Finite Group Databasebeta

← #504196

Subgroups — #504196

Order 512.

579 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26065C211yesmaximal, characteristic, complemented
2#23027C211yesmaximal, characteristic
3#13537C211yesmaximal, characteristic, complemented
4#13516C211yesmaximal, characteristic, complemented
5#13476C211yesmaximal, characteristic, complemented
6#13363C211yesmaximal, characteristic, complemented
7#13276C211yesmaximal, characteristic, complemented
8#13112C211yesmaximal, characteristic, complemented
9#13061C211yesmaximal, characteristic
10#12991C211yesmaximal, characteristic
11#12935C211yesmaximal, characteristic, complemented
12#12921C211yesmaximal, characteristic, complemented
13#12913C211yesmaximal, characteristic, complemented
14#12811C211yesmaximal, characteristic, complemented
15#12557C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2440—21no—
23#2363—21no—
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33#2083—21no—
34#2057—21no—
35#1970C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1926—21no—
40#1875—21no—
41#1864—21no—
42#1852—21no—
43#1845C2211yescharacteristic
44#1808C2211yescharacteristic
45C4×#107C2211yescharacteristic
46C8×C22⋊C4C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×C22⋊Q8—41no—
63C2×#121—21no—
64C2×C4×D4—21no—
65C2×#117—21no—
66C22×C4⋊C4—21no—
67C2×M32—22no—
68C2×M32—41no—
69C2×M32—41no—
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74#500—22no—
75#500—41no—
76#466—22no—
77#466Q814yescomplemented
78C4×D8—22no—
79C4×D8—41no—
80#427—22no—
81#427—41no—
82C2×#107—22no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—41no—
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#106—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×C22⋊C8D411yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#398—41no—
102#395—41no—
103#390—21no—
104#390—41no—
105#389—21no—
106#389—21no—
107#389—21no—
108#388—41no—
109#386—41no—
110#385—41no—
111#384—21no—
112#380—21no—
113C4×C4⋊C4—21no—
114C4×C22⋊C4—41no—
115C4×C22⋊C4D411yescharacteristic
116#367—41no—
117#366—22no—
118C4⋊C16—41no—
119C4⋊C16—41no—
120#357—22no—
121#357—41no—
122#357—41no—
123#357—41no—
124C22⋊C16—22no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#336D411yescharacteristic
131#336D411yescharacteristic
132C22×Q8—41no—
133C22×D4—21no—
134C22×D4—41no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4C2×D411yescharacteristic
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8C2×Q812yes—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×Q812yes—
152C22×C8#4112yes—
153C22×C8C2411yescharacteristic
154C4⋊Q8—42no—
155C4⋊Q8—81no—
156C22⋊Q8—84no—
157#121—24no—
158#121—42no—
159C4×D4—42no—
160C4×D4—81no—
161#117—42no—
162#117—81no—
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—21no—
170C2×C4⋊C4—42no—
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⋊C4D811yescharacteristic
186C2×C22⋊C4D811yescharacteristic
187C2×C42—41no—
188C2×C42—41no—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191M32—44no—
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×C16—21no—
202C2×C16—21no—
203C2×C16—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×Q812yes—
219#107#4112yes—
220#107#4112yes—
221#106—42no—
222#102—24no—
223#102—42no—
224#102—81no—
225C22⋊C8—22no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95D811yescharacteristic
236#95D811yescharacteristic
237C24—41no—
238C24C2×D811yescharacteristic
239C2×Q8—82no—
240C2×Q8—82no—
241C2×Q8—161no—
242C2×D4—22no—
243C2×D4—44no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—161no—
250C2×D4—161no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—161no—
267C22×C4—161no—
268C22×C4C22⋊Q812yes—
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4#13712yes—
275C22×C4C22×D411yescharacteristic
276D8—22no—
277D8—22no—
278D8—41no—
279D8—41no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8C22⋊Q812yes—
293C2×C8C22×Q811yescharacteristic
294C2×C8C2×#4111yescharacteristic
295C2×C8C2×#4111yescharacteristic
296C2×C8C2×#4111yescharacteristic
297C2×C8H(5,𝔽2)11yescharacteristic
298C2×C8H(5,𝔽2)11yescharacteristic
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
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—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—161no—
319C4⋊C4—161no—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—44no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C22⋊C4—161no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—82no—
333C42—161no—
334C16—41no—
335C16—41no—
336C16—41no—
337C16—41no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—42no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23#47412yes—
352C23C22×D811yescharacteristic
353Q8—164no—
354D4—42no—
355D4—42no—
356D4—81no—
357D4—81no—
358D4—164no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
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—81no—
396C2×C4—164no—
397C2×C4—164no—
398C2×C4#47412yes—
399C2×C4#50912yes—
400C2×C4#51012yes—
401C2×C4C2×C22⋊Q811yescharacteristic
402C2×C4#54611yescharacteristic
403C2×C4#54611yescharacteristic
404C2×C4C2×#13711yescharacteristic
405C2×C4#57711yescharacteristic
406C2×C4#57711yescharacteristic
407C8—22no—
408C8—41no—
409C8—41no—
410C8#52911yescharacteristic
411C8#53811yescharacteristic
412C8#55411yescharacteristic
413C8#55411yescharacteristic
414C22—22no—
415C22—22no—
416C22—22no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22—161no—
425C22—161no—
426C22—161no—
427C22—161no—
428C22C2×#47411yescharacteristic
429C22#329411yescharacteristic
430C22#329411yescharacteristic
431C22C2×#50911yescharacteristic
432C22C2×#51011yescharacteristic
433C22#341511yescharacteristic
434C22#341611yescharacteristic
435C4—22no—
436C4—22no—
437C4—22no—
438C4—41no—
439C4—41no—
440C4—82no—
441C4—82no—
442C4—82no—
443C4—161no—
444C4—161no—
445C4—161no—
446C4—161no—
447C4#245011yescharacteristic
448C4#308411yescharacteristic
449C4#331611yescharacteristic
450C4#332011yescharacteristic
451C2—42no—
452C2—162no—
453C2#2640811yescharacteristic
454C2#3362911yescharacteristic
455C2#3363111yescharacteristic
456C2#3387311yescharacteristic
457C2#3388711yescharacteristic
458C2#3389211yescharacteristic
459C2#3390011yescharacteristic
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. #26065C2semidirect
3. #13537C2semidirect
4. #13516C2semidirect
5. #13476C2semidirect
6. #13363C2semidirect
7. #13276C2semidirect
8. #13112C2semidirect
11. #12935C2semidirect
12. #12921C2semidirect
13. #12913C2semidirect
14. #12811C2semidirect
15. #12557C2semidirect
77. #466Q8semidirect

Maximal subgroups

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

subgroupindex
#260652
#230272
#135372
#135162
#134762
#133632
#132762
#131122
#130612
#129912
#129352
#129212
#129132
#128112
#125572