Finite Group Databasebeta

← #498237

Subgroups — #498237

Order 512.

532 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33647C211yesmaximal, characteristic
2#23019C211yesmaximal, characteristic
3#22620C211yesmaximal, characteristic, complemented
4#13174C211yesmaximal, characteristic, complemented
5#13153C211yesmaximal, characteristic, complemented
6#13143C211yesmaximal, characteristic, complemented
7#13119C211yesmaximal, characteristic, complemented
8#13118C211yesmaximal, characteristic, complemented
9#13065C211yesmaximal, characteristic
10#12955C211yesmaximal, characteristic, complemented
11#12928C211yesmaximal, characteristic, complemented
12#12677C211yesmaximal, characteristic, complemented
13#12650C211yesmaximal, characteristic, complemented
14#12616C211yesmaximal, characteristic, complemented
15#12569C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17#3050C2211yescharacteristic
18C2×#444C2211yescharacteristic
19#2933—21no—
20C22×#108C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2347—21no—
23#2237C412yescomplemented
24#2237C412yescomplemented
25#2234C412yescomplemented
26#2234C412yescomplemented
27#2159C2211yescharacteristic
28#2159C2211yescharacteristic
29C2×#365C2211yescharacteristic
30C2×#365C2211yescharacteristic
31C2×#360C2211yescharacteristic
32C2×#360C2211yescharacteristic
33#2139C2211yescharacteristic
34#2139C2211yescharacteristic
35#2112C2211yescharacteristic
36C2×#347C2211yescharacteristic
37C2×#347C2211yescharacteristic
38#1975C2211yescharacteristic
39#1946C2211yescharacteristic
40#1943C2211yescharacteristic
41#1943C2211yescharacteristic
42#1929—21no—
43#1920—21no—
44#1920—21no—
45#1871—21no—
46#1866—21no—
47#1825—21no—
48#1807C2211yescharacteristic
49#1776C2211yescharacteristic
50#1776C2211yescharacteristic
51#1773C2211yescharacteristic
52#1752C2211yescharacteristic
53#1387C2211yescharacteristic
54#1387C2211yescharacteristic
55#1387C2211yescharacteristic
56#1387C2211yescharacteristic
57#1386C2211yescharacteristic
58#1386C2211yescharacteristic
59#1383C2211yescharacteristic
60#1383C2211yescharacteristic
61#1375C2211yescharacteristic
62#1375C2211yescharacteristic
63C2×#135C2311yescharacteristic
64C2×#131—21no—
65C22×M16—21no—
66C23×C8C2311yescharacteristic
67#525—21no—
68C2×#117—21no—
69C2×M32—21no—
70C2×M32—22no—
71C2×M32—41no—
72C2×M32C2311yescharacteristic
73C2×M32C2311yescharacteristic
74C2×M32C2311yescharacteristic
75C2×M32C2311yescharacteristic
76C22×C16—21no—
77#467C2×C411yescharacteristic
78#467C2×C411yescharacteristic
79#466C2×C411yescharacteristic
80#466C2×C411yescharacteristic
81#465—21no—
82#465—21no—
83#444—22no—
84#444—41no—
85#436—41no—
86#435—41no—
87#435—41no—
88C8×D4—41no—
89C2×#108—22no—
90C2×#108—21no—
91C2×#108—41no—
92C2×#108C2311yescharacteristic
93C2×#108C2311yescharacteristic
94#427—21no—
95#427—21no—
96#427C2×C412yes—
97#427C2×C412yes—
98#427C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#107C2311yescharacteristic
101C2×#106—21no—
102C2×#106C2311yescharacteristic
103C2×#106C2311yescharacteristic
104#424—21no—
105C2×#104—21no—
106C2×#104—21no—
107#416—21no—
108#408—21no—
109C2×C22⋊C8D411yescharacteristic
110#405—21no—
111C2×#97C2311yescharacteristic
112C2×#97C2311yescharacteristic
113#389—21no—
114#389—21no—
115#388—41no—
116#388—41no—
117#383—41no—
118#380—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—41no—
122C4×C22⋊C4D411yescharacteristic
123#367C2×C412yes—
124#367C2×C412yes—
125#366C2×C412yes—
126#366C2×C412yes—
127#365—22no—
128#365—41no—
129#360—22no—
130#360—41no—
131#358C2×C412yes—
132#358C2×C412yes—
133#357C2×C412yes—
134#357C2×C412yes—
135C22⋊C16C2×C412yes—
136C22⋊C16C2×C412yes—
137#347—22no—
138#347—41no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#339—21no—
145#339—21no—
146#336D411yescharacteristic
147#336D411yescharacteristic
148C2×#41—21no—
149C23×C4C2×D411yescharacteristic
150#135—21no—
151#135—21no—
152#135—21no—
153#135—21no—
154C2×Q16C22×C411yescharacteristic
155C2×SD16—21no—
156C2×D8C22×C411yescharacteristic
157#131—42no—
158#131—81no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—41no—
162C2×M16—41no—
163C2×M16—42no—
164C2×M16—81no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—41no—
170C22×C8—41no—
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8#4111yescharacteristic
177C22×C8#4111yescharacteristic
178C22×C8C2411yescharacteristic
