Finite Group Databasebeta

← #507153

Subgroups — #507153

Order 512.

549 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#28015C211yesmaximal, characteristic, complemented
2#26254C211yesmaximal, characteristic
3#13505C211yesmaximal, characteristic, complemented
4#13474C211yesmaximal, characteristic, complemented
5#13449C211yesmaximal, characteristic, complemented
6#13328C211yesmaximal, characteristic
7#13238C211yesmaximal, characteristic, complemented
8#13238C211yesmaximal, characteristic, complemented
9#13202C211yesmaximal, characteristic
10#13149C211yesmaximal, characteristic, complemented
11#13148C211yesmaximal, characteristic, complemented
12#13130C211yesmaximal, characteristic, complemented
13#13014C211yesmaximal, characteristic, complemented
14#12663C211yesmaximal, characteristic, complemented
15#12637C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C2×#493C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2511—21no—
21#2445—21no—
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#346C2211yescharacteristic
34C2×#346C2211yescharacteristic
35C2×C4×C16C2211yescharacteristic
36#2072—21no—
37#2068—21no—
38#2054—21no—
39#2041—21no—
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1970C2211yescharacteristic
43#1969C2211yescharacteristic
44#1969C2211yescharacteristic
45#1969C2211yescharacteristic
46#1942C2211yescharacteristic
47#1942C2211yescharacteristic
48#1928—21no—
49#1879—21no—
50#1845C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×D8—21no—
61C22×D8C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C2×#127—21no—
64C2×#123—41no—
65C2×C4×D4—21no—
66C2×SD32—22no—
67C2×SD32—22no—
68C2×SD32—22no—
69C2×SD32—41no—
70C2×SD32—41no—
71C2×SD32—41no—
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#493—22no—
77#493—41no—
78C4×D8—22no—
79C4×D8—41no—
80C2×#107—21no—
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×C4⋊C8D411yescharacteristic
87C2×#103—21no—
88C2×#103—21no—
89C2×#103—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#400—41no—
98#400—41no—
99#394—41no—
100#394—41no—
101#391—21no—
102#391—21no—
103#391—21no—
104#390—21no—
105#390—21no—
106#390—21no—
107#389—21no—
108#389—21no—
109#387—41no—
110#387—41no—
111#386—41no—
112#384—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4D411yescharacteristic
115#366—41no—
116#366—41no—
117C4⋊C16—22no—
118#358—22no—
119#358—41no—
120#357—22no—
121#357—22no—
122#357—22no—
123#357—22no—
124#357—41no—
125#357—41no—
126#346—22no—
127#346—41no—
128C4×C16—41no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×Q8—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—41no—
138C22×D4—41no—
139C23×C4—41no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—42no—
158C2×D8—42no—
159C2×D8—81no—
160C2×D8#4112yes—
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2411yescharacteristic
165#127—42no—
166#127—81no—
167#123—84no—
168C4×D4—42no—
169C4×D4—81no—
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⋊C4D811yescharacteristic
180C2×C4⋊C4D811yescharacteristic
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C42—41no—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192SD32—44no—
193SD32—44no—
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—21no—
203C2×C16#4112yes—
204C2×C16#4112yes—
205C2×C16#4112yes—
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—81no—
217C4⋊C8—22no—
218#103—42no—
219#103—81no—
220#103—81no—
221#102—42no—
222#102—81no—
223#102—81no—
224#102—81no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
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—
243C24—41no—
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—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—82no—
263C2×D4—82no—
264C2×D4—82no—
265C2×D4—82no—
266C2×D4—161no—
267C2×D4—161no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—161no—
277C22×C4C2×D811yescharacteristic
278C22×C4C2×D811yescharacteristic
279C22×C4C2×D811yescharacteristic
280C22×C4C2×D811yescharacteristic
281C22×C4C2×D811yescharacteristic
282C22×C4C2×D811yescharacteristic
283C22×C4C22×D411yescharacteristic
284Q16—41no—
285Q16—41no—
286Q16—41no—
287Q16—41no—
288D8—22no—
289D8—22no—
290D8—41no—
291D8—41no—
292D8—41no—
293D8—41no—
294D8—41no—
295D8—41no—
296D8—84no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8#13512yes—
306C2×C8C2×#4111yescharacteristic
307C2×C8C2×#4111yescharacteristic
308C2×C8C2×#4111yescharacteristic
309C2×C8C2×#4111yescharacteristic
310C2×C8H(5,𝔽2)11yescharacteristic
311C2×C8H(5,𝔽2)11yescharacteristic
312C4⋊C4—22no—
313C4⋊C4—22no—
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—41no—
323C4⋊C4—41no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—161no—
327C4⋊C4—161no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—161no—
333C22⋊C4—161no—
334C22⋊C4—161no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—161no—
342C16—22no—
343C16—22no—
344C16—22no—
345C16—41no—
346C23—42no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23C22×D811yescharacteristic
361Q8—81no—
362Q8—81no—
363Q8—81no—
364Q8—81no—
365D4—42no—
366D4—42no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—81no—
371D4—81no—
372D4—81no—
373D4—164no—
374D4—164no—
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—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
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—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—164no—
410C2×C4#50912yes—
411C2×C4#51012yes—
412C2×C4#53211yescharacteristic
413C2×C4#54611yescharacteristic
414C2×C4#55011yescharacteristic
415C2×C4C2×#13511yescharacteristic
416C2×C4#57611yescharacteristic
417C2×C4#57711yescharacteristic
418C8—22no—
419C8—41no—
420C8—41no—
421C8#52911yescharacteristic
422C8#53811yescharacteristic
423C8#55111yescharacteristic
424C8#55311yescharacteristic
425C22—82no—
426C22—82no—
427C22—82no—
428C22—161no—
429C22—161no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22—161no—
437C22—161no—
438C22—161no—
439C22#316911yescharacteristic
440C22#329411yescharacteristic
441C22#339111yescharacteristic
442C22C2×#50911yescharacteristic
443C22C2×#51011yescharacteristic
444C22#341511yescharacteristic
445C22#341611yescharacteristic
446C4—22no—
447C4—22no—
448C4—41no—
449C4—41no—
450C4—41no—
451C4—41no—
452C4—161no—
453C4—161no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4—161no—
458C4#255411yescharacteristic
459C4#317911yescharacteristic
460C4#329711yescharacteristic
461C4#339411yescharacteristic
462C2—162no—
463C2—321no—
464C2#2887311yescharacteristic
465C2#3374411yescharacteristic
466C2#3375111yescharacteristic
467C2#3387711yescharacteristic
468C2#3388111yescharacteristic
469C2#3396611yescharacteristic
470C2#3396711yescharacteristic
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. #28015C2semidirect
3. #13505C2semidirect
4. #13474C2semidirect
5. #13449C2semidirect
7. #13238C2semidirect
8. #13238C2semidirect
10. #13149C2semidirect
11. #13148C2semidirect
12. #13130C2semidirect
13. #13014C2semidirect
14. #12663C2semidirect
15. #12637C2semidirect

Maximal subgroups

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

subgroupindex
#280152
#262542
#135052
#134742
#134492
#133282
#132382
#132022
#131492
#131482
#131302
#130142
#126632
#126372