Finite Group Databasebeta

← #495835

Subgroups — #495835

Order 512.

680 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33594C211yesmaximal, characteristic
2#33463C212yesmaximal, complemented
3#21705C211yesmaximal, characteristic, complemented
4C4×#443C211yesmaximal, characteristic, complemented
5#13259C211yesmaximal, characteristic, complemented
6#13182C211yesmaximal, characteristic, complemented
7#13137C212yesmaximal, complemented
8#13027C212yesmaximal, complemented
9#12822C211yesmaximal, characteristic, complemented
10C4×#364C211yesmaximal, characteristic, complemented
11#12646C212yesmaximal, complemented
12#3414C2212yes—
13#3147C2211yescharacteristic
14C2×#443—21no—
15C2×#443C2212yes—
16C2×#443C2211yescharacteristic
17C2×#443C2211yescharacteristic
18C4×#135C2211yescharacteristic
19#2931—21no—
20C4×#131—21no—
21#2214C412yescomplemented
22#2213C412yescomplemented
23#2212C414yescomplemented
24#2179—24no—
25#2178—24no—
26#2161C2212yes—
27C2×#364C2211yescharacteristic
28C2×#364C2211yescharacteristic
29#2148C2212yes—
30#2145C2212yes—
31#2143C2212yes—
32#2143C2212yes—
33C4×M32C2212yes—
34#1974—21no—
35#1974C2212yes—
36#1962—21no—
37#1848C2211yescharacteristic
38C4×#108C2211yescharacteristic
39#1772C2211yescharacteristic
40#1772C2211yescharacteristic
41#1764—21no—
42#1764—21no—
43C4×#104—21no—
44C42×C8C2211yescharacteristic
45#1389C2212yes—
46#1388C2212yes—
47#1376C2212yes—
48C2×#135—21no—
49C2×#135C2312yes—
50C2×#135C2311yescharacteristic
51C2×#131—21no—
52C2×#131—21no—
53C2×#131—22no—
54#532—21no—
55C4×#41—21no—
56#510—22no—
57#510C2×C414yes—
58#509—22no—
59#509C2×C414yes—
60#508—24no—
61C2×M32C2312yes—
62C2×M32C2312yes—
63#495—21no—
64#495C2×C412yescomplemented
65#494C2×C411yescharacteristic, complemented
66#493C2×C411yescharacteristic, complemented
67#492—21no—
68#443—22no—
69#443—22no—
70#443—21no—
71#443—21no—
72#443—21no—
73#443—21no—
74#443—22no—
75#443—22no—
76#443—41no—
77#443—41no—
78#443—41no—
79#443—41no—
80#443C2×C412yes—
81#443C2×C412yes—
82#443D412yescomplemented
83#443D412yescomplemented
84C4×Q16—21no—
85C4×SD16—22no—
86C4×D8—21no—
87#436—42no—
88C8×D4—42no—
89C2×#108—21no—
90C2×#108C2312yes—
91C2×#108C2311yescharacteristic
92C2×#108C2311yescharacteristic
93#427C2311yescharacteristic
94#423—21no—
95C2×#104—21no—
96C2×#104—21no—
97C2×#104—22no—
98C2×#104—22no—
99#416—21no—
100#408—22no—
101#405—21no—
102#405—21no—
103#405—41no—
104#405—41no—
105C4×M16—21no—
106C4×M16—41no—
107C2×C4×C8—41no—
108C2×C4×C8D411yescharacteristic
109C2×C4×C8D411yescharacteristic
110C2×C4×C8D411yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C43D411yescharacteristic
115#368—24no—
116#364—21no—
117#364—21no—
118#364C2×C412yescomplemented
119#364C2×C412yescomplemented
120#362C2×C414yescomplemented
121#361C2×C414yescomplemented
122#360—24no—
123#359—24no—
124#359—24no—
125#346—22no—
126C4×C16—22no—
127#341—21no—
128#339—21no—
129#339—21no—
130C2×#41—21no—
131C2×#41—22no—
132#135—22no—
133#135—21no—
134#135—21no—
135#135—22no—
136#135—22no—
