Finite Group Databasebeta

← #504264

Subgroups — #504264

Order 512.

532 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2231C211yesmaximal, characteristic, complemented
2C2×#2230C211yesmaximal, characteristic, complemented
3#28810C211yesmaximal, characteristic, complemented
4#23179C211yesmaximal, characteristic
5#13467C211yesmaximal, characteristic, complemented
6#13467C211yesmaximal, characteristic, complemented
7#13372C211yesmaximal, characteristic, complemented
8#13371C211yesmaximal, characteristic, complemented
9#13255C211yesmaximal, characteristic
10#13211C211yesmaximal, characteristic
11#13184C211yesmaximal, characteristic, complemented
12#13163C211yesmaximal, characteristic, complemented
13#12950C211yesmaximal, characteristic, complemented
14#12950C211yesmaximal, characteristic, complemented
15C4×#365C211yesmaximal, characteristic, complemented
16C2×#499C2211yescharacteristic
17C2×#498C2211yescharacteristic
18#3147C2211yescharacteristic
19C2×#443C2211yescharacteristic
20C2×#443C2211yescharacteristic
21#2935—21no—
22#2550—21no—
23#2231—24no—
24#2230—24no—
25C2×#365C2211yescharacteristic
26C2×#365C2211yescharacteristic
27C2×#365C2211yescharacteristic
28C2×#365C2211yescharacteristic
29C2×#364C2211yescharacteristic
30C2×#364C2211yescharacteristic
31#2150C2211yescharacteristic
32C2×#360C2211yescharacteristic
33C2×#360C2211yescharacteristic
34C2×#360C2211yescharacteristic
35C2×#360C2211yescharacteristic
36C2×#360C2211yescharacteristic
37C2×#360C2211yescharacteristic
38C2×#360C2211yescharacteristic
39C2×#360C2211yescharacteristic
40C4×M32C2211yescharacteristic
41#2059—21no—
42#2057—21no—
43#1980—21no—
44#1974C2211yescharacteristic
45#1974C2211yescharacteristic
46#1949—21no—
47#1941C2211yescharacteristic
48#1941C2211yescharacteristic
49#1876—21no—
50#1876—21no—
51#1846C2211yescharacteristic
52C4×#108C2211yescharacteristic
53C4×#106C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55#1383C2211yescharacteristic
56#1383C2211yescharacteristic
57#1383C2211yescharacteristic
58#1383C2211yescharacteristic
59#1376C2211yescharacteristic
60#1376C2211yescharacteristic
61C2×#135—21no—
62C2×#135C2311yescharacteristic
63C2×#135C2311yescharacteristic
64C2×#131—21no—
65C2×#131—21no—
66#532—21no—
67C2×C4⋊Q8—21no—
68C2×#125—21no—
69C2×M32C2311yescharacteristic
70C2×M32C2311yescharacteristic
71C2×M32C2311yescharacteristic
72C2×M32C2311yescharacteristic
73#499—22no—
74#499—41no—
75#498—22no—
76#498—41no—
77#495—21no—
78#495Q811yescharacteristic, complemented
79#495Q811yescharacteristic, complemented
80#494Q811yescharacteristic, complemented
81#493Q811yescharacteristic, complemented
82#492—21no—
83#443—22no—
84#443—22no—
85#443—41no—
86#443—41no—
87#436—41no—
88#435—41no—
89#435—41no—
90C8×D4—41no—
91C2×#108C2311yescharacteristic
92C2×#108C2311yescharacteristic
93C2×#107—21no—
94C2×#107—21no—
95C2×#106—21no—
96C2×#106C2311yescharacteristic
97C2×#106C2311yescharacteristic
98C2×#106C2311yescharacteristic
99C2×#106C2311yescharacteristic
100C2×C4⋊C8D411yescharacteristic
101C2×#104—21no—
102C2×#104—21no—
103C2×#104—21no—
104C2×#104—21no—
105#408—21no—
106#408—21no—
107C4×M16—21no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#398—41no—
112#395—41no—
113#391—41no—
114#389—21no—
115#389—21no—
116#389—41no—
117#382—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4D411yescharacteristic
120#365—22no—
121#365—22no—
122#365—22no—
123#365D414yescomplemented
124#364—22no—
125#364—22no—
126C4⋊C16—21no—
127C4⋊C16—21no—
128#360—22no—
129#360—22no—
130#360—22no—
131#360—22no—
132#360—22no—
133#360—22no—
134#360—41no—
135#360—41no—
136#346—21no—
137C4×C16—21no—
138#341—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#336D411yescharacteristic
143#336D411yescharacteristic
144C2×#41—21no—
145C2×#41—21no—
146C22×Q8—41no—
147#135—21no—
148#135—21no—
149#135—21no—
150#135—21no—
151#135—21no—
152#135—21no—
153#135—21no—
154#135—21no—
155#135—81no—
156#135—81no—
157C2×Q16—21no—
158C2×Q16C2×Q811yescharacteristic
159C2×Q16C2×Q811yescharacteristic
160C2×SD16—21no—
161C2×SD16—21no—
162C2×SD16—21no—
163C2×SD16—21no—
164C2×D8—21no—
165C2×D8C2×Q811yescharacteristic
166C2×D8C2×Q811yescharacteristic
167#131—42no—
168#131—42no—
169#131—81no—
170#131—81no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—41no—
174C2×M16—41no—
175C22×C8—41no—
176C22×C8—41no—
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2411yescharacteristic
181C4⋊Q8—21no—
182C4⋊Q8—42no—
183C4⋊Q8—81no—
184#127—21no—
185#125—42no—
186#125—81no—
187#124—21no—
188#124—21no—
189#121—41no—
190#121—41no—
191C4×D4—41no—
