Finite Group Databasebeta

← #507237

Subgroups — #507237

Order 512.

544 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27789C211yesmaximal, characteristic, complemented
2#25032C211yesmaximal, characteristic
3#13494C211yesmaximal, characteristic, complemented
4#13483C211yesmaximal, characteristic, complemented
5#13365C211yesmaximal, characteristic, complemented
6#13305C211yesmaximal, characteristic
7#13281C211yesmaximal, characteristic, complemented
8#13237C211yesmaximal, characteristic, complemented
9#13148C211yesmaximal, characteristic, complemented
10#13111C211yesmaximal, characteristic, complemented
11#13060C211yesmaximal, characteristic, complemented
12#13059C211yesmaximal, characteristic
13#13014C211yesmaximal, characteristic, complemented
14C4×#357C211yesmaximal, characteristic, complemented
15C16×C22⋊C4C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22#2489—21no—
23#2420—21no—
24C2×#367C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×C22⋊C16C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33C2×C4×C16C2211yescharacteristic
34#2073—21no—
35#2035—21no—
36#2034—21no—
37#2026—21no—
38#1970C2211yescharacteristic
39#1969C2211yescharacteristic
40#1969C2211yescharacteristic
41#1942C2211yescharacteristic
42#1927—21no—
43#1864—21no—
44#1846C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C22⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×#121—21no—
65C2×C4×D4—21no—
66C2×SD32—22no—
67C2×SD32—22no—
68C2×SD32—41no—
69C22×C16—22no—
70C22×C16—41no—
71C22×C16—41no—
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#499—22no—
77#499—41no—
78#467—22no—
79#467—41no—
80#466—22no—
81#466—41no—
82C4×D8—22no—
83C4×D8—41no—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#106—21no—
91C2×#106—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×C22⋊C8D411yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#400—41no—
102#397—41no—
103#397—41no—
104#394—41no—
105#391—21no—
106#391—21no—
107#390—21no—
108#389—21no—
109#388—41no—
110#386—41no—
111#382—21no—
112#382—41no—
113#381—41no—
114#380—21no—
115#380—21no—
116C4×C4⋊C4—21no—
117C4×C22⋊C4D411yescharacteristic
118#367—22no—
119#358—22no—
120#358—41no—
121#358—41no—
122#357—22no—
123#357—22no—
124#357—41no—
125C22⋊C16—22no—
126C4×C16—41no—
127C4×C16—41no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×Q8—21no—
135C22×D4—21no—
136C22×D4—41no—
137C23×C4—41no—
138C23×C4C2×D411yescharacteristic
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8#4112yes—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2411yescharacteristic
161#125—42no—
162#125—81no—
163C22⋊Q8—42no—
164C22⋊Q8—81no—
165#121—42no—
166#121—81no—
167C4×D4—42no—
168C4×D4—81no—
169C2×C4⋊C4—21no—
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×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
191SD32—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—44no—
203C2×C16#4112yes—
204C2×C16#4112yes—
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—21no—
216#107#4112yes—
217#106—42no—
218#106—81no—
219#103—42no—
220#103—42no—
221#103—81no—
222#102—42no—
223#102—42no—
224#102—42no—
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#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95D811yescharacteristic
241#95D811yescharacteristic
242C24—41no—
243C24C2×D811yescharacteristic
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×D4—22no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—82no—
257C2×D4—82no—
258C2×D4—161no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—161no—
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C2×D811yescharacteristic
278C22×C4C2×D811yescharacteristic
279C22×C4C2×D811yescharacteristic
280C22×C4C22×D411yescharacteristic
281Q16—41no—
282Q16—41no—
283Q16—41no—
284Q16—41no—
285D8—22no—
286D8—22no—
287D8—41no—
288D8—41no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8#13512yes—
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—41no—
310C4⋊C4—41no—
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—161no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—161no—
331C22⋊C4—161no—
332C22⋊C4—161no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—82no—
338C16—22no—
339C16—22no—
340C16—41no—
341C16—41no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—42no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23#50812yes—
356C23C22×D811yescharacteristic
357Q8—81no—
358Q8—81no—
359Q8—81no—
360Q8—81no—
361D4—42no—
362D4—42no—
363D4—81no—
364D4—81no—
365D4—164no—
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—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
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—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—164no—
406C2×C4#50812yes—
407C2×C4#53211yescharacteristic
408C2×C4#54611yescharacteristic
409C2×C4#54611yescharacteristic
410C2×C4C2×#13511yescharacteristic
411C2×C4#57711yescharacteristic
412C2×C4#57711yescharacteristic
413C8—22no—
414C8—41no—
415C8—41no—
416C8#52911yescharacteristic
417C8#53811yescharacteristic
418C8#55111yescharacteristic
419C8#55911yescharacteristic
420C22—22no—
421C22—22no—
422C22—22no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22#316911yescharacteristic
435C22#329411yescharacteristic
436C22#329411yescharacteristic
437C22C2×#50811yescharacteristic
438C22C2×#50811yescharacteristic
439C22#341611yescharacteristic
440C22#341611yescharacteristic
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—41no—
446C4—82no—
447C4—161no—
448C4—161no—
449C4—161no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4#255711yescharacteristic
454C4#317311yescharacteristic
455C4#330111yescharacteristic
456C4#330311yescharacteristic
457C2—42no—
458C2—162no—
459C2#2897511yescharacteristic
460C2#3373511yescharacteristic
461C2#3374311yescharacteristic
462C2#3386311yescharacteristic
463C2#3386311yescharacteristic
464C2#3386811yescharacteristic
465C2#3387011yescharacteristic
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. #27789C2semidirect
3. #13494C2semidirect
4. #13483C2semidirect
5. #13365C2semidirect
7. #13281C2semidirect
8. #13237C2semidirect
9. #13148C2semidirect
10. #13111C2semidirect
11. #13060C2semidirect
13. #13014C2semidirect
14. C4×#357C2semidirect
15. C16×C22⋊C4C2semidirect

Maximal subgroups

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

subgroupindex
#277892
#250322
#134942
#134832
#133652
#133052
#132812
#132372
#131482
#131112
#130602
#130592
#130142
C4×#3572
C16×C22⋊C42