Finite Group Databasebeta

← #510869

Subgroups — #510869

Order 512.

515 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28228C211yesmaximal, characteristic
2#27660C211yesmaximal, characteristic
3#13538C211yesmaximal, characteristic, complemented
4#13506C211yesmaximal, characteristic, complemented
5#13494C211yesmaximal, characteristic, complemented
6#13494C211yesmaximal, characteristic, complemented
7#13382C211yesmaximal, characteristic, complemented
8#13382C211yesmaximal, characteristic, complemented
9#13062C211yesmaximal, characteristic, complemented
10#12994C211yesmaximal, characteristic, complemented
11#12977C211yesmaximal, characteristic
12#12938C211yesmaximal, characteristic, complemented
13#12936C211yesmaximal, characteristic, complemented
14#12921C211yesmaximal, characteristic, complemented
15#12836C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2538—21no—
23#2485—21no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35#2088—21no—
36#2077—21no—
37#2036—21no—
38#2036—21no—
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1845C2211yescharacteristic
42#1825—21no—
43#1808C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic
48#1803—21no—
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×D4D411yescharacteristic
66C2×#117—21no—
67C2×M32—22no—
68C2×M32—41no—
69C2×M32—41no—
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74#467—22no—
75#467—22no—
76#467—41no—
77#467—41no—
78C4×Q16—41no—
79C4×Q16—41no—
80C8×D4—41no—
81C8×D4—41no—
82#427—41no—
83#427—41no—
84C2×#107—21no—
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#106—21no—
93C2×C4⋊C8D411yescharacteristic
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×C22⋊C8D411yescharacteristic
98C2×C22⋊C8D411yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#397—41no—
101#396—41no—
102#396—41no—
103#394—41no—
104#390—41no—
105#389—21no—
106#389—21no—
107#389—41no—
108#388—41no—
109#387—41no—
110#384—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#367—41no—
117#367—41no—
118#367—41no—
119#366—41no—
120#358—22no—
121#358—22no—
122#358—41no—
123#358—41no—
124C22⋊C16—22no—
125C22⋊C16—41no—
126C22⋊C16—41no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134C22×Q8—21no—
135C22×D4D811yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16#4112yes—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2411yescharacteristic
160C22⋊Q8—42no—
161C22⋊Q8—42no—
162C22⋊Q8—81no—
163C22⋊Q8—81no—
164C4×Q8—42no—
165C4×Q8—42no—
166C4×D4—41no—
167C4×D4—41no—
168#117—42no—
169#117—42no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4D811yescharacteristic
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4D811yescharacteristic
194C2×C22⋊C4D811yescharacteristic
195C2×C42—41no—
196C2×C42—41no—
197C2×C42C2×D411yescharacteristic
198M32—44no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16#4112yes—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—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—81no—
226#106—42no—
227C4⋊C8—41no—
228#103—42no—
229#103—42no—
230#103—42no—
231C22⋊C8—41no—
232C22⋊C8—41no—
233C4×C8—21no—
234C4×C8—21no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242C24C2×D811yescharacteristic
243C24C2×D811yescharacteristic
244C2×Q8—22no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×D4—22no—
251C2×D4—41no—
252C2×D4—41no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C22×D411yescharacteristic
278Q16—22no—
279Q16—22no—
280Q16—41no—
281Q16—41no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8C2×#4111yescharacteristic
300C2×C8C2×#4111yescharacteristic
301C2×C8H(5,𝔽2)11yescharacteristic
302C2×C8H(5,𝔽2)11yescharacteristic
303C2×C8H(5,𝔽2)11yescharacteristic
304C2×C82−1+411yescharacteristic
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—161no—
326C4⋊C4—161no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—161no—
334C22⋊C4—161no—
335C22⋊C4—161no—
336C22⋊C4—161no—
337C42—21no—
338C42—21no—
339C42—82no—
340C42—82no—
341C16—22no—
342C16—41no—
343C16—41no—
344C16—41no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23C22×D811yescharacteristic
360Q8—42no—
361Q8—42no—
362Q8—81no—
363Q8—81no—
364D4—44no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
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#51012yes—
409C2×C4#53411yescharacteristic
410C2×C4#54611yescharacteristic
411C2×C4#54711yescharacteristic
412C2×C4#57511yescharacteristic
413C2×C4#57711yescharacteristic
414C2×C4#57811yescharacteristic
415C8—41no—
416C8—41no—
417C8—41no—
418C8#53811yescharacteristic
419C8#53911yescharacteristic
420C8#55311yescharacteristic
421C8#56211yescharacteristic
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22#318611yescharacteristic
437C22#329411yescharacteristic
438C22#333411yescharacteristic
439C22C2×#51011yescharacteristic
440C22#341411yescharacteristic
441C22#341611yescharacteristic
442C22#341611yescharacteristic
443C4—22no—
444C4—41no—
445C4—41no—
446C4—41no—
447C4—82no—
448C4—161no—
449C4—161no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4—161no—
454C4—161no—
455C4#270811yescharacteristic
456C4#320111yescharacteristic
457C4#331511yescharacteristic
458C4#334811yescharacteristic
459C2—81no—
460C2—81no—
461C2#3199411yescharacteristic
462C2#3376211yescharacteristic
463C2#3377011yescharacteristic
464C2#3388411yescharacteristic
465C2#3389611yescharacteristic
466C2#3391711yescharacteristic
467C2#3392311yescharacteristic
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. #13538C2semidirect
4. #13506C2semidirect
5. #13494C2semidirect
6. #13494C2semidirect
7. #13382C2semidirect
8. #13382C2semidirect
9. #13062C2semidirect
10. #12994C2semidirect
12. #12938C2semidirect
13. #12936C2semidirect
14. #12921C2semidirect
15. #12836C2semidirect

Maximal subgroups

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

subgroupindex
#282282
#276602
#135382
#135062
#134942
#133822
#130622
#129942
#129772
#129382
#129362
#129212
#128362