Finite Group Databasebeta

← #501906

Subgroups — #501906

Order 512.

499 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27667C211yesmaximal, characteristic
2#22668C211yesmaximal, characteristic
3#13541C211yesmaximal, characteristic, complemented
4#13510C211yesmaximal, characteristic, complemented
5#13501C211yesmaximal, characteristic, complemented
6#13501C211yesmaximal, characteristic, complemented
7#13114C211yesmaximal, characteristic, complemented
8#13114C211yesmaximal, characteristic, complemented
9#12985C211yesmaximal, characteristic
10#12931C211yesmaximal, characteristic, complemented
11#12928C211yesmaximal, characteristic, complemented
12#12875C211yesmaximal, characteristic, complemented
13#12871C211yesmaximal, characteristic, complemented
14#12831C211yesmaximal, characteristic, complemented
15C2×#1387C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17D4×M16—21no—
18C2×#435C2211yescharacteristic
19C2×#430C2211yescharacteristic
20C22×#108C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2486—21no—
23C2×#365C2211yescharacteristic
24C2×#365C2211yescharacteristic
25C2×#365C2211yescharacteristic
26C2×#365C2211yescharacteristic
27C2×#349C2211yescharacteristic
28C2×#349C2211yescharacteristic
29#2112C2211yescharacteristic
30#2087—21no—
31#2076—21no—
32#1950—21no—
33#1944C2211yescharacteristic
34#1944C2211yescharacteristic
35#1855—21no—
36#1855—21no—
37#1849C2211yescharacteristic
38#1823—21no—
39#1811C2211yescharacteristic
40#1810C2211yescharacteristic
41#1809C2211yescharacteristic
42#1809C2211yescharacteristic
43#1807C2211yescharacteristic
44#1778C2211yescharacteristic
45#1387C2211yescharacteristic
46#1387C2211yescharacteristic
47#1387C2211yescharacteristic
48#1387C2211yescharacteristic
49#1387C2211yescharacteristic
50#1387C2211yescharacteristic
51#1387C2211yescharacteristic
52#1387C2211yescharacteristic
53#1383C2211yescharacteristic
54#1383C2211yescharacteristic
55#1383C2211yescharacteristic
56#1383C2211yescharacteristic
57#1377C2211yescharacteristic
58#1377C2211yescharacteristic
59C22×M16—21no—
60C22×M16—21no—
61C22×M16C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×#123—41no—
64C2×C4×D4D411yescharacteristic
65C22×C4⋊C4—21no—
66C2×M32—21no—
67C2×M32—22no—
68C2×M32—41no—
69C2×M32C2311yescharacteristic
70C2×M32C2311yescharacteristic
71C2×M32C2311yescharacteristic
72C2×M32C2311yescharacteristic
73C22×C16—21no—
74#436—41no—
75#435—21no—
76#435—21no—
77#435—21no—
78#435—21no—
79#435—41no—
80#435—41no—
81C8×D4—41no—
82#430—22no—
83#430—41no—
84C2×#108—22no—
85C2×#108—21no—
86C2×#108—21no—
87C2×#108—21no—
88C2×#108—41no—
89C2×#108C2311yescharacteristic
90C2×#108C2311yescharacteristic
91C2×#108C2311yescharacteristic
92C2×#108C2311yescharacteristic
93C2×#107—21no—
94C2×#107—21no—
95C2×#107—21no—
96C2×#107—21no—
97C2×#107—21no—
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#106C2311yescharacteristic
101C2×#106C2311yescharacteristic
102#423—21no—
103C2×C4⋊C8D411yescharacteristic
104#407—21no—
105#407—21no—
106C2×C22⋊C8D411yescharacteristic
107C2×C22⋊C8D411yescharacteristic
108C4×M16—21no—
109C2×#97C2311yescharacteristic
110#398—41no—
111#397—41no—
112#389—21no—
113#389—21no—
114#381—41no—
115#380—21no—
116#380—21no—
117#380—21no—
118#365—22no—
119#365—22no—
120#365—41no—
121#365—41no—
122#349—22no—
123#349—41no—
124C22⋊C16—21no—
125C22⋊C16—21no—
126#341—21no—
127#341—21no—
128#341—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134C22×D4SD1611yescharacteristic
135C23×C4—41no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×M16—21no—
139C2×M16—21no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—21no—
145C2×M16—21no—
146C2×M16—21no—
147C2×M16—21no—
148C2×M16—41no—
149C2×M16—41no—
150C2×M16—42no—
151C2×M16—42no—
152C2×M16—81no—
153C2×M16—81no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—41no—
159C22×C8—41no—
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8#4111yescharacteristic
164C22×C8#4111yescharacteristic
165C22×C8#4111yescharacteristic
166C22×C8#4111yescharacteristic
167C22×C8C2411yescharacteristic
168#123—84no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4SD1611yescharacteristic
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4SD1611yescharacteristic
189C2×C22⋊C4SD1611yescharacteristic
190C2×C42C2×D411yescharacteristic
191M32—21no—
192M32—21no—
193M32—21no—
194M32—21no—
