Finite Group Databasebeta

← #498228

Subgroups — #498228

Order 512.

530 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33645C211yesmaximal, characteristic
2#23018C211yesmaximal, characteristic
3#22606C211yesmaximal, characteristic, complemented
4#13170C211yesmaximal, characteristic, complemented
5#13152C211yesmaximal, characteristic, complemented
6#13134C211yesmaximal, characteristic, complemented
7#13113C211yesmaximal, characteristic, complemented
8#13113C211yesmaximal, characteristic, complemented
9#13063C211yesmaximal, characteristic
10#12939C211yesmaximal, characteristic, complemented
11#12922C211yesmaximal, characteristic, complemented
12#12671C211yesmaximal, characteristic, complemented
13#12642C211yesmaximal, characteristic, complemented
14#12611C211yesmaximal, characteristic, complemented
15C2×#1382C211yesmaximal, characteristic, complemented
16C23×C16C2211yescharacteristic
17#3050C2211yescharacteristic
18C2×#443C2211yescharacteristic
19#2933—21no—
20#2924—21no—
21C22×#108C2211yescharacteristic
22C2×#427C2211yescharacteristic
23#2237C411yescharacteristic, complemented
24#2237C411yescharacteristic, complemented
25#2236C411yescharacteristic, complemented
26#2235C411yescharacteristic, complemented
27#2234C411yescharacteristic, complemented
28#2234C411yescharacteristic, complemented
29#2233C411yescharacteristic, complemented
30#2232C411yescharacteristic, complemented
31C2×#368C2211yescharacteristic
32C2×#368C2211yescharacteristic
33#2158C2211yescharacteristic
34#2158C2211yescharacteristic
35C2×#359C2211yescharacteristic
36C2×#359C2211yescharacteristic
37#2138C2211yescharacteristic
38#2138C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40C2×#346C2211yescharacteristic
41C2×#346C2211yescharacteristic
42#1974C2211yescharacteristic
43#1945C2211yescharacteristic
44#1945C2211yescharacteristic
45#1945C2211yescharacteristic
46#1929—21no—
47#1922—21no—
48#1922—21no—
49#1871—21no—
50#1866—21no—
51#1826—21no—
52#1810C2211yescharacteristic
53#1772C2211yescharacteristic
54#1772C2211yescharacteristic
55#1772C2211yescharacteristic
56C8×C22⋊C4C2211yescharacteristic
57#1382C2211yescharacteristic
58#1382C2211yescharacteristic
59#1382C2211yescharacteristic
60#1382C2211yescharacteristic
61#1382C2211yescharacteristic
62#1382C2211yescharacteristic
63#1381C2211yescharacteristic
64#1381C2211yescharacteristic
65#1374C2211yescharacteristic
66#1374C2211yescharacteristic
67C2×#135C2311yescharacteristic
68C2×#131—21no—
69C22×M16—21no—
70C23×C8C2311yescharacteristic
71#525—21no—
72C2×#117—21no—
73C22×C16—21no—
74C22×C16—21no—
75C22×C16—22no—
76C22×C16—41no—
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81#467C2×C411yescharacteristic
82#467C2×C411yescharacteristic
83#466C2×C411yescharacteristic
84#466C2×C411yescharacteristic
85#465—21no—
86#465—21no—
87#443—22no—
88#443—41no—
89#436—41no—
90#435—41no—
91#435—41no—
92C8×D4—41no—
93#433—41no—
94#433—41no—
95#432—41no—
96#431—41no—
97C2×#108—22no—
98C2×#108—21no—
99C2×#108—41no—
100C2×#108C2311yescharacteristic
101C2×#108C2311yescharacteristic
102C2×#108C2311yescharacteristic
103C2×#108C2311yescharacteristic
104#427—21no—
105#427—21no—
106#427C2×C411yescharacteristic
107#427C2×C411yescharacteristic
108#427C2311yescharacteristic
109#427C2311yescharacteristic
110#427C2311yescharacteristic
111C2×#107C2×C411yescharacteristic
112C2×#107C2×C411yescharacteristic
113C2×#106—21no—
