Finite Group Databasebeta

← #495000

Subgroups — #495000

Order 512.

551 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33468C211yesmaximal, characteristic
2#21296C211yesmaximal, characteristic, complemented
3#20513C211yesmaximal, characteristic, complemented
4#13140C211yesmaximal, characteristic, complemented
5#13137C211yesmaximal, characteristic, complemented
6#13117C211yesmaximal, characteristic, complemented
7#13115C211yesmaximal, characteristic, complemented
8#13027C211yesmaximal, characteristic
9#13024C211yesmaximal, characteristic
10#12951C211yesmaximal, characteristic, complemented
11#12944C211yesmaximal, characteristic, complemented
12#12678C211yesmaximal, characteristic, complemented
13#12651C211yesmaximal, characteristic, complemented
14#12649C211yesmaximal, characteristic, complemented
15#12596C211yesmaximal, characteristic, complemented
16#3414C2211yescharacteristic
17C2×#444C2211yescharacteristic
18C2×#444C2211yescharacteristic
19#2943C2211yescharacteristic
20C4×#135C2211yescharacteristic
21#2922—21no—
22#2875—21no—
23AGL(1,ℤ/16)C414yescomplemented
24#2180C414yescomplemented
25#2161C2211yescharacteristic
26#2155C2211yescharacteristic
27#2155C2211yescharacteristic
28#2148C2211yescharacteristic
29#2148C2211yescharacteristic
30#2145C2211yescharacteristic
31#2145C2211yescharacteristic
32#2143C2211yescharacteristic
33#2143C2211yescharacteristic
34C2×#347C2211yescharacteristic
35C2×#347C2211yescharacteristic
36#1975C2211yescharacteristic
37#1974C2211yescharacteristic
38#1946C2211yescharacteristic
39#1945C2211yescharacteristic
40#1938—21no—
41#1937—21no—
42#1778C2211yescharacteristic
43C4×#108C2211yescharacteristic
44#1773C2211yescharacteristic
45#1773C2211yescharacteristic
46#1764—21no—
47#1764—21no—
48#1759—21no—
49#1759—21no—
50#1726C2211yescharacteristic
51#1389C2211yescharacteristic
52#1389C2211yescharacteristic
53#1388C2211yescharacteristic
54#1388C2211yescharacteristic
55#1385C2211yescharacteristic
56#1384C2211yescharacteristic
57#1377C2211yescharacteristic
58#1377C2211yescharacteristic
59#1376C2211yescharacteristic
60#1376C2211yescharacteristic
61C2×#135C2311yescharacteristic
62C2×#135C2311yescharacteristic
63C2×#131—21no—
64C2×#131—21no—
65C4×#41—21no—
66C4×#41—21no—
67#510—21no—
68#510C2×C412yes—
69#509—21no—
70#509C2×C412yes—
71#508—22no—
72C2×M32C2311yescharacteristic
73C2×M32C2311yescharacteristic
74C2×M32C2311yescharacteristic
75C2×M32C2311yescharacteristic
76#444—22no—
77#444—22no—
78#444—22no—
79#444C2×C414yes—
80#442C2×C412yescomplemented
81#441C2×C412yescomplemented
82#440—22no—
83C4×Q16—21no—
84C4×SD16—22no—
85C4×D8—21no—
86#433—42no—
87#431—42no—
88C2×#108C2311yescharacteristic
89C2×#108C2311yescharacteristic
90C2×#108C2311yescharacteristic
91C2×#108C2311yescharacteristic
92#427C2311yescharacteristic
93#427C2311yescharacteristic
94#424—21no—
95#424—21no—
96C2×#104—21no—
97C2×#104—21no—
98C2×#104—21no—
99C2×#104—21no—
100#416—21no—
101#416—21no—
102#408—21no—
103#408—21no—
104#405—21no—
105#405—21no—
106#405—41no—
107#405—41no—
108C4×M16—21no—
109C4×M16—21no—
110C4×M16—41no—
111C2×#97—41no—
112C2×#97D411yescharacteristic
113C2×#97C2311yescharacteristic
114C2×#97C2311yescharacteristic
115C2×C4×C8D411yescharacteristic
116C2×C4×C8D411yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#376D411yescharacteristic
119#368—22no—
120#365—22no—
121#365C2×C414yescomplemented
122#362—21no—
123#362C2×C412yescomplemented
124#361—21no—
125#361C2×C412yescomplemented
126#360—22no—
127#360C2×C414yes—
128#359—22no—
129#359—22no—
130#347—22no—
131#347C2×C414yescomplemented
132#341—21no—
133#341—21no—
134#339—21no—
135#339—21no—
136#339—21no—
137#339—21no—
138C2×#41—21no—
139C2×#41—21no—
140#135—21no—
141#135—21no—
142#135—21no—
143#135—21no—
144#135—21no—
145#135—21no—
146#135C22×C412yes—
147#135#4112yes—
148C2×Q16C22×C411yescharacteristic
149C2×Q16#4111yescharacteristic
150C2×SD16—21no—
151C2×SD16—21no—
152C2×D8C22×C411yescharacteristic
153C2×D8#4111yescharacteristic
154#131—24no—
155#131—24no—
156#131—42no—
157#131—42no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—41no—
163C2×M16—41no—
164C22×C8—41no—
165C22×C8—41no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2411yescharacteristic
170C4×Q8—22no—
171C4×Q8—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—42no—
