Finite Group Databasebeta

← #501924

Subgroups — #501924

Order 512.

579 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28226C211yesmaximal, characteristic
2D4×#107C211yesmaximal, characteristic
3#13494C211yesmaximal, characteristic, complemented
4#13492C211yesmaximal, characteristic, complemented
5#13383C211yesmaximal, characteristic, complemented
6#13382C211yesmaximal, characteristic, complemented
7#13190C211yesmaximal, characteristic
8#13178C211yesmaximal, characteristic, complemented
9#12993C211yesmaximal, characteristic, complemented
10#12993C211yesmaximal, characteristic, complemented
11#12890C211yesmaximal, characteristic, complemented
12#12867C211yesmaximal, characteristic, complemented
13#12866C211yesmaximal, characteristic, complemented
14#12812C211yesmaximal, characteristic, complemented
15#12638C211yesmaximal, characteristic, complemented
16C2×#467C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2536—21no—
23D4×C4⋊C4—21no—
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×C22⋊C16C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×#346C2211yescharacteristic
36#2035—21no—
37#2035—21no—
38#1947—21no—
39#1852—21no—
40#1852—21no—
41#1846C2211yescharacteristic
42#1845C2211yescharacteristic
43#1808C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1805—21no—
48C4×#107C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×D4—41no—
66C2×C4×D4D411yescharacteristic
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73#467—22no—
74#467—22no—
75#467—41no—
76#467—41no—
77C4×Q16—22no—
78C4×Q16—22no—
79C8×D4—22no—
80C8×D4—22no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—41no—
87C2×#107—41no—
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#106—21no—
96C2×C4⋊C8D411yescharacteristic
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×C22⋊C8D411yescharacteristic
101C2×C22⋊C8D411yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#398—41no—
104#394—41no—
105#389—41no—
106#388—41no—
107#388—41no—
108#385—41no—
109#385—41no—
110#384—21no—
111#382—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116C4×C4⋊C4—21no—
117#367—41no—
118#366—41no—
119C4⋊C16—22no—
120#358—22no—
121#358—22no—
122#358—41no—
123#358—41no—
124C22⋊C16—41no—
125C22⋊C16—41no—
126C22⋊C16—41no—
127C22⋊C16—41no—
128#346—22no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134C22×Q8—21no—
135C22×D4D811yescharacteristic
136C23×C4—41no—
137C23×C4—41no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16#4112yes—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2411yescharacteristic
162C22⋊Q8—42no—
163C22⋊Q8—42no—
164C22⋊Q8—81no—
165C22⋊Q8—81no—
166C4×Q8—42no—
167C4×Q8—42no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—84no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—81no—
188C2×C4⋊C4—81no—
189C2×C4⋊C4D811yescharacteristic
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⋊C4D811yescharacteristic
197C2×C22⋊C4D811yescharacteristic
198C2×C42—41no—
199C2×C42—41no—
200C2×C42C2×D411yescharacteristic
201C2×C16—21no—
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—21no—
212C2×C16—21no—
213#107—21no—
214#107—21no—
215#107—21no—
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—42no—
227#107—42no—
228#107—81no—
229#107#4112yes—
230#106—81no—
231C4⋊C8—22no—
232#103—42no—
233#103—42no—
234#103—42no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C4×C8#4112yes—
238C4×C8#4112yes—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246C24C2×D811yescharacteristic
247C24C2×D811yescharacteristic
248C2×Q8—22no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—161no—
282C22×C4—161no—
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C22×D411yescharacteristic
288Q16—22no—
289Q16—22no—
290Q16—41no—
291Q16—41no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8#12312yes—
309C2×C8C2×#4111yescharacteristic
310C2×C8C2×#4111yescharacteristic
311C2×C8C2×#4111yescharacteristic
312C2×C8C2×#4111yescharacteristic
313C2×C8H(5,𝔽2)11yescharacteristic
314C2×C8H(5,𝔽2)11yescharacteristic
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
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—82no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—161no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C42—82no—
346C42—82no—
347C42#12312yes—
348C42#13712yes—
349C16—41no—
350C16—41no—
351C16—41no—
352C16—41no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23#50912yes—
366C23#51012yes—
367C23C22×D811yescharacteristic
368Q8—42no—
369Q8—42no—
370Q8—81no—
371Q8—81no—
372D4—28no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
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—164no—
416C2×C4—164no—
417C2×C4#48012yes—
418C2×C4#48012yes—
419C2×C4C2×#12311yescharacteristic
420C2×C4#54611yescharacteristic
421C2×C4#54611yescharacteristic
422C2×C4C2×#13711yescharacteristic
423C2×C4#57711yescharacteristic
424C2×C4#57711yescharacteristic
425C8—22no—
426C8—41no—
427C8—41no—
428C8#52911yescharacteristic
429C8#54011yescharacteristic
430C8#55311yescharacteristic
431C8#55511yescharacteristic
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22C2×#48011yescharacteristic
447C22#329411yescharacteristic
448C22#329411yescharacteristic
449C22C2×#50911yescharacteristic
450C22C2×#51011yescharacteristic
451C22#341611yescharacteristic
452C22#341711yescharacteristic
453C4—22no—
454C4—22no—
455C4—41no—
456C4—41no—
457C4—82no—
458C4—82no—
459C4—161no—
460C4—161no—
461C4—161no—
462C4—161no—
463C4—161no—
464C4—161no—
465C4#242011yescharacteristic
466C4#309911yescharacteristic
467C4#332911yescharacteristic
468C4#333111yescharacteristic
469C2—42no—
470C2—42no—
471C2#2496511yescharacteristic
472C2#3364611yescharacteristic
473C2#3365111yescharacteristic
474C2#3387511yescharacteristic
475C2#3387711yescharacteristic
476C2#3387911yescharacteristic
477C2#3388311yescharacteristic
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
3. #13494C2semidirect
4. #13492C2semidirect
5. #13383C2semidirect
6. #13382C2semidirect
8. #13178C2semidirect
9. #12993C2semidirect
10. #12993C2semidirect
11. #12890C2semidirect
12. #12867C2semidirect
13. #12866C2semidirect
14. #12812C2semidirect
15. #12638C2semidirect

Maximal subgroups

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

subgroupindex
#282262
D4×#1072
#134942
#134922
#133832
#133822
#131902
#131782
#129932
#128902
#128672
#128662
#128122
#126382