Finite Group Databasebeta

← #510839

Subgroups — #510839

Order 512.

504 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27992C211yesmaximal, characteristic, complemented
2#26254C211yesmaximal, characteristic
3#13516C211yesmaximal, characteristic, complemented
4#13482C211yesmaximal, characteristic, complemented
5#13476C211yesmaximal, characteristic, complemented
6#13474C211yesmaximal, characteristic, complemented
7#13474C211yesmaximal, characteristic, complemented
8#13407C211yesmaximal, characteristic
9#13252C211yesmaximal, characteristic
10#13160C211yesmaximal, characteristic, complemented
11#13149C211yesmaximal, characteristic, complemented
12#13149C211yesmaximal, characteristic, complemented
13#13109C211yesmaximal, characteristic, complemented
14#12937C211yesmaximal, characteristic, complemented
15#12813C211yesmaximal, characteristic, complemented
16C2×#493C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18#2508—21no—
19#2445—21no—
20C2×#367C2211yescharacteristic
21C2×#366C2211yescharacteristic
22C2×C4⋊C16C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#346C2211yescharacteristic
34#2101—21no—
35#2088—21no—
36#2054—21no—
37#2041—21no—
38#1970C2211yescharacteristic
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1920—21no—
47#1879—21no—
48#1846C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8—21no—
60C22×D8C2311yescharacteristic
61C22×D8C2311yescharacteristic
62C2×#127—21no—
63C2×#123—41no—
64C2×C4×D4—21no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69#493—22no—
70#493—41no—
71C4×D8—22no—
72C4×D8—41no—
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#106—21no—
80C2×C4⋊C8D411yescharacteristic
81C2×#102—21no—
82C2×#102—21no—
83C2×#102—21no—
84C2×#102—21no—
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88#400—41no—
89#398—41no—
90#394—41no—
91#394—41no—
92#390—21no—
93#390—21no—
94#390—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#387—41no—
101#386—41no—
102#382—21no—
103#382—41no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4D411yescharacteristic
106#367—41no—
107#366—22no—
108C4⋊C16—41no—
109C4⋊C16—41no—
110C4⋊C16—41no—
111#357—22no—
112#357—22no—
113#357—22no—
114#357—41no—
115#357—41no—
116#357—41no—
117#357—41no—
118#357—41no—
119#346—41no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—41no—
131C22×D4—41no—
132C23×C4—41no—
133C2×D8—21no—
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—21no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—42no—
145C2×D8—42no—
146C2×D8—81no—
147C2×D8#4112yes—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2411yescharacteristic
152#127—42no—
153#127—81no—
154#123—84no—
155C4×D4—42no—
156C4×D4—81no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4D811yescharacteristic
169C2×C4⋊C4D811yescharacteristic
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C42—41no—
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107#4112yes—
205#106—42no—
206C4⋊C8—41no—
207#102—42no—
208#102—81no—
209#102—81no—
210#102—81no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95D811yescharacteristic
226#95D811yescharacteristic
227C24—41no—
228C24—41no—
229C2×D4—22no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—82no—
242C2×D4—82no—
243C2×D4—82no—
244C2×D4—82no—
245C2×D4—161no—
246C2×D4—161no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—161no—
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C22×D411yescharacteristic
263D8—22no—
264D8—22no—
265D8—41no—
266D8—41no—
267D8—41no—
268D8—41no—
269D8—41no—
270D8—41no—
271D8—84no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8C2×#4111yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C2×C8H(5,𝔽2)11yescharacteristic
284C2×C8H(5,𝔽2)11yescharacteristic
285C2×C82−1+411yescharacteristic
286C2×C82−1+411yescharacteristic
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—161no—
304C4⋊C4—161no—
305C4⋊C4—161no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—161no—
311C22⋊C4—161no—
312C22⋊C4—161no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—82no—
320C16—41no—
321C16—41no—
322C16—41no—
323C16—41no—
324C23—42no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23C22×D811yescharacteristic
339D4—42no—
340D4—42no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—81no—
345D4—81no—
346D4—81no—
347D4—164no—
348D4—164no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
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—164no—
385C2×C4#50912yes—
386C2×C4#53711yescharacteristic
387C2×C4#54611yescharacteristic
388C2×C4#54811yescharacteristic
389C2×C4#57511yescharacteristic
390C2×C4#57711yescharacteristic
391C2×C4#57711yescharacteristic
392C8—41no—
393C8—41no—
394C8—41no—
395C8#53911yescharacteristic
396C8#54011yescharacteristic
397C8#55211yescharacteristic
398C8#56311yescharacteristic
399C22—82no—
400C22—82no—
401C22—82no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22—161no—
407C22—161no—
408C22—161no—
409C22—161no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22#324511yescharacteristic
414C22#329411yescharacteristic
415C22#335811yescharacteristic
416C22C2×#50911yescharacteristic
417C22#341411yescharacteristic
418C22#341511yescharacteristic
419C22#341611yescharacteristic
420C4—22no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—82no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4—161no—
432C4#270711yescharacteristic
433C4#326011yescharacteristic
434C4#332311yescharacteristic
435C4#337111yescharacteristic
436C2—162no—
437C2—321no—
438C2#3196211yescharacteristic
439C2#3381711yescharacteristic
440C2#3382311yescharacteristic
441C2#3387711yescharacteristic
442C2#3388611yescharacteristic
443C2#3393311yescharacteristic
444C2#3393811yescharacteristic
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. #27992C2semidirect
3. #13516C2semidirect
4. #13482C2semidirect
5. #13476C2semidirect
6. #13474C2semidirect
7. #13474C2semidirect
10. #13160C2semidirect
11. #13149C2semidirect
12. #13149C2semidirect
13. #13109C2semidirect
14. #12937C2semidirect
15. #12813C2semidirect

Maximal subgroups

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

subgroupindex
#279922
#262542
#135162
#134822
#134762
#134742
#134072
#132522
#131602
#131492
#131092
#129372
#128132