Finite Group Databasebeta

← #505550

Subgroups — #505550

Order 512.

548 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28014C211yesmaximal, characteristic, complemented
2#22682C211yesmaximal, characteristic, complemented
3#13539C211yesmaximal, characteristic, complemented
4#13537C211yesmaximal, characteristic, complemented
5#13482C211yesmaximal, characteristic, complemented
6#13476C211yesmaximal, characteristic, complemented
7#13149C211yesmaximal, characteristic, complemented
8#13110C211yesmaximal, characteristic, complemented
9#13105C211yesmaximal, characteristic, complemented
10#13012C211yesmaximal, characteristic
11#13012C211yesmaximal, characteristic
12#13012C211yesmaximal, characteristic
13#12937C211yesmaximal, characteristic, complemented
14#12811C211yesmaximal, characteristic, complemented
15#12637C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2512—21no—
21#2349—21no—
22C2×#367C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×#357C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#346C2211yescharacteristic
31#2071—21no—
32#2067—21no—
33#1970C2211yescharacteristic
34#1970C2211yescharacteristic
35#1970C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1925—21no—
40#1864—21no—
41#1864—21no—
42#1846C2211yescharacteristic
43#1845C2211yescharacteristic
44#1804—21no—
45C4×#107C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×#126—41no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×D16—22no—
66C2×D16—22no—
67C2×D16—22no—
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C4×D8—22no—
73C4×D8—22no—
74C4×D8—22no—
75C4×D8—22no—
76C4×D8—41no—
77C4×D8—41no—
78C2×#107—21no—
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#106—21no—
86C2×C4⋊C8D411yescharacteristic
87C2×#102—21no—
88C2×#102—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#397—41no—
97#396—41no—
98#390—21no—
99#390—21no—
100#390—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#388—41no—
105#385—41no—
106#385—41no—
107#384—21no—
108#382—21no—
109#380—41no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4D411yescharacteristic
112C2×#95—41no—
113#367—22no—
114C4⋊C16—41no—
115#357—22no—
116#357—22no—
117#357—22no—
118#357—22no—
119#357—41no—
120#357—41no—
121#346—41no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×D4—21no—
129C22×D4—21no—
130C23×C4—41no—
131C23×C4—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#4112yes—
143C2×D8#4112yes—
144C2×D8#4112yes—
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2411yescharacteristic
149#126—84no—
150C4×D4—24no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—81no—
155C4×D4—81no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4D811yescharacteristic
168C2×C4⋊C4D811yescharacteristic
169C2×C22⋊C4—41no—
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×C22⋊C4—41no—
178C2×C42—41no—
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182D16—44no—
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—
193C2×C16—21no—
194C2×C16—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—21no—
205#107—21no—
206#107—81no—
207#107#4112yes—
208#106—81no—
209C4⋊C8—41no—
210#102—24no—
211#102—42no—
212#102—42no—
213#102—42no—
214#102—81no—
215#102—81no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95D811yescharacteristic
229#95D811yescharacteristic
230C24—41no—
231C24—41no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C22×C4—44no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—161no—
256C22×C4—161no—
257C22×C4—161no—
258C22×C4—161no—
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C22×D411yescharacteristic
266D8—22no—
267D8—22no—
268D8—22no—
269D8—41no—
270D8—41no—
271D8—41no—
272D8—41no—
273D8#12614yescomplemented
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8C2×#4111yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C8H(5,𝔽2)11yescharacteristic
288C2×C82−1+411yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—22no—
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—41no—
302C4⋊C4—41no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—161no—
306C4⋊C4—161no—
307C4⋊C4—161no—
308C22⋊C4—44no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C22⋊C4—161no—
316C22⋊C4—161no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—82no—
324C16—41no—
325C16—41no—
326C16—41no—
327C16—41no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23C22×D811yescharacteristic
343D4—24no—
344D4—42no—
345D4—42no—
346D4—42no—
347D4—81no—
348D4—81no—
349D4—81no—
350D4—81no—
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—21no—
367C2×C4—21no—
368C2×C4—42no—
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—81no—
385C2×C4—81no—
386C2×C4—164no—
387C2×C4—164no—
388C2×C4—164no—
389C2×C4#50912yes—
390C2×C4#52511yescharacteristic
391C2×C4#54611yescharacteristic
392C2×C4#54811yescharacteristic
393C2×C4#57511yescharacteristic
394C2×C4#57711yescharacteristic
395C2×C4#57711yescharacteristic
396C8—41no—
397C8—41no—
398C8—41no—
399C8C2×#12611yescharacteristic
400C8#53911yescharacteristic
401C8#55211yescharacteristic
402C8#55311yescharacteristic
403C22—44no—
404C22—82no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—82no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22—161no—
414C22—161no—
415C22—161no—
416C22—161no—
417C22#310011yescharacteristic
418C22#329411yescharacteristic
419C22#335811yescharacteristic
420C22C2×#50911yescharacteristic
421C22#341411yescharacteristic
422C22#341511yescharacteristic
423C22#341611yescharacteristic
424C4—22no—
425C4—41no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—82no—
431C4—161no—
432C4—161no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4#248111yescharacteristic
437C4#312511yescharacteristic
438C4#330711yescharacteristic
439C4#337611yescharacteristic
440C2—84no—
441C2—321no—
442C2#2755011yescharacteristic
443C2#3367511yescharacteristic
444C2#3367811yescharacteristic
445C2#3387311yescharacteristic
446C2#3388911yescharacteristic
447C2#3393311yescharacteristic
448C2#3394311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
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. #28014C2semidirect
2. #22682C2semidirect
3. #13539C2semidirect
4. #13537C2semidirect
5. #13482C2semidirect
6. #13476C2semidirect
7. #13149C2semidirect
8. #13110C2semidirect
9. #13105C2semidirect
13. #12937C2semidirect
14. #12811C2semidirect
15. #12637C2semidirect
273. D8#126semidirect

Maximal subgroups

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

subgroupindex
#280142
#226822
#135392
#135372
#134822
#134762
#131492
#131102
#131052
#130122
#129372
#128112
#126372