Finite Group Databasebeta

← #505345

Subgroups — #505345

Order 512.

555 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#25412C211yesmaximal, characteristic
2#22612C211yesmaximal, characteristic, complemented
3#13516C211yesmaximal, characteristic, complemented
4#13408C211yesmaximal, characteristic, complemented
5#13382C211yesmaximal, characteristic, complemented
6#13380C211yesmaximal, characteristic
7#13254C211yesmaximal, characteristic, complemented
8#13147C211yesmaximal, characteristic, complemented
9#13131C211yesmaximal, characteristic, complemented
10#13120C211yesmaximal, characteristic, complemented
11#13109C211yesmaximal, characteristic, complemented
12#12993C211yesmaximal, characteristic, complemented
13#12991C211yesmaximal, characteristic
14#12936C211yesmaximal, characteristic, complemented
15C4×#358C211yesmaximal, characteristic, complemented
16C2×#495C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C22×#107C2211yescharacteristic
21#2426—21no—
22#2345—21no—
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37#2042—21no—
38#2000—21no—
39#1970C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1921—21no—
44#1921—21no—
45#1880—21no—
46#1824—21no—
47#1808C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C8×C22⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×SD16—21no—
61C22×D8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×#124—21no—
64C2×C22⋊Q8—21no—
65C2×#121—21no—
66C2×C4×Q8—21no—
67C22×C4⋊C4—21no—
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72#495—22no—
73#495—41no—
74#467—22no—
75#467—41no—
76#466—22no—
77#466—41no—
78C4×Q16—22no—
79C4×Q16—41no—
80C2×#107—21no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#103—21no—
90C2×#103—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×C22⋊C8D411yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#396—41no—
97#394—41no—
98#390—21no—
99#390—41no—
100#389—21no—
101#389—21no—
102#389—21no—
103#388—41no—
104#387—41no—
105#387—41no—
106#380—21no—
107#380—41no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C22⋊C4—41no—
111C4×C22⋊C4D411yescharacteristic
112#367—41no—
113#366—22no—
114C4⋊C16—41no—
115#358—22no—
116#358—22no—
117#358—22no—
118#358—41no—
119#357—22no—
120#357—22no—
121#357—41no—
122#357—41no—
123C22⋊C16—41no—
124C22⋊C16—41no—
125C4×C16—41no—
126#340—21no—
127#340—21no—
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#4112yes—
145C2×SD16—42no—
146C2×SD16—42no—
147C2×SD16—81no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2411yescharacteristic
164#124—42no—
165#124—81no—
166C22⋊Q8—42no—
167C22⋊Q8—81no—
168#121—42no—
169#121—81no—
170C4×Q8—42no—
171C4×Q8—81no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4D811yescharacteristic
194C2×C22⋊C4D811yescharacteristic
195C2×C42—41no—
196C2×C42—41no—
197C2×C42—41no—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212#107—24no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107#4112yes—
223#107#4112yes—
224#107#4112yes—
225#103—42no—
226#103—42no—
227#102—42no—
228#102—81no—
229C22⋊C8—41no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
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—22no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×D4—41no—
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—44no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—161no—
274C22×C4—161no—
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C2×D811yescharacteristic
278C22×C4C2×D811yescharacteristic
279C22×C4C2×D811yescharacteristic
280C22×C4C22×D411yescharacteristic
281Q16—22no—
282Q16—22no—
283Q16—41no—
284Q16—41no—
285SD16—84no—
286D8—41no—
287D8—41no—
288D8—41no—
289D8—41no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8C2×#4111yescharacteristic
304C2×C8C2×#4111yescharacteristic
305C2×C8C2×#4111yescharacteristic
306C2×C8C2×#4111yescharacteristic
307C2×C8H(5,𝔽2)11yescharacteristic
308C2×C8H(5,𝔽2)11yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—44no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—161no—
325C4⋊C4—161no—
326C4⋊C4—161no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—161no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—82no—
341C42—161no—
342C42—161no—
343C16—41no—
344C16—41no—
345C16—41no—
346C16—41no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23#50812yes—
361C23C22×D811yescharacteristic
362Q8—42no—
363Q8—42no—
364Q8—81no—
365Q8—81no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—164no—
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—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
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—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—164no—
411C2×C4#51012yes—
412C2×C4#52511yescharacteristic
413C2×C4#54611yescharacteristic
414C2×C4#54611yescharacteristic
415C2×C4#57511yescharacteristic
416C2×C4#57711yescharacteristic
417C2×C4#57711yescharacteristic
418C8—41no—
419C8—41no—
420C8—41no—
421C8#52911yescharacteristic
422C8#54011yescharacteristic
423C8#55311yescharacteristic
424C8#55511yescharacteristic
425C22—22no—
426C22—22no—
427C22—22no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22—161no—
437C22—161no—
438C22—161no—
439C22#310011yescharacteristic
440C22#329411yescharacteristic
441C22#329411yescharacteristic
442C22C2×#50811yescharacteristic
443C22C2×#51011yescharacteristic
444C22#341511yescharacteristic
445C22#341611yescharacteristic
446C4—22no—
447C4—41no—
448C4—41no—
449C4—41no—
450C4—41no—
451C4—82no—
452C4—82no—
453C4—82no—
454C4—82no—
455C4—161no—
456C4—161no—
457C4—161no—
458C4#247611yescharacteristic
459C4#312611yescharacteristic
460C4#331711yescharacteristic
461C4#332311yescharacteristic
462C2—42no—
463C2—321no—
464C2#2739011yescharacteristic
465C2#3366811yescharacteristic
466C2#3367811yescharacteristic
467C2#3386411yescharacteristic
468C2#3386811yescharacteristic
469C2#3388311yescharacteristic
470C2#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
2. #22612C2semidirect
3. #13516C2semidirect
4. #13408C2semidirect
5. #13382C2semidirect
7. #13254C2semidirect
8. #13147C2semidirect
9. #13131C2semidirect
10. #13120C2semidirect
11. #13109C2semidirect
12. #12993C2semidirect
14. #12936C2semidirect
15. C4×#358C2semidirect

Maximal subgroups

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

subgroupindex
#254122
#226122
#135162
#134082
#133822
#133802
#132542
#131472
#131312
#131202
#131092
#129932
#129912
#129362
C4×#3582