Finite Group Databasebeta

← #505270

Subgroups — #505270

Order 512.

539 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27791C211yesmaximal, characteristic
2#22611C211yesmaximal, characteristic, complemented
3#13492C211yesmaximal, characteristic, complemented
4#13474C211yesmaximal, characteristic, complemented
5#13451C211yesmaximal, characteristic, complemented
6#13380C211yesmaximal, characteristic
7#13275C211yesmaximal, characteristic, complemented
8#13147C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic, complemented
10#13130C211yesmaximal, characteristic, complemented
11#13120C211yesmaximal, characteristic, complemented
12#13109C211yesmaximal, characteristic, complemented
13#12991C211yesmaximal, characteristic
14#12919C211yesmaximal, characteristic, complemented
15C4×#366C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C22×#107C2211yescharacteristic
19#2491—21no—
20#2344—21no—
21C2×#366C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31C2×C22⋊C16C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33#2033—21no—
34#2023—21no—
35#1970C2211yescharacteristic
36#1970C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1925—21no—
40#1918—21no—
41#1918—21no—
42#1823—21no—
43#1808C2211yescharacteristic
44#1808C2211yescharacteristic
45C4×#107C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C22⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×#124—41no—
63C2×#121—21no—
64C2×#117—41no—
65C22×C4⋊C4—21no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#500—22no—
71#500—41no—
72#466—22no—
73#466—41no—
74C2×#107—21no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#107—21no—
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#102—21no—
86C2×#102—21no—
87C2×#102—21no—
88C2×C22⋊C8D411yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#394—41no—
92#393—41no—
93#390—21no—
94#390—21no—
95#389—21no—
96#389—21no—
97#388—41no—
98#387—41no—
99#387—41no—
100#381—41no—
101#380—21no—
102#380—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C22⋊C4D411yescharacteristic
107#366—22no—
108#366—22no—
109#366—22no—
110#366—41no—
111C4⋊C16—41no—
112#357—22no—
113#357—22no—
114#357—22no—
115#357—41no—
116C22⋊C16—22no—
117C22⋊C16—41no—
118C4×C16—22no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×Q8—41no—
129C22×D4—21no—
130C22×D4—41no—
131C23×C4—41no—
132C23×C4C2×D411yescharacteristic
133C2×D8—21no—
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—21no—
138C2×D8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2411yescharacteristic
149C4⋊Q8—42no—
150C4⋊Q8—81no—
151#124—84no—
152#121—42no—
153#121—81no—
154#117—84no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4D811yescharacteristic
176C2×C22⋊C4D811yescharacteristic
177C2×C42—41no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16#4112yes—
194#107—24no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—42no—
207#107#4112yes—
208#107#4112yes—
209#107#4112yes—
210#102—42no—
211#102—42no—
212#102—81no—
213C22⋊C8—41no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95D811yescharacteristic
226#95D811yescharacteristic
227C24—41no—
228C24C2×D811yescharacteristic
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8—161no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—82no—
239C2×D4—82no—
240C2×D4—161no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—44no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C22×D411yescharacteristic
263D8—41no—
264D8—41no—
265D8—41no—
266D8—41no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8C2×#4111yescharacteristic
281C2×C8C2×#4111yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8H(5,𝔽2)11yescharacteristic
285C2×C82−1+411yescharacteristic
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—44no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C22⋊C4—41no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—161no—
312C22⋊C4D1614yes—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—82no—
318C42—82no—
319C42—82no—
320C16—22no—
321C16—41no—
322C16—41no—
323C16—41no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23C2×D1612yes—
338C23C22×D811yescharacteristic
339Q8—164no—
340D4—81no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—164no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—88no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4C2×D1612yes—
385C2×C4C2×D1612yes—
386C2×C4#52511yescharacteristic
387C2×C4#54611yescharacteristic
388C2×C4#54711yescharacteristic
389C2×C4#57511yescharacteristic
390C2×C4#57711yescharacteristic
391C2×C4#57811yescharacteristic
392C8—41no—
393C8—41no—
394C8—41no—
395C8#52911yescharacteristic
396C8#54211yescharacteristic
397C8#55511yescharacteristic
398C8#55611yescharacteristic
399C22—22no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—161no—
407C22—161no—
408C22—161no—
409C22—161no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22#310011yescharacteristic
414C22#329411yescharacteristic
415C22#333411yescharacteristic
416C22C22×D1611yescharacteristic
417C22#341411yescharacteristic
418C22#341611yescharacteristic
419C22#341711yescharacteristic
420C4—22no—
421C4—22no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—82no—
426C4—82no—
427C4—82no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4—161no—
432C4#247411yescharacteristic
433C4#312111yescharacteristic
434C4#332711yescharacteristic
435C4#334811yescharacteristic
436C2—42no—
437C2—321no—
438C2#2732611yescharacteristic
439C2#3366011yescharacteristic
440C2#3368111yescharacteristic
441C2#3386111yescharacteristic
442C2#3389411yescharacteristic
443C2#3390111yescharacteristic
444C2#3391511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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
2. #22611C2semidirect
3. #13492C2semidirect
4. #13474C2semidirect
5. #13451C2semidirect
7. #13275C2semidirect
8. #13147C2semidirect
9. #13130C2semidirect
10. #13130C2semidirect
11. #13120C2semidirect
12. #13109C2semidirect
14. #12919C2semidirect
15. C4×#366C2semidirect

Maximal subgroups

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

subgroupindex
#277912
#226112
#134922
#134742
#134512
#133802
#132752
#131472
#131302
#131202
#131092
#129912
#129192
C4×#3662