Finite Group Databasebeta

← #501875

Subgroups — #501875

Order 512.

524 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27660C211yesmaximal, characteristic
2#22656C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13507C211yesmaximal, characteristic, complemented
5#13494C211yesmaximal, characteristic, complemented
6#13492C211yesmaximal, characteristic, complemented
7#13112C211yesmaximal, characteristic, complemented
8#13109C211yesmaximal, characteristic, complemented
9#12976C211yesmaximal, characteristic
10#12921C211yesmaximal, characteristic, complemented
11#12919C211yesmaximal, characteristic, complemented
12#12868C211yesmaximal, characteristic, complemented
13#12867C211yesmaximal, characteristic, complemented
14#12834C211yesmaximal, characteristic, complemented
15#12556C211yesmaximal, characteristic, complemented
16C23×C16C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C22×#107C2211yescharacteristic
21#2485—21no—
22#2352—21no—
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C22⋊C16C2211yescharacteristic
28C2×C22⋊C16C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30#2088—21no—
31#2077—21no—
32#1949—21no—
33#1942C2211yescharacteristic
34#1942C2211yescharacteristic
35#1854—21no—
36#1853—21no—
37#1846C2211yescharacteristic
38#1825—21no—
39#1808C2211yescharacteristic
40#1808C2211yescharacteristic
41#1808C2211yescharacteristic
42#1808C2211yescharacteristic
43#1808C2211yescharacteristic
44C4×#107C2211yescharacteristic
45#1380C2211yescharacteristic
46#1380C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×#117—21no—
63C2×#117—21no—
64C22×C4⋊C4—21no—
65C22×C16—22no—
66C22×C16—22no—
67C22×C16—41no—
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C8×D4—22no—
73C8×D4—22no—
74#427—22no—
75#427—22no—
76#427—22no—
77#427—41no—
78C2×#107—22no—
79C2×#107—21no—
80C2×#107—41no—
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×C4⋊C8D411yescharacteristic
93C2×C22⋊C8D411yescharacteristic
94C2×C22⋊C8D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#397—41no—
97#394—41no—
98#389—21no—
99#389—21no—
100#388—41no—
101#388—41no—
102#387—41no—
103#387—41no—
104#386—41no—
105#382—21no—
106#380—21no—
107#380—21no—
108#380—21no—
109#380—21no—
110#380—21no—
111C4×C4⋊C4—21no—
112C4×C22⋊C4—41no—
113#367—41no—
114#366—22no—
115#366—22no—
116#366—41no—
117C22⋊C16—22no—
118C22⋊C16—22no—
119C22⋊C16—41no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128C22×D4D811yescharacteristic
129C23×C4—41no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8#4112yes—
146C22×C8#4112yes—
147C22×C8C2411yescharacteristic
148C4×D4—22no—
149C4×D4—22no—
150#117—42no—
151#117—42no—
152#117—42no—
153#117—81no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
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—42no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—81no—
167C2×C4⋊C4D811yescharacteristic
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×C16—21no—
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—44no—
193C2×C16#4112yes—
194#107—21no—
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—21no—
207#107—21no—
208#107—21no—
209#107—42no—
210#107#4112yes—
211#106—42no—
212#106—42no—
213#106—81no—
214C4⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C4×C8—21no—
218C4×C8—21no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229C24C2×D811yescharacteristic
230C24C2×D811yescharacteristic
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—41no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—161no—
255C22×C4#12312yes—
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4#13712yes—
261C22×C4C22×D411yescharacteristic
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8#12312yes—
279C2×C8C2×#4111yescharacteristic
280C2×C8C2×#4111yescharacteristic
281C2×C8C2×#4111yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C2×C8H(5,𝔽2)11yescharacteristic
284C2×C82−1+411yescharacteristic
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
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—41no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—161no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—161no—
312C22⋊C4—161no—
313C22⋊C4—161no—
314C42—21no—
315C42—21no—
316C42—82no—
317C42—82no—
318C42—161no—
319C16—22no—
320C16—41no—
321C16—41no—
322C16—41no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23#48012yes—
336C23#50812yes—
337C23C22×D811yescharacteristic
338D4—44no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
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—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
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—164no—
383C2×C4#48012yes—
384C2×C4C2×#12311yescharacteristic
385C2×C4#54611yescharacteristic
386C2×C4#54711yescharacteristic
387C2×C4C2×#13711yescharacteristic
388C2×C4#57711yescharacteristic
389C2×C4#57811yescharacteristic
390C8—22no—
391C8—41no—
392C8—41no—
393C8#52911yescharacteristic
394C8#53911yescharacteristic
395C8#55111yescharacteristic
396C8#55211yescharacteristic
397C22—22no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22C2×#48011yescharacteristic
412C22#329411yescharacteristic
413C22#333411yescharacteristic
414C22C2×#50811yescharacteristic
415C22#341411yescharacteristic
416C22#341511yescharacteristic
417C22#341711yescharacteristic
418C4—22no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—82no—
423C4—161no—
424C4—161no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4#241911yescharacteristic
431C4#309711yescharacteristic
432C4#331211yescharacteristic
433C4#334411yescharacteristic
434C2—42no—
435C2—42no—
436C2#2493311yescharacteristic
437C2#3364511yescharacteristic
438C2#3365211yescharacteristic
439C2#3386711yescharacteristic
440C2#3388611yescharacteristic
441C2#3390611yescharacteristic
442C2#3391611yescharacteristic
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. #13539C2semidirect
4. #13507C2semidirect
5. #13494C2semidirect
6. #13492C2semidirect
7. #13112C2semidirect
8. #13109C2semidirect
10. #12921C2semidirect
11. #12919C2semidirect
12. #12868C2semidirect
13. #12867C2semidirect
14. #12834C2semidirect
15. #12556C2semidirect

Maximal subgroups

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

subgroupindex
#276602
#226562
#135392
#135072
#134942
#134922
#131122
#131092
#129762
#129212
#129192
#128682
#128672
#128342
#125562