Finite Group Databasebeta

← #502386

Subgroups — #502386

Order 512.

616 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2228C211yesmaximal, characteristic, complemented
2C2×D4×C16C211yesmaximal, characteristic, complemented
3#28708C211yesmaximal, characteristic, complemented
4#28225C211yesmaximal, characteristic
5#13494C211yesmaximal, characteristic, complemented
6#13493C211yesmaximal, characteristic, complemented
7#13381C211yesmaximal, characteristic, complemented
8#13380C211yesmaximal, characteristic, complemented
9#13281C211yesmaximal, characteristic, complemented
10#13178C211yesmaximal, characteristic, complemented
11#13059C211yesmaximal, characteristic, complemented
12#13059C211yesmaximal, characteristic, complemented
13#12835C211yesmaximal, characteristic, complemented
14#12834C211yesmaximal, characteristic, complemented
15C4×#357C211yesmaximal, characteristic, complemented
16C23×C16C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23#2562—21no—
24#2535—21no—
25#2228—24no—
26D4×C16—24no—
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×C22⋊C16C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38C2×C4×C16C2211yescharacteristic
39#2077—21no—
40#2077—21no—
41#2034—21no—
42#2034—21no—
43#1989—21no—
44#1846C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic
48#1808C2211yescharacteristic
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51#1804—21no—
52C4×#107C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×#125—21no—
65C2×#121—21no—
66C2×#121—21no—
67C2×C4×D4—21no—
68C2×C4×D4D411yescharacteristic
69C22×C16—22no—
70C22×C16—22no—
71C22×C16—22no—
72C22×C16—22no—
73C22×C16—41no—
74C22×C16—41no—
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79#499—22no—
80#499—22no—
81#466—22no—
82#466—22no—
83#466—41no—
84#466—41no—
85C4×D8—22no—
86C4×D8—22no—
87C8×D4—22no—
88C8×D4—22no—
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#106—21no—
97C2×#106—21no—
98C2×C4⋊C8D411yescharacteristic
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×C22⋊C8D411yescharacteristic
103C2×C22⋊C8D411yescharacteristic
104C2×C4×C8C2311yescharacteristic
105#400—41no—
106#397—41no—
107#394—41no—
108#392—41no—
109#388—41no—
110#388—41no—
111#386—41no—
112#386—41no—
113#382—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120C4×C4⋊C4—21no—
121#367—22no—
122#367—41no—
123#366—22no—
124#366—41no—
125C4⋊C16D414yescomplemented
126#357—22no—
127#357—22no—
128#357—22no—
129#357—22no—
130C22⋊C16—22no—
131C22⋊C16—22no—
132C4×C16—22no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140C22×D4—21no—
141C22×D4—41no—
142C22×D4—41no—
143C22×D4D811yescharacteristic
144C23×C4—41no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8#4112yes—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2411yescharacteristic
169#125—42no—
170#125—42no—
171#121—42no—
172#121—42no—
173#121—81no—
174#121—81no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—42no—
178C4×D4—42no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4D811yescharacteristic
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4D811yescharacteristic
199C2×C22⋊C4D811yescharacteristic
200C2×C42—41no—
201C2×C42C2×D411yescharacteristic
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—44no—
212C2×C16—44no—
213C2×C16C2×D412yes—
214C2×C16C2×D412yes—
215C2×C16#4112yes—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107—21no—
230#106—42no—
231#106—81no—
232C4⋊C8—22no—
233#102—42no—
234#102—42no—
235#102—42no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C4×C8C2×D412yes—
239C4×C8#4112yes—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249C24—41no—
250C24C2×D811yescharacteristic
251C24C2×D811yescharacteristic
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—82no—
262C2×D4—82no—
263C2×D4—82no—
264C2×D4—82no—
265C2×D4—161no—
266C2×D4—161no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—161no—
288C22×C4—161no—
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C2×D811yescharacteristic
292C22×C4C2×D811yescharacteristic
293C22×C4C22×D411yescharacteristic
294D8—22no—
295D8—22no—
296D8—41no—
297D8—41no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8#12112yes—
315C2×C8C22×D411yescharacteristic
316C2×C8C2×#4111yescharacteristic
317C2×C8C2×#4111yescharacteristic
318C2×C8C2×#4111yescharacteristic
319C2×C8H(5,𝔽2)11yescharacteristic
320C2×C82−1+411yescharacteristic
321C4⋊C4—22no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—161no—
346C22⋊C4—161no—
347C22⋊C4—161no—
348C22⋊C4—161no—
349C42—161no—
350C42#12112yes—
351C42#13512yes—
352C16—22no—
353C16—22no—
354C16—22no—
355C16—41no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—42no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23#50812yes—
376C23#50812yes—
377C23C22×D811yescharacteristic
378D4—28no—
379D4—42no—
380D4—42no—
381D4—81no—
382D4—81no—
383D4—164no—
384D4—164no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—164no—
425C2×C4#46612yes—
426C2×C4#46612yes—
427C2×C4C2×#12111yescharacteristic
428C2×C4#54611yescharacteristic
429C2×C4#54711yescharacteristic
430C2×C4C2×#13511yescharacteristic
431C2×C4#57711yescharacteristic
432C2×C4#57811yescharacteristic
433C8—22no—
434C8—41no—
435C8—41no—
436C8#52911yescharacteristic
437C8#54211yescharacteristic
438C8#54611yescharacteristic
439C8#54811yescharacteristic
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—161no—
458C22—161no—
459C22—161no—
460C22—161no—
461C22C2×#46611yescharacteristic
462C22#329411yescharacteristic
463C22#333411yescharacteristic
464C22C2×#50811yescharacteristic
465C22C2×#50811yescharacteristic
466C22#341511yescharacteristic
467C22#341611yescharacteristic
468C4—22no—
469C4—22no—
470C4—41no—
471C4—41no—
472C4—161no—
473C4—161no—
474C4—161no—
475C4—161no—
476C4—161no—
477C4—161no—
478C4—161no—
479C4#242711yescharacteristic
480C4#306311yescharacteristic
481C4#330211yescharacteristic
482C4#334011yescharacteristic
483C2—42no—
484C2—42no—
485C2—162no—
486C2#2533811yescharacteristic
487C2#3360011yescharacteristic
488C2#3361211yescharacteristic
489C2#3386311yescharacteristic
490C2#3386411yescharacteristic
491C2#3390411yescharacteristic
492C2#3390611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#28225C2
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×#2228C2semidirect
2. C2×D4×C16C2semidirect
3. #28708C2semidirect
5. #13494C2semidirect
6. #13493C2semidirect
7. #13381C2semidirect
8. #13380C2semidirect
9. #13281C2semidirect
10. #13178C2semidirect
11. #13059C2semidirect
12. #13059C2semidirect
13. #12835C2semidirect
14. #12834C2semidirect
15. C4×#357C2semidirect
125. C4⋊C16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22282
C2×D4×C162
#287082
#282252
#134942
#134932
#133812
#133802
#132812
#131782
#130592
#128352
#128342
C4×#3572