Finite Group Databasebeta

← #512266

Subgroups — #512266

Order 512.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33138C211yesmaximal, characteristic
2#20455C211yesmaximal, characteristic, complemented
3#13131C211yesmaximal, characteristic, complemented
4#13130C211yesmaximal, characteristic, complemented
5#13130C211yesmaximal, characteristic, complemented
6#13130C211yesmaximal, characteristic, complemented
7#13112C211yesmaximal, characteristic, complemented
8#13112C211yesmaximal, characteristic, complemented
9#13110C211yesmaximal, characteristic, complemented
10#13109C211yesmaximal, characteristic, complemented
11#12992C211yesmaximal, characteristic
12#12991C211yesmaximal, characteristic
13#12921C211yesmaximal, characteristic, complemented
14#12919C211yesmaximal, characteristic, complemented
15#12845C211yesmaximal, characteristic, complemented
16C2×#466C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C2×#405C2211yescharacteristic
20#2624—21no—
21#2274—21no—
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31C2×C22⋊C16C2211yescharacteristic
32#1970C2211yescharacteristic
33#1970C2211yescharacteristic
34#1942C2211yescharacteristic
35#1942C2211yescharacteristic
36#1936—21no—
37#1936—21no—
38#1895—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×D8C2311yescharacteristic
60C22×M16D411yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#121—21no—
63C2×#117—21no—
64C2×#117D411yescharacteristic
65C22×C4⋊C4—21no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#466—22no—
71#466—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×#102—21no—
86C2×#102—21no—
87C2×#102—21no—
88#405—22no—
89#405—22no—
90C2×#97D411yescharacteristic
91C2×#97D411yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#399—41no—
95#397—41no—
96#394—41no—
97#394—41no—
98#390—21no—
99#390—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#366—22no—
113#366—22no—
114#366—22no—
115#357—22no—
116#357—22no—
117#357—22no—
118#357—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×D4—21no—
129C22×D4—41no—
130C23×C4—41no—
131C23×C4C2×D411yescharacteristic
132C2×D8—21no—
133C2×D8—21no—
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—21no—
138C2×M16—22no—
139C2×M16—41no—
140C2×M16—41no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8#4112yes—
149C22×C8#4112yes—
150C22×C8C2411yescharacteristic
151#121—42no—
152#121—42no—
153#117—22no—
154#117—22no—
155#117—24no—
156#117—42no—
157C2×C4⋊C4—24no—
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—21no—
165C2×C4⋊C4—42no—
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⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4D811yescharacteristic
179C2×C22⋊C4D811yescharacteristic
180C2×C42—41no—
181C2×C42—41no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199#107—24no—
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—21no—
210#107#4112yes—
211#107#4112yes—
212#107#4112yes—
213#107#4112yes—
214#106—24no—
215#102—42no—
216#102—42no—
217#102—42no—
218#97—22no—
219#97—22no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231C24—41no—
232C24C2×D811yescharacteristic
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—82no—
240C2×D4—161no—
241C2×D4—161no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—44no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—161no—
255C22×C4—161no—
256C22×C4#12312yes—
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4#13712yes—
263C22×C4C22×D411yescharacteristic
264D8—41no—
265D8—41no—
266D8—41no—
267D8—41no—
268M16—44no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8#12312yes—
282C2×C8C2×#4111yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C8C2×#4111yescharacteristic
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—44no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—161no—
306C4⋊C4—161no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—44no—
310C22⋊C4—82no—
311C22⋊C4—161no—
312C22⋊C4—161no—
313C22⋊C4—161no—
314C22⋊C4—161no—
315C22⋊C4—161no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—44no—
321C42—161no—
322C42—161no—
323C42—161no—
324C42—161no—
325C16—41no—
326C16—41no—
327C16—41no—
328C16—41no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23#48012yes—
343C23C22×D811yescharacteristic
344D4—81no—
345D4—81no—
346D4—81no—
347D4—81no—
348D4—164no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
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×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—164no—
389C2×C4#48012yes—
390C2×C4C2×#12311yescharacteristic
391C2×C4#52511yescharacteristic
392C2×C4#52511yescharacteristic
393C2×C4C2×#13711yescharacteristic
394C2×C4#57511yescharacteristic
395C2×C4#57511yescharacteristic
396C8—22no—
397C8—41no—
398C8—41no—
399C8#52911yescharacteristic
400C8#52911yescharacteristic
401C8#52911yescharacteristic
402C8#52911yescharacteristic
403C22—22no—
404C22—22no—
405C22—22no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22—161no—
414C22—161no—
415C22—161no—
416C22—161no—
417C22C2×#48011yescharacteristic
418C22#310011yescharacteristic
419C22#310011yescharacteristic
420C22#341511yescharacteristic
421C22#341511yescharacteristic
422C22#341611yescharacteristic
423C22#341711yescharacteristic
424C4—22no—
425C4—41no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—82no—
430C4—82no—
431C4—82no—
432C4—82no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4#285311yescharacteristic
437C4#309311yescharacteristic
438C4#311711yescharacteristic
439C4#311711yescharacteristic
440C2—42no—
441C2—321no—
442C2#3309411yescharacteristic
443C2#3365011yescharacteristic
444C2#3365211yescharacteristic
445C2#3367911yescharacteristic
446C2#3368111yescharacteristic
447C2#3368211yescharacteristic
448C2#3368411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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
2. #20455C2semidirect
3. #13131C2semidirect
4. #13130C2semidirect
5. #13130C2semidirect
6. #13130C2semidirect
7. #13112C2semidirect
8. #13112C2semidirect
9. #13110C2semidirect
10. #13109C2semidirect
13. #12921C2semidirect
14. #12919C2semidirect
15. #12845C2semidirect

Maximal subgroups

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

subgroupindex
#331382
#204552
#131312
#131302
#131122
#131102
#131092
#129922
#129912
#129212
#129192
#128452