114#424—21no—
115#424—21no—
116#424—21no—
117C2×#104—21no—
118C2×#104—21no—
119#416—21no—
120#408—21no—
121#407—21no—
122C2×C22⋊C8D411yescharacteristic
123#405—21no—
124#405—21no—
125#405—21no—
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C4×C22⋊C4D411yescharacteristic
129#368—22no—
130#368—41no—
131#367C2×C411yescharacteristic
132#367C2×C411yescharacteristic
133#367C2×C411yescharacteristic
134#367C2×C411yescharacteristic
135#366C2×C411yescharacteristic
136#366C2×C411yescharacteristic
137#366C2×C411yescharacteristic
138#366C2×C411yescharacteristic
139#359—22no—
140#359—41no—
141#358C2×C411yescharacteristic
142#358C2×C411yescharacteristic
143#358C2×C411yescharacteristic
144#358C2×C411yescharacteristic
145#357C2×C411yescharacteristic
146#357C2×C411yescharacteristic
147#357C2×C411yescharacteristic
148#357C2×C411yescharacteristic
149C22⋊C16C2×C411yescharacteristic
150C22⋊C16C2×C411yescharacteristic
151C22⋊C16C2×C411yescharacteristic
152C22⋊C16C2×C411yescharacteristic
153#346—22no—
154#346—41no—
155#341—21no—
156#339—21no—
157#339—21no—
158#339—21no—
159#339—21no—
160#339—21no—
161#339—21no—
162#336D411yescharacteristic
163#336D411yescharacteristic
164C2×#41—21no—
165C23×C4C2×D411yescharacteristic
166#135—21no—
167#135—21no—
168#135—21no—
169#135—21no—
170C2×Q16C22×C411yescharacteristic
171C2×SD16—21no—
172C2×D8C22×C411yescharacteristic
173#131—42no—
174#131—81no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—41no—
180C2×M16—41no—
181C2×M16—42no—
182C2×M16—81no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—41no—
188C22×C8—41no—
189C22×C8C22×C411yescharacteristic
190C22×C8C22×C411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8#4111yescharacteristic
195C22×C8#4111yescharacteristic
196C22×C8C2411yescharacteristic
197#123—41no—
198C22⋊Q8—21no—
199C22⋊Q8—21no—
200#121—21no—
201#121—21no—
202#120—41no—
203C4×D4—41no—
204C4×D4—41no—
205#117—21no—
206#117—21no—
207#117—21no—
208#117—21no—
209#117—21no—
210#117—41no—
211#117—41no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—41no—
216C2×C42—41no—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C16—21no—
220C2×C16—21no—
221C2×C16—21no—
222C2×C16—21no—
223C2×C16—21no—
224C2×C16—21no—
225C2×C16—21no—
226C2×C16—21no—
227C2×C16—21no—
228C2×C16—42no—
229C2×C16—42no—
230C2×C16C22×C411yescharacteristic
231C2×C16C22×C411yescharacteristic
232C2×C16C22×C411yescharacteristic
233C2×C16C22×C411yescharacteristic
234C2×C16#4112yes—
235#108—21no—
236#108—21no—
237#108—21no—
238#108—21no—
239#108—21no—
240#108—21no—
241#108—21no—
242#108—42no—
243#108#4112yes—
244#107—21no—
245#107—21no—
246#107C22×C411yescharacteristic
247#107C22×C411yescharacteristic
248#107C22×C411yescharacteristic
249#107C22×C411yescharacteristic
250#107C22×C411yescharacteristic
251#107C22×C411yescharacteristic
252#106—21no—
253#106—21no—
254#106—21no—
255#106—21no—
256#106—21no—
257C4⋊C8—41no—
258C4⋊C8—41no—
259C4⋊C8—41no—
260C4⋊C8—41no—
261C4⋊C8—41no—
262C4⋊C8—41no—
263#104—42no—
264#104—81no—
265#103—21no—
266#103—21no—
267#102—21no—
268#102—21no—
269C22⋊C8—41no—
270C22⋊C8—41no—
271C22⋊C8—41no—
272C22⋊C8—41no—
273C22⋊C8D811yescharacteristic
274C22⋊C8D811yescharacteristic
275C22⋊C8D811yescharacteristic
276C22⋊C8D811yescharacteristic
277#97—41no—
278#97—41no—
279#97—41no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—41no—
285C4×C8—41no—
286C4×C8—41no—
287#95—21no—
288C24#13511yescharacteristic