137#135—22no—
138#135—41no—
139#135—41no—
140#135—41no—
141#135—41no—
142#135C22×C412yes—
143#135C2×D412yes—
144C2×Q16—21no—
145C2×Q16C22×C412yes—
146C2×Q16#4111yescharacteristic
147C2×SD16—22no—
148C2×SD16—21no—
149C2×SD16—22no—
150C2×D8—21no—
151C2×D8C22×C412yes—
152C2×D8#4111yescharacteristic
153#131—22no—
154#131—22no—
155#131—22no—
156#131—22no—
157#131—41no—
158#131—41no—
159#131—41no—
160#131—41no—
161#131—42no—
162#131—42no—
163#131—42no—
164#131—42no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—22no—
168C2×M16—41no—
169C2×M16—41no—
170C2×M16—42no—
171C22×C8—41no—
172C22×C8—41no—
173C22×C8—42no—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2411yescharacteristic
178C4⋊Q8—21no—
179#127—21no—
180#124—22no—
181#121—42no—
182C4×Q8—22no—
183C4×D4—22no—
184C4×D4—42no—
185C4×D4—42no—
186#117—21no—
187#117—41no—
188C2×C42—41no—
189C2×C42—41no—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193Q32—24no—
194SD32—24no—
195SD32—24no—
196D16—24no—
197M32—22no—
198M32—22no—
199M32—22no—
200M32C22×C414yes—
201C2×C16—22no—
202C2×C16—22no—
203#108—22no—
204#108—21no—
205#108—21no—
206#108—22no—
207#108—41no—
208#108—41no—
209#108C22×C412yes—
210#108C2×D412yes—
211#107—21no—
212#106#4112yes—
213C4⋊C8—42no—
214#104—22no—
215#104—22no—
216#104—22no—
217#104—22no—
218#104—42no—
219#104—42no—
220#104—42no—
221#104—42no—
222#103—22no—
223#102—22no—
224C22⋊C8—44no—
225#97—41no—
226#97—41no—
227#97—41no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—41no—
235C4×C8—41no—
236C4×C8—41no—
237C4×C8D811yescharacteristic
238C4×C8D811yescharacteristic
239C4×C8D811yescharacteristic
240C4×C8D811yescharacteristic
241C4×C8C22×C411yescharacteristic
242C4×C8C22×C411yescharacteristic
243C4×C8C2×D411yescharacteristic
244C4×C8C2×D411yescharacteristic
245C4×C8#4111yescharacteristic
246C4×C8#4111yescharacteristic
247C4×C8#4111yescharacteristic
248C4×C8#4111yescharacteristic
249#41—22no—
250#41—22no—
251#41—41no—
252#41—41no—
253#41—42no—
254#41—42no—
255#41—42no—
256#41—42no—
257C2×Q8—22no—
258C2×Q8—21no—
259C2×D4—22no—
260C2×D4—21no—
261C2×D4—42no—
262C2×D4—41no—
263C2×D4—42no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4C22×D411yescharacteristic
271Q16—22no—
272Q16—22no—
273Q16—21no—
274Q16—41no—
275Q16—41no—
276Q16C4×D412yes—
277SD16—22no—
278SD16—22no—
279SD16—42no—
280SD16—42no—
281SD16—42no—
282SD16—42no—
283D8—22no—
284D8—22no—
285D8—21no—
286D8—41no—
287D8—41no—
288D8C4×D412yes—
289M16—22no—
290M16—22no—
291M16—41no—
292M16—41no—
293M16—42no—
294M16—42no—
295M16—42no—
296M16—42no—
297M16—42no—
298M16—42no—
299M16—82no—
300M16—82no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—82no—
312C2×C8—82no—
313C2×C8C4×D411yescharacteristic
314C2×C8C4×D411yescharacteristic
315C2×C8#12111yescharacteristic
316C2×C8#12111yescharacteristic
317C2×C8C2×D811yescharacteristic
318C2×C8C2×D811yescharacteristic
319C2×C8C2×D811yescharacteristic
320C2×C8C2×D811yescharacteristic
321C2×C8#13511yescharacteristic
322C2×C8#13511yescharacteristic