192C4×D4—41no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C42—41no—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207M32—21no—
208M32—21no—
209M32—21no—
210M32—21no—
211M32—21no—
212M32—21no—
213M32—21no—
214M32C2×D412yes—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219#108—21no—
220#108—21no—
221#108—21no—
222#108C2×D412yes—
223#107—42no—
224#107—81no—
225#106—21no—
226#106—21no—
227#106—21no—
228#106—21no—
229#106—21no—
230#106—21no—
231#106—42no—
232#106C2×D412yes—
233#106#4112yes—
234C4⋊C8SD1611yescharacteristic
235C4⋊C8SD1611yescharacteristic
236C4⋊C8SD1611yescharacteristic
237C4⋊C8SD1611yescharacteristic
238#104—42no—
239#104—42no—
240#104—81no—
241#104—81no—
242C22⋊C8—41no—
243C22⋊C8—41no—
244C22⋊C8—41no—
245C22⋊C8—41no—
246#97—41no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—41no—
252C4×C8C2×Q811yescharacteristic
253C4×C8C2×Q811yescharacteristic
254C4×C8#4111yescharacteristic
255C4×C8#4111yescharacteristic
256#95—21no—
257#95—41no—
258#95—41no—
259#95—41no—
260#95—41no—
261#95—41no—
262#41—41no—
263#41—41no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269#41—41no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—82no—
273C2×Q8—82no—
274C2×Q8—161no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4AGL(1,ℤ/8)11yescharacteristic
290C22×C4AGL(1,ℤ/8)11yescharacteristic
291C22×C4C22×D411yescharacteristic
292Q16—21no—
293Q16—21no—
294Q16—21no—
295Q16—21no—
296Q16—81no—
297SD16—41no—
298SD16—41no—
299SD16—41no—
300SD16—41no—
301SD16—81no—
302SD16—81no—
303D8—21no—
304D8—21no—
305D8—21no—
306D8—21no—
307D8—81no—
308M16—22no—
309M16—41no—
310M16—41no—
311M16—41no—
312M16—82no—
313M16—82no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—82no—
323C2×C8—82no—
324C2×C8C22⋊Q811yescharacteristic
325C2×C8C22⋊Q811yescharacteristic
326C2×C8C2×SD1611yescharacteristic
327C2×C8C2×SD1611yescharacteristic
328C2×C8C2×SD1611yescharacteristic
329C2×C8C2×SD1611yescharacteristic
330C2×C8C22×D411yescharacteristic
331C2×C8C22×Q811yescharacteristic
332C2×C8C2×#4111yescharacteristic
333C2×C8C2×#4111yescharacteristic
334C2×C8H(5,𝔽2)11yescharacteristic
335C2×C82−1+411yescharacteristic
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4—161no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—161no—
361C42C22⋊Q811yescharacteristic
362C42C22⋊Q811yescharacteristic
363C42C2×SD1611yescharacteristic
364C42C2×SD1611yescharacteristic
365C16—22no—
366C16—41no—
367C16—41no—
368C16—41no—
369C23—41no—
370C23—41no—
371C23C2×AGL(1,ℤ/8)11yescharacteristic
372Q8—41no—
373Q8—41no—
374Q8—41no—
375Q8—41no—
376Q8—164no—
377D4—41no—
378D4—41no—
379D4—41no—
380D4—41no—
381D4—82no—
382D4—81no—
383D4—81no—
384D4—81no—
385D4—81no—
386D4—161no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4#47611yescharacteristic
414C2×C4#47611yescharacteristic
415C2×C4#47611yescharacteristic
416C2×C4#47611yescharacteristic
417C2×C4C2×C22⋊Q811yescharacteristic
418C2×C4D4211yescharacteristic
419C2×C4#54811yescharacteristic
420C2×C4C22×SD1611yescharacteristic
421C2×C4#57611yescharacteristic
422C2×C4#57711yescharacteristic
423C8—21no—
424C8—21no—
425C8—21no—
426C8—21no—
427C8—21no—
428C8—21no—
429C8—42no—
430C8—81no—
431C8#53211yescharacteristic
432C8#54211yescharacteristic
433C8D4×Q811yescharacteristic
434C8#55411yescharacteristic
435C22—21no—
436C22—21no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22C2×#47611yescharacteristic
444C22#327411yescharacteristic
445C22#337311yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—42no—
451C4—81no—
452C4—82no—
453C4—82no—
454C4—81no—
455C4—81no—
456C4—161no—
457C4—161no—
458C4#245211yescharacteristic
459C4#308111yescharacteristic
460C4D4×SD1611yescharacteristic
461C4#336211yescharacteristic
462C2—21no—
463C2—21no—
464C2—161no—
465C2—161no—
466C2#2646411yescharacteristic
467C2#3363711yescharacteristic
468C2#3363811yescharacteristic
subgroupG/H
center, socleC22C2×#476
commutator, Frattini, cosocleC22×C8C24
prime radical#3147C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ 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. C2×#2231C2semidirect
2. C2×#2230C2semidirect
3. #28810C2semidirect
5. #13467C2semidirect
6. #13467C2semidirect
7. #13372C2semidirect
8. #13371C2semidirect
11. #13184C2semidirect
12. #13163C2semidirect
13. #12950C2semidirect
14. #12950C2semidirect
15. C4×#365C2semidirect
78. #495Q8semidirect
79. #495Q8semidirect
80. #494Q8semidirect
81. #493Q8semidirect
123. #365D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22312
C2×#22302
#288102
#231792
#134672
#133722
#133712
#132552
#132112
#131842
#131632
#129502
C4×#3652