Finite Group Databasebeta

← #504331

Subgroups — #504331

Order 512.

563 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
1#28832C211yesmaximal, characteristic, complemented
2#23185C211yesmaximal, characteristic
3#13528C211yesmaximal, characteristic, complemented
4#13526C211yesmaximal, characteristic, complemented
5#13476C211yesmaximal, characteristic, complemented
6#13474C211yesmaximal, characteristic, complemented
7#13252C211yesmaximal, characteristic
8#13206C211yesmaximal, characteristic
9#13178C211yesmaximal, characteristic, complemented
10#13160C211yesmaximal, characteristic, complemented
11#12937C211yesmaximal, characteristic, complemented
12#12937C211yesmaximal, characteristic, complemented
13#12813C211yesmaximal, characteristic, complemented
14#12811C211yesmaximal, characteristic, complemented
15#12674C211yesmaximal, characteristic, complemented
16C2×#493C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19#2570—21no—
20#2366—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#346C2211yescharacteristic
37#2085—21no—
38#2084—21no—
39#1982—21no—
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1949—21no—
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1879—21no—
46#1879—21no—
47#1846C2211yescharacteristic
48#1846C2211yescharacteristic
49#1845C2211yescharacteristic
50C4×#107C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8—21no—
60C22×D8C2311yescharacteristic
61C22×D8C2311yescharacteristic
62C2×#127—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—41no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#493—22no—
71#493Q814yes—
72C4×D8—22no—
73C4×D8—22no—
74C4×D8—41no—
75C4×D8—41no—
76C2×#107—21no—
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#106—21no—
84C2×#106—21no—
85C2×C4⋊C8D411yescharacteristic
86C2×#102—21no—
87C2×#102—21no—
88C2×#102—21no—
89C2×#102—21no—
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93#400—41no—
94#398—41no—
95#390—21no—
96#390—21no—
97#390—41no—
98#389—21no—
99#389—21no—
100#387—41no—
101#387—41no—
102#386—41no—
103#386—41no—
104#384—21no—
105#382—21no—
106#382—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4D411yescharacteristic
110#367—22no—
111#367—41no—
112#366—22no—
113#366—41no—
114C4⋊C16—22no—
115C4⋊C16—41no—
116C4⋊C16—41no—
117#357—22no—
118#357—22no—
119#357—41no—
120#357—41no—
121#357—41no—
122#357—41no—
123#357—41no—
124#357—41no—
125#346—22no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#336D411yescharacteristic
131#336D411yescharacteristic
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—41no—
135C22×D4—41no—
136C23×C4—41no—
137C23×C4—41no—
138C2×D8—24no—
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×D8—21no—
148C2×D8—21no—
149C2×D8—81no—
150C2×D8—81no—
151C2×D8C2×Q812yes—
152C2×D8C2×Q812yes—
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2411yescharacteristic
157#127—24no—
158#127—42no—
159C4×D4—42no—
160C4×D4—42no—
161C4×D4—81no—
162C4×D4—81no—
163C4×D4—84no—
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—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4D811yescharacteristic
175C2×C4⋊C4D811yescharacteristic
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×C42—41no—
183C2×C42—41no—
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C16—21no—
188C2×C16—21no—
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#4112yes—
199#107—21no—
200#107—21no—
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—81no—
211#107#4112yes—
212#106—81no—
213#106—81no—
214C4⋊C8—22no—
215#102—42no—
216#102—42no—
217#102—81no—
218#102—81no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8C2×Q812yes—
224C4×C8#4112yes—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95D811yescharacteristic
234#95D811yescharacteristic
235C24—41no—
236C24—41no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—82no—
252C2×D4—82no—
253C2×D4—161no—
254C2×D4—161no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—161no—
266C22×C4—161no—
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C22×D411yescharacteristic
274D8—22no—
275D8—22no—
276D8—22no—
277D8—22no—
278D8—41no—
279D8—41no—
280D8—41no—
281D8—41no—
282D8—84no—
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×C8C22⋊Q812yes—
292C2×C8C22×Q811yescharacteristic
293C2×C8C2×#4111yescharacteristic
294C2×C8C2×#4111yescharacteristic
295C2×C8C2×#4111yescharacteristic
296C2×C8H(5,𝔽2)11yescharacteristic
297C2×C8H(5,𝔽2)11yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—22no—
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—41no—
311C4⋊C4—41no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—161no—
315C4⋊C4—161no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22⋊C4—161no—
321C22⋊C4—161no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—82no—
327C42—161no—
328C42C22⋊Q812yes—
329C42#13712yes—
330C16—22no—
331C16—41no—
332C16—41no—
333C16—41no—
334C23—42no—
335C23—42no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23C22×D811yescharacteristic
349D4—42no—
350D4—42no—
351D4—42no—
352D4—42no—
353D4—88no—
354D4—81no—
355D4—81no—
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—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—42no—
375C2×C4—81no—
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—164no—
393C2×C4—164no—
394C2×C4#47412yes—
395C2×C4#47412yes—
396C2×C4#50912yes—
397C2×C4C2×C22⋊Q811yescharacteristic
398C2×C4#54611yescharacteristic
399C2×C4#55011yescharacteristic
400C2×C4C2×#13711yescharacteristic
401C2×C4#57611yescharacteristic
402C2×C4#57711yescharacteristic
403C8—22no—
404C8—41no—
405C8—41no—
406C8#52911yescharacteristic
407C8#54011yescharacteristic
408C8#55411yescharacteristic
409C8#55411yescharacteristic
410C22—82no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—161no—
417C22—161no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22—161no—
423C22—161no—
424C22C2×#47411yescharacteristic
425C22#329411yescharacteristic
426C22#339111yescharacteristic
427C22C2×#50911yescharacteristic
428C22#341411yescharacteristic
429C22#341511yescharacteristic
430C22#341611yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—82no—
438C4—161no—
439C4—161no—
440C4—161no—
441C4—161no—
442C4—161no—
443C4#245411yescharacteristic
444C4#308111yescharacteristic
445C4#333311yescharacteristic
446C4#340011yescharacteristic
447C2—162no—
448C2—162no—
449C2#2652611yescharacteristic
450C2#3362711yescharacteristic
451C2#3363111yescharacteristic
452C2#3387611yescharacteristic
453C2#3387711yescharacteristic
454C2#3396611yescharacteristic
455C2#3396811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#493C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28832C2semidirect
3. #13528C2semidirect
4. #13526C2semidirect
5. #13476C2semidirect
6. #13474C2semidirect
9. #13178C2semidirect
10. #13160C2semidirect
11. #12937C2semidirect
12. #12937C2semidirect
13. #12813C2semidirect
14. #12811C2semidirect
15. #12674C2semidirect

Maximal subgroups

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

subgroupindex
#288322
#231852
#135282
#135262
#134762
#134742
#132522
#132062
#131782
#131602
#129372
#128132
#128112
#126742