323C2×C8AGL(1,ℤ/8)11yescharacteristic
324C2×C8AGL(1,ℤ/8)11yescharacteristic
325C2×C8C23×C411yescharacteristic
326C2×C8C22×D411yescharacteristic
327C2×C8C2×#4111yescharacteristic
328C2×C8C2×#4111yescharacteristic
329C2×C8C2×#4111yescharacteristic
330C2×C8C2×#4111yescharacteristic
331C4⋊C4—22no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—44no—
336C22⋊C4—41no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—42no—
341C42—44no—
342C42—41no—
343C42C4×D411yescharacteristic
344C42C4×D411yescharacteristic
345C42#12111yescharacteristic
346C42#12111yescharacteristic
347C42C2×D811yescharacteristic
348C42C2×D811yescharacteristic
349C16—24no—
350C16—42no—
351C23—41no—
352C23—42no—
353C23#57511yescharacteristic
354Q8—22no—
355Q8—41no—
356Q8—42no—
357Q8—42no—
358D4—22no—
359D4—42no—
360D4—42no—
361D4—41no—
362D4—42no—
363D4—42no—
364D4—84no—
365D4—82no—
366D4—82no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—84no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4C2×C4×D411yescharacteristic
385C2×C4C2×#12111yescharacteristic
386C2×C4#52511yescharacteristic
387C2×C4C22×D811yescharacteristic
388C2×C4C2×#13511yescharacteristic
389C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
390C8—41no—
391C8—42no—
392C8—42no—
393C8—82no—
394C8C4×D811yescharacteristic
395C8C4×D811yescharacteristic
396C8C4×D811yescharacteristic
397C8C4×D811yescharacteristic
398C8#45911yescharacteristic
399C8#45911yescharacteristic
400C8#45911yescharacteristic
401C8#45911yescharacteristic
402C8C2×C4×D411yescharacteristic
403C8C4×#4111yescharacteristic
404C8#52911yescharacteristic
405C8#53211yescharacteristic
406C22—41no—
407C22—42no—
408C22—82no—
409C22—84no—
410C22—82no—
411C22C4×AGL(1,ℤ/8)11yescharacteristic
412C22#303511yescharacteristic
413C22#310011yescharacteristic
414C4—22no—
415C4—22no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—42no—
421C4—82no—
422C4—82no—
423C4C4×#12111yescharacteristic
424C4C2×C4×D811yescharacteristic
425C4C2×#45911yescharacteristic
426C4#310611yescharacteristic
427C2—21no—
428C2—21no—
429C2—21no—
430C2—82no—
431C2—162no—
432C2C4×#45911yescharacteristic
subgroupG/H
centerC8#459
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC2C4×#459
prime radical#33594C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C8 ◁ C22×C8 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33463C2semidirect
3. #21705C2semidirect
4. C4×#443C2semidirect
5. #13259C2semidirect
6. #13182C2semidirect
7. #13137C2semidirect
8. #13027C2semidirect
9. #12822C2semidirect
10. C4×#364C2semidirect
11. #12646C2semidirect
21. #2214C4semidirect
22. #2213C4semidirect
23. #2212C4semidirect
64. #495C2×C4semidirect
65. #494C2×C4semidirect
66. #493C2×C4semidirect
82. #443D4semidirect
83. #443D4semidirect
118. #364C2×C4semidirect
119. #364C2×C4semidirect
120. #362C2×C4semidirect
121. #361C2×C4semidirect

Maximal subgroups

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

subgroupindex
#335942
#334632
#217052
C4×#4432
#132592
#131822
#131372
#130272
#128222
C4×#3642
#126462