Finite Group Databasebeta

← #505497

Subgroups — #505497

Order 512.

615 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27840C211yesmaximal, characteristic, complemented
2#22451C211yesmaximal, characteristic
3#13494C211yesmaximal, characteristic, complemented
4#13492C211yesmaximal, characteristic, complemented
5#13480C211yesmaximal, characteristic, complemented
6#13474C211yesmaximal, characteristic, complemented
7#13147C211yesmaximal, characteristic, complemented
8#13131C211yesmaximal, characteristic, complemented
9#13120C211yesmaximal, characteristic, complemented
10#12938C211yesmaximal, characteristic, complemented
11#12936C211yesmaximal, characteristic, complemented
12#12812C211yesmaximal, characteristic, complemented
13#12733C211yesmaximal, characteristic
14#12733C211yesmaximal, characteristic
15C4×#358C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19#2497—21no—
20#2341—21no—
21C2×#367C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×#358C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic
35#2100—21no—
36#2076—21no—
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1921—21no—
42#1880—21no—
43#1880—21no—
44#1845C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic
48#1788—21no—
49C4×#107C2211yescharacteristic
50C8×C22⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×#41—21no—
60C2×#135—22no—
61C2×#135—22no—
62C2×#135—22no—
63C2×#135—22no—
64C2×#135—41no—
65C2×#135—41no—
66C22×Q16C2311yescharacteristic
67C22×SD16—21no—
68C22×D8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C4×Q16—22no—
77C4×Q16—22no—
78C4×Q16—41no—
79C4×Q16—41no—
80C2×#107—21no—
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#103—21no—
88C2×#103—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×C22⋊C8D411yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#400—41no—
95#396—41no—
96#390—21no—
97#390—21no—
98#390—41no—
99#390—41no—
100#389—21no—
101#389—21no—
102#388—41no—
103#387—41no—
104#384—21no—
105#380—21no—
106#380—21no—
107#380—21no—
108#380—41no—
109C4×C4⋊C4—21no—
110C4×C22⋊C4—41no—
111C4×C22⋊C4D411yescharacteristic
112#367—41no—
113#366—41no—
114C4⋊C16—41no—
115#358—22no—
116#358—22no—
117#358—22no—
118#358—22no—
119#357—22no—
120#357—22no—
121#357—41no—
122#357—41no—
123C22⋊C16—41no—
124C22⋊C16—41no—
125C4×C16—41no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C2×#41—24no—
135C2×#41—42no—
136C2×#41—42no—
137C2×#41—42no—
138C2×#41—81no—
139C2×#41—81no—
140C22×Q8—21no—
141C22×D4—21no—
142C22×D4—41no—
143C23×C4—41no—
144C23×C4C2×D411yescharacteristic
145#135—44no—
146#135—44no—
147#135—44no—
148#135—44no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16#4112yes—
153C2×Q16#4112yes—
154C2×Q16#4112yes—
155C2×SD16—24no—
156C2×SD16—42no—
157C2×SD16—42no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8#4112yes—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2411yescharacteristic
174C4×Q8—42no—
175C4×Q8—42no—
176C4×Q8—81no—
177C4×Q8—81no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊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⋊C4D811yescharacteristic
198C2×C22⋊C4D811yescharacteristic
199C2×C42—41no—
200C2×C42—41no—
201C2×C42—41no—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
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—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—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—21no—
227#107—21no—
228#107—42no—
229#103—42no—
230#103—42no—
231#102—81no—
232#102—81no—
233C22⋊C8—41no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95D811yescharacteristic
245#95D811yescharacteristic
246C24—41no—
247C24C2×D811yescharacteristic
248#41—48no—
249#41—84no—
250#41—84no—
251#41—84no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×D4—22no—
259C2×D4—44no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—82no—
266C2×D4—82no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—44no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C22×D411yescharacteristic
289Q16—22no—
290Q16—22no—
291Q16—22no—
292Q16#12614yes—
293SD16—48no—
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—21no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8C2×#4111yescharacteristic
312C2×C8C2×#4111yescharacteristic
313C2×C8C2×#4111yescharacteristic
314C2×C8C2×#4111yescharacteristic
315C2×C8H(5,𝔽2)11yescharacteristic
316C2×C8H(5,𝔽2)11yescharacteristic
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—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—161no—
333C4⋊C4—161no—
334C4⋊C4—161no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—161no—
340C22⋊C4—161no—
341C22⋊C4—161no—
342C22⋊C4—161no—
343C22⋊C4—161no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—82no—
349C42—82no—
350C42—161no—
351C16—41no—
352C16—41no—
353C16—41no—
354C16—41no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—42no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23C22×D811yescharacteristic
370Q8—24no—
371Q8—42no—
372Q8—42no—
373Q8—42no—
374D4—42no—
375D4—42no—
376D4—88no—
377D4—81no—
378D4—81no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—88no—
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—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4#50812yes—
419C2×C4#51012yes—
420C2×C4#52511yescharacteristic
421C2×C4#54611yescharacteristic
422C2×C4#54611yescharacteristic
423C2×C4#57511yescharacteristic
424C2×C4#57711yescharacteristic
425C2×C4#57711yescharacteristic
426C8—41no—
427C8—41no—
428C8—41no—
429C8C2×#12611yescharacteristic
430C8#53811yescharacteristic
431C8#55111yescharacteristic
432C8#55111yescharacteristic
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22—161no—
447C22#310011yescharacteristic
448C22#329411yescharacteristic
449C22#329411yescharacteristic
450C22C2×#50811yescharacteristic
451C22C2×#51011yescharacteristic
452C22#341511yescharacteristic
453C22#341611yescharacteristic
454C4—22no—
455C4—22no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—44no—
460C4—161no—
461C4—161no—
462C4—161no—
463C4—161no—
464C4—161no—
465C4—161no—
466C4#248011yescharacteristic
467C4#312311yescharacteristic
468C4#331111yescharacteristic
469C4#332311yescharacteristic
470C2—81no—
471C2—162no—
472C2#2751811yescharacteristic
473C2#3366711yescharacteristic
474C2#3367911yescharacteristic
475C2#3386811yescharacteristic
476C2#3387211yescharacteristic
477C2#3388311yescharacteristic
478C2#3390011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×#135C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #27840C2semidirect
3. #13494C2semidirect
4. #13492C2semidirect
5. #13480C2semidirect
6. #13474C2semidirect
7. #13147C2semidirect
8. #13131C2semidirect
9. #13120C2semidirect
10. #12938C2semidirect
11. #12936C2semidirect
12. #12812C2semidirect
15. C4×#358C2semidirect

Maximal subgroups

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

subgroupindex
#278402
#224512
#134942
#134922
#134802
#134742
#131472
#131312
#131202
#129382
#129362
#128122
#127332
C4×#3582