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293C2×Q8—21no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4C4×D411yescharacteristic
308C22×C4C4×D411yescharacteristic
309C22×C4C2×D811yescharacteristic
310C22×C4C2×D811yescharacteristic
311C22×C4#13511yescharacteristic
312C22×C4C22×D411yescharacteristic
313Q16—21no—
314Q16—21no—
315SD16—41no—
316SD16—41no—
317D8—21no—
318D8—21no—
319M16—22no—
320M16—41no—
321M16—41no—
322M16—41no—
323M16—41no—
324M16—41no—
325M16—41no—
326M16—41no—
327M16—82no—
328M16—82no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—82no—
348C2×C8—82no—
349C2×C8C4×D411yescharacteristic
350C2×C8C4×D411yescharacteristic
351C2×C8C2×D811yescharacteristic
352C2×C8C2×D811yescharacteristic
353C2×C8C2×D811yescharacteristic
354C2×C8C2×D811yescharacteristic
355C2×C8C23×C411yescharacteristic
356C2×C8C2×#4111yescharacteristic
357C2×C8C2×#4111yescharacteristic
358C2×C8C2×#4111yescharacteristic
359C2×C8H(5,𝔽2)11yescharacteristic
360C2×C82−1+411yescharacteristic
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C22⋊C4—42no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C16—21no—
391C16—21no—
392C16—21no—
393C16—21no—
394C16—22no—
395C16—41no—
396C23—21no—
397C23—22no—
398C23—21no—
399C23—41no—
400C23—41no—
401C23C4×D811yescharacteristic
402C23C4×D811yescharacteristic
403C23C2×#13511yescharacteristic
404Q8—41no—
405Q8—41no—
406D4—41no—
407D4—41no—
408D4—81no—
409D4—82no—
410D4—81no—
411D4—161no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—161no—
439C2×C4C4×D811yescharacteristic
440C2×C4C4×D811yescharacteristic
441C2×C4C2×C4×D411yescharacteristic
442C2×C4#54611yescharacteristic
443C2×C4#54711yescharacteristic
444C2×C4C22×D811yescharacteristic
445C2×C4#57711yescharacteristic
446C2×C4#57811yescharacteristic
447C8—21no—
448C8—21no—
449C8—21no—
450C8—21no—
451C8—21no—
452C8—21no—
453C8—42no—
454C8—81no—
455C8—81no—
456C8—81no—
457C8C4×#4111yescharacteristic
458C8#52011yescharacteristic
459C8#55111yescharacteristic
460C8#55211yescharacteristic
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—42no—
469C22—42no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22#217912yes—
474C22C2×C4×D811yescharacteristic
475C22#330011yescharacteristic
476C22#333711yescharacteristic
477C4—21no—
478C4—21no—
479C4—81no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4#235911yescharacteristic
489C4#294711yescharacteristic
490C4#329411yescharacteristic
491C4#333411yescharacteristic
492C2—21no—
493C2—21no—
494C2—22no—
495C2—41no—
496C2—161no—
497C2#2291611yescharacteristic
498C2C2×#217911yescharacteristic
499C2#3346411yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22C2×C4×D8
prime radical#3050C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22606C2semidirect
4. #13170C2semidirect
5. #13152C2semidirect
6. #13134C2semidirect
7. #13113C2semidirect
8. #13113C2semidirect
10. #12939C2semidirect
11. #12922C2semidirect
12. #12671C2semidirect
13. #12642C2semidirect
14. #12611C2semidirect
15. C2×#1382C2semidirect
23. #2237C4semidirect
24. #2237C4semidirect
25. #2236C4semidirect
26. #2235C4semidirect
27. #2234C4semidirect
28. #2234C4semidirect
29. #2233C4semidirect
30. #2232C4semidirect

Maximal subgroups

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

subgroupindex
#336452
#230182
#226062
#131702
#131522
#131342
#131132
#130632
#129392
#129222
#126712
#126422
#126112
C2×#13822