Finite Group Databasebeta

← #495836

Subgroups — #495836

Order 512.

666 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33594C211yesmaximal, characteristic
2#33463C212yesmaximal, complemented
3#21707C211yesmaximal, characteristic, complemented
4#20514C211yesmaximal, characteristic, complemented
5#13259C211yesmaximal, characteristic, complemented
6#13182C211yesmaximal, characteristic, complemented
7#13137C212yesmaximal, complemented
8#13027C212yesmaximal, complemented
9#12822C211yesmaximal, characteristic, complemented
10#12656C211yesmaximal, characteristic, complemented
11#12646C212yesmaximal, complemented
12#3414C2212yes—
13#3147C2211yescharacteristic
14C2×#444—21no—
15C2×#443C2212yes—
16C2×#443C2211yescharacteristic
17C2×#443C2211yescharacteristic
18C4×#135C2211yescharacteristic
19#2932—21no—
20#2875—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#1975—21no—
35#1974C2212yes—
36#1963—21no—
37#1848C2211yescharacteristic
38C4×#108C2211yescharacteristic
39#1772C2211yescharacteristic
40#1772C2211yescharacteristic
41#1765—21no—
42#1765—21no—
43#1759—21no—
44#1726C2211yescharacteristic
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#444—41no—
69#444—41no—
70#444—41no—
71#444—41no—
72#443—22no—
73#443—22no—
74#443—21no—
75#443—21no—
76#443—22no—
77#443—22no—
78#443—22no—
79#443C2×C412yes—
80#443C2×C412yes—
81#443D414yescomplemented
82C4×Q16—21no—
83C4×SD16—22no—
84C4×D8—21no—
85#435—44no—
86C2×#108—21no—
87C2×#108C2312yes—
88C2×#108C2311yescharacteristic
89C2×#108C2311yescharacteristic
90#427C2311yescharacteristic
91#423—21no—
92C2×#104—21no—
93C2×#104—21no—
94C2×#104—22no—
95C2×#104—22no—
96#416—21no—
97#408—22no—
98#405—21no—
99#405—21no—
100#405—41no—
101#405—41no—
102C4×M16—21no—
103C4×M16—41no—
104C2×#97—41no—
105C2×#97D411yescharacteristic
106C2×#97D411yescharacteristic
107C2×#97D411yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#376D411yescharacteristic
112#368—24no—
113#364—21no—
114#364—21no—
115#364C2×C412yescomplemented
116#364C2×C412yescomplemented
117#362C2×C414yescomplemented
118#361C2×C414yescomplemented
119#360—24no—
120#359—24no—
121#359—24no—
122#346—22no—
123C4×C16—22no—
124#341—21no—
125#339—21no—
126#339—21no—
127C2×#41—21no—
128C2×#41—22no—
129#135—22no—
130#135—21no—
131#135—21no—
132#135—22no—
133#135—22no—
134#135—22no—
135#135—41no—
136#135—41no—
137#135—41no—
138#135—41no—
139#135C22×C412yes—
140#135C2×D412yes—
141C2×Q16—21no—
142C2×Q16C22×C412yes—
143C2×Q16#4111yescharacteristic
144C2×SD16—22no—
145C2×SD16—21no—
146C2×SD16—22no—
147C2×D8—21no—
148C2×D8C22×C412yes—
149C2×D8#4111yescharacteristic
150#131—22no—
151#131—22no—
152#131—24no—
153#131—41no—
154#131—41no—
155#131—42no—
156#131—42no—
157#131—42no—
158#131—42no—
159#131—42no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—22no—
163C2×M16—41no—
164C2×M16—41no—
165C2×M16—42no—
166C22×C8—41no—
167C22×C8—41no—
168C22×C8—42no—
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2411yescharacteristic
173C4⋊Q8—21no—
174#127—21no—
175#124—22no—
176#121—42no—
177C4×Q8—22no—
178C4×D4—22no—
179C4×D4—42no—
180C4×D4—42no—
181#117—21no—
182#117—41no—
183C2×C42—41no—
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187Q32—24no—
188SD32—24no—
189SD32—24no—
190D16—24no—
191M32—22no—
192M32—22no—
193M32—22no—
194M32C22×C414yes—
195C2×C16—22no—
196C2×C16—22no—
197#108—22no—
198#108—21no—
199#108—21no—
200#108—22no—
201#108—41no—
202#108—41no—
203#108C22×C412yes—
204#108C2×D412yes—
205#107—21no—
206#106#4112yes—
207C4⋊C8—42no—
208#104—22no—
209#104—22no—
210#104—24no—
211#104—42no—
212#104—42no—
213#104—42no—
214#104—42no—
215#103—22no—
216#102—22no—
217C22⋊C8—44no—
218#97—21no—
219#97—21no—
220#97—22no—
221#97—41no—
222#97—41no—
223#97—41no—
224#97D811yescharacteristic
225#97D811yescharacteristic
226#97D811yescharacteristic
227#97D811yescharacteristic
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—41no—
231C4×C8—41no—
232C4×C8—41no—
233C4×C8C22×C411yescharacteristic
