—21no—
33#1916—21no—
34#1862C411yescharacteristic
35#1862C411yescharacteristic
36#1842C2211yescharacteristic
37#1802C411yescharacteristic
38#1797—21no—
39C4×#107C2211yescharacteristic
40C4×#107C2211yescharacteristic
41#1770C2211yescharacteristic
42C8×C4⋊C4C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45#1767C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×#102C2211yescharacteristic
50#1744C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
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—
74#446—44no—
75#431—44no—
76C2×#107C2×C411yescharacteristic
77C2×#107C2×C411yescharacteristic
78C2×#107C2×C411yescharacteristic
79C2×#107C2×C411yescharacteristic
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2×C411yescharacteristic
85C2×C4⋊C8C2×C411yescharacteristic
86C2×C4⋊C8D411yescharacteristic
87C2×C4⋊C8D411yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×#102C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×#97—21no—
101C2×#97—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#391—21no—
112#389—21no—
113#389C2×C411yescharacteristic
114#389C2×C411yescharacteristic
115#384C2×C411yescharacteristic
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C4⋊C4C2311yescharacteristic
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C43C2311yescharacteristic
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336C2×C411yescharacteristic
144#336C2×C411yescharacteristic
145#336D411yescharacteristic
146#336D411yescharacteristic
147#335—22no—
148#335—22no—
149D4⋊C8—22no—
150D4⋊C8—22no—
151#322—22no—
152C82—22no—
153C22×Q8—21no—
154C22×D4C22×C411yescharacteristic
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8C22×C411yescharacteristic
165C22×C8C22×C411yescharacteristic
166C22×C8C22×C411yescharacteristic
167C22×C8C22×C411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C4⋊Q8—22no—
173C4⋊Q8—41no—
174C4×Q8—42no—
175C4×Q8—42no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—41no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42#4111yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42C2411yescharacteristic
213#107—22no—
214#107—22no—
215#107—22no—
216#107M1614yes—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—41no—
227C4⋊C8—42no—
228C4⋊C8—42no—
229C4⋊C8—42no—
230C4⋊C8—81no—
231C4⋊C8D814yes—
232C4⋊C8#4112yes—
233#102—22no—
234#102—22no—
235#102—22no—
236#102—22no—
237C22⋊C8—42no—
238C22⋊C8—42no—
239C22⋊C8—81no—
240C22⋊C8—81no—
241#97—42no—
242#97—42no—
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—21no—
253C4×C8—21no—
254C4×C8—42no—
255C4×C8—42no—
256C4×C8—42no—
257C4×C8#4112yes—
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—42no—
267C2×Q8—42no—
268C2×Q8—81no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4#13112yes—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—81no—
294C22×C4C4×D411yescharacteristic
295C22×C4C4×D411yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4C4×D411yescharacteristic
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—22no—
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×C8—41no—
340C2×C8C2×M1612yes—
341C2×C8C2×D812yes—
342C2×C8C2×D812yes—
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—21no—
353C4⋊C4—21no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4C2×M1612yes—
360C4⋊C4C2×M1612yes—
361C4⋊C4#13112yes—
362C22⋊C4—24no—
363C22⋊C4—42no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—42no—
381C42—42no—
382C42C2×D812yes—
383C42#13512yes—
384C42C2×#4111yescharacteristic
385C42C2×#4111yescharacteristic
386C422−1+411yescharacteristic
387C422−1+411yescharacteristic
388C23—22no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23C2×C4×D411yescharacteristic
396Q8—84no—
397D4—22no—
398D4—22no—
399D4—41no—
400D4—41no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—48no—
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—84no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4C4×D812yes—
441C2×C4C4×D812yes—
442C2×C4C4×#4111yescharacteristic
443C2×C4#51911yescharacteristic
444C2×C4#54711yescharacteristic
445C2×C4#54811yescharacteristic
446C2×C4#55211yescharacteristic
447C2×C4#55611yescharacteristic
448C2×C4C22×M1611yescharacteristic
449C2×C4C2×#13111yescharacteristic
450C2×C4C2×#13111yescharacteristic
451C2×C4#56811yescharacteristic
452C2×C4C22×D811yescharacteristic
453C2×C4C2×#13511yescharacteristic
454C2×C4#57711yescharacteristic
455C2×C4#57811yescharacteristic
456C8—22no—
457C8—22no—
458C8—22no—
459C8—42no—
460C8—41no—
461C8—41no—
462C8—42no—
463C8—41no—
464C8—41no—
465C8—41no—
466C8—81no—
467C8—81no—
468C8—81no—
469C8—81no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22#236211yescharacteristic
478C22#292911yescharacteristic
479C22D4×M1611yescharacteristic
480C22C2×C4×D811yescharacteristic
481C22#294711yescharacteristic
482C22#295511yescharacteristic
483C22#295711yescharacteristic
484C4—21no—
485C4—21no—
486C4—22no—
487C4—22no—
488C4—41no—
489C4—42no—
490C4—42no—
491C4—41no—
492C4—42no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—81no—
497C4#296511yescharacteristic
498C4#296811yescharacteristic
499C4#298411yescharacteristic
500C4#298511yescharacteristic
501C4#333411yescharacteristic
502C4#334011yescharacteristic
503C4#335811yescharacteristic
504C4#336411yescharacteristic
505C2—82no—
506C2#1790011yescharacteristic
507C2#1792111yescharacteristic
508C2M16×D811yescharacteristic
509C2#1805311yescharacteristic
510C2#2305211yescharacteristic
511C2#2306011yescharacteristic
512C2#2308211yescharacteristic
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. #23151C2semidirect
4. #9874C2semidirect
5. #9604C2semidirect
10. #9533C2semidirect
11. #9533C2semidirect
12. #8291C2semidirect
13. C8×#107C2semidirect
14. #8264C2semidirect

Maximal subgroups

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

subgroupindex
#231512
#227952
C4×#4742
#98742
#96042
#95582
#95542
#95452
#95412
#95332
#82912
C8×#1072
#82642
#81742