Finite Group Databasebeta

← #512268

Subgroups — #512268

Order 512.

535 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33139C211yesmaximal, characteristic
2#20455C211yesmaximal, characteristic
3#13133C211yesmaximal, characteristic, complemented
4#13132C211yesmaximal, characteristic, complemented
5#13132C211yesmaximal, characteristic, complemented
6#13132C211yesmaximal, characteristic, complemented
7#13112C211yesmaximal, characteristic, complemented
8#13111C211yesmaximal, characteristic, complemented
9#13110C211yesmaximal, characteristic, complemented
10#13110C211yesmaximal, characteristic, complemented
11#12994C211yesmaximal, characteristic
12#12993C211yesmaximal, characteristic
13#12921C211yesmaximal, characteristic
14#12920C211yesmaximal, characteristic
15#12845C211yesmaximal, characteristic
16C2×#467C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C2×#405C2211yescharacteristic
20#2625—21no—
21#2274—21no—
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31C2×C22⋊C16C2211yescharacteristic
32#1969C2211yescharacteristic
33#1969C2211yescharacteristic
34#1942C2211yescharacteristic
35#1942C2211yescharacteristic
36#1935—21no—
37#1935—21no—
38#1894—21no—
39#1808C2211yescharacteristic
40#1776—21no—
41#1776—21no—
42C4×#107C2211yescharacteristic
43C4×#107C2211yescharacteristic
44C4×#107C2211yescharacteristic
45C4×#107C2211yescharacteristic
46#1736—21no—
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×M16D411yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C22⋊Q8—21no—
63C2×#117—21no—
64C2×#117D411yescharacteristic
65C22×C4⋊C4—21no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#467—22no—
71#467—22no—
72#427—22no—
73#427—22no—
74C2×#107—21no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#106—21no—
85C2×#103—21no—
86C2×#103—21no—
87C2×#103—21no—
88#405—22no—
89#405—22no—
90C2×#97D411yescharacteristic
91C2×#97D411yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#400—41no—
95#398—41no—
96#397—41no—
97#397—41no—
98#391—21no—
99#391—21no—
100#389—21no—
101#389—21no—
102#380—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—41no—
108C4×C4⋊C4—41no—
109C4×C22⋊C4—41no—
110C4×C22⋊C4—41no—
111#367—22no—
112#367—22no—
113#367—22no—
114#366—22no—
115#358—22no—
116#358—22no—
117#358—22no—
118#358—22no—
119C22⋊C16—22no—
120C22⋊C16—22no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128C22×Q8—21no—
129C23×C4—41no—
130C23×C4C2×D411yescharacteristic
131C2×Q16—21no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×M16—22no—
138C2×M16—41no—
139C2×M16—41no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8#4112yes—
148C22×C8#4112yes—
149C22×C8C2411yescharacteristic
150C22⋊Q8—42no—
151C22⋊Q8—42no—
152#117—22no—
153#117—22no—
154#117—24no—
155#117—42no—
156C2×C4⋊C4—24no—
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—21no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4D811yescharacteristic
170C2×C4⋊C4D811yescharacteristic
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4D811yescharacteristic
175C2×C22⋊C4D811yescharacteristic
176C2×C42—41no—
177C2×C42—41no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42—41no—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
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—21no—
194C2×C16—21no—
195#107—24no—
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#4112yes—
207#107#4112yes—
208#107#4112yes—
209#107#4112yes—
210#106—24no—
211#103—42no—
212#103—42no—
213#103—42no—
214#97—22no—
215#97—22no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
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—
229#95—41no—
230#95—41no—
231C24C2×D811yescharacteristic
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—44no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—161no—
252C22×C4—161no—
253C22×C4#12312yes—
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4#13712yes—
260C22×C4C22×D411yescharacteristic
261Q16—41no—
262Q16—41no—
263Q16—41no—
264Q16—41no—
265M16—44no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8#12312yes—
279C2×C8C2×#4111yescharacteristic
280C2×C8C2×#4111yescharacteristic
281C2×C8C2×#4111yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—44no—
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—161no—
304C4⋊C4—161no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—44no—
308C22⋊C4—82no—
309C22⋊C4—161no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—44no—
315C42—161no—
316C42—161no—
317C42—161no—
318C42—161no—
319C16—41no—
320C16—41no—
321C16—41no—
322C16—41no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23#48012yes—
330C23C22×D811yescharacteristic
331Q8—81no—
332Q8—81no—
333Q8—81no—
334Q8—81no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
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—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—81no—
358C2×C4—81no—
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—164no—
378C2×C4#48012yes—
379C2×C4C2×#12311yescharacteristic
380C2×C4#52511yescharacteristic
381C2×C4#52511yescharacteristic
382C2×C4C2×#13711yescharacteristic
383C2×C4#57511yescharacteristic
384C2×C4#57511yescharacteristic
385C8—22no—
386C8—41no—
387C8—41no—
388C8#52911yescharacteristic
389C8#52911yescharacteristic
390C8#52911yescharacteristic
391C8#52911yescharacteristic
392C22—22no—
393C22—22no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22C2×#48011yescharacteristic
400C22#310011yescharacteristic
401C22#310011yescharacteristic
402C22#341611yescharacteristic
403C22#341611yescharacteristic
404C22#341611yescharacteristic
405C22#341711yescharacteristic
406C4—22no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—82no—
412C4—82no—
413C4—82no—
414C4—82no—
415C4—161no—
416C4—161no—
417C4—161no—
418C4—161no—
419C4#285311yescharacteristic
420C4#309311yescharacteristic
421C4#311711yescharacteristic
422C4#311711yescharacteristic
423C2—42no—
424C2#3309411yescharacteristic
425C2#3365111yescharacteristic
426C2#3365311yescharacteristic
427C2#3368011yescharacteristic
428C2#3368111yescharacteristic
429C2#3368311yescharacteristic
430C2#3368511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13133C2semidirect
4. #13132C2semidirect
5. #13132C2semidirect
6. #13132C2semidirect
7. #13112C2semidirect
8. #13111C2semidirect
9. #13110C2semidirect
10. #13110C2semidirect

Maximal subgroups

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

subgroupindex
#331392
#204552
#131332
#131322
#131122
#131112
#131102
#129942
#129932
#129212
#129202
#128452