234C4×C8C22×C411yescharacteristic
235C4×C8C2×D412yes—
236C4×C8#4111yescharacteristic
237C4×C8#4111yescharacteristic
238C4×C8#4112yes—
239#95—41no—
240#41—22no—
241#41—22no—
242#41—41no—
243#41—41no—
244#41—42no—
245#41—42no—
246#41—42no—
247#41—42no—
248C2×Q8—22no—
249C2×Q8—21no—
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—42no—
253C2×D4—41no—
254C2×D4—42no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4C22×D411yescharacteristic
262Q16—22no—
263Q16—22no—
264Q16—21no—
265Q16—41no—
266Q16—41no—
267Q16C4×D412yes—
268SD16—22no—
269SD16—22no—
270SD16—42no—
271SD16—42no—
272SD16—42no—
273SD16—42no—
274D8—22no—
275D8—22no—
276D8—21no—
277D8—41no—
278D8—41no—
279D8C4×D412yes—
280M16—22no—
281M16—22no—
282M16—41no—
283M16—41no—
284M16—42no—
285M16—42no—
286M16—44no—
287M16—42no—
288M16—42no—
289M16—82no—
290M16—82no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—44no—
299C2×C8—42no—
300C2×C8—82no—
301C2×C8—82no—
302C2×C8C4×D411yescharacteristic
303C2×C8C4×D411yescharacteristic
304C2×C8#12112yes—
305C2×C8C2×D811yescharacteristic
306C2×C8C2×D811yescharacteristic
307C2×C8C2×D811yescharacteristic
308C2×C8C2×D811yescharacteristic
309C2×C8#13511yescharacteristic
310C2×C8#13511yescharacteristic
311C2×C8AGL(1,ℤ/8)12yes—
312C2×C8C23×C411yescharacteristic
313C2×C8C22×D411yescharacteristic
314C2×C8C2×#4111yescharacteristic
315C2×C8C2×#4111yescharacteristic
316C2×C8C2×#4111yescharacteristic
317C2×C8C2×#4111yescharacteristic
318C4⋊C4—22no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—44no—
323C22⋊C4—41no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—42no—
328C42—41no—
329C42C4×D411yescharacteristic
330C42C4×D411yescharacteristic
331C42#12112yes—
332C42C2×D811yescharacteristic
333C42C2×D811yescharacteristic
334C16—24no—
335C16—42no—
336C23—41no—
337C23—42no—
338C23#57511yescharacteristic
339Q8—22no—
340Q8—41no—
341Q8—42no—
342Q8—42no—
343D4—22no—
344D4—42no—
345D4—42no—
346D4—41no—
347D4—42no—
348D4—42no—
349D4—82no—
350D4—82no—
351D4—162no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—162no—
368C2×C4C2×C4×D411yescharacteristic
369C2×C4C2×#12111yescharacteristic
370C2×C4#52511yescharacteristic
371C2×C4C22×D811yescharacteristic
372C2×C4C2×#13511yescharacteristic
373C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
374C8—41no—
375C8—42no—
376C8—42no—
377C8—82no—
378C8C4×D811yescharacteristic
379C8C4×D811yescharacteristic
380C8C4×D811yescharacteristic
381C8C4×D811yescharacteristic
382C8#45912yes—
383C8#45912yes—
384C8C2×C4×D411yescharacteristic
385C8C4×#4111yescharacteristic
386C8#52911yescharacteristic
387C8#53211yescharacteristic
388C22—41no—
389C22—42no—
390C22—82no—
391C22—84no—
392C22—82no—
393C22C4×AGL(1,ℤ/8)11yescharacteristic
394C22#303511yescharacteristic
395C22#310011yescharacteristic
396C4—41no—
397C4—41no—
398C4—42no—
399C4—41no—
400C4—41no—
401C4—42no—
402C4—82no—
403C4—82no—
404C4C4×#12111yescharacteristic
405C4C2×C4×D811yescharacteristic
406C4C2×#45911yescharacteristic
407C4#310611yescharacteristic
408C2—21no—
409C2—21no—
410C2—21no—
411C2—82no—
412C2—162no—
413C2C4×#45911yescharacteristic
subgroupG/H
centerC4C2×#459
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC2C4×#459
prime radical#33594C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C8 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33463C2semidirect
3. #21707C2semidirect
4. #20514C2semidirect
5. #13259C2semidirect
6. #13182C2semidirect
7. #13137C2semidirect
8. #13027C2semidirect
9. #12822C2semidirect
10. #12656C2semidirect
11. #12646C2semidirect
21. #2214C4semidirect
22. #2213C4semidirect
23. #2212C4semidirect
64. #495C2×C4semidirect
65. #494C2×C4semidirect
66. #493C2×C4semidirect
81. #443D4semidirect
115. #364C2×C4semidirect
116. #364C2×C4semidirect
117. #362C2×C4semidirect
118. #361C2×C4semidirect

Maximal subgroups

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

subgroupindex
#335942
#334632
#217072
#205142
#132592
#131822
#131372
#130272
#128222
#126562
#126462