179#123—41no—
180C22⋊Q8—21no—
181C22⋊Q8—21no—
182#121—21no—
183#121—21no—
184#120—41no—
185C4×D4—41no—
186C4×D4—41no—
187#117—22no—
188#117—22no—
189#117—21no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C42—41no—
202C2×C42—41no—
203C2×C42—41no—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206M32—21no—
207M32—21no—
208M32—21no—
209M32—21no—
210M32—21no—
211M32—21no—
212M32—21no—
213M32—42no—
214M32#4112yes—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—42no—
218C2×C16C22×C412yes—
219C2×C16C22×C412yes—
220#108—21no—
221#108—21no—
222#108—21no—
223#108—42no—
224#108#4112yes—
225#107—21no—
226#107—21no—
227#107C22×C412yes—
228#107C22×C412yes—
229#107C22×C411yescharacteristic
230#107C22×C411yescharacteristic
231#106—22no—
232#106—22no—
233#106—21no—
234#106—21no—
235#106—21no—
236#106—21no—
237#106—21no—
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—21no—
247C22⋊C8—21no—
248C22⋊C8—41no—
249C22⋊C8—41no—
250#97—21no—
251#97—21no—
252#97—21no—
253#97—21no—
254#97—41no—
255C4×C8—41no—
256#95—21no—
257#95—41no—
258#95—41no—
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×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4C4×D411yescharacteristic
284C22×C4C4×D411yescharacteristic
285C22×C4AGL(1,ℤ/8)11yescharacteristic
286C22×C4AGL(1,ℤ/8)11yescharacteristic
287C22×C4AGL(1,ℤ/8)11yescharacteristic
288C22×C4C22×D411yescharacteristic
289Q16—21no—
290Q16—21no—
291SD16—41no—
292SD16—41no—
293D8—21no—
294D8—21no—
295M16—22no—
296M16—41no—
297M16—41no—
298M16—41no—
299M16—82no—
300M16—82no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
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—41no—
318C2×C8—41no—
319C2×C8—82no—
320C2×C8—82no—
321C2×C8C4×D411yescharacteristic
322C2×C8C4×D411yescharacteristic
323C2×C8C23×C411yescharacteristic
324C2×C8C2×#4111yescharacteristic
325C2×C8C2×#4111yescharacteristic
326C2×C8C2×#4111yescharacteristic
327C2×C8H(5,𝔽2)11yescharacteristic
328C2×C82−1+411yescharacteristic
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C22⋊C4—41no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—81no—
351C22⋊C4—161no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—41no—
357C42—42no—
358C42—42no—
359C42—161no—
360C42—161no—
361C16—22no—
362C16—22no—
363C16—22no—
364C16—41no—
365C23—21no—
366C23—21no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23#44211yescharacteristic
371C23#44211yescharacteristic
372C23C2×AGL(1,ℤ/8)11yescharacteristic
373Q8—41no—
374Q8—41no—
375D4—41no—
376D4—41no—
377D4—82no—
378D4—81no—
379D4—81no—
380D4—161no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
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—81no—
414C2×C4#44211yescharacteristic
415C2×C4#44211yescharacteristic
416C2×C4C2×C4×D411yescharacteristic
417C2×C4#54611yescharacteristic
418C2×C4#54711yescharacteristic
419C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
420C2×C4#57711yescharacteristic
421C2×C4#57811yescharacteristic
422C8—21no—
423C8—21no—
424C8—41no—
425C8—41no—
426C8—42no—
427C8—81no—
428C8C4×#4111yescharacteristic
429C8#52011yescharacteristic
430C8#55111yescharacteristic
431C8#55211yescharacteristic
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22C2×#44211yescharacteristic
446C22#330811yescharacteristic
447C22#334511yescharacteristic
448C4—21no—
449C4—21no—
450C4—81no—
451C4—81no—
452C4—81no—
453C4—82no—
454C4—81no—
455C4—82no—
456C4—81no—
457C4—161no—
458C4—161no—
459C4#235911yescharacteristic
460C4#294711yescharacteristic
461C4#331011yescharacteristic
462C4#334311yescharacteristic
463C2—21no—
464C2—21no—
465C2—41no—
466C2—41no—
467C2—161no—
468C2#2293811yescharacteristic
469C2#3346811yescharacteristic
470C2#3346911yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#3050C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22620C2semidirect
4. #13174C2semidirect
5. #13153C2semidirect
6. #13143C2semidirect
7. #13119C2semidirect
8. #13118C2semidirect
10. #12955C2semidirect
11. #12928C2semidirect
12. #12677C2semidirect
13. #12650C2semidirect
14. #12616C2semidirect
15. #12569C2semidirect
23. #2237C4semidirect
24. #2237C4semidirect
25. #2234C4semidirect
26. #2234C4semidirect

Maximal subgroups

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

subgroupindex
#336472
#230192
#226202
#131742
#131532
#131432
#131192
#131182
#130652
#129552
#129282
#126772
#126502
#126162
#125692