Finite Group Databasebeta

← #505135

Subgroups — #505135

Order 512.

549 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#26255C211yesmaximal, characteristic, complemented
2#22807C211yesmaximal, characteristic
3#13530C211yesmaximal, characteristic, complemented
4#13481C211yesmaximal, characteristic, complemented
5#13474C211yesmaximal, characteristic
6#13327C211yesmaximal, characteristic
7#13253C211yesmaximal, characteristic, complemented
8#13238C211yesmaximal, characteristic
9#13133C211yesmaximal, characteristic, complemented
10#13109C211yesmaximal, characteristic, complemented
11#13105C211yesmaximal, characteristic, complemented
12#13014C211yesmaximal, characteristic
13#13014C211yesmaximal, characteristic
14#13013C211yesmaximal, characteristic
15#12638C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C2×#494C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21#2446—21no—
22#2354—21no—
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#346C2211yescharacteristic
35#2056—21no—
36#2043—21no—
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1969C2211yescharacteristic
40#1969C2211yescharacteristic
41#1969C2211yescharacteristic
42#1942C2211yescharacteristic
43#1925—21no—
44#1925—21no—
45#1878—21no—
46#1845C2211yescharacteristic
47#1802—21no—
48C4×#107C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16—21no—
60C22×Q16C2311yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C2×C4⋊Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×D4—21no—
66C2×#117—41no—
67C2×SD32—22no—
68C2×SD32—22no—
69C2×SD32—22no—
70C2×SD32—22no—
71C2×SD32—41no—
72C2×SD32—41no—
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77#494—22no—
78#494—41no—
79C4×Q16—22no—
80C4×Q16—41no—
81C4×D8—22no—
82C4×D8—22no—
83C2×#107—21no—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×#103—21no—
91C2×#103—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#400—41no—
101#398—41no—
102#391—21no—
103#391—21no—
104#391—21no—
105#390—21no—
106#390—21no—
107#389—21no—
108#388—41no—
109#388—41no—
110#387—41no—
111#387—41no—
112#384—21no—
113#380—41no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4D411yescharacteristic
117#366—22no—
118#366—41no—
119C4⋊C16—41no—
120#358—22no—
121#358—22no—
122#358—22no—
123#358—41no—
124#358—41no—
125#358—41no—
126#357—41no—
127#357—41no—
128#346—41no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×Q8—21no—
135C22×Q8—21no—
136C22×D4—21no—
137C23×C4—41no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—42no—
150C2×Q16—42no—
151C2×Q16—81no—
152C2×Q16#4112yes—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8#4112yes—
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2411yescharacteristic
163C4⋊Q8—42no—
164C4⋊Q8—81no—
165C4×Q8—42no—
166C4×Q8—81no—
167C4×D4—24no—
168C4×D4—42no—
169#117—84no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4D811yescharacteristic
182C2×C4⋊C4D811yescharacteristic
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C42—41no—
189C2×C42—41no—
190C2×C42—41no—
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194SD32—44no—
195SD32—44no—
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—81no—
218#107#4112yes—
219C4⋊C8—41no—
220#103—42no—
221#103—81no—
222#103—81no—
223#103—81no—
224#102—24no—
225#102—81no—
226#102—81no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95D811yescharacteristic
243#95D811yescharacteristic
244C24—41no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×D4—22no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C22×C4—44no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C2×D811yescharacteristic
278C22×C4C2×D811yescharacteristic
279C22×C4C22×D411yescharacteristic
280Q16—22no—
281Q16—22no—
282Q16—41no—
283Q16—41no—
284Q16—41no—
285Q16—41no—
286Q16—41no—
287Q16—41no—
288Q16—84no—
289D8—22no—
290D8—22no—
291D8—41no—
292D8—41no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8C2×#4111yescharacteristic
303C2×C8C2×#4111yescharacteristic
304C2×C8C2×#4111yescharacteristic
305C2×C8C2×#4111yescharacteristic
306C2×C8H(5,𝔽2)11yescharacteristic
307C2×C8H(5,𝔽2)11yescharacteristic
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
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—41no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—161no—
326C22⋊C4—44no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—161no—
330C22⋊C4—161no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—82no—
338C42—82no—
339C42—82no—
340C16—22no—
341C16—41no—
342C16—41no—
343C16—41no—
344C23—42no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23C22×D811yescharacteristic
352Q8—42no—
353Q8—42no—
354Q8—81no—
355Q8—81no—
356Q8—81no—
357Q8—81no—
358Q8—81no—
359Q8—81no—
360D4—42no—
361D4—42no—
362D4—81no—
363D4—81no—
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—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—88no—
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#50912yes—
403C2×C4#51012yes—
404C2×C4#52511yescharacteristic
405C2×C4#54611yescharacteristic
406C2×C4#55011yescharacteristic
407C2×C4#57511yescharacteristic
408C2×C4#57611yescharacteristic
409C2×C4#57711yescharacteristic
410C8—41no—
411C8—41no—
412C8—41no—
413C8#52911yescharacteristic
414C8#54011yescharacteristic
415C8#55111yescharacteristic
416C8#55111yescharacteristic
417C22—82no—
418C22—82no—
419C22—82no—
420C22—161no—
421C22—161no—
422C22—161no—
423C22—161no—
424C22#310011yescharacteristic
425C22#329411yescharacteristic
426C22#339111yescharacteristic
427C22C2×#50911yescharacteristic
428C22C2×#51011yescharacteristic
429C22#341611yescharacteristic
430C22#341711yescharacteristic
431C4—22no—
432C4—22no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—82no—
438C4—82no—
439C4—161no—
440C4—161no—
441C4—161no—
442C4—161no—
443C4—161no—
444C4#247111yescharacteristic
445C4#311911yescharacteristic
446C4#331911yescharacteristic
447C4#340111yescharacteristic
448C2—162no—
449C2#2719011yescharacteristic
450C2#3367011yescharacteristic
451C2#3368011yescharacteristic
452C2#3387511yescharacteristic
453C2#3389611yescharacteristic
454C2#3396411yescharacteristic
455C2#3396511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
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. #26255C2semidirect
3. #13530C2semidirect
4. #13481C2semidirect
7. #13253C2semidirect
9. #13133C2semidirect
10. #13109C2semidirect
11. #13105C2semidirect
15. #12638C2semidirect

Maximal subgroups

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

subgroupindex
#262552
#228072
#135302
#134812
#134742
#133272
#132532
#132382
#131332
#131092
#131052
#130142
#130132
#126382