195M32—21no—
196M32—21no—
197M32—21no—
198M32—42no—
199M32#4112yes—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—42no—
205#108—21no—
206#108—21no—
207#108—21no—
208#108—21no—
209#108—21no—
210#108—21no—
211#108—21no—
212#108—42no—
213#108—42no—
214#108—81no—
215#108#4112yes—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—41no—
221#107—41no—
222#106—21no—
223#106—21no—
224#106—21no—
225#106—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—41no—
229C4⋊C8—41no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232C22⋊C8—21no—
233C22⋊C8—21no—
234C22⋊C8—41no—
235C22⋊C8—41no—
236C22⋊C8—41no—
237C22⋊C8—41no—
238#97—21no—
239#97—21no—
240#97—41no—
241C4×C8—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247C24C2×SD1611yescharacteristic
248C24C2×SD1611yescharacteristic
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—42no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4#12311yescharacteristic
271C22×C4#12311yescharacteristic
272C22×C4C2×SD1611yescharacteristic
273C22×C4C2×SD1611yescharacteristic
274C22×C4C2×SD1611yescharacteristic
275C22×C4C2×SD1611yescharacteristic
276C22×C4AGL(1,ℤ/8)11yescharacteristic
277C22×C4AGL(1,ℤ/8)11yescharacteristic
278C22×C4C22×D411yescharacteristic
279M16—22no—
280M16—41no—
281M16—41no—
282M16—41no—
283M16—41no—
284M16—41no—
285M16—41no—
286M16—41no—
287M16—41no—
288M16—41no—
289M16—41no—
290M16—41no—
291M16—82no—
292M16—82no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
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—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—82no—
314C2×C8—82no—
315C2×C8#12311yescharacteristic
316C2×C8#12311yescharacteristic
317C2×C8C2×#4111yescharacteristic
318C2×C8C2×#4111yescharacteristic
319C2×C8C2×#4111yescharacteristic
320C2×C8C2×#4111yescharacteristic
321C2×C8H(5,𝔽2)11yescharacteristic
322C2×C82−1+411yescharacteristic
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—81no—
336C4⋊C4—81no—
337C22⋊C4—21no—
338C22⋊C4—21no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C42—21no—
345C42—21no—
346C16—22no—
347C16—41no—
348C16—41no—
349C16—41no—
350C23—21no—
351C23—22no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23#48111yescharacteristic
360C23#48111yescharacteristic
361C23C22×SD1611yescharacteristic
362D4—42no—
363D4—81no—
364C2×C4—21no—
365C2×C4—21no—
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—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
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—162no—
398C2×C4—162no—
399C2×C4#48111yescharacteristic
400C2×C4#48111yescharacteristic
401C2×C4C2×#12311yescharacteristic
402C2×C4#54611yescharacteristic
403C2×C4#54711yescharacteristic
404C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
405C2×C4#57711yescharacteristic
406C2×C4#57811yescharacteristic
407C8—21no—
408C8—21no—
409C8—41no—
410C8—41no—
411C8—42no—
412C8—81no—
413C8—81no—
414C8—81no—
415C8#52911yescharacteristic
416C8#53911yescharacteristic
417C8#55111yescharacteristic
418C8#55211yescharacteristic
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—42no—
427C22—42no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—81no—
432C22—81no—
433C22#223812yes—
434C22C2×#48111yescharacteristic
435C22#329811yescharacteristic
436C22#333511yescharacteristic
437C4—21no—
438C4—21no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4—161no—
443C4—161no—
444C4—161no—
445C4—161no—
446C4#241911yescharacteristic
447C4#309711yescharacteristic
448C4#331011yescharacteristic
449C4#334311yescharacteristic
450C2—21no—
451C2—21no—
452C2—22no—
453C2—41no—
454C2—81no—
455C2#2493711yescharacteristic
456C2C2×#223811yescharacteristic
457C2#3365811yescharacteristic
subgroupG/H
center, socleC22C2×#481
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D4SD16
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#435 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #13541C2semidirect
4. #13510C2semidirect
5. #13501C2semidirect
6. #13501C2semidirect
7. #13114C2semidirect
8. #13114C2semidirect
10. #12931C2semidirect
11. #12928C2semidirect
12. #12875C2semidirect
13. #12871C2semidirect
14. #12831C2semidirect
15. C2×#1387C2semidirect

Maximal subgroups

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

subgroupindex
#276672
#226682
#135412
#135102
#135012
#131142
#129852
#129312
#129282
#128752
#128712
#128312
C2×#13872