Finite Group Databasebeta

← #504127

Subgroups — #504127

Order 512.

567 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2230C211yesmaximal, characteristic, complemented
2C22×#365C211yesmaximal, characteristic, complemented
3#26060C211yesmaximal, characteristic, complemented
4#23018C211yesmaximal, characteristic
5#13533C211yesmaximal, characteristic, complemented
6#13521C211yesmaximal, characteristic, complemented
7#13467C211yesmaximal, characteristic, complemented
8#13457C211yesmaximal, characteristic, complemented
9#13186C211yesmaximal, characteristic, complemented
10#13116C211yesmaximal, characteristic, complemented
11#13065C211yesmaximal, characteristic
12#13001C211yesmaximal, characteristic
13#12950C211yesmaximal, characteristic, complemented
14#12928C211yesmaximal, characteristic, complemented
15#12914C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C2×#498C2211yescharacteristic
18#3050C2211yescharacteristic
19C2×#443C2211yescharacteristic
20#2933—21no—
21C22×#108C2211yescharacteristic
22C22×#106C2211yescharacteristic
23#2440—21no—
24#2230—24no—
25C2×#365—24no—
26C2×#365C2211yescharacteristic
27C2×#365C2211yescharacteristic
28C2×#365C2211yescharacteristic
29C2×#365C2211yescharacteristic
30C2×#365C2211yescharacteristic
31C2×#365C2211yescharacteristic
32C2×#364C2211yescharacteristic
33C2×#364C2211yescharacteristic
34C2×#360C2211yescharacteristic
35C2×#360C2211yescharacteristic
36C2×#360C2211yescharacteristic
37C2×#360C2211yescharacteristic
38#2112C2211yescharacteristic
39#2084—21no—
40#2058—21no—
41#1974C2211yescharacteristic
42#1945C2211yescharacteristic
43#1941C2211yescharacteristic
44#1941C2211yescharacteristic
45#1929—21no—
46#1871—21no—
47#1866—21no—
48#1853—21no—
49#1844C2211yescharacteristic
50#1807C2211yescharacteristic
51#1772C2211yescharacteristic
52C8×C22⋊C4C2211yescharacteristic
53#1386C2211yescharacteristic
54#1386C2211yescharacteristic
55#1386C2211yescharacteristic
56#1386C2211yescharacteristic
57#1383C2211yescharacteristic
58#1383C2211yescharacteristic
59#1376C2211yescharacteristic
60#1376C2211yescharacteristic
61C2×#135C2311yescharacteristic
62C2×#131—21no—
63C22×M16—21no—
64C23×C8C2311yescharacteristic
65C2×C4⋊Q8—21no—
66#525—21no—
67C2×C22⋊Q8—41no—
68C22×C4⋊C4—21no—
69C2×M32—21no—
70C2×M32—22no—
71C2×M32—22no—
72C2×M32C2311yescharacteristic
73C2×M32C2311yescharacteristic
74C2×M32C2311yescharacteristic
75C2×M32C2311yescharacteristic
76C22×C16—21no—
77#498—22no—
78#498—41no—
79#467Q811yescharacteristic, complemented
80#467Q811yescharacteristic, complemented
81#466Q811yescharacteristic, complemented
82#466Q811yescharacteristic, complemented
83#465—21no—
84#465—21no—
85#443—22no—
86#443—41no—
87#436—41no—
88#435—41no—
89#435—41no—
90C8×D4—41no—
91C2×#108—22no—
92C2×#108—21no—
93C2×#108—41no—
94C2×#108C2311yescharacteristic
95C2×#108C2311yescharacteristic
96#427C2311yescharacteristic
97C2×#106—22no—
98C2×#106—21no—
99C2×#106—21no—
100C2×#106—21no—
101C2×#106—41no—
102C2×#106C2311yescharacteristic
103C2×#106C2311yescharacteristic
104C2×#106C2311yescharacteristic
105C2×#106C2311yescharacteristic
106#424—21no—
107C2×#104—21no—
108C2×#104—21no—
109#416—21no—
110#408—21no—
111C2×C22⋊C8D411yescharacteristic
112#405—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#398—41no—
116#395—41no—
117#389—21no—
118#389—21no—
119#385—41no—
120#384—21no—
121#380—21no—
122C4×C22⋊C4D411yescharacteristic
123#365—22no—
124#365—22no—
125#365—22no—
126#365—22no—
127#365—41no—
128#365D414yescomplemented
129#364—22no—
130#364—22no—
131#360—22no—
132#360—22no—
133#360—22no—
134#360—41no—
135C22⋊C16—21no—
136C22⋊C16—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#339—21no—
141#339—21no—
142#336D411yescharacteristic
143#336D411yescharacteristic
144C2×#41—21no—
145C22×Q8—41no—
146C23×C4—41no—
147C23×C4C2×D411yescharacteristic
148#135—21no—
149#135—21no—
150#135—21no—
151#135—21no—
152C2×Q16C2×Q811yescharacteristic
153C2×SD16—21no—
154C2×D8C2×Q811yescharacteristic
155#131—42no—
156#131—81no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16—41no—
160C2×M16—41no—
161C2×M16—42no—
162C2×M16—81no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—41no—
168C22×C8—41no—
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×Q811yescharacteristic
173C22×C8C2×Q811yescharacteristic
174C22×C8#4111yescharacteristic
175C22×C8#4111yescharacteristic
176C22×C8C2411yescharacteristic
177C4⋊Q8—42no—
178C4⋊Q8—81no—
179#123—41no—
180C22⋊Q8—21no—
181C22⋊Q8—21no—
182C22⋊Q8—84no—
183#121—21no—
184#121—21no—
185#120—41no—
186C4×D4—41no—
187C4×D4—41no—
