Finite Group Databasebeta

← #506264

Subgroups — #506264

Order 512.

627 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2176C211yesmaximal, characteristic, complemented
2C2×#2168C211yesmaximal, characteristic
3#28228C211yesmaximal, characteristic
4#28226C211yesmaximal, characteristic
5#13494C211yesmaximal, characteristic, complemented
6#13494C211yesmaximal, characteristic, complemented
7#13306C211yesmaximal, characteristic, complemented
8#13306C211yesmaximal, characteristic, complemented
9#13062C211yesmaximal, characteristic, complemented
10#13062C211yesmaximal, characteristic, complemented
11#13060C211yesmaximal, characteristic, complemented
12#13060C211yesmaximal, characteristic, complemented
13#12890C211yesmaximal, characteristic, complemented
14#12814C211yesmaximal, characteristic, complemented
15#12812C211yesmaximal, characteristic, complemented
16C22×Q32C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C23×C16C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×#467C2211yescharacteristic
21C2×#467C2211yescharacteristic
22C2×#467C2211yescharacteristic
23C2×C4×Q16C2211yescharacteristic
24C2×C4×Q16C2211yescharacteristic
25C2×C8×D4C2211yescharacteristic
26#2538—21no—
27#2536—21no—
28#2176—24no—
29#2168—24no—
30C2×#367C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#358C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40C2×#346C2211yescharacteristic
41#2036—21no—
42#2036—21no—
43#2035—21no—
44#2035—21no—
45#1846C2211yescharacteristic
46#1845C2211yescharacteristic
47#1808C2211yescharacteristic
48#1808C2211yescharacteristic
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51#1805—21no—
52#1803—21no—
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×Q16C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×D4D411yescharacteristic
72C2×Q32—22no—
73C2×Q32—22no—
74C2×Q32—22no—
75C2×M32—22no—
76C2×M32—41no—
77C2×M32—41no—
78C22×C16—22no—
79C22×C16—41no—
80C22×C16—41no—
81C22×C16C2311yescharacteristic
82C22×C16C2311yescharacteristic
83C22×C16C2311yescharacteristic
84C22×C16C2311yescharacteristic
85#467—22no—
86#467—22no—
87#467—22no—
88#467—22no—
89#467—41no—
90#467—41no—
91#467—41no—
92#467—41no—
93C4×Q16—22no—
94C4×Q16—22no—
95C4×Q16—22no—
96C4×Q16—22no—
97C8×D4—22no—
98C8×D4—22no—
99C2×#107—21no—
100C2×#107C2311yescharacteristic
101C2×#107C2311yescharacteristic
102C2×#107C2311yescharacteristic
103C2×#107C2311yescharacteristic
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#106—21no—
107C2×C4⋊C8D411yescharacteristic
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×#103—21no—
113C2×#103—21no—
114C2×C22⋊C8D411yescharacteristic
115C2×C22⋊C8D411yescharacteristic
116C2×C4×C8C2311yescharacteristic
117#398—41no—
118#396—41no—
119#396—41no—
120#394—41no—
121#390—41no—
122#389—41no—
123#388—41no—
124#388—41no—
125#384—21no—
126#382—21no—
127#380—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131#367—22no—
132C4⋊C16—22no—
133#358—22no—
134#358—22no—
135#358—41no—
136#358—41no—
137#358—41no—
138#358—41no—
139C22⋊C16—22no—
140C22⋊C16—22no—
141#346D414yes—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×D4D811yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16—21no—
156C2×Q16—21no—
157C2×Q16—21no—
158C2×Q16—21no—
159C2×Q16—21no—
160C2×Q16—21no—
161C2×Q16#4112yes—
162C2×Q16#4112yes—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2411yescharacteristic
179C22⋊Q8—42no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183C22⋊Q8—81no—
184C22⋊Q8—81no—
185C22⋊Q8—81no—
186C22⋊Q8—81no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×Q8—42no—
190C4×Q8—42no—
191C4×D4—22no—
192C4×D4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4D811yescharacteristic
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4D811yescharacteristic
