Finite Group Databasebeta

← #509534

Subgroups — #509534

Order 512.

542 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2255C211yesmaximal, characteristic, complemented
2C2×#2233C211yesmaximal, characteristic
3#27790C211yesmaximal, characteristic
4#26065C211yesmaximal, characteristic, complemented
5#13526C211yesmaximal, characteristic, complemented
6#13493C211yesmaximal, characteristic, complemented
7#13474C211yesmaximal, characteristic, complemented
8#13453C211yesmaximal, characteristic, complemented
9#13381C211yesmaximal, characteristic
10#13363C211yesmaximal, characteristic, complemented
11#13177C211yesmaximal, characteristic, complemented
12#13149C211yesmaximal, characteristic, complemented
13#13121C211yesmaximal, characteristic, complemented
14#13110C211yesmaximal, characteristic, complemented
15#12919C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C22×#107C2211yescharacteristic
20#2490—21no—
21#2440—21no—
22#2255—24no—
23#2233—24no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×C22⋊C16C2211yescharacteristic
37C2×#346C2211yescharacteristic
38#2083—21no—
39#2057—21no—
40#2034—21no—
41#2025—21no—
42#1970C2211yescharacteristic
43#1970C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1926—21no—
47#1852—21no—
48#1845C2211yescharacteristic
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C22⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C4⋊Q8—21no—
64C2×#125—21no—
65C2×C22⋊Q8—41no—
66C2×#121—21no—
67C22×C4⋊C4—21no—
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72#500—22no—
73#500—41no—
74#499—22no—
75#499—41no—
76#466—22no—
77#466—41no—
78C2×#107—22no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—41no—
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#106—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×C22⋊C8D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#398—41no—
98#397—41no—
99#395—41no—
100#394—41no—
101#390—21no—
102#390—21no—
103#390—41no—
104#389—21no—
105#389—21no—
106#388—41no—
107#385—41no—
108#384—21no—
109#380—21no—
110#380—21no—
111C4×C4⋊C4—21no—
112C4×C22⋊C4D411yescharacteristic
113#367—22no—
114#367—22no—
115#367D414yes—
116#366—22no—
117#366—22no—
118#366—41no—
119C4⋊C16—22no—
120#357—22no—
121#357—22no—
122#357—22no—
123#357—41no—
124C22⋊C16—22no—
125C22⋊C16—22no—
126#346—22no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×Q8—41no—
135C22×D4—21no—
136C22×D4—41no—
137C23×C4—41no—
138C23×C4C2×D411yescharacteristic
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2411yescharacteristic
155C4⋊Q8—42no—
156C4⋊Q8—81no—
157#125—42no—
158#125—81no—
159C22⋊Q8—84no—
160#121—42no—
161#121—81no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—81no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4D811yescharacteristic
187C2×C22⋊C4D811yescharacteristic
188C2×C42—41no—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16C2×D412yes—
202C2×C16#4112yes—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—42no—
216#107—42no—
217#107—81no—
218#107C2×D412yes—
219#107C2×D412yes—
220#106—42no—
221#102—42no—
222#102—42no—
223#102—81no—
224C22⋊C8—22no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95D811yescharacteristic
235#95D811yescharacteristic
236C24—41no—
237C24C2×D811yescharacteristic
238C2×Q8—82no—
239C2×Q8—82no—
240C2×Q8—161no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—82no—
248C2×D4—82no—
249C2×D4—161no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—161no—
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C22×D411yescharacteristic
272D8—41no—
273D8—41no—
274D8—41no—
275D8—41no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8#13712yes—
289C2×C8C22×D411yescharacteristic
290C2×C8C2×#4111yescharacteristic
291C2×C8H(5,𝔽2)11yescharacteristic
292C2×C82−1+411yescharacteristic
293C2×C82−1+411yescharacteristic
294C2×C82−1+411yescharacteristic
295C4⋊C4—22no—
296C4⋊C4—22no—
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—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—161no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—161no—
323C22⋊C4—161no—
324C22⋊C4—161no—
325C22⋊C4—161no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—161no—
331C16—22no—
332C16—22no—
333C16—41no—
334C16—41no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23#51012yes—
349C23C22×D811yescharacteristic
350Q8—164no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—81no—
355D4—164no—
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—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
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—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—164no—
396C2×C4#50912yes—
397C2×C4#53611yescharacteristic
398C2×C4D4211yescharacteristic
399C2×C4#54711yescharacteristic
400C2×C4C2×#13711yescharacteristic
401C2×C4#57611yescharacteristic
402C2×C4#57811yescharacteristic
403C8—22no—
404C8—41no—
405C8—41no—
406C8#53611yescharacteristic
407C8#54211yescharacteristic
408C8#54811yescharacteristic
409C8#56311yescharacteristic
410C22—22no—
411C22—22no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—161no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22—161no—
423C22—161no—
424C22#322611yescharacteristic
425C22D4×D811yescharacteristic
426C22#333411yescharacteristic
427C22C2×#50911yescharacteristic
428C22C2×#51011yescharacteristic
429C22#341511yescharacteristic
430C22#341611yescharacteristic
431C4—22no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—82no—
437C4—82no—
438C4—161no—
439C4—161no—
440C4—161no—
441C4—161no—
442C4—161no—
443C4#267411yescharacteristic
444C4#324311yescharacteristic
445C4#327011yescharacteristic
446C4#335711yescharacteristic
447C2—42no—
448C2—321no—
449C2#3077311yescharacteristic
450C2#3380511yescharacteristic
451C2#3380711yescharacteristic
452C2#3383211yescharacteristic
453C2#3384011yescharacteristic
454C2#3391211yescharacteristic
455C2#3391311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2255C2semidirect
4. #26065C2semidirect
5. #13526C2semidirect
6. #13493C2semidirect
7. #13474C2semidirect
8. #13453C2semidirect
10. #13363C2semidirect
11. #13177C2semidirect
12. #13149C2semidirect
13. #13121C2semidirect
14. #13110C2semidirect
15. #12919C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22552
C2×#22332
#277902
#260652
#135262
#134932
#134742
#134532
#133812
#133632
#131772
#131492
#131212
#131102
#129192