188#117—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—81no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—41no—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205M32—24no—
206M32—21no—
207M32—21no—
208M32—21no—
209M32—21no—
210M32—21no—
211M32—21no—
212M32C2×D412yes—
213M32#4112yes—
214C2×C16—24no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219#108—21no—
220#108—21no—
221#108—21no—
222#108—42no—
223#108C2×D412yes—
224#107C2×Q811yescharacteristic
225#107C2×Q811yescharacteristic
226#106—21no—
227#106—21no—
228#106—21no—
229#106—21no—
230#106—21no—
231#106—21no—
232#106—21no—
233#106—21no—
234#106—42no—
235#106—42no—
236#106—81no—
237#106C2×D412yes—
238C4⋊C8—41no—
239C4⋊C8—41no—
240#104—42no—
241#104—81no—
242#103—21no—
243#103—21no—
244#102—21no—
245#102—21no—
246C22⋊C8—41no—
247C22⋊C8—41no—
248C22⋊C8SD1611yescharacteristic
249C22⋊C8SD1611yescharacteristic
250C22⋊C8SD1611yescharacteristic
251C22⋊C8SD1611yescharacteristic
252#97—41no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—41no—
258#95—21no—
259#95—41no—
260#95—41no—
261#95—41no—
262C24AGL(1,ℤ/8)11yescharacteristic
263#41—41no—
264#41—41no—
265#41—41no—
266#41—41no—
267C2×Q8—21no—
268C2×Q8—82no—
269C2×Q8—82no—
270C2×Q8—161no—
271C2×D4—21no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—161no—
289C22×C4C22⋊Q811yescharacteristic
290C22×C4C22⋊Q811yescharacteristic
291C22×C4C2×SD1611yescharacteristic
292C22×C4C2×SD1611yescharacteristic
293C22×C4AGL(1,ℤ/8)11yescharacteristic
294C22×C4C22×D411yescharacteristic
295Q16—21no—
296Q16—21no—
297SD16—41no—
298SD16—41no—
299D8—21no—
300D8—21no—
301M16—22no—
302M16—41no—
303M16—41no—
304M16—41no—
305M16—82no—
306M16—82no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—82no—
324C2×C8—82no—
325C2×C8C22⋊Q811yescharacteristic
326C2×C8C22⋊Q811yescharacteristic
327C2×C8C2×SD1611yescharacteristic
328C2×C8C2×SD1611yescharacteristic
329C2×C8C2×SD1611yescharacteristic
330C2×C8C2×SD1611yescharacteristic
331C2×C8C22×D411yescharacteristic
332C2×C8C22×Q811yescharacteristic
333C2×C8C2×#4111yescharacteristic
334C2×C8C2×#4111yescharacteristic
335C2×C8H(5,𝔽2)11yescharacteristic
336C2×C82−1+411yescharacteristic
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—22no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—161no—
355C22⋊C4—24no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—82no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—41no—
367C16—22no—
368C16—22no—
369C16—41no—
370C16—41no—
371C23—21no—
372C23—22no—
373C23—21no—
374C23—41no—
375C23—41no—
376C23#47611yescharacteristic
377C23#47611yescharacteristic
378C23C2×AGL(1,ℤ/8)11yescharacteristic
379Q8—41no—
380Q8—41no—
381Q8—164no—
382D4—41no—
383D4—41no—
384D4—81no—
385D4—82no—
386D4—81no—
387D4—161no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—164no—
422C2×C4#47611yescharacteristic
423C2×C4#47611yescharacteristic
424C2×C4C2×C22⋊Q811yescharacteristic
425C2×C4D4211yescharacteristic
426C2×C4#54711yescharacteristic
427C2×C4C22×SD1611yescharacteristic
428C2×C4#57711yescharacteristic
429C2×C4#57711yescharacteristic
430C8—21no—
431C8—21no—
432C8—21no—
433C8—21no—
434C8—21no—
435C8—21no—
436C8—42no—
437C8—81no—
438C8#52511yescharacteristic
439C8#53911yescharacteristic
440C8D4×Q811yescharacteristic
441C8#55411yescharacteristic
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—42no—
450C22—42no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22#223012yescomplemented
455C22C2×#47611yescharacteristic
456C22#326511yescharacteristic
457C22#334511yescharacteristic
458C4—21no—
459C4—21no—
460C4—42no—
461C4—42no—
462C4—82no—
463C4—81no—
464C4—81no—
465C4—81no—
466C4—161no—
467C4—161no—
468C4—161no—
469C4#244911yescharacteristic
470C4#308411yescharacteristic
471C4D4×SD1611yescharacteristic
472C4#333511yescharacteristic
473C2—21no—
474C2—21no—
475C2—22no—
476C2—41no—
477C2—161no—
478C2#2633811yescharacteristic
479C2C2×#223011yescharacteristic
480C2#3363811yescharacteristic
subgroupG/H
center, socleC22C2×#476
commutator, Frattini, cosocleC22×C8C24
prime radical#3050C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2230C2semidirect
2. C22×#365C2semidirect
3. #26060C2semidirect
5. #13533C2semidirect
6. #13521C2semidirect
7. #13467C2semidirect
8. #13457C2semidirect
9. #13186C2semidirect
10. #13116C2semidirect
13. #12950C2semidirect
14. #12928C2semidirect
15. #12914C2semidirect
79. #467Q8semidirect
80. #467Q8semidirect
81. #466Q8semidirect
82. #466Q8semidirect
128. #365D4semidirect
454. C22#2230semidirect

Maximal subgroups

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

subgroupindex
C2×#22302
C22×#3652
#260602
#230182
#135332
#135212
#134672
#134572
#131862
#131162
#130652
#130012
#129502
#129282
#129142