217C2×C22⋊C4D811yescharacteristic
218C2×C42—41no—
219C2×C42—41no—
220C2×C42C2×D411yescharacteristic
221Q32—28no—
222M32—44no—
223C2×C16—21no—
224C2×C16—21no—
225C2×C16—21no—
226C2×C16—21no—
227C2×C16—21no—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—44no—
234C2×C16C2×D412yes—
235C2×C16C2×D412yes—
236#107—21no—
237#107—21no—
238#107—21no—
239#107—21no—
240#107—21no—
241#107—21no—
242#107—21no—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—81no—
249#106—81no—
250C4⋊C8—22no—
251#103—42no—
252#103—42no—
253#103—42no—
254#103—42no—
255#103—42no—
256#103—42no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C4×C8C2×D412yes—
260C4×C8#4112yes—
261#95—41no—
262#95—41no—
263#95—41no—
264#95—41no—
265#95—41no—
266#95—41no—
267C24C2×D811yescharacteristic
268C24C2×D811yescharacteristic
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—41no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4C2×D811yescharacteristic
305C22×C4C2×D811yescharacteristic
306C22×C4C2×D811yescharacteristic
307C22×C4C2×D811yescharacteristic
308C22×C4C22×D411yescharacteristic
309Q16—22no—
310Q16—22no—
311Q16—22no—
312Q16—22no—
313Q16—41no—
314Q16—41no—
315Q16—41no—
316Q16—41no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
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#12112yes—
334C2×C8C22×D411yescharacteristic
335C2×C8C2×#4111yescharacteristic
336C2×C8C2×#4111yescharacteristic
337C2×C8C2×#4111yescharacteristic
338C2×C8H(5,𝔽2)11yescharacteristic
339C2×C8H(5,𝔽2)11yescharacteristic
340C4⋊C4—22no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—161no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—82no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C22⋊C4—161no—
370C22⋊C4—161no—
371C42—82no—
372C42—82no—
373C42#12112yes—
374C42#13512yes—
375C16—22no—
376C16—22no—
377C16—41no—
378C16—41no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23#51012yes—
393C23C22×D811yescharacteristic
394Q8—42no—
395Q8—42no—
396Q8—42no—
397Q8—42no—
398Q8—81no—
399Q8—81no—
400Q8—81no—
401Q8—81no—
402D4—44no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4#46612yes—
446C2×C4#46612yes—
447C2×C4C2×#12111yescharacteristic
448C2×C4#54611yescharacteristic
449C2×C4#54611yescharacteristic
450C2×C4C2×#13511yescharacteristic
451C2×C4#57711yescharacteristic
452C2×C4#57711yescharacteristic
453C8—22no—
454C8—41no—
455C8—41no—
456C8#52511yescharacteristic
457C8#53511yescharacteristic
458C8#55311yescharacteristic
459C8#55311yescharacteristic
460C22—22no—
461C22—22no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22C2×#46611yescharacteristic
475C22#329411yescharacteristic
476C22#329411yescharacteristic
477C22C2×#51011yescharacteristic
478C22#341411yescharacteristic
479C22#341611yescharacteristic
480C22#341711yescharacteristic
481C4—22no—
482C4—22no—
483C4—41no—
484C4—41no—
485C4—82no—
486C4—82no—
487C4—161no—
488C4—161no—
489C4—161no—
490C4—161no—
491C4—161no—
492C4—161no—
493C4#253511yescharacteristic
494C4#306011yescharacteristic
495C4#330311yescharacteristic
496C4#330311yescharacteristic
497C2—42no—
498C2—81no—
499C2#2814011yescharacteristic
500C2#3360611yescharacteristic
501C2#3361311yescharacteristic
502C2#3387911yescharacteristic
503C2#3388011yescharacteristic
504C2#3388311yescharacteristic
505C2#3388411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2176C2semidirect
5. #13494C2semidirect
6. #13494C2semidirect
7. #13306C2semidirect
8. #13306C2semidirect
9. #13062C2semidirect
10. #13062C2semidirect
11. #13060C2semidirect
12. #13060C2semidirect
13. #12890C2semidirect
14. #12814C2semidirect
15. #12812C2semidirect

Maximal subgroups

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

subgroupindex
C2×#21762
C2×#21682
#282282
#282262
#134942
#133062
#130622
#130602
#128902
#128142
#128122