Finite Group Databasebeta

← #510635

Subgroups — #510635

Order 512.

503 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27660C211yesmaximal, characteristic
2#24779C211yesmaximal, characteristic
3#13516C211yesmaximal, characteristic, complemented
4#13493C211yesmaximal, characteristic, complemented
5#13475C211yesmaximal, characteristic, complemented
6#13383C211yesmaximal, characteristic, complemented
7#13382C211yesmaximal, characteristic, complemented
8#13382C211yesmaximal, characteristic, complemented
9#13190C211yesmaximal, characteristic
10#13167C211yesmaximal, characteristic, complemented
11#13150C211yesmaximal, characteristic, complemented
12#12994C211yesmaximal, characteristic, complemented
13#12921C211yesmaximal, characteristic, complemented
14#12921C211yesmaximal, characteristic, complemented
15#12868C211yesmaximal, characteristic, complemented
16C2×#467C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C2×#427C2211yescharacteristic
21#2485—21no—
22#2415—21no—
23C2×#367C2211yescharacteristic
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×#346C2211yescharacteristic
37#2088—21no—
38#2077—21no—
39#2066—21no—
40#2035—21no—
41#1969C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1873—21no—
46#1825—21no—
47#1808C2211yescharacteristic
48#1808C2211yescharacteristic
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C4×D4D411yescharacteristic
65C2×#117—21no—
66C2×#117—21no—
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71#467—22no—
72#467—41no—
73#467—41no—
74#467—41no—
75C8×D4—41no—
76C8×D4—41no—
77#427—22no—
78#427—41no—
79#427—41no—
80#427—41no—
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#106—21no—
89C2×#106—21no—
90C2×C4⋊C8D411yescharacteristic
91C2×#103—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×C22⋊C8D411yescharacteristic
95C2×C22⋊C8D411yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#397—41no—
98#397—41no—
99#394—41no—
100#394—41no—
101#391—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—41no—
106#388—41no—
107#388—41no—
108#387—41no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#367—22no—
114#367—41no—
115#366—41no—
116#366—41no—
117C4⋊C16—41no—
118#358—22no—
119#358—41no—
120#358—41no—
121#358—41no—
122C22⋊C16—22no—
123C22⋊C16—22no—
124C22⋊C16—41no—
125C22⋊C16—41no—
126#346—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—21no—
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—42no—
163C22⋊Q8—81no—
164C4×D4—41no—
165C4×D4—41no—
166#117—42no—
167#117—42no—
168#117—42no—
169#117—81no—
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⋊C4D811yescharacteristic
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
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⋊C4D811yescharacteristic
192C2×C22⋊C4D811yescharacteristic
193C2×C42—41no—
194C2×C42—41no—
195C2×C42C2×D411yescharacteristic
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
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#4112yes—
208#107—21no—
209#107—21no—
210#107—21no—
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#4112yes—
222#106—42no—
223#106—42no—
224C4⋊C8—41no—
225#103—42no—
226#103—42no—
227#103—81no—
228C22⋊C8—41no—
229C22⋊C8—41no—
230C4×C8—21no—
231C4×C8—21no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241C24C2×D811yescharacteristic
242C24C2×D811yescharacteristic
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—21no—
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—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4C22×D411yescharacteristic
277Q16—41no—
278Q16—41no—
279Q16—41no—
280Q16—41no—
281C2×C8—21no—
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—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8C2×#4111yescharacteristic
299C2×C8C2×#4111yescharacteristic
300C2×C8H(5,𝔽2)11yescharacteristic
301C2×C8H(5,𝔽2)11yescharacteristic
302C2×C8H(5,𝔽2)11yescharacteristic
303C2×C82−1+411yescharacteristic
304C4⋊C4—22no—
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—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—161no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—161no—
333C22⋊C4—161no—
334C42—21no—
335C42—21no—
336C42—82no—
337C42—82no—
338C16—22no—
339C16—41no—
340C16—41no—
341C16—41no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23C22×D811yescharacteristic
357Q8—81no—
358Q8—81no—
359Q8—81no—
360Q8—81no—
361D4—44no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
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—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—81no—
384C2×C4—81no—
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#53411yescharacteristic
407C2×C4#54611yescharacteristic
408C2×C4#54711yescharacteristic
409C2×C4#57511yescharacteristic
410C2×C4#57711yescharacteristic
411C2×C4#57811yescharacteristic
412C8—41no—
413C8—41no—
414C8—41no—
415C8#54011yescharacteristic
416C8#54211yescharacteristic
417C8#55311yescharacteristic
418C8#56211yescharacteristic
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22#318611yescharacteristic
434C22#329411yescharacteristic
435C22#333411yescharacteristic
436C22#341411yescharacteristic
437C22#341411yescharacteristic
438C22#341611yescharacteristic
439C22#341711yescharacteristic
440C4—41no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—82no—
445C4—161no—
446C4—161no—
447C4—161no—
448C4—161no—
449C4—161no—
450C4—161no—
451C4—161no—
452C4#270111yescharacteristic
453C4#320111yescharacteristic
454C4#332711yescharacteristic
455C4#335111yescharacteristic
456C2—81no—
457C2—81no—
458C2#3177011yescharacteristic
459C2#3376311yescharacteristic
460C2#3376711yescharacteristic
461C2#3388211yescharacteristic
462C2#3388811yescharacteristic
463C2#3391611yescharacteristic
464C2#3391711yescharacteristic
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. #13516C2semidirect
4. #13493C2semidirect
5. #13475C2semidirect
6. #13383C2semidirect
7. #13382C2semidirect
8. #13382C2semidirect
10. #13167C2semidirect
11. #13150C2semidirect
12. #12994C2semidirect
13. #12921C2semidirect
14. #12921C2semidirect
15. #12868C2semidirect

Maximal subgroups

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

subgroupindex
#276602
#247792
#135162
#134932
#134752
#133832
#133822
#131902
#131672
#131502
#129942
#129212
#128682