175C4×D4—42no—
176#117—21no—
177#117—21no—
178#117—41no—
179#117—41no—
180C2×C42—41no—
181C2×C42—41no—
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185Q32—22no—
186SD32—22no—
187SD32—22no—
188D16—22no—
189M32—21no—
190M32—21no—
191M32—21no—
192M32—21no—
193M32—21no—
194M32—21no—
195M32C22×C412yes—
196M32C22×C412yes—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201#108—21no—
202#108—21no—
203#108—21no—
204#108—21no—
205#108—21no—
206#108—21no—
207#108C22×C412yes—
208#108#4112yes—
209#107—21no—
210#107—21no—
211#106C22×C412yes—
212#106#4112yes—
213C4⋊C8—42no—
214C4⋊C8—42no—
215#104—24no—
216#104—24no—
217#104—42no—
218#104—41no—
219#104—41no—
220#103—22no—
221#103—22no—
222#102—22no—
223#102—22no—
224C22⋊C8—42no—
225C22⋊C8—42no—
226#97—21no—
227#97—21no—
228#97—21no—
229#97—21no—
230#97—41no—
231#97—41no—
232#97—41no—
233#97—41no—
234#97C22×C412yes—
235#97#4112yes—
236C4×C8—22no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—41no—
240C4×C8—41no—
241C4×C8—41no—
242C4×C8—41no—
243C4×C8—41no—
244#95—41no—
245#41—22no—
246#41—22no—
247#41—41no—
248#41—41no—
249#41—41no—
250#41—41no—
251#41—41no—
252#41—41no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C2×D4—41no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4C22×D411yescharacteristic
267Q16—21no—
268Q16—21no—
269Q16—21no—
270Q16#11712yes—
271SD16—22no—
272SD16—22no—
273SD16—41no—
274SD16—41no—
275D8—21no—
276D8—21no—
277D8—21no—
278D8#11712yes—
279M16—22no—
280M16—22no—
281M16—44no—
282M16—44no—
283M16—41no—
284M16—41no—
285M16—41no—
286M16—41no—
287M16—41no—
288M16—41no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—44no—
297C2×C8—44no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8C4×D412yes—
303C2×C8AGL(1,ℤ/8)12yes—
304C2×C8C23×C411yescharacteristic
305C2×C8C2×#4111yescharacteristic
306C2×C8C2×#4111yescharacteristic
307C2×C8C2×#4111yescharacteristic
308C2×C8C2×#4111yescharacteristic
309C2×C8C2×#4111yescharacteristic
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—42no—
323C42—42no—
324C42—41no—
325C42—41no—
326C42C4×D412yes—
327C42AGL(1,ℤ/8)11yescharacteristic
328C42AGL(1,ℤ/8)11yescharacteristic
329C16—22no—
330C16—22no—
331C16—41no—
332C16—41no—
333C23—41no—
334C23—41no—
335C23#57511yescharacteristic
336Q8—22no—
337Q8—41no—
338Q8—41no—
339Q8—41no—
340D4—22no—
341D4—42no—
342D4—42no—
343D4—41no—
344D4—41no—
345D4—41no—
346D4—81no—
347D4—81no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—84no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—162no—
365C2×C4C2×C4×D411yescharacteristic
366C2×C4#52511yescharacteristic
367C2×C4#52511yescharacteristic
368C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
369C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
370C2×C4#57511yescharacteristic
371C8—22no—
372C8—22no—
373C8—42no—
374C8—41no—
375C8—42no—
376C8—81no—
377C8—81no—
378C8C2×#11711yescharacteristic
379C8C4×#4111yescharacteristic
380C8#52911yescharacteristic
381C8#52911yescharacteristic
382C22—41no—
383C22—42no—
384C22—82no—
385C22—82no—
386C22—81no—
387C22C4×AGL(1,ℤ/8)11yescharacteristic
388C22#311111yescharacteristic
389C22#311411yescharacteristic
390C4—41no—
391C4—42no—
392C4—41no—
393C4—42no—
394C4—81no—
395C4—82no—
396C4—82no—
397C4—81no—
398C4#229011yescharacteristic
399C4C4×AGL(1,ℤ/8)11yescharacteristic
400C4#310911yescharacteristic
401C4#311511yescharacteristic
402C2—21no—
403C2—21no—
404C2—21no—
405C2—82no—
406C2—161no—
407C2#2112211yescharacteristic
subgroupG/H
centerC4#3109
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC2#21122
prime radical#3414C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21296C2semidirect
3. #20513C2semidirect
4. #13140C2semidirect
5. #13137C2semidirect
6. #13117C2semidirect
7. #13115C2semidirect
10. #12951C2semidirect
11. #12944C2semidirect
12. #12678C2semidirect
13. #12651C2semidirect
14. #12649C2semidirect
15. #12596C2semidirect
23. AGL(1,ℤ/16)C4semidirect
24. #2180C4semidirect
80. #442C2×C4semidirect
81. #441C2×C4semidirect
121. #365C2×C4semidirect
123. #362C2×C4semidirect
125. #361C2×C4semidirect
131. #347C2×C4semidirect

Maximal subgroups

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

subgroupindex
#334682
#212962
#205132
#131402
#131372
#131172
#131152
#130272
#130242
#129512
#129442
#126782
#126512
#126492
#125962