Finite Group Databasebeta

← #511526

Subgroups — #511526

Order 512.

530 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28267C211yesmaximal, characteristic, complemented
2#28263C211yesmaximal, characteristic, complemented
3#13528C211yesmaximal, characteristic, complemented
4#13527C211yesmaximal, characteristic, complemented
5#13507C211yesmaximal, characteristic, complemented
6#13505C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic, complemented
8#13474C211yesmaximal, characteristic, complemented
9#13263C211yesmaximal, characteristic, complemented
10#13235C211yesmaximal, characteristic
11#13166C211yesmaximal, characteristic, complemented
12#13012C211yesmaximal, characteristic
13#13012C211yesmaximal, characteristic
14#12937C211yesmaximal, characteristic, complemented
15#12937C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2580—21no—
21#2538—21no—
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic
35#2067—21no—
36#2067—21no—
37#2065—21no—
38#2065—21no—
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1970C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1846C2211yescharacteristic
48#1845C2211yescharacteristic
49#1804—21no—
50#1802—21no—
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×C22⋊Q8—41no—
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×D16—41no—
66C2×D16—41no—
67C2×D16Q814yescomplemented
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C8×Q8—41no—
73C8×Q8—41no—
74C4×D8—41no—
75C4×D8—41no—
76C4×D8—41no—
77C4×D8—41no—
78C2×#107—21no—
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#106—21no—
86C2×C4⋊C8D411yescharacteristic
87C2×C4⋊C8D411yescharacteristic
88C2×C4⋊C8D411yescharacteristic
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#399—41no—
99#396—41no—
100#396—41no—
101#393—41no—
102#390—21no—
103#390—21no—
104#390—21no—
105#390—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#385—41no—
111#385—41no—
112#384—21no—
113#382—21no—
114#380—41no—
115#367—22no—
116#367—41no—
117#367—41no—
118#366—41no—
119#366—41no—
120C4⋊C16—41no—
121#357—22no—
122#357—22no—
123#357—41no—
124#357—41no—
125#357—41no—
126#357—41no—
127C4×C16—41no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132C22×Q8D811yescharacteristic
133C22×D4—21no—
134C22×D4—21no—
135C23×C4—41no—
136C23×C4—41no—
137C2×D8—21no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8C2×Q812yes—
148C2×D8C2×Q812yes—
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2411yescharacteristic
153C22⋊Q8—84no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×D4—42no—
157C4×D4—42no—
158C4×D4—42no—
159C4×D4—42no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4D811yescharacteristic
173C2×C4⋊C4D811yescharacteristic
174C2×C4⋊C4D811yescharacteristic
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
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×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188D16—44no—
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—
199C2×C16C2×Q812yes—
200C2×C16#4112yes—
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—21no—
211#107—21no—
212#107—21no—
213#107—81no—
214#106—81no—
215C4⋊C8—41no—
216C4⋊C8—41no—
217C4⋊C8—41no—
218#102—42no—
219#102—42no—
220#102—81no—
221#102—81no—
222#102—81no—
223#102—81no—
224C4×C8—21no—
225C4×C8—21no—
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#95—41no—
236C24—41no—
237C24—41no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—161no—
264C22×C4—161no—
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C22×D411yescharacteristic
272D8—22no—
273D8—22no—
274D8—22no—
275D8—22no—
276D8—41no—
277D8—41no—
278D8—41no—
279D8—41no—
280C2×C8—21no—
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×C8C22×Q811yescharacteristic
290C2×C8C2×#4111yescharacteristic
291C2×C8H(5,𝔽2)11yescharacteristic
292C2×C8H(5,𝔽2)11yescharacteristic
293C2×C8H(5,𝔽2)11yescharacteristic
294C2×C82−1+411yescharacteristic
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—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—161no—
314C4⋊C4—161no—
315C4⋊C4—161no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—82no—
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—21no—
331C42—21no—
332C16—22no—
333C16—22no—
334C16—41no—
335C16—41no—
336C23—42no—
337C23—42no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23C22×D811yescharacteristic
351Q8—44no—
352D4—42no—
353D4—42no—
354D4—42no—
355D4—42no—
356D4—81no—
357D4—81no—
358D4—81no—
359D4—81no—
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—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
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—164no—
395C2×C4—164no—
396C2×C4#50812yes—
397C2×C4#50912yes—
398C2×C4#53411yescharacteristic
399C2×C4D4×Q811yescharacteristic
400C2×C4#55011yescharacteristic
401C2×C4#57511yescharacteristic
402C2×C4#57611yescharacteristic
403C2×C4#57811yescharacteristic
404C8—41no—
405C8—41no—
406C8—41no—
407C8#53911yescharacteristic
408C8#54311yescharacteristic
409C8#55411yescharacteristic
410C8#56211yescharacteristic
411C22—82no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—161no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22—161no—
423C22—161no—
424C22—161no—
425C22#318611yescharacteristic
426C22Q8×D811yescharacteristic
427C22#339111yescharacteristic
428C22C2×#50811yescharacteristic
429C22C2×#50911yescharacteristic
430C22#341511yescharacteristic
431C22#341611yescharacteristic
432C4—22no—
433C4—22no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—161no—
439C4—161no—
440C4—161no—
441C4—161no—
442C4—161no—
443C4—161no—
444C4#273211yescharacteristic
445C4#320311yescharacteristic
446C4#338911yescharacteristic
447C4#340211yescharacteristic
448C2—162no—
449C2—162no—
450C2#3264811yescharacteristic
451C2#3375411yescharacteristic
452C2#3376411yescharacteristic
453C2#3394711yescharacteristic
454C2#3395111yescharacteristic
455C2#3396111yescharacteristic
456C2#3396611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28267C2semidirect
2. #28263C2semidirect
3. #13528C2semidirect
4. #13527C2semidirect
5. #13507C2semidirect
6. #13505C2semidirect
7. #13476C2semidirect
8. #13474C2semidirect
9. #13263C2semidirect
11. #13166C2semidirect
14. #12937C2semidirect
15. #12937C2semidirect
67. C2×D16Q8semidirect

Maximal subgroups

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

subgroupindex
#282672
#282632
#135282
#135272
#135072
#135052
#134762
#134742
#132632
#132352
#131662
#130122
#129372